Resilient Multi-Agent State Estimation for Smart City Traffic: A Systems Engineering Approach to Emission Mitigation

Abstract

Uninterrupted traffic flow monitoring is a prerequisite for optimal resource allocation and minimizing vehicular emissions in smart cities. However, centralized traffic management architectures are highly vulnerable to single points of failure. When structural sensor malfunctions occur, the resulting network unobservability paralyzes dynamic signalization, triggering cascading traffic congestion, extended idling times, and severe greenhouse gas emissions. To address this cyber-ecological vulnerability, we propose the Hybrid Multi-Agent State Estimation (H-MASE) protocol, a fully decentralized decision-support framework designed from an applied systems reliability engineering perspective. By deploying PSAs and VLAs directly onto IoT-enabled edge devices at smart intersections, H-MASE leverages a hop-by-hop edge computing topology to collaboratively track macroscopic route flow dynamics. Mathematically, this distributed estimation process is formulated as a network-wide least-squares convex optimization problem, where local projection operators function as exact Distributed Gradient Descent steps to minimize the global residual sum of squares. The distributed consensus mechanism acts as a spatial variance reduction tool, effectively dampening measurement noise and stochastic demand fluctuations. Furthermore, we introduce an autonomous anomaly detection logic that isolates severe structural faults rapidly, which is mathematically structured to prevent false alarms under bounded disturbance conditions. Numerical simulations demonstrate that the protocol yields a highly resilient optimality gap (e.g., a Root Mean Square Error of merely 0.81 vehicles per estimated state) even under catastrophic hardware failures. Ultimately, H-MASE provides a robust, fail-safe data foundation for sustainable urban logistics and green-wave signalization, ensuring that smart cities maintain ecological resilience and optimal resource utilization under severe structural disruptions.

1. Introduction

Rapid urbanization and the exponential surge in vehicle ownership have transformed urban traffic networks into highly congested macroscopic service systems. This congestion represents a major bottleneck in sustainable urban logistics and operations management. Beyond the severe economic losses and supply chain delays, the transportation sector remains a primary driver of greenhouse gas (GHG) emissions [1]. Inefficient idling at intersections and stop-and-go traffic waves act as massive resource sinks, contributing substantially to local air pollution and global climate change [2]. To minimize the impact of pollutant emissions and unsustainable fuel consumption in urban areas, a paradigm shift toward advanced Intelligent Transportation Systems (ITS) has become mandatory. Harnessing multi-agent technologies to manage urban traffic forms a fertile research area for improving transportation efficiency, environmental care, and safety in smart cities [3].

Central to this smart mobility mission is Traffic State Estimation (TSE), which serves as the foundational data-driven decision-support system for next-generation ITS. It seeks to infer operational parameters—such as queue lengths, flow capacities, and spatial resource utilization—from sparse observational data [4]. High-fidelity TSE is the essential cyber-physical backbone for downstream operations research applications, including network-wide eco-routing and green-wave signal optimization, which are vital for mitigating the environmental impacts of recurrent congestion events [5].

However, despite the proliferation of Internet of Things (IoT) sensors, conventional traffic monitoring architectures are severely encumbered by their centralized nature. From a systems reliability engineering perspective, central servers introduce a severe single-point-of-failure vulnerability [6]. When unexpected structural sensor malfunctions or communication link failures occur, centralized observers frequently fail to adapt, leading to network-wide unobservability. In the physical realm, this cyber-failure paralyzes dynamic traffic light controllers, forcing them into suboptimal, fixed-cycle fallback operations. This operational degradation instantly triggers cascading artificial bottlenecks, extended vehicular idling, and a drastic surge in localized emissions [7]. Therefore, ensuring the continuous, fail-safe operation of TSE under catastrophic hardware disruptions is not merely a computational challenge, but a ecological imperative for smart cities. Transitioning from vulnerable cloud-based architectures to resilient, IoT-enabled edge computing paradigms is essential to maintain uninterrupted traffic intelligence.

