From Assets and Processes to Service Ecosystems: A Hierarchical Digital Twin Framework for Knowledge Representation

Abstract

Digital twins (DTs) have become a central paradigm for modeling cyber–physical systems and digital infrastructures, yet the term is applied to very different representations—from physical assets to operational processes and service environments. This ambiguity obscures how the various DT interpretations relate to one another and at which level knowledge can be represented and extracted. This paper develops a conceptual and mathematical framework that treats asset-centric, process-centric, and service-centric DTs as successive levels of system abstraction. DTs are modeled as mappings between real-world entities and their digital representations, and the three paradigms are connected through explicit cross-layer dependencies, with service-centric twins shown to form a distinct level that cannot be reduced to asset and process descriptions alone; the framework is then extended to the ecosystem level as a digital service ecosystem twin. Because each level fixes the entities, features, and relations available to data-driven methods, the framework also specifies where machine-learning and knowledge-extraction tasks operate within layered DT architectures. The approach is illustrated and validated for structural and cross-layer consistency through a smart-city electricity ecosystem, providing a unified basis for interpreting the evolution of DTs toward service-oriented digital ecosystems.

1. Introduction

1.1. Background and Motivation

Digital Twin (DT) technologies have become an important paradigm for modeling and managing complex cyber–physical systems [1]. Originally introduced in aerospace engineering and product lifecycle management, DTs enable the creation of digital representations of real-world systems that support monitoring, simulation, and decision making throughout the system lifecycle [2].

Over the past decade, DT concepts have expanded to numerous application domains, including manufacturing, smart cities, transportation systems, energy infrastructures, and healthcare [3]. In these contexts, DTs are commonly used to integrate physical infrastructures with digital models capable of analyzing system states, predicting system behavior, and supporting operational optimization.

Despite this rapid development, the term digital twin is used in the literature to describe heterogeneous types of system representations. In some studies, DTs primarily represent physical assets, such as machines, infrastructure components, or sensors [4]. In other works, the concept refers to models of operational processes, including workflows, system interactions, and data flows [5]. More recently, DT approaches have begun to extend toward service environments, where digital models represent services delivered by digital infrastructures [6].

These different interpretations suggest that DT technologies are evolving toward increasingly higher levels of system abstraction. However, the literature still lacks a unified theoretical framework capable of clearly distinguishing different DT paradigms and explaining their relationships.

This observation motivates the development of a conceptual and mathematical framework capable of describing the evolution of DT paradigms from asset-level representations to service ecosystem models.

The central research problem is therefore not simply whether these DT interpretations coexist in the literature, but how they can be theoretically distinguished and formally related within a common system representation logic. Addressing this problem is essential for moving from fragmented DT terminology toward a more rigorous and scalable architectural understanding.

A key conceptual insight of this work is that these paradigms can be interpreted as the result of a purpose-driven evolution of system representations. In real-world systems, technological infrastructures are typically created to satisfy specific goals or societal needs. Physical assets are deployed to achieve these goals, operational processes emerge to coordinate the use of these assets, and service ecosystems subsequently arise to deliver value to users and stakeholders. This conceptual progression can be summarized as

$$\mathrm{Purpose}\to \mathrm{Assets}\to \mathrm{Processes}\to \mathrm{Services}.$$

DTs can therefore be understood as digital representations corresponding to different levels of this structure.

1.2. Related Work

While the DT term is widely used, the literature reveals several distinct conceptual interpretations that correspond to different types of system representations. In this section, prior research is analyzed through the lens of the three DT paradigms introduced in this paper: asset-centric, process-centric, and service-centric digital twins.

Early conceptualizations of DTs were introduced within aerospace and manufacturing research, where digital models were used to represent aircraft components, industrial equipment, and infrastructure systems [6,7,8,9]. These studies emphasized the importance of continuous synchronization between the physical system and its digital representation through real-time data integration.

Subsequent work expanded this perspective by incorporating IoT technologies, data analytics, and machine learning into DT architectures. The study [10] proposed a five-dimensional model of DTs that integrates physical entities, virtual models, services, data, and connections. Similarly, Grieves and Vickers [11] described DTs as an extension of product lifecycle management systems, enabling continuous monitoring and optimization of physical products.

Research in industrial systems and smart infrastructure continues to develop this asset-centric perspective, focusing on predictive maintenance, condition monitoring, and lifecycle optimization [12,13,14,15]. These approaches typically model DTs as representations of individual assets or tightly coupled systems, emphasizing physical state estimation and operational performance.

While asset-level modeling provides detailed information about individual system components, many real-world systems are defined primarily by interactions and operational processes rather than by isolated assets. This observation has led to the development of process-oriented DTs, which focus on modeling workflows, system interactions, and operational dynamics.

In manufacturing and logistics systems, DTs have been used to simulate production processes, optimize workflows, and analyze system performance [16,17,18]. These models often integrate discrete-event simulations, system dynamics models, and data-driven analytics to capture the behavior of complex operational processes.

Similarly, DTs have been applied to transportation networks, supply chains, and urban infrastructure systems where interactions among multiple components play a central role [19,20,21]. In these contexts, the DT represents a system-level operational model that integrates data from multiple assets and processes.

Research on system-of-systems DTs further extends this approach by modeling interconnected infrastructures such as smart cities or industrial ecosystems [22,23,24]. These studies emphasize the importance of representing interactions among heterogeneous subsystems and coordinating multiple DTs within integrated operational environments.

More recent developments in DT research have begun to shift the focus from operational systems toward service ecosystems and value creation processes. In service-oriented architectures, the primary objective is not simply to model assets or processes but to understand how services are delivered, coordinated, and optimized within complex socio-technical systems.

Service science and digital platform research have introduced concepts such as service ecosystems, where multiple actors interact to co-create value through shared infrastructures and digital platforms [25,26,27]. These ideas have increasingly influenced DT architectures in domains such as smart cities, healthcare, and urban mobility.

Several studies have proposed DT models capable of representing complex service environments. For example, studies [28,29] explored DT approaches for smart cities, where urban infrastructures, services, and citizen interactions form large-scale digital ecosystems. Similarly, discussions of the industrial metaverse and digital platform ecosystems emphasize the integration of multiple DTs across service networks [30,31,32].

Recent literature shows that DT research is expanding rapidly into real-world infrastructures and large-scale operational environments. A recent broad review by Iliuţă et al. emphasizes that the field has moved far beyond its original manufacturing-centered scope and now includes diverse applied domains requiring more mature analytical interpretation [33]. Liu et al. further show that current DT research is increasingly discussed in terms of capability maturity, implementation logic, and the structured evolution of DT functionality [34].

This tendency is especially visible in urban and built-environment contexts. Mousavi et al. review DT applications in the built environment and highlight persistent challenges related to lifecycle integration, interoperability, and operational coordination across complex real systems [35]. In the smart-city domain, Huzzat et al. describe DTs as increasingly important for the coordinated management of heterogeneous urban infrastructures and services [36]. Sacoto-Cabrera et al. additionally show that recent smart-city DT research is strongly connected with IoT- and AI-enabled orchestration of multiple urban subsystems rather than isolated asset monitoring alone [37].

Parallel developments are also reported in domain-specific infrastructure systems. Aghazadeh Ardebili et al. show that DTs in smart energy systems are now being studied not only as monitoring tools, but also as design and management instruments for complex energy environments [38]. Wu et al. similarly demonstrate that DT research in transportation infrastructure is increasingly oriented toward integrated representation of roads, bridges, and mobility-related operational processes, while still facing challenges of scalability and architecture [39].

A further relevant body of literature concerns metamodeling, model-based systems engineering (MBSE), and cyber–physical systems (CPS). Metamodeling and model-driven engineering address abstraction levels, modeling languages, and relations between models and metamodels [40]. MBSE treats models as primary artifacts for system specification, architecture definition, analysis, verification, traceability, and lifecycle management [22]. CPS research provides the cyber–physical foundation of DTs by focusing on the interaction between physical processes, sensing, computation, communication, and control [30]. These streams are important for DT research because they clarify how complex systems can be represented, structured, traced, and connected to physical operation. At the same time, they usually do not place service delivery, actors, service dependencies, and ecosystem-level value at the center of the DT representation. This observation further motivates the need for a representation logic that connects cyber–physical asset and process models with service- and ecosystem-level knowledge structures.

Taken together, the reviewed studies provide important insights into DT implementations at the asset, process, and service levels. At the same time, this body of literature remains fragmented in two important respects. First, these directions are typically developed as parallel research streams rather than as elements of a shared theoretical structure. Second, most studies focus primarily on application domains, architectures, or functional capabilities, while providing limited critical explanation of how different DT paradigms are structurally related across levels of abstraction. For this reason, the purpose of the present review is not merely to classify existing studies, but to identify the central research gap addressed in this paper: the absence of a unified conceptual and mathematical framework capable of relating asset-centric, process-centric, and service-centric DTs within a single hierarchical interpretive model.

1.3. Research Gap, Contributions and Paper Structure

The research gap identified in this paper follows directly from the above review. Although prior studies provide substantial contributions to asset-level, process-level, and emerging service-oriented DT research, they do not yet offer a unified formal account of the structural relationships among these paradigms. As a result, the evolution of DT representations across increasing levels of system abstraction remains insufficiently theorized.

Despite the rapid expansion of DT research, the literature still reveals several conceptual and methodological limitations. Most existing studies adopt an asset-oriented perspective, in which DTs are primarily interpreted as digital representations of physical devices or infrastructure components. Although such models enable detailed monitoring and lifecycle analysis of assets, they provide only a partial representation of complex socio-technical systems.

A second limitation concerns process-level modeling. Some studies extend DT architectures to represent operational workflows and system interactions. However, the conceptual distinction between asset-level and process-level DT representations is rarely formalized within a unified theoretical framework.

A third and more fundamental gap concerns the representation of service ecosystems. In many contemporary digital infrastructures, including smart cities, energy systems, and digital platforms, the primary objective of system operation is increasingly defined by the services delivered to users and the value generated within service ecosystems. Nevertheless, the literature still lacks a clear theoretical framework that formally distinguishes asset-centric DT (ACDT), process-centric DT (PCDT) and service-centric DT (SCDT) paradigms and explains their structural relationships.

This paper addresses these gaps by developing a unified conceptual and mathematical framework for the hierarchical interpretation of DT paradigms. Rather than treating asset-oriented, process-oriented, and service-oriented DTs as loosely related or application-dependent variants, the paper formalizes them as distinct but structurally connected levels of system representation. Within this framework, service-centric DTs are defined as an ontologically distinct representation level that cannot be fully reduced to asset-level or process-level models alone. The framework is further extended to the ecosystem level through the formal interpretation of digital service ecosystem twins. Because each level fixes the entities, relations, and observable variables that downstream methods can act on, this hierarchical interpretation also determines the level at which machine-learning models operate and at which knowledge is represented and extracted from a DT, which is the perspective developed further in Section 4.

The objective of this study is not to implement and benchmark a specific ML model, but to define the representational structure required for such models to operate coherently across asset, process, service, and ecosystem levels.

The added value of the proposed formulation lies in establishing the starting point for a new kind of digital-twin modelling rather than in presenting a mature technological implementation. The framework defines how assets, processes, services, actors, dependencies, and ecosystem-level value structures can be represented within a single hierarchical logic. This is intended to provide a formal modelling substrate on which future machine-learning, knowledge-extraction, and decision-support methods can be developed. At the present stage, the framework should therefore be understood as a low technology readiness level (low-TRL) conceptual and mathematical formulation: it clarifies the representational objects, cross-layer relations, assumptions, and traceability principles required for subsequent computational development, but it does not yet claim operational deployment, empirical ML validation, or software-level maturity.

