Dynamic Risk Transmission in Public–Private Partnership Projects: A Causality-Informed Network Framework

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The proposed causality-informed dynamic risk transmission framework can support lifecycle risk governance in PPP infrastructure projects by quantifying directed risk influences and identifying upstream initiators (RTII) for early, targeted intervention.

Abstract

Interdependent factors in complex engineering systems propagate and amplify through directed influence networks rather than evolving independently. This study proposes a causality-informed dynamic transmission framework and applies it to risk propagation in Public–Private Partnership (PPP) infrastructure projects. Directed inter-factor influences are quantified using the Decision-Making Trial and Evaluation Laboratory (DEMATEL) method to construct a directed influence structure. Each factor’s endogenous evolution is modeled by a logistic trajectory (slow accumulation–rapid escalation–gradual saturation), while an external-transfer component separates internal dynamics from transmission-induced increments. A five-dimension, 21-factor PPP risk system is developed from the literature and failure cases. Here, “causality-informed” denotes expert-elicited directional assumptions, not formal counterfactual causal identification. The results show that transfer-driven increments concentrate mainly in technical/engineering and economic dimensions, whereas political/legal risks exhibit smaller increments; such increments are most evident for preparation-related and all-stage factors and cluster around the rapid-rise interval. We further propose the Risk Transmission Initiation Index (RTII) to identify likely upstream initiators with cascade-triggering potential, enabling prevention-oriented risk control in coupled engineering systems.

1. Introduction

Interdependent factors in coupled project and engineering systems seldom evolve in isolation; instead, they influence one another and may propagate and amplify through directed networks, as evidenced in PPP risk propagation research adopting a network perspective [1]. To structure such interdependencies, decision-oriented causal mapping tools are frequently used; for example, the Decision-Making Trial and Evaluation Laboratory (DEMATEL) has been integrated with related prioritization schemes to identify critical drivers and solution priorities in production settings [2]. Propagation-network thinking is also well established in engineering manufacturing, where machining error propagation has been modeled as a complex network to reveal how upstream deviations transmit to downstream outcomes [3]. In PPP practice, the consequences of accumulated and interacting risks are often manifested through renegotiations and broader partnership frictions, indicating that localized disturbances can escalate into systemic disruptions over time [4]. Complementarily, propagation-control studies in precision assembly show that managing transmitted deviations is essential for system stability [5], and optimization under dual uncertainties further illustrates the need for dynamic, decision-oriented responses when cascading operational impacts are plausible [6]. Building on these insights, this study examines how PPP risk factors transmit and intensify over time in a coupled project environment and develops a causality-informed dynamic transmission framework.

PPP refers to a long-term contractual arrangement in which the government competitively selects a private partner to participate in the financing, construction, operation, and maintenance of public infrastructure or services. Large-scale PPP infrastructure projects face complex and evolving risk environments due to long concession periods, multi-actor governance structures, and sensitivity to policy, market, and operational uncertainties. Accordingly, PPP risks tend to accumulate, interact, and propagate across project stages, making it insufficient to treat risks as isolated items; rather, it becomes necessary to clarify directed influence pathways and the dynamics through which risks intensify across the lifecycle [1].

A substantial body of PPP research has focused on risk identification, assessment, and allocation. Empirical work has examined risk assessment practices and allocation patterns in PPP projects [7], while other studies have applied multi-criteria and expert-based approaches to evaluate PPP risks under uncertainty [8] and to identify critical risk factors in specific PPP sectors such as waste-to-energy projects [9]. In parallel, DEMATEL-based influence structuring is often combined with multi-criteria decision-making frameworks (e.g., ANP and VIKOR) to support strategy selection when interdependencies cannot be ignored [10]. These efforts have advanced the understanding of PPP risk structures and prioritization, yet they typically remain stage-specific or essentially static in how risk is represented.

Despite this progress, synthesis studies indicate that mainstream PPP scholarship still emphasizes governance, performance, and risk management themes, with risks frequently modeled as discrete factors rather than as an evolving, interacting system [11,12]. Systematic reviews focusing on PPP risks similarly show that prevailing approaches concentrate on identifying, assessing, and allocating risks, whereas time-dependent accumulation, interaction, and propagation mechanisms receive comparatively limited attention [13,14]. Even studies examining how risks are managed during PPP formation tend to document managerial responses without explicitly modeling nonlinear transmission and amplification processes [15]. Meanwhile, the high incidence of PPP renegotiations underscores the long-term and inherently incomplete nature of PPP contracts, implying that risks can be activated and reshaped by sequential shocks and strategic behaviors over time—features that static assessments are ill suited to capture [16].

To address these gaps, we develop a lifecycle-oriented dynamic framework for risk transmission in PPP infrastructure projects that links an expert-elicited directed influence structure to time-dependent propagation processes and system-level outcomes. The framework quantifies inter-risk directional influences to establish a computable transmission backbone and decomposes risk evolution into intrinsic growth and transmitted increments, thereby enabling the identification of likely upstream initiators with cascade-triggering potential and supporting prevention-oriented lifecycle risk governance. The framework is designed for mechanism-informed risk governance and initiator prioritization rather than statistical identification of causal effect sizes.

2. Literature Review and Research Gap

PPP risk research has expanded into a large, multidisciplinary body of work spanning project management, construction engineering, public administration, and infrastructure finance. Bibliometric and scientometric reviews indicate that, rather than converging on a single paradigm, the field has developed several dominant streams, including risk identification and assessment, risk allocation and sharing, analytical models and techniques, and governance-oriented mitigation responses [17,18]. Consistent with these mappings, the literature reflects a mature foundation of largely static risk assessment traditions alongside a growing—though still fragmented—interest in system-oriented perspectives that move beyond factor lists and rankings to explain inter-risk interdependence and lifecycle dynamics [17,18].

Within these streams, a substantial body of work advances PPP risk management through structured assessment and allocation frameworks that identify salient risks, elicit expert judgments, and generate prioritized risk lists or allocation recommendations for decision making. In parallel, contract-design research emphasizes that risk allocation is not merely a technical exercise but a governance problem shaped by bounded rationality and long-term uncertainty; accordingly, “optimal” allocation principles may diverge from allocations that are feasible under market capacity, financing constraints, and institutional conditions [19,20]. Representative studies operationalize this logic using multi-dimensional risk registers and weighting-based assessment schemes to support risk sharing and response planning under long-term concession arrangements [21]. This allocative emphasis is increasingly reflected in standardized risk matrices and guidance tools designed to translate empirical experience into bankable contracting conventions for specific sectors and asset classes [22].

More recently, studies have moved beyond independent-factor views by explicitly modeling risk interactions and “risk paths”—that is, sequences through which risks influence broader success outcomes—and by using latent-variable or pathway-oriented approaches to reveal interaction effects that static rankings may mask [23]. Complementing these efforts, methods for structuring interdependencies—such as integrating fuzzy network weighting with interpretive structural modeling—have been widely adopted to uncover driving–dependence hierarchies among PPP risks and to map multilayer influence structures in complex project settings [24]. However, systematic evidence suggests that much of this work still operationalizes interdependence as a largely static influence pattern, with comparatively fewer studies explicitly linking dependency structures to lifecycle evolution, feedback loops, and time-varying exposure under real-world project shocks [25,26].

Related advances in project risk science formalize propagation by representing projects as coupled systems (e.g., schedule–risk couplings) and simulating how localized disturbances percolate across layers, creating emergent cross-stage exposure that cannot be inferred from factor-level importance alone [27]. In addition, simulation-based risk interdependency network models combine causal structuring with stochastic (e.g., Monte Carlo) simulation to evaluate risks not only by marginal impact but also by propagation loss and systemic influence, thereby supporting treatment decisions under dynamic interaction effects [28]. Complementing these approaches, machine-learning studies increasingly treat risk as an emergent outcome of nonlinear, interdependent interactions within project systems, enabling more proactive, performance-oriented monitoring than conventional expert weighting and static rankings [29].