Recent literature highlights that mitigating sensor-induced traffic disruptions and optimizing intersection flows are important for sustainable urban mobility management. Empirical studies demonstrate that suboptimal traffic light synchronizations and inefficient flows at complex physical junctions, such as multi-arm intersections and large roundabouts, drastically exacerbate vehicular pollutant emissions and fuel consumption [8,9]. Consequently, establishing robust and fail-safe traffic state estimation is essential to directly curtail GHG emissions by enabling dynamic, sustainable travel route planning and eco-routing protocols [10]. Furthermore, to achieve these sustainability-oriented urban traffic optimizations without the immense energy and bandwidth consumption of centralized cloud servers, recent advancements strongly advocate for the deployment of decentralized, multi-agent frameworks [11]. By bridging this gap, the proposed framework embeds distributed convex optimization directly into IoT edge devices (e.g., roadside units). This applied edge-computing approach explicitly aligns robust multi-agent mathematics with the practical, cyber-ecological resilience required by modern smart transport systems.

2. Related Work

To overcome the inherent fragility and computational bottlenecks of centralized monitoring, researchers are increasingly pivoting toward distributed architectures [6]. In these frameworks, the urban traffic network is modeled as a multi-agent system, where autonomous agents collaborate over a decentralized communication topology without relying on a central server [6]. This paradigm shift not only ensures greater expansibility and structural robustness against local failures but also allows the system to approach global optimization through hop-by-hop local interactions and consensus mechanisms [3,6].

A particularly effective strategy for managing the immense complexity of metropolitan grids is the implementation of hierarchical multi-agent frameworks [12]. By decomposing a large-scale network into manageable layers—such as region agents, intersection agents, and turning-movement agents—these hierarchical structures effectively reduce the problem’s dimensionality and enhance learning efficiency across the system [12]. Recent advancements have further integrated these structures with game-theoretic models, such as the Nash-Stackelberg hierarchical game, to systematically resolve goal conflicts between primary and secondary intersections while ensuring overall network stability [13]. Furthermore, to ensure that these distributed controllers remain sensitive to both immediate and long-term dynamics, modern algorithms now employ multi-granularity fusion [14]. This allows agents to hierarchically learn both fine-grained current step-states and coarse-grained historical trends, facilitating the derivation of more physically consistent traffic-signal policies [14]. To address the real-time constraints of rapidly evolving urban environments, broad reinforcement learning has emerged as a high-speed alternative to traditional deep architectures, utilizing ridge regression to achieve fast remodeling and adaptation [15].

Origin-Destination (O-D) matrix estimation is widely recognized as one of the most important and challenging components of TSE. To address this problem, various methodologies have been developed, ranging from traditional mathematical programming to modern data-driven approaches. For instance, Englezou et al. proposed a nonlinear mathematical program utilizing a signalized path-based cell transmission model to capture detailed traffic dynamics more accurately [16]. Similarly, Dey et al. performed statistical O-D flow estimation using network tomography based on link counts alone, notably requiring no historical trip data to achieve accurate flow predictions [17]. With the integration of next-generation data sources, O-D estimation has become increasingly complex and data-centric. Wei et al. evaluated a data-driven network assignment method that replaces traditional traffic assignment with GPS probe data to achieve consistent O-D and link flow estimation across urban networks [18]. In a related vein, Vahidi and Shafahi presented a least squares model that fuses vehicle identification data from Automatic License Plate Recognition (ALPR) cameras with link counts, establishing temporal and spatial dependencies through variance-covariance matrices [19]. To manage the computational burden of large-scale networks, Pourhassan et al. developed a heuristic approach utilizing zone-aggregate traffic profiles to support the development of large-scale dynamic traffic assignment models [20]. Meanwhile, Peng et al. introduced a physics-constrained deep learning framework that incorporates GraphSAGE architecture to capture intricate network topology relationships, enhancing robustness against missing data scenarios [21]. Directly addressing the need for decentralized management, Etemadnia and Abdelghany proposed a distributed recursive heuristic that partitions the study area into subareas and merges local estimates through a hierarchical multi-threading mechanism to reduce processing time [22]. Despite these advancements, a significant reliability gap remains in the literature. Most existing frameworks either rely on a centralized optimization authority or lack the dynamic fault-tolerance necessary to withstand structural sensor failures. Furthermore, while modern data-driven and deep learning models exhibit high adaptability, they act as black-box systems that lack formal, worst-case mathematical guarantees regarding estimation stability and error bounds.