The contribution of the paper is fourfold:

• It provides a formal distinction between asset-centric, process-centric, and service-centric DT paradigms and establishes a hierarchical interpretation of these paradigms as progressively higher levels of abstraction in the digital representation of complex socio-technical systems.
• It develops a mathematical formalization of their structural relationships through mappings, explicit modeling assumptions, constructions, and conditional propositions that clarify their composability, emergence, and ontological non-reducibility.
• It extends the proposed framework to the ecosystem level by showing how interacting services, actors, and value-generation mechanisms can be represented as digital service ecosystem twins.
• It defines a reference representation layer for future machine-learning and knowledge-extraction tasks by specifying the entities, features, relations, and cross-layer traceability conditions required for such tasks, without claiming empirical ML implementation in the present study.

Taken together, the conceptual interpretation, mathematical formalization, and hierarchical architecture presented in this paper establish a unified theoretical framework for understanding the evolution of DT paradigms from asset-oriented representations to service ecosystem models.

The proposed framework is illustrated through a smart city electricity ecosystem, demonstrating how asset infrastructure, operational processes, services, and ecosystem interactions can be represented within a unified DT architecture.

Accordingly, the novelty of this study lies not in presenting another domain-specific digital twin application, but in providing a generalized theoretical framework that explains how different DT paradigms can be distinguished, related, and integrated within a single hierarchical representation model.

The remainder of the paper is organized as follows. Section 2 introduces the conceptual foundations of DT paradigms and discusses their ontological evolution together with the proposed mathematical framework. Section 3 presents the results of the study through a smart city electricity ecosystem case study, including asset-, process-, and service-level taxonomies and the cross-layer assets–processes–services (APS) matrix. Section 4 discusses the theoretical implications of the proposed framework, including the role of SCDTs and the orchestration of multiple DT paradigms within complex digital infrastructures. Section 5 concludes the paper by summarizing the main findings and outlining directions for future research.

2. Materials and Methods

2.1. Ontological Evolution of Digital Twin Paradigms

The diversity of DT implementations observed in the literature reflects a deeper evolution in the way real-world systems are represented in digital environments. Rather than emerging simultaneously, different DT paradigms correspond to successive layers of system abstraction.

The evolution of DT technologies can be interpreted as a purpose-driven transformation of system representation. As system objectives become more complex, the focus of digital representation shifts from individual physical assets to operational processes and eventually to service ecosystems. This conceptual progression is illustrated in Figure 1.

Figure 1 illustrates the conceptual transition from asset-oriented system representation toward service-oriented digital ecosystems. The starting point of this evolution is the purpose or objective that motivates system monitoring, optimization, or control. To achieve this purpose, physical assets and infrastructure components must first be represented digitally, leading to the development of ACDTs.

As systems grow in complexity, assets become interconnected through operational workflows. Digital representations must therefore capture not only individual components but also the processes that link them. This requirement leads to the emergence of PCDTs.

Modern socio-technical systems increasingly operate as service ecosystems, where value is created through interactions between services, actors, and digital platforms. At this level, system representation shifts from processes to services themselves, giving rise to SCDTs. This progression reflects a gradual increase in the abstraction level of DT representations.

To clarify the conceptual differences between the three DT paradigms introduced above,

Table 1. Comparative Characteristics of Digital Twin Paradigms.

Characteristic	Asset-Centric Digital Twin	Process-Centric Digital Twin	Service-Centric Digital Twin
Primary ontological referent	Physical assets and infrastructure components	Operational processes and workflows	Services and service ecosystems
System abstraction level	Component level	Operational system level	Ecosystem level
Typical real-world entities	Devices, machines, sensors, infrastructure objects	Workflows, operational interactions, control processes	Services, actors, and service interactions
Digital representation focus	Digital replica of physical assets	Digital model of system operations	Digital representation of service ecosystems
Mathematical representation	Mapping between asset set and digital models	Mapping between process set and digital process models	Mapping between service ecosystem and digital ecosystem model
Typical structural model	Component-based system model	Workflow or process network	Directed graph of services and actors
Typical DT architecture layer	Physical infrastructure layer	Operational process layer	Service ecosystem layer
Primary evaluation metrics	Asset state, reliability, operational performance	Process efficiency, workflow performance	Service value, ecosystem performance
Role in DT hierarchy	Foundational layer of DT architectures	Integration layer linking multiple assets	Highest abstraction layer representing value creation
Relation to other paradigms	Provides data sources for higher-level models	Integrates multiple asset-level twins	Emerges from coordinated processes and actors
Relation to theoretical framework	Basis of Asset-Centric mapping	Intermediate layer of DT composition	Ontologically distinct DT paradigm

summarizes their main characteristics, including ontological referents, abstraction levels, modeling structures, and architectural roles. The comparison highlights the progressive transition from asset-level representations toward service ecosystem modeling that underlies the ontological evolution of DT paradigms.

The comparison presented in Table 1 highlights that the three Digital Twin paradigms correspond to progressively increasing levels of system abstraction and correspond to different classes of ontological referents.

2.2. Formal Representation Framework for Hierarchical Digital Twins

The conceptual distinctions between ACDTs, PCDTs and SCDTs can be formalized using a mathematical framework that captures the relationships between physical systems and their digital representations.

The mathematical interpretation of DT paradigms can be expressed through a hierarchy of mappings between elements of the physical system and their corresponding digital representations. The general structure of these mappings is illustrated in Figure 2.

Three specialized mappings correspond to the three DT paradigms considered in this study: ACDTs, PCDTs and SCDTs. These mappings form a hierarchical structure that reflects the increasing abstraction level of system representation.

The purpose of the mathematical framework developed in this section is not merely to provide symbolic definitions of different DT paradigms, but to explain how these paradigms emerge as progressively higher levels of system representation. In this sense, the equations that follow should be interpreted as a formal description of hierarchical evolution: (1) the DT is represented in its most general form as a mapping between real-world entities and digital representations; (2) this general mapping is specialized to assets, then to operational processes formed by interacting assets, (3) to services and service ecosystems generated by coordinated processes. Accordingly, the mathematical role of the framework is to demonstrate that the evolution from asset-centric to process-centric and service-centric DTs is not terminological, but structural.

Let $R$ denote the set of real-world referents represented by a digital twin and let $D$ denote the corresponding set of digital representations. The general digital twin mapping is denoted by

$$DT:R\to D$$

Within the hierarchical framework developed in this paper, four classes of referents are distinguished. The set of physical assets is denoted by $A$, the set of operational processes by $P$, the set of services by $S$, and the service ecosystem by $E$. The set of actors involved in service provision, coordination, regulation, or consumption is denoted by $U$. Correspondingly, the digital representation spaces at the asset, process, service, and ecosystem levels are denoted by $D_A$, $D_P$, $D_S$, and $D_E$, respectively.

The specialized digital twin mappings are defined as follows:

$${DT}_A:A\to D_A$$

$${DT}_P:P\to D_P$$

$${DT}_S:S\to D_S$$

$${DT}_E:E\to D_E$$

Here, ${DT}_A$, ${DT}_P$, ${DT}_S$ and ${DT}_E$ denote the asset-centric, the process-centric, the service-centric and digital service ecosystem twin mapping, respectively. This notation is used consistently throughout the remainder of the paper.

The cross-layer relations among referents are represented by dependency sets. The asset-process dependency relation is denoted by

$$E_{AP}\subseteq A\times P$$

where $\left(a,p\right)\in E_{AP}$ means that asset $a$ participates in, supports, or enables process $p$.

The process-service dependency relation is denoted by

$$E_{PS}\subseteq P\times S$$

where $\left(p,s\right)\in E_{PS}$ means that process $p$ supports or enables service $s$. Service-level dependencies are denoted by

$$E_{SS}\subseteq S\times S,$$

where $\left(s_i,s_j\right)\in E_{SS}$ indicates that service $s_j$ depends on, interacts with, or is influenced by service $s_i$. The set of actor-service relations may be denoted by

$$E_{US}\subseteq U\times S,$$

where $\left(u,s\right)\in E_{US}$ means that actor $u$ provides, regulates, consumes, or otherwise participates in service $s$.

For a process $p\in P$, let

$${\Gamma }_P\left(p\right)=\{a\in A:\left(a,p\right)\in E_{AP}\}$$

denote the set of assets participating in process $p$. Similarly, for a service $s\in \mathcal{S}$, let

$${\Gamma }_S\left(s\right)=\{p\in P:\left(p,s\right)\in E_{PS}\}$$

denote the set of processes enabling service $s$. These component sets are used to define cross-layer traceability between adjacent representational levels.

The paper uses the symbol $\preccurlyeq$ to denote layered embedding or representational dependency. Thus,

$${DT}_A\preccurlyeq {DT}_P\preccurlyeq {DT}_S\preccurlyeq {DT}_E$$

means that higher-level representations preserve traceable links to the lower-level representations from which they are composed. For adjacent levels, this relation is defined through projection operators. For example, ${DT}_A\preccurlyeq {DT}_P$ holds if there exists a projection

$${\pi }_{P\to A}:D_P\to 2^{D_A}$$

such that, for every process $p\in P$,

$${\pi }_{P\to A}\left[{DT}_P\left(p\right)\right]=\{{DT}_A\left(a\right):a\in {\Gamma }_P\left(p\right)\}.$$

Analogously, ${DT}_P\preccurlyeq {DT}_S$ holds if there exists a projection

$${\pi }_{S\to P}:D_S\to 2^{D_P}$$

such that, for every service $s\in S$,

$${\pi }_{S\to P}\left[{DT}_S\left(s\right)\right]=\{{DT}_P\left(p\right):p\in {\Gamma }_S\left(s\right)\}.$$

The ecosystem-level projection ${\pi }_{E\to S}$ is defined in the same way for services composing the service ecosystem. These projections provide the formal basis for cross-layer traceability in the proposed hierarchy.

The framework relies on the following modeling assumptions.

Assumption 1.

Asset observability. Each relevant asset $a\in A$ has observable, measurable, or otherwise representable state variables that can be included in an asset-level digital representation ${DT}_A\left(a\right)$.

Assumption 2.

Process composition. Operational processes are composed of interactions among assets. Therefore, each process $p\in P$ is associated with a non-empty set of participating assets ${\Gamma }_P\left(p\right)\subseteq A$.

Assumption 3.

Service composition. Services are enabled by operational processes and delivered to or through actors. Therefore, each service $s\in S$ is associated with a set of enabling processes ${\Gamma }_S\left(s\right)\subseteq P$ and may be connected with one or more actors from $\mathcal{U}$.

Assumption 4.

Service-level structural independence. Actor roles, service dependencies, and value-generation mechanisms are not, in general, fully determined by asset and process layers alone. Two systems may therefore have the same asset and process structures while differing at the service or ecosystem level.

Assumption 5.

Faithfulness of service-level representation. The service-centric digital twin mapping ${DT}_S$ is assumed to be faithful to relevant service-level differences. In other words, if two service configurations differ in actors, dependencies, or value-generation mechanisms that are relevant to the modeled ecosystem, their service-level digital representations should also differ.

Assumption 6.

Cross-layer traceability. Higher-level representations preserve recoverable references to their lower-level components through projection operators. This assumption does not imply that higher-level representations are reducible to lower-level ones; it only requires that the lower-level components participating in a higher-level representation remain identifiable.

These assumptions define the modeling scope of the proposed formal framework. Definitions introduced below specify the digital twin mappings at each level, constructions explain how layered representations can be formed under the stated assumptions, and propositions describe conditional consequences of these assumptions. This separation is used to avoid treating modeling premises as independently derived mathematical results.