While the above studies have strengthened risk prioritization and dependency mapping in PPP settings, a clear gap remains between inter-risk structure and transmission mechanisms. Recent advances in project and infrastructure risk science increasingly model risk as a propagating, cross-stage phenomenon—for example, multilayer network formulations that represent coupled stakeholder–schedule systems and simulate how local disturbances spread across layers [30], as well as lifecycle-oriented dynamic interdependency networks that embed phase evolution and propagation behavior into risk assessment and treatment [31]. In parallel, data-driven studies treat risk interactions and cross-stage exposure as first-order drivers of performance, emphasizing nonlinearities and interaction effects that conventional static rankings may fail to detect [32]. However, these advances have not been translated into the PPP infrastructure domain in a form that is simultaneously causally defensible, lifecycle-consistent, and decision-operational—i.e., capable of separating propagation initiators from downstream manifestations rather than conflating both in static importance rankings. This points to a structural gap in PPP risk scholarship: an absence of an integrated modeling logic that (i) grounds transmission in an interpretable transmission backbone; (ii) represents time-dependent accumulation, amplification, and threshold-like escalation; and (iii) supports governance by identifying upstream trigger risks as intervention levers. The persistence of renegotiation in mature PPP programs further underscores the practical salience of this gap, because long-horizon contracts are necessarily incomplete, and shocks, political timing, and bargaining incentives can endogenously reshape exposures over time—conditions under which transmission, amplification, and threshold effects become governance-critical [33,34].

3. Materials and Methods

3.1. Risk Identification and Classification

In this study, risk is defined as an uncertain event or condition that can negatively affect PPP project performance objectives (cost, schedule, service performance, or financial viability) over the project lifecycle. This definition is used consistently for risk identification, screening, and classification. To parameterize the subsequent dynamic transmission model, we construct a structured PPP project risk register through a systematic literature synthesis and expert validation process. The procedure produces a labeled risk-factor set, as defined in Equation (1):

$$\mathrm{F}={\left\{\left(f_i, d_i,s_i\right)\right\}}_{i=1}^N ,$$

(1)

where $\mathrm{F}$ denotes the labeled risk-factor set, ${\mathit{f}}_{\mathit{i}}$ denotes a risk factor, ${\mathit{d}}_{\mathit{i}}$ denotes its thematic dimension label, and ${\mathit{s}}_{\mathit{i}}$ denotes its lifecycle phase label.

3.1.1. Risk Elicitation and Standardization, and Expert Screening

We first elicit candidate PPP risks from representative studies and authoritative sources in the PPP risk literature. Overlapping items are consolidated by merging synonymous statements and rephrasing them into standardized and operationally interpretable risk descriptions, yielding an initial candidate set.

Domain experts then screened the candidate risks against two inclusion criteria. First, a risk had to be observable or inferable at some point during the PPP life cycle. Second, it had to be performance-relevant, i.e., plausibly affecting project outcomes via at least one of the following pathways: contractual governance, financing and cash flow, construction delivery, or operational service provision. Items failing either criterion were removed or reformulated to improve interpretability and contextual fit. In total, nine experts participated in the elicitation and scoring process. They were drawn from four stakeholder groups: academia (E1–E3), government agencies with PPP implementation experience (E4–E5), construction firms (E6–E7), and full-cycle PPP consulting practice (E8–E9). All experts had at least five years of PPP-related research or practical experience. Expert scores were elicited through a structured protocol and aggregated to form the DEMATEL direct-influence matrix; consistency checks were implemented in the same workflow to support reproducibility.

3.1.2. Two-Layer Labeling by Dimension and Phase

Each retained risk was labeled along two axes to support both conceptual interpretation and temporal positioning in the dynamic model. Thematic labels captured the dominant source of the risk, classifying each item into one of five dimensions: policy and regulation (PO), social environment (SO), economic market (EC), project technology (TE), and organizational management (MA).

In parallel, each risk was assigned a lifecycle label based on its primary occurrence window and managerial relevance: Preparation, Construction, or Operation. Risks that persist across multiple phases or cannot be attributed to a single phase were classified as All-stage. Accordingly, the phase categories used in the subsequent analysis are Preparation, Construction, Operation, and All-stage.

3.2. Directed Influence Quantification of Risk Interactions

This study quantifies the direction and strength of interactions among PPP project risks using the Decision-Making Trial and Evaluation Laboratory (DEMATEL) method. The procedure is applied at two hierarchical levels: (i) the dimension level (five risk dimensions) and (ii) the factor level (21 risk factors). The computational workflow is identical across levels; only node indexing (dimension vs. factor) and matrix size differ.

3.2.1. Expert Elicitation and Direct-Relation Matrix

For each ordered pair ($i\to j$), experts assess the extent to which node $\mathrm{i}$ influences node $j$ on a five-point scale (e.g., 0 = no influence, 1 = low, 2 = moderate, 3 = high, 4 = very high). The five-point scale was adopted to keep expert judgments consistent and easy to apply without losing basic intensity discrimination. For expert $k\in \left\{1,\dots ,K\right\}$, the assessments form a direct-relation matrix, as shown in Equation (2):

$$A^{\left(k\right) }={\left[{a_{ij}}^{(k)}\right]}_{n\ast n}, a_{ii}^{\left(k\right)}=0,$$

(2)

where $\mathrm{n}$ denotes the number of nodes at the analyzed level.

3.2.2. Inter-Rater Agreement Assessment

To ensure the reliability of expert judgments ${a_{ij}}^{(k)}$ used to construct the direct-relation matrices $A^{\left(k\right) }$, inter-rater agreement was evaluated using Fleiss’ kappa. The coefficient was computed based on the 0–4 categorical ratings independently assigned by the nine experts to each ordered pair ($i\to j$). The analysis was conducted separately at the dimension level (five dimensions; 20 ordered pairs) and the factor level (21 risk factors; 420 ordered pairs). The strength of agreement was interpreted according to the benchmark classification proposed by Landis and Koch [35].

3.2.3. Normalization and Total-Relation Matrix

In this study, arithmetic averaging is used because all experts are assigned equal weight, which keeps the aggregation transparent and reproducible.

The group-aggregated direct-relation matrix is obtained by arithmetic averaging, as given in Equation (3):

$$A={\displaystyle \frac{1}{K}}{\displaystyle \sum }_{k=1}^K{A}^{\left(k\right)}.$$

(3)

To ensure convergence of indirect-effect accumulation, $A$ is normalized by the maximum row sum, as shown in Equation (4):

$$s=\underset{i}{\mathit{max}}{{\displaystyle \sum }}_{j=1}^n{a}_{ij}, {\rm X}={\displaystyle \frac{1}{s}}A$$

(4)

This normalization guarantees that the spectral radius $\rho \left({\rm X}\right)<1$, so the inverse ${\left({\rm I}-{\rm X}\right)}^{-1}$ exists.

The total-relation matrix $\mathsf{{\rm T}}$, capturing both direct and indirect effects, is then computed using Equation (5):

$${\rm T}={\rm X}{\left({\rm I}-{\rm X}\right)}^{-1}$$

(5)

Thus, DEMATEL captures network-wide interdependencies beyond pairwise links via the total-relation matrix $\mathsf{{\rm T}}$, while temporal feedback is handled in the subsequent dynamic transmission simulation.