The relationships among DT paradigms can now be expressed through a limited number of conditional propositions, constructions, and traceability results.

Proposition 1.

Conditional Non-Reducibility of Service-Centric Digital Twins.

Under Assumptions 4 and 5, a service-centric digital twin is not reconstructively reducible to asset-centric and process-centric digital twins alone. More precisely, say that $D{T}_S$ is reconstructively reducible to $D{T}_A$ and $D{T}_P$ if there exists a reconstruction operator

$$\rho :D_A\times D_P\to D_S$$

such that, for every admissible service configuration, the service-level representation can be recovered from the asset- and process-level representations alone

$$D{T}_S\left(s\right)=\rho \left[D{T}_A\left(\mathcal{A}\right),D{T}_P\left(\mathcal{P}\right)\right]$$

Under Assumption 4, actor roles, service dependencies, and value-generation mechanisms are not fully determined by the asset and process layers. Therefore, two service configurations may share the same asset set $A$, the same process set $P$, and the same asset–process structure, while differing in actors, service dependencies, or value-generation mechanisms. Under Assumption 5, $D{T}_S$ is faithful to such service-level differences. Consequently, the corresponding service-level digital representations must differ, even though the asset- and process-level representations are identical.

If a reconstruction operator $\rho$ based only on $D{T}_A$ and $D{T}_P$ existed, it would return the same service-level representation for both configurations, because its inputs would be identical. This contradicts the faithfulness of $D{T}_S$. Therefore, under the stated assumptions, $D{T}_S$ cannot be reconstructively reduced to a finite composition of $D{T}_A$ and $D{T}_P$ alone.

This proposition should be interpreted as a conditional modeling result. It does not claim that service-level irreducibility holds for all possible systems, but only that it follows when service-level actors, dependencies, and value structures are not fully determined by lower-level asset and process structures.

Construction 1.

Layered Composition of Digital Twin Representations.

A layered digital twin representation can be constructed when the asset-, process-, service-, and ecosystem-level mappings are connected through explicit cross-layer dependency relations and projection operators.

Let $D{T}_A,D{T}_P,D{T}_S$ and $D{T}_E$ be the digital twin mappings at the four representational levels. The hierarchy is constructed by requiring the following layered embedding relations:

$$D{T}_A\preccurlyeq D{T}_P$$

$$D{T}_P\preccurlyeq D{T}_S$$

$$D{T}_S\preccurlyeq D{T}_E$$

The first relation means that each process-level representation preserves traceable links to the asset-level representations participating in that process. Formally, for a process $p\in P$,

$${\pi }_{P\to A}\left[D{T}_P\left(p\right)\right]=\{D{T}_A\left(a\right):a\in {\Gamma }_P\left(p\right)\}$$

where ${\Gamma }_P\left(p\right)$ is the set of assets participating in process $p$.

The second relation means that each service-level representation preserves traceable links to the process-level representations enabling that service. For a service $s\in S$

$${\pi }_{S\to P}\left[D{T}_S\left(s\right)\right]=\{D{T}_P\left(p\right):p\in {\Gamma }_S\left(s\right)\}$$

where ${\Gamma }_S\left(s\right)$ is the set of processes enabling service $s$.

The third relation means that the ecosystem-level representation preserves traceable links to the service-level representations composing the service ecosystem. If ${\Gamma }_E\left(e\right)$ denotes the set of services forming ecosystem $e\in E$, then

$${\pi }_{E\to S}\left[D{T}_S\left(s\right)\right]=\{D{T}_S\left(s\right):s\in {\Gamma }_E\left(e\right)\}$$

Thus, the layered composition of digital twins is not modeled as set inclusion between mappings. It is modeled as a system of projection-based representational dependencies between adjacent levels

$$D{T}_A\preccurlyeq D{T}_P\preccurlyeq D{T}_S\preccurlyeq D{T}_E$$

This construction provides the formal basis for cross-layer traceability in the proposed digital twin hierarchy.

Modeling Principle 1. Service-Oriented Representation Shift.

When the primary objective of system modeling, evaluation, or decision making is defined in terms of service outcomes, user value, actor coordination, or ecosystem performance, the appropriate digital twin representation shifts from asset-level or process-level modeling toward service-centric or ecosystem-level modeling.

This principle explains why service-centric digital twins are introduced as a separate representational level. Asset-centric digital twins are appropriate when the main modeling objective concerns the state, reliability, or performance of physical components. Process-centric digital twins are appropriate when the main objective concerns workflow coordination, operational efficiency, or process-level interactions. Service-centric digital twins become necessary when the main objective concerns service delivery, service dependencies, stakeholder value, or ecosystem-level outcomes.

This principle is not a theorem in the strict mathematical sense. It is a modeling rule that guides the choice of representational level according to the purpose of the digital twin.

Construction 2.

Asset-Centric Digital Twin Representation.

Under Assumption 1, an asset-centric digital twin representation can be constructed for any system containing identifiable assets with observable or representable state variables.

Let $A=\{a_1,a_2,\dots ,a_n\}$ be the set of physical assets in the system. For each asset $a_i\in A$, let $x_{a_i}\left(t\right)$ denote its observed or estimated state at time $t$. The asset-level digital representation may include this state together with structural parameters, configuration information, operational status, and performance indicators.

The asset-centric digital twin mapping is then defined as

$$D{T}_A:A\to D_A$$

where

$$D{T}_A\left(a_i\right)=d_{a_i}, d_{a_i}\in D_A$$

Here, $d_{a_i}$ denotes the digital representation of asset $a_i$. This construction establishes the foundational layer of the hierarchy, because asset-level representations provide the primary digital descriptions from which higher-level process and service representations can be composed.

Construction 3.

Process-Centric Digital Twin Representation.

Under Assumption 2, a process-centric digital twin representation can be constructed when operational processes are formed by interactions among assets.

Let $P$ denote the set of operational processes in the system. For each process $p\in P$, the set of participating assets is defined by

$${\Gamma }_P\left(p\right)=\{a\in \mathcal{A}:\left(a,p\right)\in E_{AP}\}$$

A process-level representation must therefore include not only a description of the process itself, but also the asset-level representations participating in that process. The process-centric digital twin mapping is defined as

$$D{T}_P:P\to D_P$$

where

$$D{T}_P\left(p\right)=d_p, d_p\in D_P$$

The representation $d_p$ may include workflow logic, operational rules, data flows, control dependencies, performance indicators, and references to the asset-level digital representations

$$D{T}_A\left(a\right):a\in {\Gamma }_P$$

Thus, the process-centric digital twin is constructed over interactions among assets rather than over isolated physical components. This explains why $D{T}_P$ is layered above $D{T}_A$, while still preserving traceable links to asset-level representations

$$D{T}_A\preccurlyeq D{T}_P$$

Construction 4.

Service-Centric Digital Twin Representation.

Under Assumptions 3–5, a service-centric digital twin representation can be constructed when operational processes enable services delivered to or through actors, and when service-level dependencies or value-generation mechanisms are relevant to system evaluation.

Let $S$ denote the set of services and $U$ the set of actors involved in service provision, regulation, coordination, or consumption. For each service $s\in S$, the set of enabling processes is defined by

$${\Gamma }_S\left(s\right)=\{p\in P:\left(p,s\right)\in E_{PS}\}$$

The service-centric digital twin mapping is defined as

$$D{T}_S:S\to D_S$$

where

$$D{T}_S\left(s\right)=d_s, d_s\in D_S$$

The representation $d_s$ may include service objectives, service status, service quality indicators, actor roles, dependency relations, value indicators, and references to the process-level digital representations

$$D{T}_P\left(p\right):p\in {\Gamma }_S\left(s\right)$$

At the ecosystem level, services, actors, and service dependencies can be represented as

$$e=\left(A,U,E_{PS},E_{SS},E_{US},V\right)$$

where $E_{SS}$ denotes service–service dependencies, $E_{US}$ denotes actor–service relations, and $V$ denotes the value-generation structure or ecosystem-level value function. The ecosystem-level digital twin mapping is then defined as

$$D{T}_E:\mathcal{E}\to D_E$$

where

$$D{T}_E\left(e\right)=d_e, d_e\in D_E$$

This construction establishes the service-centric and ecosystem-level layers of the proposed hierarchy:

$$D{T}_P\preccurlyeq D{T}_S\preccurlyeq D{T}_E$$

The service-centric representation is therefore not treated as a simple aggregation of asset and process representations. It is constructed as a higher-level representation that preserves traceability to lower-level processes while also introducing actor relations, service dependencies, and value-generation mechanisms.

Theorem 1.

Cross-Layer Traceability of the Hierarchical Digital Twin Representation.

Let $D{T}_A$, $D{T}_P$, $D{T}_S$, and $D{T}_E$ be digital twin mappings satisfying the layered embedding relations

$$D{T}_A\preccurlyeq D{T}_P\preccurlyeq D{T}_S\preccurlyeq D{T}_E$$

Assume that the corresponding projection operators exist

$${\pi }_{P\to A}:D_P\to 2^{D_A}$$

$${\pi }_{S\to P}:D_S\to 2^{D_P}$$

$${\pi }_{E\to S}:D_E\to 2^{D_S}$$

Then the hierarchical digital twin representation is cross-layer traceable. In particular, asset-level representations participating in an ecosystem-level representation can be recovered through the composition of projection operators.

Because the projection operators are set-valued, define their lifted composition as follows. For a set $Q_S\subseteq D_S$

$${\pi }_{S\to P}^{\mathrm{*}}\left(Q_S\right)=\bigcup _{d_s\in Q_S}{\pi }_{S\to P}\left(d_s\right)$$

Similarly, for a set $Q_P\subseteq D_P$

$${\pi }_{P\to A}^{\mathrm{*}}\left(Q_P\right)=\bigcup _{{\pi }_{S\to P}\left(d_s\right)}{\pi }_{P\to A}\left(d_p\right)$$

The composed projection from the ecosystem level to the asset level is therefore

$${\pi }_{E\to A}={\pi }_{P\to A}^{\mathrm{*}}\circ {\pi }_{S\to P}^{\mathrm{*}}\circ {\pi }_{E\to S}$$

For any ecosystem $e\in E$, this gives

$${\pi }_{E\to A}\left[D{T}_E\left(e\right)\right]=\bigcup _{d_s\in {\pi }_{E\to S}\left[D{T}_E\left(e\right)\right]} \bigcup _{d_p\in {\pi }_{S\to P}\left(d_s\right)}{\pi }_{P\to A}\left(d_p\right)$$

Thus, every asset-level representation participating in the asset–process–service chains of the ecosystem can be recovered from the ecosystem-level representation through successive projections.

Proof of Theorem 1.

 

By the definition of layered embedding, $D{T}_S\preccurlyeq D{T}_E$ implies that the ecosystem-level representation $D{T}_E\left(e\right)$ preserves recoverable references to the service-level representations composing ecosystem $e$. Therefore, applying ${\pi }_{E\to S}$ to $D{T}_E\left(e\right)$ returns the set of service-level digital representations associated with that ecosystem.

Similarly, $D{T}_P\preccurlyeq D{T}_S$ implies that each service-level representation preserves recoverable references to the process-level representations enabling the corresponding service. Applying ${\pi }_{S\to P}$ to each recovered service-level representation returns the set of process-level representations associated with that service.

$D{T}_A\preccurlyeq D{T}_P$ implies that each process-level representation preserves recoverable references to the asset-level representations participating in that process. Applying ${\pi }_{P\to A}$ to each recovered process-level representation returns the corresponding asset-level representations.