3.2.4. Driving, Dependence, Centrality, and Causality

Based on the total-relation matrix ${\rm T}={\left[t_{ij}\right]}_{n\ast n}$, the driving power of node $\mathrm{i}$ (i.e., the overall influence exerted on other nodes) is defined as the row sum in Equation (6):

$$P_i={\displaystyle \sum }_{j=1}^n{t}_{ij}$$

(6)

The dependence of node $i$ (i.e., the overall influence received from other nodes) is defined as the column sum in Equation (7):

$$R_i={\displaystyle \sum }_{j=1}^n{t}_{ji}$$

(7)

Accordingly, the prominence (centrality) $C_i$ and net causality $E_i$ are given in Equation (8):

$$C_i=P_i+R_i, E_i=P_i-R_i$$

(8)

A larger $C_i$ indicates that node $\mathrm{i}$ is more prominent (i.e., more centrally involved) in the risk interaction network. The sign of $E_i$ further distinguishes causal roles: $E_i>0$ (equivalently, $P_i>R_i$) implies that node $\mathrm{i}$ belongs to the net-cause group and exerts more influence on other nodes than it receives, whereas $E_i<0$ ($P_i<R_i$) indicates that node $i$ is in the net-effect group and is influenced more than it influences others.

3.2.5. Importance Index and Normalized Weights

To obtain a single scalar measure of overall importance for subsequent modeling, we define an importance index $b_i$ as the squared Euclidean magnitude of the two-dimensional vector ($P_i$, $R_i$), as given in Equation (9):

$$b_i={P_i}^2+{R_i}^2$$

(9)

This index captures the overall interaction intensity of node $\mathrm{i}$ by jointly accounting for its driving power ($P_i$) and dependence (${\mathrm{R}}_i$), thereby avoiding the loss of information that may arise when these two components are considered separately. The corresponding normalized weight is computed using Equation (10):

$${\omega }_i={\displaystyle \frac{b_i}{{{\displaystyle \sum }}_{i=1}^n{b}_i}}, {{\displaystyle \sum }}_{i=1}^n{\omega }_i=1$$

(10)

The resulting weight vector $\omega =\left({\omega }_1,\dots ,{\omega }_n\right)$ is used to aggregate system-level risk and/or parameterize subsequent simulations.

For transparency and reproducibility, we clarify how DEMATEL outputs are operationalized in the dynamic simulation. DEMATEL-derived directional influence metrics are normalized and mapped to growth-and saturation-related parameters.

3.3. Dynamic Risk Transmission Model

To capture the lifecycle evolution and propagation of PPP project risks, we develop a dynamic transmission model that integrates nonlinear self-propagation and coupled diffusion (external transmission) within a unified framework. The model is designed to reproduce the characteristic nonlinear evolution of risks—slow accumulation in the early stage, rapid diffusion after entering a critical region, and saturation in the late stage—while allowing risks to propagate through an empirically quantified causal network.

3.3.1. Notation and Directed Transmission Network

Let $\mathcal{F}={\left\{\left(f_i\right)\right\}}_{i=1}^N$ denote the set of risk factors identified in Section 3.1. Let $x_i\left(t\right)\in \left[0, 1\right]$ denote the normalized risk state (or output intensity) of risk-factor $f_i$ at time $\mathrm{t}$. Inter-risk causal influences are represented by a directed weighted matrix $W=\left[w_{ij}\right]$, where $w_{ij}\ge 0$ denotes the transmission strength from risk $i$ to risk $j$. The matrix $\mathrm{W}$ is parameterized based on the directed influence quantification results in Section 3.2 and normalized to ensure comparability across risks.

3.3.2. Nonlinear Self-Propagation (Within-Risk Dynamics)—Construction Based on the Logistic Function

To describe the endogenous escalation of an individual risk factor over the project lifecycle, we adopt the logistic function, which is widely used to model S-shaped growth processes featuring three regimes: (i) slow accumulation in the early stage, (ii) accelerated escalation around a critical turning region, and (iii) saturation in the late stage. Figure 1 visualizes this canonical profile and highlights the turning (activation) region that separates gradual buildup from rapid escalation. The different background colors indicate the gradual buildup stage and the rapid escalation stage.

Specifically, for risk $i$, the standard logistic form is given in Equation (11):

$$x_i\left(t\right)={\displaystyle \frac{L_i}{1+exp\left(-k_i\left(t-t_{0,i}\right)\right)}},$$

(11)

where $L_i$ is the saturation level (upper bound), $k_i$ is the steepness (growth sensitivity) controlling how sharply the curve transitions around the turning region, and $t_{0, i}$ is the midpoint time at which the risk reaches $L_i∕2$.

PPP infrastructure risks are interdependent across contractual, financial, and operational interfaces; therefore, within-risk escalation is modeled with a logistic form as a parsimonious representation of the observed slow-accumulation–accelerated-growth–gradual-saturation pattern, while network measures parameterize cross-risk heterogeneity in prominence and directional influence.

Based on the DEMATEL-derived influence matrix (Section 3.2), we compute driving power $P_i$ and dependence $R_i$ (Equations (6) and (7)) and then derive prominence/centrality $C_i$ and net causality $E_i$ (Equation (8)). We interpret ${\mathrm{P}}_{\mathrm{i}}$ as the system-wide exposure and accumulation capacity of risk $\mathrm{i}$ and thus set the logistic saturation level as $L_i=P_i$. Net causality $R_i$ captures the driving tendency of risk $\mathrm{i}$; therefore, the logistic steepness is parameterized by its magnitude to ensure nonnegativity, i.e., $k_i\leftarrow \left|R_i\right|$ (with scaling if needed). Because the logistic steepness must be non-negative, we use $\left|R_i\right|$ to quantify escalation intensity, while the sign of ${\mathrm{R}}_{\mathrm{i}}$ is retained to interpret whether a risk is predominantly causal ($R_i>0$) or resultant ($R_i<0$) in the PPP network. This mapping is adopted as a parsimonious identification strategy to preserve cross-risk comparability and parameter identifiability under limited observations rather than to claim a unique structural form.

For notational convenience, we denote the within-risk logistic output by $F_i\left(\mathrm{t}\right)$ as defined in Equation (12), and will be used as the risk potential in Section 3.3.3.

$$F_i\left(t\right)={\displaystyle \frac{P_i}{1+exp\left(-\left|R_i\right|\left(t-t_{0,i}\right)\right)}}$$

(12)

3.3.3. Coupled Diffusion (External Transmission) Model in PPP Risk Networks

To characterize cross-risk transmission in PPP projects, we adopt a heat-conduction analogy in which “risk potential” diffuses from higher to lower levels along direct coupling channels. In heat transfer, Fourier’s law is given in Equation (13):

$$q=-\lambda \Delta T$$

(13)

In Equation (12), $\mathrm{q}$ denotes the heat-flux density (heat transferred per unit area and unit time), $\mathsf{\lambda }$ is the thermal conductivity measuring the material’s ability to conduct heat, and $\Delta T$ is the temperature gradient in space. The negative sign indicates that heat flows spontaneously from higher temperature to lower temperature.

PPP projects form tightly coupled socio-technical systems in which risks interact through contractual, financial, and operational interfaces. Let $F_i\left(t\right)$ denote the within-risk potential of factor $i$ at time $t$, given by the nonlinear self-propagation model in Section 3.3.2. Under the heat-conduction analogy of Equation (13), the risk potential $F$ plays the role of temperature $T$, the cross-risk diffusion $T_{ij}$ corresponds to flux, and the coupling intensity $c_{ij}$ corresponds to conductivity.

Accordingly, external transmission along the PPP risk network is modeled as diffusion driven by potential differences. The transmission from risk factor $i$ to $j$ is defined in Equation (14) as

$$T_{ij}\left(t\right)=c_{ij}\left(F_i\left(t\right)-F_j\left(t\right)\right), c_{ij}\ge 0$$

(14)

where $c_{ij}$ denotes the DEMATEL-based direct coupling intensity from $i$ to $j$. The sign is determined by the potential difference $F_i\left(t\right)-F_j\left(t\right)$, which naturally induces diffusion from higher to lower potential.

Because the dynamic potentials $F_i\left(t\right)$ are nonlinear and would require repeated evaluation in simulation, we adopt a tractable approximation by replacing the instantaneous potential contrast with the DEMATEL net-causality contrast, as shown in Equation (15):

$$F_i\left(t\right)-F_j\left(t\right)=R_i-R_j$$

(15)

Thus, the implemented transmission term becomes time-invariant, as given in Equation (16):

$$T_{ij}=c_{ij}\left(R_i-R_j\right)$$

(16)

Accordingly, the overall risk output of factor $\mathrm{j}$ is the sum of its internal evolution and the total external transmission received from its upstream neighbors, as given in Equation (17):

$$G_j\left(t\right)=F_j\left(t\right)+T_j$$

(17)

where $T_j$ denotes the total external transmission received by factor $\mathrm{j}$.

From an interpretive perspective, the directed transmission links can be read as propagation paths through which localized shocks accumulate into cascade chains [36].

3.3.4. Construction of Aggregated Risk Trajectories

Dimension-Wise Aggregation

Let ${\mathcal{F}}_d$ denote the set of risks in dimension $d\in \left\{\mathrm{PO}, \mathrm{SO}, \mathrm{EC}, \mathrm{TE}, \mathrm{MA}\right\}$. According to Equation (16), the factor-level risk output can be decomposed into an internal component $F_j\left(t\right)$ (logistic-type evolution) and an external transmission component $T_{ij}\left(t\right)$. In the case study, the external transmission contribution is treated as time-invariant. We therefore aggregate the internal component by pointwise summation, as given in Equation (18).

$$F_{sum, d}\left(t\right)={\displaystyle \sum }_{j\in {\mathcal{F}}_d}{F}_j\left(t\right)$$

(18)

Because $F_{sum, d}\left(t\right)$ is a sum of nonlinear logistic trajectories, we refit a logistic curve to ${\widehat{F}}_d\left(t\right)$ using nonlinear least squares, as shown in Equation (19):

$${\widehat{F}}_d\left(t\right)={\displaystyle \frac{{\widehat{P}}_d}{1+exp\left(-\left|{\widehat{R}}_d\right|\left(t-{\widehat{t}}_{0, d}\right)\right)}}$$

(19)

By contrast, since $T_{ij}$ is time-invariant in the case study, its aggregation is obtained by summation without refitting. Specifically, the external transmission received by factor $\mathrm{j}$ is defined as the column sum over its nonzero entries, as given in Equation (20):

$$T_j={\displaystyle \sum }_{i\in U\left(j\right)}{T}_{ij}$$

(20)

where $U\left(j\right)$ denotes the set of upstream risks with directed links to $j$ (i.e., $i\to j$). Aggregating $T_j$ over all risks in dimension $d$ yields the dimension-level external component, as given in Equation (21):

$$T_d={{\displaystyle \sum }}_{j\in {\mathcal{F}}_d}{T}_j={{\displaystyle \sum }}_{j\in {\mathcal{F}}_d}{{\displaystyle \sum }}_{i\in U\left(j\right)}{T}_{ij}$$

(21)

Stage-Wise Decomposition

Let ${\mathcal{F}}_s$ denote the set of risks assigned to category $s\in \left\{Preparation, Construction, Operation, all stage\right\}$. Analogously to the dimension-wise aggregation, the stage-level internal component is aggregated as given in Equation (22):

$$F_{sum, s}\left(t\right)={\displaystyle \sum }_{j\in {\mathcal{F}}_s}{F}_j\left(t\right)$$

(22)

and is then refitted by a logistic curve to give Equation (23):

$${\widehat{F}}_s\left(t\right)={\displaystyle \frac{{\widehat{P}}_s}{1+exp\left(-\left|{\widehat{R}}_s\right|\left(t-{\widehat{t}}_{0, s}\right)\right)}}$$

(23)

The stage-level external component is obtained by summing the received external transmission over risks in ${\mathcal{F}}_{\mathrm{s}}$, as given in Equation (24):

$$T_s={{\displaystyle \sum }}_{j\in {\mathcal{F}}_s}{T}_j={{\displaystyle \sum }}_{j\in {\mathcal{F}}_s}{{\displaystyle \sum }}_{i\in U\left(j\right)}{T}_{ij}$$

(24)

3.3.5. Key Risk Identification via RTII (Initiator-Oriented Ratio Index)

According to Equation (16), the risk output of factor $\mathrm{j}$ is decomposed into an internal component $F_j\left(t\right)$ with a logistic-type evolution and an external transmission component. In the case study, the external contribution is treated as time-invariant and summarized by the received external transmission $T_j$. The parameter $P_j$ denotes the estimated saturation level of the internal logistic component $F_j\left(t\right)$, obtained via nonlinear least-squares refitting. To identify initiator-type risks, we define the Risk Transmission Initiation Index (RTII) as, in Equation (25) as:

$$RTI{I}_j={\displaystyle \frac{P_j}{T_j+\tau }}$$

(25)

Here, $T_j$ represents the external transmission received by factor $j$, whereas $P_j$ captures the long-run magnitude implied by the internal logistic evolution. The ratio in Equation (25) is initiator-oriented: when $T_j$ is small (or zero), the observed magnitude is less dependent on external transmission, suggesting that the risk is more likely to lie near the beginning of propagation chains. The constant $\tau$ avoids division-by-zero when $T_j=0$ (we set $\tau =0.1$ in the empirical study).

For comparability across risks, we standardize RTII using a reference-risk scaling. Specifically, we select a representative risk $f_{ref}$ as the normalization benchmark and define the standardized index in Equation (26) as

$$RTI{I}_{std, j}={\displaystyle \frac{RTI{I}_j}{RTI{I}_{f_{ref}}}}$$

(26)

such that $RTI{I}_{std, ref}=1$. The reference risk $f_{ref}$ is specified in the empirical analysis.

4. Empirical Study and Results

This section demonstrates the proposed causality-informed nonlinear transmission framework through an empirical PPP infrastructure project. We first introduce the case context and data sources and then present the project’s risk profile in terms of thematic dimensions and life-cycle phases. Next, we report the estimated directed influence structure obtained from DEMATEL and interpret its key structural features. Finally, we analyze the dynamic transmission outcomes of the coupled nonlinear model, highlighting threshold-driven escalation patterns and identifying the initiating risks that disproportionately trigger cascading effects.

4.1. Risk Register Development and Data Sources

The study is situated in a representative PPP infrastructure setting characterized by a long concession horizon, multi-actor governance, and stage-dependent risk exposure across preparation, construction, and operation. The contractual arrangement specifies standard responsibilities and performance requirements, providing an appropriate context to examine both causal risk interdependencies and time-dependent propagation mechanisms.

The risk register was constructed by running two parallel evidence streams—(i) a structured review of PPP risk studies and (ii) an extraction of risk manifestations from documented PPP project failure cases—and triangulating them to finalize the 21 risk factors; the complete traceability mapping is provided in Appendix A 

Table A1. Traceability of risk factors: evidence from literature review and documented failure cases.