Taking the union over all services and processes in the ecosystem gives the composed projection ${\pi }_{E\to A}$. Therefore, the ecosystem-level representation is cross-layer traceable to the asset level through the composition of adjacent projection operators. □

Theorem 1 does not claim that higher-level digital twins are reducible to lower-level ones. Rather, it establishes a weaker and more practically relevant property: higher-level representations can preserve traceable references to the lower-level representations from which they are composed. This traceability is essential for hierarchical digital twin architectures because it allows service-level outcomes and ecosystem-level knowledge to be interpreted in relation to the assets and processes that support them.

2.3. Methodological Interpretation of Digital Twin Hierarchy

The mathematical framework presented in Section 2.2 establishes the hierarchical relationships between asset-centric, process-centric, and service-centric digital twins and their ecosystem-level extension. In practical DT architectures, these paradigms do not appear as independent models but as integrated layers of system representation [21,22,23,24].

Figure 3 illustrates the hierarchical inclusion of DT paradigms, where each higher-level representation integrates and extends the lower-level DT models.

At the foundational level, ACDTs represent individual physical assets and infrastructure components. These models capture the operational states of devices, sensors, and infrastructure elements and provide the primary data sources for higher-level DT representations.

When multiple assets interact through operational workflows, their relationships give rise to PCDTs, which describe system operations such as monitoring processes, data flows, and control mechanisms. Process-level DTs therefore integrate multiple asset-level models into coherent operational system representations.

Operational processes, in turn, enable the delivery of services to users and stakeholders. At this level, system representation shifts toward SCDTs, which capture services, service interactions, and service performance within the system.

In many contemporary digital infrastructures, services operate within broader ecosystems involving multiple actors, services, and institutional relationships. Such environments can be represented through DSET, which provide ecosystem-level DT representations.

The structural characteristics and roles of the considered paradigms are summarized in

Table 2. Hierarchical structure of digital twin paradigms.

Level of Representation	Digital Twin Paradigm	Primary System Elements	Modeling Focus	Role in the Hierarchy
Level 1	Asset-centric digital twins	Physical assets, sensors, devices, infrastructure components	Representation of physical state, structure, and performance	Foundational representation layer
Level 2	Process-centric digital twins	Operational workflows, monitoring processes, communication processes	Modeling operational interactions and system workflows	Integration of multiple asset-level twins
Level 3	Service-centric digital twins	Services, service providers, service consumers	Representation of service operations and interactions	Modeling service delivery and operational value
Level 4	Digital service ecosystem twins	Service ecosystems, actor networks, governance structures	Modeling ecosystem-level interactions and value creation	Ecosystem-level integration of multiple services

, which highlights the progression of DT representations from physical infrastructure to service ecosystems.

As shown in Table 2, DT paradigms correspond to increasing levels of abstraction and system integration. Each higher-level paradigm extends the representational scope of the previous one, moving from physical infrastructure to operational processes, service interactions, and finally ecosystem-level value creation. This hierarchical interpretation provides the conceptual basis for applying the proposed DT framework to complex infrastructures such as smart city energy systems, which are examined in the following section.

The proposed three-level composition should be understood as an architectural representation logic rather than as a fixed software implementation pattern. In practice, its implementation requires a layered integration of (a) asset-level digital representations, (b) process-level coordination models built on asset interactions, and (c) service-level models capturing actor-oriented service delivery and ecosystem value. Accordingly, the required architectural solution is not a single monolithic DT, but a multi-layer structure in which different representational levels are orchestrated through explicit mappings and cross-layer dependencies. Within the scope of this paper, the viability of this concept is demonstrated at the level of formal consistency and case-based structural instantiation rather than by numerical simulation or experimental benchmarking.

2.4. Model Interpretation and Implementation Logic

The proposed mathematical framework is intended as a conceptual system-modeling instrument rather than as a numerical simulation model in the narrow sense. Its implementation consists in identifying the relevant system elements at each representational level and assigning them to the corresponding DT paradigm.

In practical terms, implementation of the model involves four steps.

Step 1. The physical infrastructure of the considered system is decomposed into identifiable assets such as sensors, devices, communication components, computing nodes, and infrastructure units. These elements form the basis of the asset-centric DT layer.

Step 2. The interactions among these assets are analyzed to identify operational processes such as data acquisition, transmission, monitoring, control, and automation. These processes define the process-centric DT layer.

Step 3. The operational outcomes of these processes are interpreted in terms of services delivered to end users, operators, institutions, or market participants. These services define the service-centric DT layer.

Step 4. The relationships among services, actors, and dependencies are structured into an ecosystem-level representation, which forms the basis of the digital service ecosystem twin.

Thus, implementation of the model does not require that all layers be developed as software modules at once. Rather, the framework provides a systematic procedure for moving from infrastructure-level representation toward ecosystem-level representation. In the present study in Section 3, this logic is illustrated through the smart city electricity ecosystem, where assets, processes, services, and ecosystem interactions are mapped step by step into the proposed hierarchy.

3. Results

3.1. Smart City Electricity Ecosystem Case Study

To illustrate the practical implications of the proposed DT paradigm framework, this section presents a case study based on a smart-city electricity consumption and billing ecosystem. The example demonstrates how the three DT paradigms can be progressively constructed within a real-world urban infrastructure [28,29].

The considered system represents a simplified electricity accounting and management infrastructure in a smart city. In such an environment, electricity consumption is monitored through distributed metering devices installed in residential and commercial buildings. These devices automatically transmit consumption data through communication networks to centralized data-processing platforms, where operational monitoring, forecasting, and billing processes are performed. At a higher level, these processes enable a variety of digital services, including automated billing, tariff optimization, consumption forecasting, and energy management.

The layered transformation of this physical infrastructure into different DT paradigms is illustrated in Figure 4.

The figure illustrates how a physical electricity monitoring infrastructure in a smart city can be represented through three DT paradigms. At the lowest level, physical devices such as smart meters and communication infrastructure are represented through ACDTs capturing device states and performance. At the operational level, data flows and interactions between system components form PCDTs that model monitoring and control processes. At the ecosystem level, the system enables digital services such as billing, forecasting, tariff optimization, and energy management, which are represented through SCDTs capturing interactions among actors, services, and value creation mechanisms.

From the perspective of the paper’s stated contributions, the significance of this case study lies in the fact that it provides a single system context within which all three proposed DT paradigms can be demonstrated. Rather than treating asset-centric, process-centric, and service-centric DTs as separate conceptual possibilities, the case study shows how they emerge within one unified socio-technical system. In this sense, the case study functions as a results-level validation of the hierarchical interpretation proposed in the theoretical part of the paper.

3.2. System Purpose and Operational Context

The initial purpose of the system is the accurate accounting and efficient management of electricity consumption in an urban environment. Traditional manual measurement and billing systems often lead to delayed data collection, operational inefficiencies, and limited analytical capabilities.

The introduction of smart metering infrastructure allows electricity consumption to be monitored continuously and transmitted automatically to centralized data processing systems. As a result, the system evolves from a simple measurement infrastructure into a data-driven energy management ecosystem [28].

This transformation creates the conditions for the emergence of multiple DT representations corresponding to different levels of system abstraction.

3.3. Asset-Centric Digital Twin Taxonomy for the Smart City Electricity Ecosystem

The ACDT paradigm focuses on representing the physical infrastructure that forms the foundation of the smart city electricity ecosystem. In this paradigm, each physical component of the system is associated with a digital representation that captures its operational state, configuration, and performance characteristics.

Formally, the asset layer of the system can be represented as a set of physical components $R_A=\{A_1,A_2,\dots ,A_n\}$, whose digital representations constitute the ACDT structure $D{T}_A:R_A\to D_A$, where $R_A$ denotes the set of physical assets and $D_A$ denotes the corresponding set of digital representations.

The internal structure of the asset layer is illustrated in Figure 5, which presents a detailed taxonomy of asset-level components in the smart city electricity ecosystem.

The taxonomy organizes the infrastructure into six major categories that collectively define the physical foundation of the digital electricity ecosystem:

• Measurement and sensing infrastructure, which includes smart electricity meters, industrial energy meters, grid monitoring sensors, and transformer monitoring devices responsible for capturing electricity consumption and grid parameters.
• Communication infrastructure, which enables the transmission of measurement data and operational signals across the system through local communication interfaces, Internet of Things communication networks, network routing systems, and communication security components.
• Data processing infrastructure, which provides the computational environment required for storing, aggregating, and analyzing electricity consumption data through edge computing systems, centralized computing platforms, data storage systems, and data management infrastructure.
• Electricity distribution infrastructure, which represents the physical energy system itself and includes power generation sources, transmission networks, distribution networks, and grid control infrastructure responsible for maintaining system stability.
• Supporting infrastructure, which ensures the reliable operation of the ecosystem through environmental monitoring systems, operational support infrastructure such as backup power supply units, and cybersecurity infrastructure.
• Asset Digital Twin infrastructure, which represents the digital counterparts of physical system components, including device-level digital twins and infrastructure-level digital twins.

Taken together, these categories form a structured representation of the physical infrastructure that underlies the ACDT paradigm. Each component of this taxonomy corresponds to a class of physical assets whose operational states are captured and maintained within the digital twin environment.

However, while ACDTs provide detailed representations of individual infrastructure elements, they do not explicitly describe the operational interactions among assets. These interactions arise when assets participate in coordinated workflows such as data acquisition, communication, and system monitoring. Modeling these interactions requires a higher level of abstraction, which is addressed by PCDTs, discussed in the following section.

As a result, the asset-level taxonomy does more than describe the physical infrastructure of the smart city electricity ecosystem. It provides the first concrete result supporting the proposed framework by showing how the general DT mapping can be instantiated at the foundational asset level. This directly supports the contribution that the paper makes in distinguishing asset-centric DTs as the first layer of the proposed hierarchy.

3.4. Process-Centric Digital Twin Taxonomy for the Smart City Electricity Ecosystem

While Asset-Centric Digital Twins represent the physical infrastructure of the smart city electricity ecosystem, the PCDT paradigm focuses on modeling the operational workflows that connect these infrastructure components. These workflows define how measurement data are collected, transmitted, processed, monitored, and ultimately used for decision making and automation within the electricity management system.

Formally, the process layer of the system can be represented as a set of operational processes $R_P=\{P_1,P_2,\dots ,P_m\}$, whose digital representations constitute the PCDT structure $D{T}_P:R_P\to D_P$, where $R_P$ denotes the set of operational processes and $D_P$ denotes their digital representations within the DT environment.

The internal structure of the process layer is illustrated in Figure 6, which presents a detailed taxonomy of process-level components in the smart city electricity ecosystem.

The taxonomy organizes the operational workflows of the system into seven major process groups that collectively describe the dynamic behavior of the electricity monitoring ecosystem.

• The first category corresponds to data acquisition processes, which include electricity consumption measurement, sensor data collection, and distributed data synchronization processes responsible for generating and aligning measurement data across the system.
• The second category represents data transmission processes, which enable the transfer of measurement data from distributed devices to data processing platforms. These processes include local data transmission workflows, wide-area communication processes, and data routing mechanisms within the communication infrastructure.
• The third category consists of data processing processes, which transform raw measurement data into usable information. This category includes data aggregation processes, data cleaning and validation workflows, and analytical processing mechanisms such as consumption pattern analysis and demand forecasting.
• The fourth category includes monitoring processes, which continuously observe the operational state of the electricity infrastructure. These processes involve infrastructure monitoring, operational monitoring, and event detection mechanisms used to identify anomalies or system failures.
• The fifth category represents control and decision processes, which enable system operators to manage the electricity infrastructure through operational control workflows, automated control mechanisms, and decision support processes supporting planning and optimization.
• The sixth category includes operational automation processes, which automate routine operational tasks within the electricity ecosystem. These processes include automated meter reading workflows, billing preparation processes, and reporting and notification services.
• The seventh category represents process-level digital twin coordination processes, which integrate multiple operational workflows into coordinated system behavior. These processes include asset-to-process integration mechanisms and workflow orchestration processes that synchronize multiple processes across the digital twin environment.