Code	Risk Factor	Definition (Brief)	Literature Review Sources 1	Documented Failure/Underperformance Case Sources 2
$f_1$	Policy and legal change risk	Policy/law shifts that alter obligations, tariffs, or compliance costs	[7,13,16,19,38,39]	C4; C5
$f_2$	Government approval delay risk	Delays in permits/approvals that postpone start or disrupt sequencing	[13,19,21,39]	C1; C2
$f_3$	Government credit risk	Risk of government non-performance due to fiscal stress/credit constraints	[7,13,16,19,26,40]	C2; C3
$f_4$	Government supervision/regulatory risk	Regulatory enforcement uncertainty, discretion, or compliance shocks	[7,9,13,19,21,39]	—
$f_5$	Public opposition risk	Social resistance, legitimacy loss, protests, or acceptance problems	[7,13,41,42,43]	C5
$f_6$	Environmental risk	Environmental constraints, EIA uncertainty, climate/extreme-event exposure	[7,13,19,26,43]	C5
$f_7$	Land acquisition risk	Land assembly, resettlement disputes, and right-of-way delays	[7,13,19,26,43]	—
$f_8$	Revenue (income) risk	Revenue shortfall versus forecast (tolls, tariffs, availability linked demand)	[7,13,19,44,45,46]	C4
$f_9$	Inflation risk	Input price and indexation mismatch that erodes real returns	[7,13,19,27,45]	—
$f_{10}$	Financing risk	Debt/equity raising difficulties; refinancing and interest-rate shocks	[7,13,19,26,47]	C7; C2; C6
$f_{11}$	Government payment arrears risk	Delayed/withheld availability payments or service-fee payments	[7,13,16,19,40,48]	C10
$f_{12}$	Market demand risk	Volume/usage uncertainty affecting throughput and cashflow	[7,13,19,44,45,46]	C2
$f_{13}$	Parallel-project competition risk	Competing routes/assets dilute demand and revenue	[7,13,19,25,44]	C8; C9
$f_{14}$	Design risk	Design deficiencies or scope/design changes causing rework and delay	[7,19,38,43,47,49]	C7; C6
$f_{15}$	Organizational risk	Internal capability gaps, coordination failures, weak governance	[7,9,13,41,47]	C1; C7; C3
$f_{16}$	Construction duration (schedule) risk	Schedule slippage from technical/contractual/interface causes	[7,19,25,38,43,45]	C1; C4; C3; C11; C6
$f_{17}$	Construction cost overrun risk	Capex overrun during construction from scope, claims, productivity loss	[7,19,38,43,45,47]	C1; C3; C11; C6
$f_{18}$	Operation cost overrun risk	Opex exceeds plan due to maintenance, energy, labor, or performance regime	[7,13,19,26,45]	C2; C3; C6
$f_{19}$	Contract risk	Incomplete/ambiguous contract terms; claims, disputes, renegotiation triggers	[7,13,16,38,40,48]	C1
$f_{20}$	SPV fund misappropriation risk	Misuse/diversion of project-company funds; integrity and control failures	[7,9,13,47,48]	—
$f_{21}$	Partnership/cooperation risk	Breakdown in trust, opportunism, poor collaboration among partners	[7,9,13,41,42,47]	C7

¹ To improve readability, literature sources are coded. Numbers in the “Literature review sources” column indicate key references where the corresponding risk factor is identified or discussed; these numbers correspond to items in the main reference list (e.g., [7,9,13,16,19,21,25,26,38,39,40,41,42,43,44,45,46,47,48,49]). ² (Case ID mapping). All illustrative failure/underperformance cases reported in the “Documented failure/underperformance case sources” column are drawn from Pan and Jiang [50]. For readability, cases are coded as follows:

.

Table 1. Risk register with dimension and lifecycle labels.

Code	Risk Item (Risk Name)	Dimension	Project Phase
$f_1$	Policy and legal change risk	Policy & regulation	All-stage
$f_2$	Government approval delay risk	Policy & regulation	Preparation
$f_3$	Government credit risk	Policy & regulation	All-stage
$f_4$	Government supervision/regulatory risk	Policy & regulation	All-stage
$f_5$	Public opposition risk	Social environment	All-stage
$f_6$	Environmental risk	Social environment	All-stage
$f_7$	Land acquisition risk	Social environment	Preparation
$f_8$	Revenue (income) risk	Economic market	Operation
$f_9$	Inflation risk	Economic market	All-stage
$f_{10}$	Financing risk	Economic market	Preparation
$f_{11}$	Government payment arrears risk	Economic market	Construction
$f_{12}$	Market demand risk	Economic market	Operation
$f_{13}$	Parallel-project competition risk	Economic market	Operation
$f_{14}$	Design risk	Project technology	Preparation
$f_{15}$	Organizational risk	Project technology	All-stage
$f_{16}$	Construction duration (schedule) risk	Project technology	Construction
$f_{17}$	Construction cost overrun risk	Project technology	Construction
$f_{18}$	Operation cost overrun risk	Project technology	Operation
$f_{19}$	Contract risk	Organizational management	All-stage
$f_{20}$	SPV fund misappropriation risk	Organizational management	All-stage
$f_{21}$	Partnership/cooperation risk	Organizational management	All-stage

summarizes the risk register used in the empirical study. Following the identification and labeling rules in Section 3.1, each risk factor is assigned a thematic dimension (policy and regulation, social environment, economic market, project technology, or organizational management) and a primary lifecycle phase (Preparation, Construction, Operation, or All-stage). This two-axis labeling provides both conceptual interpretability and temporal positioning for the subsequent causal estimation and dynamic transmission analysis.

At the dimension level, the risk register covers five thematic categories: policy and regulation ($f_1$–$f_4$), social environment ($f_5$–$f_7$), economic market ($f_8$–$f_{13}$), project technology ($f_{14}$–$f_{18}$), and organizational management ($f_{19}$–${\mathrm{f}}_{21}$). Notably, economic-market and technology-related risks account for relatively larger shares of the register, reflecting the sensitivity of long-term PPP performance to revenue/cash-flow uncertainty and to the evolving dynamics of delivery and operation costs.

From a lifecycle perspective, risks are grouped into Preparation, Construction, and Operation, while non-phase-specific risks are coded as All-stage. Preparation risks mainly concern early institutional and technical readiness, construction risks are dominated by schedule and cost issues, and operation risks capture revenue and fiscal-payment uncertainties over the concession horizon. The All-stage indicates persistent exposure, motivating a dynamic (rather than stage-isolated) analysis in the following sections.

4.2. Inter-Rater Reliability Results

Based on the inter-rater agreement assessment described in Section 3.2.1, Fleiss’ kappa was 0.8221 at the dimension level (20 ordered pairs) and 0.7105 at the factor level (420 ordered pairs) (

Table 2. Inter-rater agreement results based on Fleiss’ kappa.

Level	Ordered Pairs	Raters	Rating Scale	Fleiss’ κ	Interpretation
Dimensions (5)	20	9	0–4	0.8221	Almost perfect
Factors (21)	420	9	0–4	0.7105	Substantial

). According to the benchmark of Landis and Koch [35], these values correspond to almost perfect and substantial agreement, respectively. The results indicate strong chance-corrected consistency among the nine experts, supporting the reliability of the aggregated influence matrices used in the subsequent DEMATEL analysis.

4.3. Estimated Directed Influence Structure Results

Using the total-relation matrix $T$ (Equation (5)), we compute driving power $P_i$ and dependence $R_i$ (Equations (6) and (7)) and then derive centrality $C_i$ and net causality $E_i$ (Equation (8)). We further compute the importance index $b_i$ and normalize it into weights ${\omega }_i$ (Equations (9) and (10)), which are subsequently used in the downstream dynamic simulations and aggregation analysis in Section 4.3.

4.3.1. Dimension-Level Causal Roles and System Prominence

At the dimension level (PO, SO, EC, TE, and MA),

Table 3. Dimension-level DEMATEL indices and weights.

Dimension	Driving ${\mathit{P}}_{\mathit{i}}$	Dependence ${\mathit{R}}_{\mathit{i}}$	Centrality ${\mathit{C}}_{\mathit{i}}$	Net causality ${\mathit{E}}_{\mathit{i}}$	Weight ${\mathit{\omega }}_{\mathit{i}}$	Role
PO	5.903	3.652	9.585	2.238	0.185	Net-cause
SO	4.768	4.694	10.041	0.074	0.193	Near-balance
EC	5.329	6.802	12.132	−1.473	0.288	Net-effect
TE	4.562	6.313	10.875	−1.751	0.227	Net-effect
MA	3.766	2.784	7.502	0.982	0.107	Net-cause

summarizes the DEMATEL indices and normalized weights. Overall, EC exhibits the highest prominence and weight, indicating that market-related risks are most centrally embedded in the interaction system. In terms of directional role, PO and MA act predominantly as upstream drivers, whereas EC and TE are mainly downstream accumulation dimensions.

Figure 2 visualizes these roles using a $\mathrm{P}-\mathrm{E}$ cause–effect map: the horizontal axis represents prominence $\mathrm{P}$, and the vertical axis represents net causality $E$. Dimensions above the zero line ($E_i>0$) are classified as net causes, while those below ($E_i<0$) are net effects. This visualization provides an intuitive structural basis for interpreting the cascade patterns reported in Section 4.4.

4.3.2. Factor-Level Key Risks for Subsequent Modeling

At the factor level (21 risks;$\mathrm{i}$ and $\mathrm{j}$ index risk factors; see Table 1), influence in the estimated interaction structure is typically concentrated: a small subset of risks accounts for a disproportionate share of overall interaction intensity. To keep the main text concise,

Table 4. Top-10 risk factors by normalized weight.

Rank	Code	Risk Factor	Driving ${\mathit{P}}_{\mathit{i}}$	Dependence ${\mathit{R}}_{\mathit{i}}$	Centrality ${\mathit{C}}_{\mathit{i}}$	Net Causality ${\mathit{E}}_{\mathit{i}}$	Weight ${\mathit{\omega }}_{\mathit{i}}$	Role
1	$f_8$	Revenue risk	1.315	4.073	5.388	−2.758	0.123	Net-effect
2	$f_7$	Land acquisition risk	1.455	3.108	4.563	−1.653	0.079	Net-effect
3	$f_{16}$	Schedule risk	1.392	2.943	4.272	−1.614	0.070	Net-effect
4	$f_3$	Government credit	2.725	1.633	4.358	1.029	0.068	Net-cause
5	$f_{10}$	Financing risk	1.792	2.312	4.104	−0.520	0.057	Net-effect
6	$f_{15}$	Organizational risk	2.469	1.421	3.890	1.048	0.054	Net-cause
7	$f_{14}$	Design risk	2.020	1.949	3.969	0.071	0.053	Near-balance
8	$f_1$	Policy/legal change	2.618	0.921	3.539	1.697	0.052	Net-cause
9	$f_{11}$	Payment arrears	1.535	2.306	3.841	−0.771	0.051	Net-effect
10	$f_{18}$	Operation cost overrun	1.423	2.338	3.761	−0.915	0.050	Net-effect

reports the top-10 risks ranked by the normalized weight ${\omega }_i$, together with their phase and dimension labels. These items constitute the most influential nodes and are therefore used as primary candidates for subsequent analysis of initiating, amplifying, and outcome-sensitive behaviors.

Beyond weight ranking, we further characterized each risk’s directional role using the net causality measure $E_i$ (upstream driver vs. downstream receiver) reported in Figure 3. To visualize this role jointly with structural prominence, Figure 3 plots each risk in the centrality–causality plane, where the horizontal axis represents centrality $P$, and the vertical axis represents causality $R$. Risks are categorized into the cause group, effect group, and a near-balance group (close to neutral directionality). This scatter view complements Table 4 by showing whether highly weighted risks tend to act as upstream initiators (high $R$), downstream receivers (low $R$), or near-balance intermediaries, thereby motivating the selection of focal risks and the interpretation of cascade pathways in the subsequent dynamic transmission experiments (Section 4.4).

4.4. Sensitivity Check of Structural Robustness

To assess robustness, we conducted a Monte Carlo sensitivity check on DEMATEL outputs at both levels (factor: 21 risks; dimension: five dimensions), using ±5%, ±10%, and ±15% perturbations (2000 runs per level).

The results in

Table 5. Monte Carlo sensitivity results under ±5%, ±10%, and ±15% perturbations (2000 runs per level).

Level	Perturbation	Mean ΔIntensity (%)	95% CI of ΔIntensity (%)	Rank Consistency (Spearman ρ)
FACTORS (21)	±5%	−0.0160	[−1.2335, 1.2592]	0.9988
FACTORS (21)	±10%	0.0083	[−2.6006, 2.4709]	0.9969
FACTORS (21)	±15%	0.0205	[−3.8496, 3.8795]	0.9947
DIMENSION (5)	±5%	−0.0206	[−2.5327, 2.4572]	1.0000
DIMENSION (5)	±10%	−0.0488	[−5.1238, 5.1124]	0.9911
DIMENSION (5)	±15%	−0.0461	[−7.3731, 7.6960]	0.9911

show that the intensity changes are centered near zero at both levels (factor mean: −0.0160% to 0.0205%; dimension mean: −0.0488% to −0.0206%). Even at ±15% perturbation, the 95% interval remains bounded (factor: [−3.8496%, 3.8795%]; dimension: [−7.3731%, 7.6960%]). Rank consistency is consistently high (Spearman ρ = 0.9947–0.9988 for factors; 0.9911–1.0000 for dimensions), with 100% top-1 retention in all settings. These results indicate that parameter perturbations mainly affect absolute magnitudes, while the core structural conclusions remain stable.

4.5. Dynamic Transmission Results

This section characterizes the dynamic risk transmission regimes implied by the proposed causality-based propagation model. Leveraging the directed influence structure established in Section 4.2, we track the temporal evolution of risk states and summarize system behavior through the aggregated output $Y\left(t\right)$, which captures escalation pace, peak intensity, and saturation dynamics. Decomposing $Y\left(t\right)$ by risk dimensions (and project stages where relevant) reveals a clear separation between upstream activation and downstream accumulation: early movements are driven by structurally upstream risks, whereas later surges are dominated by compounded impacts transmitted through the causal network. This decomposition provides a compact empirical signature of nonlinear escalation in PPP infrastructure risk systems and directly motivates differentiated governance priorities across time and dimensions.

4.5.1. Transmission Accumulation Rule

Within each dimension, risks are ordered by decreasing net causality $\mathrm{R}$, as shown in Figure 4, as follows: PO: $f_1>f_3>f_4>f_2$; SO: $f_6>f_5>f_7$; EC: $f_9>f_{12}>f_{10}>f_{11}>f_{13}>f_8$; TE: $f_{15}>f_{14}>f_{17}>f_{18}>f_{16}$; MA: $f_{21}$ > $f_{19}$ > $f_{20}$.

This ordering yields the schematic transmission backbone in Figure 4, which visualizes the dominant directed paths within each dimension and the cross-dimension links. For computation, the PO within-dimension links shown in Figure 4 are encoded into the corresponding coupling–diffusion effect matrix (

Table 6. Political/legal risk dimension coupling diffusion effect matrix.