Taken together, these categories provide a structured representation of the operational workflows that define the PCDT paradigm. Each category describes a class of processes whose execution states and interactions are captured and maintained within the digital twin system.

In contrast to ACDTs, which represent the state of individual infrastructure components, PCDTs capture the dynamic interactions between assets that define system behavior. These interactions form the operational backbone of the electricity monitoring ecosystem and enable the emergence of higher-level system functionality.

However, while PCDTs describe how infrastructure components interact within operational workflows, the ultimate purpose of the smart city electricity ecosystem lies in the delivery of digital services such as automated billing, consumption forecasting, tariff optimization, and energy management. These services emerge from coordinated processes and interactions across the electricity ecosystem.

From the viewpoint of the manuscript’s main contributions, the process-level taxonomy constitutes the second key result of the study. It demonstrates that DT representation can be extended from isolated infrastructure components toward operational workflows and system interactions. This result is important because it gives concrete form to the paper’s claim that process-centric DTs represent an intermediate abstraction level emerging from coordinated asset behavior.

Capturing these service interactions requires a further abstraction level of system representation, which is addressed by SCDTs, discussed in the following section.

3.5. Service-Centric Digital Twin Taxonomy for the Smart City Electricity Ecosystem

While ACDTs represent physical infrastructure and PCDTs model operational workflows, the SCDT paradigm focuses on representing the services that emerge from coordinated interactions among infrastructure components and operational processes.

At this level of abstraction, the smart city electricity ecosystem is interpreted as a service system, where electricity data, infrastructure capabilities, and operational workflows are integrated to generate value for consumers, grid operators, municipalities, and market participants.

Formally, the service layer of the ecosystem can be represented as a set of services $R_S=\{S_1,S_2,\dots ,S_k\}$, whose digital representations constitute the SCDT structure $D{T}_S:R_S\to D_S$, where $R_S$ denotes the set of services and $D_S$ represents their digital representations within the digital twin ecosystem.

The internal structure of the service layer is illustrated in Figure 7, which presents a detailed taxonomy of service-level components in the smart city electricity ecosystem.

The taxonomy organizes the service ecosystem into seven major service domains that collectively describe the functional capabilities of the smart electricity ecosystem.

• The first domain corresponds to consumer energy management services, which provide end users with tools for monitoring electricity consumption, optimizing energy usage, and receiving operational notifications related to electricity consumption and service availability.
• The second domain includes billing and financial services, which manage the financial transactions associated with electricity consumption. These services include automated billing mechanisms, payment processing workflows, and financial reporting services that support both consumers and electricity providers.
• The third domain represents grid management services, which support the operational management of the electricity infrastructure. These services include load monitoring mechanisms, demand forecasting services, and grid optimization tools used to maintain grid stability and operational efficiency.
• The fourth domain includes energy market services, which enable the participation of distributed energy resources and consumers in electricity markets. These services include dynamic pricing services, distributed energy integration mechanisms, and energy portfolio management tools.
• The fifth domain represents municipal and smart city services, which provide higher-level analytical capabilities for urban energy management. These services support city-level monitoring of electricity consumption, sustainability management, and infrastructure planning processes.
• The sixth domain corresponds to digital platform services, which provide the technological foundation for integrating data, applications, and service orchestration mechanisms across the electricity ecosystem.
• Finally, the seventh domain represents service ecosystem governance services, which ensure the reliable and secure operation of the electricity service ecosystem. These services include regulatory compliance mechanisms, cybersecurity services, and ecosystem governance processes that coordinate interactions among stakeholders.

Taken together, these domains define the functional architecture of the SCDT paradigm. Each category represents a class of services whose operational state, dependencies, and performance indicators are captured within the digital twin environment.

Unlike ACDTs and PCDTs, which focus on infrastructure and operational workflows, SCDTs represent value creation mechanisms within the ecosystem. At this level, digital twins no longer describe individual components or processes but instead represent the services delivered to ecosystem participants.

This shift from infrastructure modeling to service representation reflects the ontological transition. In this framework, services emerge as higher-level abstractions constructed from coordinated assets and operational processes.

Consequently, the SCDT layer provides the conceptual foundation for DSETs, where multiple services interact within a unified digital ecosystem representation.

The service-level taxonomy provides the third major result supporting the paper’s contribution. It shows how the proposed framework can represent not only infrastructure and operations, but also service delivery, actor interaction, and ecosystem-level value generation. In this respect, the result directly supports the claim that service-centric DTs form an ontologically distinct representational level and provide the basis for extension toward digital service ecosystem twins.

3.6. Cross-Layer Digital Twin Integration

The case study illustrates how the three DT paradigms form a hierarchical system of representations corresponding to increasing levels of system abstraction.

The asset, process, and service layers described in the previous sections represent different abstraction levels of the smart city electricity ecosystem. Each level corresponds to a distinct DT paradigm and a specific mathematical representation introduced earlier in the paper.

Table 3. Cross-layer mapping of DT paradigms in the smart city electricity ecosystem.

System Representation Layer	Physical System Elements	Digital Twin Paradigm	Mathematical Representation	Case Study Example
Asset layer	Smart meters, transformers, communication modules, data servers	Asset-centric digital twin	$D{T}_A:R_A\to D_A$	Device monitoring, meter status tracking
Process layer	Data acquisition, transmission workflows, monitoring loops	Process-centric digital twin	$D{T}_P:R_P\to D_P$	Data aggregation, anomaly detection, system monitoring
Service layer	Billing services, demand forecasting, tariff management	Service-centric digital twin	$D{T}_S:R_S\to D_S$	Automated billing, consumption forecasting, tariff optimization
Service ecosystem layer	Interacting services and actors across the energy ecosystem	Digital service ecosystem twin	$DSET:E\to D_E$	Smart-city electricity management platform

provides an operational interpretation of the proposed system model by linking each abstraction layer to its corresponding DT paradigm, mathematical representation, and case-study realization. In this sense, the table does not merely summarize the framework but demonstrates how the formal model can be implemented in practice through the identification of concrete system elements and their associated digital representations.

As shown in Table 3, the smart city electricity ecosystem can be interpreted as a hierarchical system in which different DT paradigms correspond to progressively higher abstraction levels of system representation.

3.7. APS-Matrix: A Knowledge-Representation Layer for the Smart City Electricity Ecosystem

The three introduced paradigms can be integrated into a unified ontological structure that captures the multi-level composition of the smart city electricity ecosystem. This structure is referred to as the APS-matrix, where APS denotes the three fundamental layers of the system: assets (A), processes (P), and services (S).

The APS-matrix therefore represents the structural relationships between physical infrastructure, operational processes, and the services that emerge from them within the ecosystem.

Formally, the APS-matrix can be defined as

$$APS=\left(A,P,S\right)$$

where $A$ denotes the set of asset-level components, $P$ denotes the set of process-level components, $S$ denotes the set of service-level components.

The APS-matrix provides a structured representation of how physical assets support operational processes and how these processes enable digital services. In this sense, it serves as a practical instantiation of the hierarchical DT relationships formalized earlier in the paper.

From an implementation perspective, the APS-matrix serves as a practical mapping tool that connects the three modeled abstraction layers. It shows which physical assets support which operational processes and which services emerge from these processes. Accordingly, the matrix can be interpreted as an implementation-oriented bridge between the formal mathematical framework and its domain-specific realization in the smart city electricity ecosystem.

Formal Interpretation

Let $A=\{A_1,A_2,\dots ,A_n\}$ be the set of asset categories, $P=\{P_1,P_2,\dots ,P_m\}$—the set of process categories, and $S=\{S_1,S_2,\dots ,S_k\}$—the set of service categories.

Then the APS-matrix can be interpreted as a structured mapping

$${\mathcal{M}}_{APS}:A\times P\to S$$

that identifies which combinations of assets and processes enable specific classes of services.

Equivalently, the matrix can be read as a layered ontological dependency structure:

$$A\Rightarrow P\Rightarrow S$$

where assets provide the material and digital infrastructure, processes coordinate their operation, and services deliver ecosystem-level functionality and value.

To illustrate the APS-matrix concept in the context of the smart city electricity ecosystem,

Table 4. APS-matrix of the smart city electricity ecosystem.

Asset-Level Components	Process-Level Components	Service-Level Components
Smart electricity meters	Electricity consumption measurement	Real-time consumption monitoring services
Smart electricity meters	Automated meter reading	Automated billing services
Smart electricity meters	Data acquisition workflows	Historical consumption reporting services
Communication gateways and IoT modules	Data transmission processes	Data platform services
Communication gateways and IoT modules	Communication routing processes	Open energy data access services
Data servers and cloud platforms	Data aggregation and storage	Energy analytics platform services
Data servers and cloud platforms	Analytical processing workflows	Consumption forecasting services
Data servers and cloud platforms	Event detection processes	High-consumption alert services
Transformers and substations	Load monitoring processes	Grid load monitoring services
Transformers and substations	Grid stability monitoring	Grid resilience analysis services
Distribution networks	Load balancing processes	Energy distribution optimization services
Distribution networks	Demand-response control processes	Dynamic tariff services
Edge computing nodes	Local processing workflows	Smart city energy management applications
Edge computing nodes	Process orchestration workflows	Cross-platform service integration services
Cybersecurity infrastructure	Security monitoring processes	Secure data exchange services
Cybersecurity infrastructure	Identity and access management processes	Ecosystem governance services
Regulatory and compliance infrastructure	Compliance monitoring processes	Regulatory compliance services
Environmental sensors	Weather-correlated data acquisition	Weather-correlated consumption forecasting services
Distributed generation units	Distributed generation coordination	Renewable energy integration services
Distributed generation units	Local energy balancing processes	Prosumer energy trading services

presents a structured mapping between asset-level infrastructure components, operational processes, and the services that emerge from their interaction. Each row of the table represents a specific dependency path connecting a physical asset to the operational processes it supports and to the services that these processes enable.

APS-matrix reveals the ontological dependencies that connect the physical, operational, and service layers of the smart city electricity system. This representation provides a practical instantiation of the APS structure.

Taken together, the results of the case study support the main contributions claimed earlier in the paper in three ways. First, they demonstrate that asset-centric, process-centric, and service-centric DTs can be identified and represented as distinct levels of system abstraction within a single socio-technical environment. Second, they show that these levels can be connected through explicit cross-layer mappings, thereby supporting the hierarchical interpretation proposed in the mathematical framework. Third, they illustrate how the transition from infrastructure representation to service ecosystem representation can be operationalized through the APS-matrix. Therefore, Section 3 provides concrete structural support for the conceptual and mathematical contribution stated in the Abstract and in Section 1.3.

3.8. Concept-Level Validation of the Proposed Framework

Within the scope of this study, the proposed DT concept is not tested through numerical simulation or software benchmarking, but through structured concept-level validation. In this context, validation means examining whether the proposed framework can consistently represent a complex socio-technical system across the asset, process, and service levels, while preserving explicit cross-layer relationships between these levels.

The smart city electricity ecosystem provides the basis for this validation. First, the framework is instantiated at the asset level through the identification of physical infrastructure components and their digital representations. Second, the framework is tested at the process level by showing that operational workflows can be systematically derived from interactions among these assets. Third, the framework is tested at the service level by demonstrating that coordinated processes can be interpreted as service structures involving actors, dependencies, and ecosystem value.