SOURCE/TARGET	f1	f3	f4	f2
f1	-	$c_{13}\left(R_1-R_3\right)$	$c_{14}\left(R_1-R_4\right)$	$c_{12}\left(R_1-R_2\right)$
f3	-	-	$c_{34}\left(R_3-R_4\right)$	$c_{32}\left(R_3-R_2\right)$
f4	-	-	-	$c_{42}\left(R_4-R_2\right)$
f2

), where rows denote source risks, and columns denote destination risks; a typical nonzero entry in the coupling–diffusion matrix takes the form $T_{ij}=c_{ij}\left(R_i-R_j\right)$, following the implemented transmission rule in Equation (16). To keep the main text readable and focused, we present the PO block as a concrete illustration; the encodings for the other dimensions are fully analogous and follow the same procedure and are therefore not repeated here.

From Table 6, column $f_1$ contains only “–”; hence, $T_1=0$. For $f_4$, the nonzero entries are $M_{14}=c_{14}\left(R_1-R_4\right)$ and $M_{34}=c_{34}\left(R_3-R_4\right)$; thus, $T_4$ = $c_{14}\left(R_1-R_4\right)+c_{34}\left(R_3-R_4\right)$.

4.5.2. Transmission Curves: Dimension- and Stage-Wise Decomposition

Figure 5 reports the dimension-wise aggregation under self-propagation (baseline) and with the received transmission term $T_d$ added, based on Equations (18)–(21). The estimated baseline saturation and growth parameters differ substantially across dimensions: EC exhibits the highest saturation of ${\widehat{P}}_d=12.14$ with ${\widehat{R}}_d=1.56$; TE follows with ${\widehat{P}}_d=10.87$1 and ${\widehat{R}}_d=1.75$; SO shows ${\widehat{P}}_d=10.05$ but an almost flat growth rate ${\widehat{R}}_d=0.88$; PO has ${\widehat{P}}_d=$9.58 with the steepest growth among the five (${\widehat{R}}_d=2.25$); and MA yields the lowest saturation of ${\widehat{P}}_d=7.50$ with ${\widehat{R}}_d=0.99$.

Adding the dimension-level received transmission $T_d$ = ${\displaystyle \sum }_{j\in {\mathcal{F}}_d}{T}_j$ = ${\displaystyle \sum }_{j\in {\mathcal{F}}_d}{\displaystyle \sum }_{i\in U\left(j\right)}{T}_{ij}$ produces a clearly heterogeneous uplift: TE shows the largest transmission contribution ($T_d=5.03$), EC the second largest ($T_d=2.89$), SO a moderate uplift ($T_d=1.00$), MA only a small increase ($T_d=0.41$), and PO exhibits no measurable uplift ($T_d=0$). Consistent with the trajectories, the transmission-enabled curves remain uniformly above the self-propagation baselines once the trajectories enter the steepening window, and the late-stage separation is largely preserved as an additive increase in the dimension-wise accumulated output.

Figure 6 shows a clear and consistent uplift of the stage-wise aggregated trajectories once the transmission component is included (green dotted) relative to the self-propagation baseline (blue/orange). The divergence is negligible in the early period but becomes most visible around the common inflection window ($t_0\approx 9-10$) across all panels, indicating that transmission effects are most salient when the stage trajectories steepen. The strength of the transmission contribution is stage-dependent: the annotated additive transmission level is largest for preparation-stage risks ($T_s=9.93$) and All-stage ($T_s=9.06$), followed by operation-stage risks ($T_s=7.08$), and is lowest for construction-stage risks ($T_s=6.20$). Baseline saturation levels also differ markedly by category (All-stage $P=27.7$; preparation $P=14.32$; construction $P=12.67$; operation $P=18.06$), while the fitted inflection times remain tightly clustered ($t_0\approx 9.2-10$), suggesting that cross-stage differences are dominated by magnitude (saturation and additive transmission) rather than timing.

4.5.3. Initiators and Transmission Amplifiers (RTII-Based Key Risk Identification)

To make the initiator–amplifier separation directly observable, we compared three representative risks with high, medium, and low centrality under baseline/self-growth versus transmission-enhanced settings. As shown in Figure 7, all three representatives follow logistic-type trajectories, while the transmission-enhanced setting yields a clear upward shift relative to the baseline/self-growth benchmark, producing an immediately visible transmission-induced separation across different network positions. This visible separation motivates a quantitative index to distinguish initiator-type risks from those that become large mainly by accumulating incoming transmission.

To distinguish risks that are simply large in magnitude from those that can structurally ignite cascades, we classified risks into initiators and amplifiers/accumulators based on the transmission decomposition in Equation (17) and the Risk Transmission Initiation Index (RTII) in Equation (25). RTII is initiator-oriented because it normalizes a factor’s internal saturation level $P_j$ by the received external transmission $T_j$, where $T_j$ is defined as the column sum of incoming transmissions (Equation (20)), and $\mathsf{\tau }$ prevents division by zero (Equation (25)). Accordingly, risks with no received transmission ($T_j=0$; Equation (20)) are regarded as strict initiators, whereas risks with very limited received transmission ($T_j>0$ but small) and high RTII are treated as initiator-like, consistent with upstream positions whose realized levels are less dependent on incoming transmission. In contrast, amplifiers/accumulators are characterized by large transmission-driven increments, manifested as a pronounced separation between self-only and transmission-enabled outcomes (Equation (17)).

Table 7. RTII ranking results (with RTIIstd).

RISK	RISK NAME	PHASE/CATEGORY	DIMENSION	RTII	RTIIstd
f15	Organizational risk	All-stage	TE	390	9.21
f1	Policy & legal change risk	All-stage	PO	357	8.43
f21	Cooperation risk	All-stage	MA	261	6.19
f6	Environmental risk	All-stage	SO	226	6.16
f9	Inflation risk	All-stage	EC	194	5.33
f12	Market demand risk	Operation	EC	91.75	4.58
f14	Design risk	Preparation	TE	13.06	2.37
f10	Financing risk	Preparation	EC	8.49	1.70
f3	Government credit risk	All-stage	PO	5.38	1.15
f11	Government payment arrears risk	Construction	EC	4.86	1.04
f18	O&M cost overrun risk	Operation	TE	4.78	1.03
f5	Public opposition risk	All-stage	SO	4.66	1.00
f19	Contract risk	All-stage	MA	3.85	0.83
f4	Government regulation risk	All-stage	PO	3.32	0.73
f13	Parallel project competition risk	Operation	EC	3.20	0.71
f17	Construction cost overrun risk	Construction	TE	2.45	0.56
f8	Revenue risk	Operation	EC	2.18	0.50
f16	Schedule delay risk	Construction	TE	2.09	0.47
f7	Land acquisition risk	Preparation	SO	1.90	0.44
f2	Approval delay risk	Preparation	PO	1.68	0.39
f20	SPV fund misappropriation risk	All-stage	MA	1.51	0.35

reports the RTII-based ranking results for identifying system-level initiators. The top-ranked risks include $f_{15}$, $f_1$, $f_{21}$, and $f_6$ (followed by $f_{12}$, $f_{14}$, and $f_{10}$), and the table further provides each item’s phase/category, dimension, and standardized RTII ($RTI{I}_{std}$; Equation (26)). These results align with the pattern observed in Figure 7: initiator-type risks exhibit limited dependence on incoming transmission (small $T_j$) yet can exert disproportionate influence through outgoing pathways. In contrast, receiver-type risks tend to have higher $T_j$ (higher received external transmission), reflecting stronger dependence on upstream inputs; consequently, their outcomes change more when transmission effects are included, consistent with their downstream positions in the directed influence structure. We additionally performed a robustness check for RTII calculation variants (scale, offset, and nonlinear transform) under ±5%, ±10%, and ±15% perturbations (2000 runs each). Across variants, ranking consistency remained acceptable to high (Spearman ρ\rhoρ from 0.7513 to 1.0000; top-8 Jaccard from 0.7633 to 1.0000), and top-3 retention was 85.95–100%. These results further support that the key RTII-based initiator characterization is structurally stable under reasonable calculation-method changes.

5. Discussion and Implications

5.1. Evidence from Aggregation Analyses: Domain Concentration and Stage-Dependent Amplification

Based on the dimension-wise aggregation in Figure 5, the pattern is clear. When external risk transmission is incorporated, the technical/engineering domain (TE) shows the largest upward shift, as measured by the external transmission contribution (=5.03), whereas the political/legal domain (PO) remains essentially unchanged, with the external transmission contribution = 0.00 (as indicated in the PO panel of Figure 5). In plain terms, external shocks tend to “show up” as engineering delivery (and economic) problems, while political/legal issues are less likely to be pushed upward by shocks coming from other domains.

The stage-wise aggregation further indicates that external transmission is not equally strong throughout the project. Early on, the baseline and transmission-enhanced trajectories are nearly overlapping; the separation becomes much more visible once the overall risk curve enters its rapid-rise segment (i.e., when the curve starts to steepen). The additive transmission level—which captures how much external transmission lifts the aggregated risk level—is highest in the Preparation stage (9.93) and for the All-stage (9.06), followed by the Operation stage (7.08), and is lowest in the Construction stage (6.20). This suggests a simple interpretation: early and system-level issues are more likely to trigger knock-on effects, whereas later-stage issues more often reflect consequences becoming heavier on an existing track.

Implications for project managers and government implementers are direct. Prioritize governance and coordination in the preparation and system-level areas (given the highest transmission levels of 9.93 and 9.06); strengthen delivery controls where impacts concentrate—especially in TE (5.03)—through tighter change control, interface management, and milestone-based early warning. When the project enters the rapid-rise period of the risk curve, increase the cadence and speed of risk response to prevent localized issues from escalating into system-wide disruption.

To ensure that the expert-based influence structure is not driven by subjective inconsistency, inter-rater agreement and robustness were evaluated. Fleiss’ kappa (Table 2) indicates substantial to almost perfect chance-corrected agreement, and sensitivity analyses (Table 5) confirm that the structural pattern and key initiator identification remain stable despite parameter perturbations.

5.2. Chain Position Matters: Why Some “Smaller” Risks Drive Larger System-Level Amplification

As shown in Figure 7, under the “doubling of the $\mathrm{P}$ value” setting, aggregated All-stage responds differently in the self-growth case versus the self-growth case with external risk transmission. For the high-centrality risk $f_{20}$ (SPV fund misappropriation), its relatively high saturation level ($\mathrm{P}$ = 7.18) makes the All-stage curve rise more visibly under self-growth, yet adding external transmission produces little additional uplift—consistent with $f_{20}$ being positioned near the back end of its within-dimension transmission order, where doubling the $\mathrm{P}$ value mainly raises its own late-stage ceiling with limited scope to propagate further. By contrast, $f_{15}$ (organizational risk) and $f_9$ (inflation risk) have smaller $\mathrm{P}$ values under self-growth (notably $f_9$: $\mathrm{P}$ = 1.94, far below $f_{20}$:$\mathrm{P}$ = 7.18) but show a much stronger uplift once external transmission is included, because they sit at the front end of their respective transmission orders: When the $\mathrm{P}$ value increases for a risk located early in the transmission sequence, the effect often does not remain confined to that risk itself; instead, it is carried forward as downstream risks accumulate, thereby lifting the overall system level. The results support this interpretation: Although $f_{15}$ has $\mathrm{P}$ = 3.90—only about 54% of $f_{20}$—under the “doubling of the PPP value + external risk transmission” case, the late-stage aggregated level becomes close to that of $f_{20}$ (approximately 45.79 vs. 47.27). This indicates that system-level amplification depends not only on the magnitude of $\mathrm{P}$ value but also strongly on whether a risk is positioned early or late in the transmission chain.

In other words, looking only at the $\mathrm{P}$ value can put too much attention on downstream high-$\mathrm{P}$-value risks like $f_{20}$ and miss early-chain risks like $f_{15}$ and $f_9$, which become more influential once external transmission is considered. This is precisely why the paper introduces $RTII$: not to repeat a “who is largest” ranking but to highlight risks with stronger initiating potential in the transmission chain. Table 7 is consistent with this logic. $f_{15}$ organizational risk ranks first ($RTI{I}_{std}$ = 9.21), followed by $f_1$ policy and legal change risk ($RTI{I}_{std}$ = 8.43) and $f_{21}$ cooperation risk ($RTI{I}_{std}$ = 6.19); ${\mathrm{f}}_9$ inflation risk is also high ($RTI{I}_{std}$ = 5.33). By contrast, risks with ${\mathrm{RTII}}_{\mathrm{std}}$ below 1 (e.g., contract-related and other downstream risks in Table 7) are more consistent with backend positions where external influences have already accumulated and the remaining role is more about absorbing consequences than initiating further propagation. Practically, $RTI{I}_{std}$ provides decision makers with a clearer basis for prioritization: focus early interventions on high-$RTI{I}_{std}$ risks located early in the transmission sequence, while applying consequence-control measures to low-$RTI{I}_{std}$ risks located late in the transmission sequence once they materialize. In addition, robustness checks on RTII calculation variants show that although absolute values may vary, the key initiator structure remains stable.

6. Conclusions

This study develops an application-oriented, dynamic risk transmission framework for Public–Private Partnership projects by integrating (i) DEMATEL-based quantification of directed inter-risk influence strength and polarity and (ii) a logistic-type nonlinear evolution model with an external transfer mechanism to capture cross-dimension escalation. The proposed framework reproduces a typical nonlinear escalation pattern (“slow accumulation–critical acceleration–gradual saturation”) at the system level. When external transfer is incorporated, the model indicates dimension-dependent amplification, with more pronounced transfer-induced escalation in the technical/engineering and economic dimensions and comparatively limited additional escalation in the political/legal dimension. In time, transfer-driven increments are most evident for preparation-stage risks and all-stage risks, suggesting that early-stage conditions can materially influence downstream exposure through cascading effects.

To enable actionable governance and engineering decision support, we further introduce the Risk Transmission Initiation Index to identify upstream initiators with high cascade-triggering potential. The Risk Transmission Initiation Index serves as an operational triage rule for PPP governance by prioritizing upstream risks for mitigation sequencing, calibrating monitoring intensity, and guiding contingency allocation based on cascade-triggering potential in the directed influence network. Based on the observed transmission dynamics, two operational implications follow: (1) early, targeted mitigation on initiators identified by the Risk Transmission Initiation Index to suppress cascade formation and (2) enhanced monitoring and control for dimensions that more readily absorb transferred risks—particularly technical/engineering and economic—implemented before and during the acceleration phase, when marginal risk growth is highest.

A key limitation of the present study is the absence of a full longitudinal single-project case, which constrains project-level external validation of practical utility. A second limitation is that the current logistic parameterization ($L_i=P_i$, $k_i=\left|R_i\right|$) is a parsimonious identification choice under limited observations rather than a unique structural specification. Future work will therefore prioritize end-to-end validation on a fully documented Public–Private Partnership project while benchmarking against commonly used risk-prioritization baselines and testing alternative parameterizations with explicit uncertainty modeling. For practice, the framework supports differentiated actions across key stakeholders: policymakers can prioritize early governance safeguards for high-transmission risks; contractors can allocate mitigation resources to upstream initiators and strengthen cross-interface control; and financing institutions can embed transmission-sensitive indicators into due diligence, covenant design, and dynamic risk monitoring [37].