The validity of the concept is further supported by the cross-layer mapping in Table 3 and by the APS-matrix in Table 4. These results show that the proposed framework can represent not only isolated DT views, but also their hierarchical integration within one coherent system representation. In this sense, the case study does not merely illustrate the framework but serves as a structured test of its representational consistency, cross-level completeness, and practical interpretability.

At the same time, this form of validation should be understood as conceptual and architectural rather than numerical or experimental. Further research may extend the present work toward simulation-based, implementation-oriented, or domain-specific empirical validation.

4. Discussion

The results presented in the previous sections demonstrate that DT technologies can be interpreted as an evolving set of paradigms corresponding to different levels of system representation. This section discusses the theoretical implications of the proposed framework, focusing on the ontological evolution of DT models, the role of service-centric digital twins, and the orchestration of multiple DT paradigms within complex digital infrastructures.

4.1. Theoretical Implications of the Ontological Digital Twin Hierarchy

One of the central contributions of this study is the interpretation of DT technologies through the concept of ontological system levels. Traditional DT research has largely focused on representing physical objects or operational processes [5,8,12,15]. However, modern digital infrastructures increasingly operate as complex socio-technical ecosystems where services, actors, and interactions define system behavior.

The framework proposed in this paper identifies three distinct but related ontological levels of DT representation:

• Asset-centric digital twins, representing individual physical entities.
• Process-centric digital twins, representing operational processes formed by interactions between assets.
• Service-centric digital twins, representing service ecosystems and value creation mechanisms.

This interpretation suggests that DT technologies are evolving from object-based representations toward ecosystem-level models. Such an evolution reflects broader trends in digital infrastructures, where system performance is increasingly evaluated in terms of service delivery and ecosystem outcomes rather than individual component performance.

From a theoretical perspective, this ontological interpretation clarifies the relationships between existing DT implementations and provides a conceptual foundation for future DT architectures.

From this perspective, the main theoretical contribution of the study is not limited to classification. The proposed framework explains why these paradigms should be interpreted as nested ontological levels of representation rather than as interchangeable labels. This distinction is important because it changes the logic of DT architecture design: service-oriented representations are not merely broader applications of asset-level twins, but require a different representational focus centered on services, actors, dependencies, and value creation.

To further clarify the methodological contribution of the proposed framework, it is useful to compare it with the principal streams of DT research discussed in the literature. Although existing studies have made substantial contributions to asset-level modeling, process-oriented system representation, and service-oriented digital environments, these approaches are often presented as separate perspectives rather than as formally related levels of abstraction.

Table 5. Comparison of representative DT approaches and the proposed hierarchical framework.

Approach Type	Primary Unit of Representation	Main Methodological Focus	Level of Abstraction	Main Limitation in Existing Literature	Difference from the Proposed Framework
Asset-centric Digital Twin approaches	Physical assets, devices, infrastructure components	State monitoring, simulation, lifecycle representation, predictive maintenance	Component level	Usually focused on isolated physical entities and their digital counterparts	Interpreted in this paper as the foundational layer of a broader hierarchical framework
Process-centric Digital Twin approaches	Operational workflows, interactions among assets, control logic	Process monitoring, workflow modeling, system coordination	Operational level	Often addresses system behavior, but rarely formalizes its relation to lower and higher DT paradigms	Interpreted in this paper as an intermediate representational layer emerging from interacting assets
Service-oriented Digital Twin approaches	Services, platforms, user interactions, value delivery structures	Service coordination, value creation, platform-based interaction	Service/ecosystem level	Often conceptually broad, but insufficiently integrated with asset- and process-level DT formalisms	Interpreted in this paper as a distinct ontological layer formally connected to lower-level DT paradigms
Proposed hierarchical framework	Assets, processes, services, and service ecosystems	Formal representation of hierarchical evolution across DT paradigms	Multi-level	—	Provides a unified conceptual and mathematical structure linking DT paradigms through mappings, propositions, and theorems

summarizes this comparison and highlights the specific difference of the framework proposed in this paper.

As shown in Table 5, the difference of the proposed framework lies not simply in discussing one more DT category, but in providing a formal logic that connects previously fragmented perspectives into a single hierarchical model. In this interpretation, asset-centric, process-centric, and service-centric DTs are not competing alternatives, but successive ontological levels of representation corresponding to increasing system abstraction. This comparison helps clarify that the contribution of the study is methodological and theoretical: it explains how different DT paradigms can be understood within one coherent representational architecture.

4.2. Comparative Positioning with Existing Digital Twin Frameworks

After the proposed framework has been formally defined, it is useful to clarify its relationship to existing streams of DT research. The purpose of this comparison is not to present the APS/DSET framework as a replacement for established DT approaches, but to specify the particular representational gap that it addresses.

Existing DT research already provides several important foundations for representing complex systems. Asset-centric DT studies emphasize physical entities, virtual models, synchronization mechanisms, and lifecycle monitoring. Five-dimensional DT models extend this view by incorporating physical entities, virtual models, services, data, and connections. Process-oriented and system-of-systems DT approaches further broaden the scope of representation toward workflows, interacting subsystems, and coordinated infrastructures. Service-oriented architectures introduce service delivery and value co-creation as important design concerns. Digital-twin ontologies and knowledge graphs provide semantic mechanisms for representing entities, relations, and reasoning structures.

However, these approaches usually emphasize different representational dimensions rather than explicitly organizing asset-, process-, service-, and ecosystem-level entities within one cross-layer hierarchy. In particular, existing frameworks often either focus on physical and operational system representation, or include services as functional elements without systematically formalizing the transition from assets to processes, from processes to services, and from services to ecosystem-level structures involving actors, dependencies, and value-generation mechanisms. The APS/DSET framework addresses this gap by defining a layered representation in which assets, processes, services, actors, dependencies, and ecosystem value are connected through explicit cross-layer relations.

Table 6. Comparison of representative DT-related approaches and the proposed APS/DSET framework.

Existing Framework or Research Stream	Represented Entities	Treatment of Services, Actors, and Value	Support for ML/Knowledge Extraction	What the Proposed APS/DSET Framework Adds
Asset-centric DT and product-lifecycle DT approaches	Physical assets, products, components, sensors, virtual models, lifecycle states	Services, actors, and value are usually secondary or external to the core asset representation	Supports condition monitoring, anomaly detection, predictive maintenance, and lifecycle analytics at the asset level	Positions asset-level DTs as the foundational layer of a broader hierarchy and connects asset states to process and service outcomes
Five-dimensional DT models	Physical entity, virtual model, service, data, and connection dimensions	Services are included as one dimension, but actor structures, service dependencies, and ecosystem-level value logic are usually not the primary object of formalization	Provides a broad architecture for data integration and intelligent services, but does not by itself define cross-layer APS relations for learning tasks	Interprets services as a distinct representational level and formalizes how asset, process, and service entities are connected through explicit dependency mappings
Process-oriented and business-process DT approaches	Workflows, operational processes, activities, events, information flows, and process performance indicators	Services may be treated as outputs of processes, but actor roles and value co-creation are not always modeled as part of the DT structure	Supports process mining, workflow optimization, bottleneck detection, and event prediction	Places process-centric DTs between asset-level and service-level representations, making process models traceable to both physical assets and service outcomes
System-of-systems DT approaches	Interconnected infrastructures, subsystems, platforms, networks, and operational environments	Services and actors may appear at the system level, but value-generation mechanisms are often implicit or domain-specific	Supports coordination, simulation, interoperability, and system-level analytics across heterogeneous DTs	Provides an explicit ontological layering in which system-of-systems integration is connected to asset–process–service dependencies and ecosystem-level service representation
Service-oriented DT and DT-for-services approaches	Services, service functions, service delivery processes, and digital service environments	Services are central; actors and value may be considered, but the lower-level asset and process foundations of services are not always formally traceable	Supports service monitoring, service optimization, and service-oriented decision support	Connects service-centric DTs to the asset and process layers that enable them and extends service representation toward DSET as an ecosystem-level structure
Digital-twin ontologies and semantic DT approaches	Classes, properties, semantic relations, domain concepts, and interoperability structures	Services, actors, and value can be represented if included in the ontology, but their hierarchical relation to assets and processes depends on the ontology design	Supports semantic interoperability, reasoning, rule-based inference, and knowledge representation	Provides a domain-independent APS schema that can guide ontology construction by specifying the required asset, process, service, actor, and value relations
Digital-twin knowledge graphs	Graph nodes and edges representing entities, states, relations, events, and sometimes processes or services	Services, actors, and value can be represented as graph entities, but the graph may not distinguish DT abstraction levels unless these levels are explicitly modeled	Supports graph reasoning, link prediction, graph-based learning, knowledge completion, and explainable inference	Defines the APS-matrix and DSET structure as a graph-ready schema for cross-layer knowledge representation and future ML/knowledge-extraction pipelines
Proposed APS/DSET framework	Assets, processes, services, actors, dependencies, and ecosystem-level value structures	Services, actors, and value are treated as explicit representational elements of the service-centric and ecosystem-level DT layers	Provides a structured basis for graph-based prediction, cross-layer attribution, anomaly propagation analysis, and knowledge-graph enrichment	Integrates asset-centric, process-centric, service-centric, and ecosystem-level DTs into one hierarchical representation with explicit cross-layer traceability

summarizes the position of the proposed framework relative to representative DT-related approaches.

The comparison shows that the proposed framework occupies a specific methodological position. It does not replace asset-centric, process-oriented, system-of-systems, service-oriented, ontological, or knowledge-graph-based DT approaches. Instead, it provides a unifying cross-layer representation that explains how these approaches can be related within a common hierarchy of system abstraction.

This comparison also clarifies the relationship between the proposed framework and machine learning or knowledge extraction. Existing ML-oriented DT studies often focus on prediction or anomaly detection at the asset or process level, while ontology- and knowledge-graph-based approaches focus on semantic representation and reasoning. The APS/DSET framework adds a cross-layer schema that specifies where entities, features, relations, and targets are located in the DT hierarchy. As a result, it can support future graph-based learning, knowledge-graph completion, cross-layer attribution, and service-level prediction tasks by making asset–process–service dependencies explicit.

4.3. Role of Service-Centric Digital Twins

The introduction of SCDTs represents a key conceptual extension of traditional DT architectures. While asset-level and process-level DTs focus on monitoring physical infrastructure and operational processes, SCDTs shift the focus toward the services delivered by digital infrastructures.

In many contemporary systems, including smart cities, energy infrastructures, transportation networks, and digital platforms, system objectives are increasingly defined in terms of service outcomes. These outcomes include service availability, service quality, user experience, and ecosystem value.

Under such conditions, DT models that focus exclusively on assets or processes may provide incomplete representations of system behavior. SCDTs address this limitation by modeling the relationships between services, actors, and service dependencies within a digital ecosystem.

The smart city energy example analyzed in Section 3 illustrates this transition. In this case, the introduction of smart meters initially enables asset-level monitoring. Subsequent integration of communication networks and data processing systems creates process-level DTs. Finally, the availability of digital infrastructure enables the creation of energy-related services such as automated billing and demand-response mechanisms, which together form a service ecosystem.

Within this context, SCDTs provide the most appropriate representation of system behavior. This interpretation also clarifies the manuscript’s original contribution. The paper does not treat service-centric digital twins as a terminological extension of existing DT models, but as a formally distinct representation layer with its own ontological referents and analytical purpose. In this sense, the study contributes to the theoretical maturation of Digital Twin research by articulating the conditions under which service-centric representations emerge as a necessary level of abstraction.

4.4. Orchestration of Digital Twin Paradigms

Although the three DT paradigms represent different levels of abstraction, they should not be interpreted as mutually exclusive models. Instead, they form complementary layers within a hierarchical DT architecture [21,22,23,24].

The integration of these paradigms can be described as DT orchestration, where multiple DT models operate simultaneously within a unified system architecture. In such architectures, asset-level DT provide data streams that feed process-level models, while process-level models provide aggregated information supporting service-level ecosystem representations.

This layered orchestration enables DT systems to capture system behavior across multiple levels of abstraction, ranging from physical infrastructure to service ecosystems.

At this point, an important practical distinction should be made between two possible architectural strategies. The first strategy relies on a set of separate DT models connected mainly through data exchange interfaces. The second strategy, proposed in this paper, relies on a coordinated multi-level framework in which asset-level, process-level, and service-level representations are interpreted within a shared cross-level logic. Although both strategies may support operational data exchange, they differ substantially in their ability to represent dependency structures, preserve semantic consistency across abstraction levels, and support system-wide interpretation and decision making. To clarify this difference,

Table 7. Interoperable separate DT models vs. integrated multi-level DT framework.

Criterion	Separate DT Models with Data Transfer	Integrated Multi-Level DT Framework
Main design logic	Independent models exchange selected data through interfaces	Multiple DT levels are coordinated through a shared cross-level representational logic
Primary objective	Local monitoring, isolated analytics, subsystem-specific optimization	Cross-level system interpretation, coordinated analysis, and ecosystem-oriented decision support
Relationship between levels	External linkage through data transfer only	Explicit structural linkage between assets, processes, services, and ecosystem outcomes
Semantic consistency across abstraction levels	Potentially fragmented; meanings may differ between models	Stronger consistency due to shared hierarchical interpretation
Traceability from asset state to service outcome	Indirect and often difficult to reconstruct	Explicitly supported through cross-layer mappings
Support for cross-layer diagnosis	Limited	Stronger, because dependencies across representational levels are retained
Support for service-oriented management	Partial and model-dependent	Built into the framework through service- and ecosystem-level representation
Architectural form	Typically distributed and loosely coupled	May remain technically distributed, but is conceptually and analytically integrated
Suitability for narrow operational tasks	High	Also suitable, but potentially more elaborate than necessary for purely local tasks
Suitability for complex socio-technical systems	Limited when system-wide coordination is required	High, especially when service outcomes and ecosystem effects must be interpreted together
Main practical limitation	Weak cross-level interpretability despite data availability	Greater modeling effort and stronger coordination requirements
Main advantage	Simplicity and modularity	Cross-level coherence, traceability, and richer decision support

compares the two approaches from a practical and architectural perspective.

As shown in Table 7, the proposed framework should not be understood as requiring a single monolithic DT implementation. Its main advantage lies instead in providing an integrated representational logic across multiple abstraction levels. In this sense, the framework goes beyond simple interoperability between isolated models by making explicit how asset states influence operational processes and how these processes affect service outcomes and ecosystem-level performance. This distinction is particularly important in complex socio-technical systems, where the practical value of a DT increasingly depends not only on data availability, but also on the ability to interpret cross-level relationships in a consistent and decision-relevant way.

4.5. Methodological Principles for Digital Twin Orchestration

The hierarchical relationship between DT paradigms identified in this study implies that practical DT architectures require mechanisms for coordinating multiple DT models operating at different levels of abstraction. This coordination can be described as DT orchestration, which integrates asset-level, process-level, and service-level models within a unified analytical framework [21,22,24].

The proposed framework suggests a methodological procedure for constructing SCDTs through progressive integration of lower-level DT models.

The orchestration process can be interpreted as a sequence of four conceptual stages.

Stage 1. Asset Digitalization

At the initial stage, physical infrastructure components are represented through ACDTs. These models capture the operational state of individual system elements and provide the primary data sources for higher-level DT models.

Formally, this stage corresponds to the mapping $D{T}_A:R_A\to D_A$.

Stage 2. Process Integration

ACDTs are integrated into operational workflows that represent system processes. PCDTs describe interactions between assets and enable monitoring and analysis of system operations.

This stage corresponds to the mapping $D{T}_P:R_P\to D_P$.

Stage 3. Service Abstraction

Operational processes are interpreted in terms of the services they enable. At this level, DT models capture relationships between services, actors, and service dependencies.

This leads to the emergence of SCDTs represented by $D{T}_S:E\to D_S$.

Stage 4. Ecosystem Integration

Finally, multiple services are integrated into a service ecosystem represented by a DSET. At this stage, DT architectures capture ecosystem-level interactions and value creation mechanisms $DSET:E\to D_E$.

This progressive orchestration process enables DT systems to evolve from component-level monitoring tools into ecosystem-level analytical platforms capable of supporting complex digital infrastructures.

4.6. Algorithmic Construction of Service-Centric Digital Twins

The theoretical framework introduced in this study suggests that SCDTs emerge from structured interactions between asset-level infrastructure and operational processes. While ACDTs represent individual system components and PCDTs capture operational workflows, SCDTs arise when system representation is oriented toward service outcomes delivered within a digital ecosystem.

This process can be formalized as the following algorithm (Algorithm 1).

Algorithm 1. Construction of Service-Centric Digital Twins

Input:

$\mathrm{Asset} \mathrm{set}: A=\{A_1,A_2,\dots ,A_n\}$

$\mathrm{Process} \mathrm{set}: P=\{P_1,P_2,\dots ,P_m\}$

$\mathrm{Service} \mathrm{set}: S=\{S_1,S_2,\dots ,S_k\}$

Output:

$\mathrm{SCDT} D{T}_S$ representing the digital service ecosystem.

• Step 1. Asset Identification

Identify the set of physical infrastructure assets participating in the system.

Construct ACDTs $D{T}_A=\{D{T}_{A_1},D{T}_{A_2},\dots ,D{T}_{A_n}\}$ representing the digital states of individual assets.

• Step 2. Process Modeling

Identify operational processes connecting assets.

Construct PCDTs $D{T}_P=(D{T}_A,\mathsf{\Phi })$, where $\mathsf{\Phi }$ represents operational workflows linking asset states.

• Step 3. Service Extraction

Identify service outcomes produced by system processes.

Define service set $S=\{S_1,S_2,\dots ,S_k\}$ representing digital services delivered by the ecosystem.

• Step 4. Service Dependency Mapping

Construct dependency mapping $\mathsf{\Psi }:P\to S$ linking operational processes to service outcomes.

• Step 5. Service Ecosystem Construction

Define service ecosystem structure $E=(S,U,\Gamma )$, where $S$—services, $U$—actors, $\Gamma$—service dependencies.

• Step 6. Service-Centric Digital Twin Formation

Construct the SCDT $D{T}_S:E\to D_E$ representing the digital model of the service ecosystem.

Algorithm 1 formalizes the conceptual transition from infrastructure-level system representation toward service-level ecosystem modeling. The algorithm operationalizes the ontological progression described in Section 2 and provides a practical methodology for constructing SCDTs in complex digital ecosystems.

The algorithm also illustrates how SCDTs naturally extend the hierarchy of DT paradigms. Once services become the primary representation layer of the system, interactions between services and actors form a broader DSET, confirming the theoretical relation introduced earlier in the paper.

4.7. Machine Learning and Knowledge Extraction in Layered Digital Twin Architectures

The hierarchical interpretation proposed in this paper has direct implications for where machine learning and knowledge extraction operate within a DT, since each representational level fixes a different unit of observation and therefore a different learning problem. At the asset level, the mapping $D{T}_A:R_A\to D_A$ defines per-component state vectors $x_i\left(t\right)$, which are the natural inputs for the supervised and unsupervised models used in condition monitoring, anomaly detection, and remaining-useful-life estimation. At the process level, the mapping $D{T}_P:R_P\to D_P$ exposes relations between assets as a process graph $G_P=\left(R_A,R_P\right)$, so that the learning tasks become relational: workflow classification, bottleneck and event detection, and process forecasting are most naturally addressed with sequence models and graph-based learning over $G_P$. At the service level, the mapping $D{T}_S:E\to D_S$ and the ecosystem tuple $E=\left(S,U,\Gamma \right)$ make services, actors, and dependencies the primary entities, so the relevant tasks shift toward demand and value forecasting, recommendation and coordination among actors, and learning over the dependency structure $\Gamma$.

The same hierarchy organizes knowledge extraction. The taxonomies introduced in Section 3 and the APS-matrix ${\mathcal{M}}_{APS}:A\times P\to S$ can be read as an explicit knowledge-representation layer in which nodes correspond to assets, processes, and services, and edges encode the asset $\Rightarrow$ process $\Rightarrow$ service dependency structure. This structure provides the schema against which extracted facts are interpreted, and it allows knowledge obtained at one level to be propagated to adjacent levels: asset-level fault evidence, for example, can be lifted through the process layer into a statement about degraded service value $V_E\left(t\right)$. In this sense the framework distinguishes data-level learning, which estimates the digital representations $D_A$, $D_P$, and $D_S$ from observations, from knowledge-level extraction, which recovers the dependencies $\Gamma$ and the APS relations that connect the levels.

The APS-matrix can also be interpreted as a machine-learning-ready graph representation. In the tabular form introduced in Section 3.7, each row describes a dependency path connecting an asset-level component, a process-level component, and a service-level component. For learning and knowledge-extraction purposes, this structure can be transformed into a heterogeneous graph in which different types of nodes correspond to the three representational layers of the proposed hierarchy.

Let the APS graph be defined as

$$G_{\mathrm{A}\mathrm{P}\mathrm{S}}=\left(V_A\cup V_P\cup V_S,\hspace{0.33em}{E}_{AP}\cup E_{PS}\cup E_{SS}\right)$$

where $V_A$ is the set of asset nodes, $V_P$ is the set of process nodes, and $V_S$ is the set of service nodes. The edge set $E_{AP}$ represents support relations between assets and processes, $E_{PS}$ represents enabling relations between processes and services, and $E_{SS}$ represents dependency or interaction relations among services. In this form, the APS-matrix is not only a descriptive cross-layer mapping, but also a structured graph schema on which learning and knowledge-extraction methods can operate.

Feature matrices can be assigned to each node type. Let

$$X_A\left(t\right)={\left[x_a\left(t\right)\right]}_{a\in V_A}$$

denote asset-level features, such as meter readings, device status, communication quality, transformer load, or cybersecurity indicators. Similarly,

$$X_P\left(t\right)={\left[x_p\left(t\right)\right]}_{p\in V_P}$$

denotes process-level features, such as data-acquisition frequency, transmission delay, aggregation completeness, event-detection rates, or workflow reliability. Finally,

$$X_S\left(t\right)={\left[x_s\left(t\right)\right]}_{s\in V_S}$$

denotes service-level features, such as billing accuracy, forecasting error, alert frequency, service availability, tariff-response performance, or user-oriented value indicators.

A generic graph-based learning task over the APS representation can then be expressed as

$${\widehat{y}}_s\left(t+h\right)=f_{\theta }\left[G_{APS},X_A\left(t\right),X_P\left(t\right),X_S\left(t\right)\right]$$

where ${\widehat{y}}_s\left(t+h\right)$ denotes a predicted service-level outcome at forecasting horizon $h$, and $f_{\theta }$ is a learning model parameterized by $\theta$. Depending on the task, ${\widehat{y}}_s\left(t+h\right)$ may represent predicted service degradation, service value, demand level, tariff-response outcome, cybersecurity risk, or the probability of disruption in a service supported by lower-level assets and processes.

This formulation clarifies the role of the APS-matrix in machine learning without claiming that a trained model is implemented in the present study. The matrix defines the representational substrate for future empirical pipelines: it specifies which entities are modeled, which cross-layer relations are available, which features can be assigned to each level, and which service-level targets can be predicted or explained. For example, a graph neural network could be used to propagate asset-level degradation signals through process dependencies toward service-level outcomes, while a knowledge-graph completion method could infer missing asset-process or process-service relations. Forecasting tasks could be evaluated using metrics such as MAE, RMSE, or MAPE, classification tasks using precision, recall, F1-score, and AUROC, and knowledge-graph completion tasks using MRR, Hits@k, or link-prediction F1-score.

Thus, the APS-matrix provides a bridge between the conceptual hierarchy developed in this paper and future computational implementations. It transforms the proposed asset-process-service representation into a graph-structured input space suitable for graph-based prediction, cross-layer attribution, anomaly propagation analysis, and knowledge-graph extraction. The empirical training and benchmarking of such models remain outside the scope of the present article and are identified as a direction for future research.

To make this methodological contribution more explicit, the APS-matrix can be interpreted not only as a conceptual mapping between system layers, but also as a reference representation for machine-learning and knowledge-extraction tasks. In this interpretation, asset-level components, process-level operations, and service-level outcomes form a heterogeneous dependency structure in which each node type defines a specific class of features and each cross-layer link defines a possible path for prediction, attribution, or knowledge propagation. The objective of the present paper is not to report a trained model or an empirical benchmark, but to specify how such models can be instantiated on the basis of the proposed hierarchical representation.

Table 8. Reference ML and knowledge-extraction instantiation over the APS matrix.

Task	Input	Representation	Output	Possible Model	Evaluation Metrics
Asset-state anomaly detection	Asset state vectors $x_a\left(t\right)$ from smart meters, gateways, servers, transformers	Asset nodes $A$	Anomaly score ${\widehat{z}}_a\left(t\right)$	Isolation Forest, autoencoder, temporal model	Precision, recall, F1, AUROC
Process disruption prediction	Asset states + asset-process links $E_{AP}$	Process graph	Probability of process degradation ${\widehat{y}}_p(t+h)$	GNN, temporal GNN, sequence model	F1, AUROC, lead time
Service-value forecasting	Asset/process states + APS links $E_{AP},E_{PS}$	Heterogeneous APS graph	Predicted service value ${\widehat{v}}_s(t+h)$	R-GCN, GAT, graph aggregation + regression	MAE, RMSE, MAPE
Service-impact attribution	Predicted degradation paths $A\to P\to S$	APS dependency paths	Ranked list of responsible assets/processes	Graph explainability, path attribution	Precision@k, Recall@k
Knowledge-graph completion	APS triples: asset–supports–process; process–enables–service	Knowledge graph	Missing or inferred APS links	KG embedding, rule-based extraction	MRR, Hits@k, link-prediction F1

therefore summarizes a reference machine-learning and knowledge-extraction instantiation over the APS-matrix. For each representative task, the table identifies the corresponding input data, representation structure, expected output, possible model class, and evaluation metrics. This makes explicit how the proposed framework can support concrete analytical pipelines while preserving the conceptual scope of the study.

Making these levels explicit addresses a recurring difficulty in applying machine learning to digital infrastructures: models trained on isolated asset or process signals often cannot be related to service-level outcomes, because the connecting structure is left implicit. The layering relation $D{T}_A\preccurlyeq D{T}_P\preccurlyeq D{T}_S\preccurlyeq D{T}_E$ defined in Section 2.2 specifies how lower-level representations are embedded into higher-level ones, and therefore how features and predictions can be aggregated upward and how service-level objectives can be attributed downward to the assets and processes responsible for them. This cross-level traceability is what allows learned models to support service- and ecosystem-oriented decision making rather than only local optimization.

The present study formalizes this structure rather than reporting trained models, and the smart-city electricity ecosystem is used to validate the representation at the level of structural and cross-layer consistency. Within this scope, the contribution to machine learning and knowledge extraction is methodological: the framework specifies the entities, features, and relations on which models and extraction methods act at each level, together with the conditions under which results obtained at one level can be combined with those at another. Instantiating concrete learning pipelines, such as graph-based service-value forecasting over $\Gamma$ or knowledge-graph extraction from the APS-matrix and benchmarking them on operational data is identified as the principal direction for empirical follow-up work.

To make the methodological role of the proposed APS representation more operational, the reference learning and knowledge-extraction logic can be summarized as an Algorithm 2. This algorithm does not describe an empirical experiment performed in the present study. It specifies how the proposed hierarchy can be transformed into a concrete analytical pipeline once operational data, labels, and validation criteria are available.

Algorithm 2. Reference APS-Based Learning and Knowledge-Extraction Pipeline

Input: APS-matrix describing asset-process-service dependencies. asset-level observations $X_A\left(t\right)$; process-level indicators $X_P\left(t\right)$; service-level indicators or target variables $X_S\left(t\right)$; optional labels $Y\left(t\right)$, such as service degradation events, service-value scores, demand levels, anomaly classes, or known APS relations. Output: predicted service-level outcome ${\widehat{y}}_s\left(t+h\right)$; inferred or completed APS relations. ranked cross-layer impact paths from assets to processes and services. task-specific performance metrics.

Step 1. APS Graph Construction

Transform the APS-matrix into a heterogeneous graph

$$G_{\mathrm{A}\mathrm{P}\mathrm{S}}=\left(V_A\cup V_P\cup V_S;\hspace{0.33em}{E}_{AP}\cup E_{PS}\cup E_{SS}\right)$$

where $V_A$, $V_P$, and $V_S$ denote asset, process, and service nodes, respectively. The edge set $E_{AP}$ represents asset-process support relations, $E_{PS}$ represents process-service enabling relations, and $E_{SS}$ represents service-level dependency or interaction relations.

Step 2. Feature Assignment

Assign feature vectors to each node type. Asset nodes receive infrastructure and operational features, such as meter readings, device status, communication quality, load level, or fault indicators. Process nodes receive workflow and operational features, such as data-transmission delay, aggregation completeness, monitoring frequency, control-event frequency, or process reliability. Service nodes receive service-level features, such as billing accuracy, forecasting error, service availability, tariff-response performance, user demand, or service-value indicators.

Step 3. Task and Target Definition

Define the analytical task to be solved on the APS graph. Depending on the application, the target may be formulated as service-value forecasting, service degradation prediction, anomaly propagation analysis, knowledge-graph completion, or cross-layer impact attribution. A generic prediction task can be expressed as

$${\widehat{y}}_s\left(t+h\right)=f_{\theta }\left[G_{\mathrm{A}\mathrm{P}\mathrm{S}},X_A\left(t\right),X_P\left(t\right),X_S\left(t\right)\right]$$

where $h$ is the prediction horizon and $f_{\theta }$ is a learning model parameterized by $\theta$.

Step 4. Model Selection

Select a model class corresponding to the task structure. Forecasting tasks may use regression, temporal models, or graph-based temporal learning. Classification tasks may use supervised learning, anomaly-detection models, or graph neural networks. Knowledge-extraction tasks may use rule-based extraction, knowledge-graph embedding, link prediction, or graph-completion methods. The choice of model depends on the availability of time-series data, labels, and validated cross-layer dependency relations.

Step 5. Model Training or Inference

Train the selected model when labeled or historical operational data are available. If labeled data are not available, apply unsupervised or semi-supervised methods to detect anomalous nodes, infer missing relations, or identify structurally important dependency paths. In both cases, the APS graph constrains the learning process by preserving cross-layer traceability between asset-level observations, process-level mechanisms, and service-level outcomes.

Step 6. Evaluation

Evaluate the output using task-specific metrics. Forecasting tasks can be evaluated using MAE, RMSE, MAPE, or similar error measures. Classification and anomaly-detection tasks can be evaluated using precision, recall, F1-score, AUROC, or AUPRC. Knowledge-graph completion and link-prediction tasks can be evaluated using MRR, Hits@k, link-prediction F1-score, or precision@k [22,40]. Cross-layer attribution can be evaluated by comparing the ranked asset-process-service paths with expert-validated dependency paths or historical incident records.

Step 7. Knowledge Extraction and Cross-Layer Interpretation

Translate the model output into interpretable cross-layer knowledge. For example, an asset-level fault pattern may be propagated through process dependencies into a predicted degradation of a billing, forecasting, or energy-management service. Similarly, missing APS links inferred by a knowledge-graph completion model may indicate previously unrecognized dependencies between infrastructure components, operational workflows, and service outcomes.

The algorithm clarifies how the proposed APS representation can support concrete machine-learning and knowledge-extraction pipelines by defining the input structure, graph representation, possible analytical targets, model classes, and evaluation criteria.

4.8. Limitations and Future Research

Despite the conceptual contributions of this study, several limitations should be acknowledged. First, the analysis presented in this work focuses primarily on conceptual and mathematical modeling of DT paradigms. While the smart city energy ecosystem provides a useful illustrative example, further empirical validation in real-world digital infrastructures would strengthen the proposed framework. The absence of an implemented ML pipeline is a deliberate scope limitation of the present paper. The proposed APS representation defines the input structure for such pipelines, but empirical training and benchmarking require operational time-series data, labeled service outcomes, and validated ground-truth dependency relations.

Second, the mathematical formalization introduced in this study represents a simplified abstraction of complex socio-technical systems. Real-world service ecosystems often involve additional layers of complexity, including regulatory constraints, economic incentives, and behavioral interactions among actors. Future research could extend the proposed framework to incorporate such factors within more comprehensive ecosystem models.

Third, the orchestration of multiple DTs within large-scale infrastructures raises significant technical challenges related to interoperability, data integration, and governance. Developing scalable architectures capable of integrating heterogeneous DT models remains an important direction for future research.

The concept of digital service ecosystem twins opens several promising research directions, including ecosystem-level simulation, value optimization, and the integration of DT technologies with artificial intelligence and data-driven decision-support systems.

5. Conclusions

This paper has examined how digital twins evolve from representations of physical assets toward models of operational processes and service ecosystems, and has argued that these interpretations are best treated as successive levels of system abstraction rather than competing labels. To this end, it developed a conceptual and mathematical framework that distinguishes asset-centric, process-centric, and service-centric DTs, models each as a mapping between real-world entities and their digital representations, and connects them through explicit cross-layer dependencies. A set of definitions, modeling assumptions, constructions, and propositions, together with a cross-layer traceability theorem, establishes service-centric twins as a distinct representational level that cannot be reduced to asset and process descriptions alone, and extends the framework to the ecosystem level as a digital service ecosystem twin.

The framework was instantiated through a smart-city electricity ecosystem, in which asset, process, and service taxonomies and the asset–process–service matrix express the dependencies linking infrastructure, operations, and services. Within the scope of this study the framework is validated for structural and cross-layer consistency rather than by numerical simulation. Because each level fixes the entities, features, and relations available to data-driven methods, the APS-matrix also provides a graph-ready schema that specifies where machine-learning and knowledge-extraction tasks, such as prediction, anomaly propagation, cross-layer attribution, and knowledge-graph completion, are most appropriately formulated.

The main contribution is therefore a unified basis for distinguishing and relating DT representational levels that are frequently conflated in the literature, with direct implications for the design of DT architectures in smart cities, energy systems, transportation networks, and digital platforms. By making explicit which level suits a given analytical task and how results propagate across levels, the framework supports the integration of infrastructure monitoring, process coordination, and service-oriented analysis. Empirical training and benchmarking of the learning pipelines defined here, together with richer ecosystem models incorporating regulatory, economic, and behavioral factors, remain the principal directions for future work.