To bridge this cyber-physical reliability gap, this paper adopts a strict Operations Research (OR) perspective. Finding the true macroscopic route flow vector $\mathbf{r}$ given distributed network observations is formulated as a network-wide least-squares convex optimization problem:

$$\underset{\mathbf{r}\in {\mathbb{R}}^n}{min}\mathcal{J}\left(\mathbf{r}\right)={\displaystyle \frac{1}{2}}\sum _{i=1}^N{\parallel M_i\mathbf{r}-b_i\parallel }_2^2$$

(1)

where $\mathbf{r}\in {\mathbb{R}}^n$ is the unknown macroscopic route-flow vector to be estimated, N represents the total number of collaborative autonomous agents, $M_i$ is the local topological sensing matrix mapping the global state to agent i, and $b_i$ denotes its corresponding local noisy measurement or tautological topological anchor. By reformulating the decentralized state estimation task through this distributed convex optimization framework, we replace heuristic black-box learning processes with provable analytical architectures. In the proposed framework, the local orthogonal projections executed by the agents correspond to exact Distributed Gradient Descent (DGD) steps aimed at minimizing their local cost functions, while the topological consensus matrix acts as the distributed penalty mechanism enforcing global feasibility.

The practical significance of the proposed Hybrid Multi-Agent State Estimation (H-MASE) framework lies in its ability to equip every localized edge node (i.e., intersection sensor) with real-time, global awareness of the entire macroscopic traffic network. By enabling an isolated intersection to autonomously compute and track the densities of all network-wide routes without relying on a central fusion center, this decentralized architecture provides significant operational advantages for smart city operations:

• Traditional traffic lights operate reactively based on localized queue lengths, inherently suffering from a myopic control perspective. In contrast, global awareness allows isolated intersections to anticipate incoming traffic flows originating from distant nodes. This predictive capability forms the foundation for proactive operations research interventions, enabling system-wide dynamic scheduling and optimal resource allocation to minimize average waiting times across the network.
• In the paradigm of Connected and Autonomous Vehicles (CAVs), relying on centralized cloud architectures for routing introduces significant round-trip latency and unnecessary energy overhead. By utilizing IoT-enabled edge nodes equipped with complete global states, CAVs can seamlessly query the nearest smart intersection via Vehicle-to-Infrastructure (V2I) communication to acquire instantaneous, network-wide route densities. This edge-level query capability directly facilitates real-time dynamic eco-routing and sustainable network flow optimization [23], thereby bypassing the need for energy-intensive cloud processing.
• Conventional Supervisory Control and Data Acquisition (SCADA) based traffic management centers are highly susceptible to communication bottlenecks, natural disasters, or cyber-attacks (e.g., DDoS). The proposed swarm-intelligence topology ensures high resilience; if a central server or a specific regional node collapses, the remaining healthy nodes continue to collaboratively monitor the global state with uncompromised accuracy.
• Continuously transmitting raw vehicle trajectories, GPS coordinates, or license plate data to a central cloud poses severe privacy risks and consumes massive bandwidth. The H-MASE protocol operates by exchanging abstracted mathematical consensus variables between adjacent physical neighbors, which inherently protects user privacy and effectively mitigates the telecommunication overhead.
• For next-generation urban logistics, including Mobility as a Service (MaaS) and crowd-shipping platforms, continuous and reliable traffic data is a prerequisite. The resilient, high-fidelity state vector provided by H-MASE serves as a robust data foundation for these smart city applications, ensuring that sustainable fleet routing and dynamic dispatching operations remain uninterrupted even during severe IoT infrastructure degradations.

The main contributions of this paper can be summarized as follows:

• Decentralized Cyber-Physical Architecture: We propose the H-MASE protocol, a novel hybrid framework combining Physical Sensor Agents (PSAs) and Virtual Logic Agents (VLAs). While PSAs process stochastic measurements, VLAs act as fault-immune topological relays. This localized, hop-by-hop distributed topology ensures that the communication graph remains firmly connected, effectively eliminating the single-point-of-failure vulnerability inherent in conventional centralized systems.
• Autonomous Fault Isolation with High Reliability: We introduce a robust, localized switching logic that instantaneously detects and isolates severe structural sensor faults. As mathematically proven and empirically validated, this mechanism is designed to theoretically avoid false alarms under bounded nominal conditions.
• Theoretical Resilience and Input-to-State Stability (ISS) Guarantees: The paper provides mathematical proofs establishing ISS. We demonstrate that as long as the surviving ensemble of healthy PSAs maintains structural observability, the fault-immune VLAs preserve network-wide consensus, preventing global error divergence even after the autonomous isolation of compromised nodes.
• Foundation for Network-Wide Optimization: From an operations research perspective, the proposed framework supplies resilient, real-time route density estimates. These high-fidelity states serve as a reliable foundational input for dynamic resource allocation, stochastic queue management and traffic signal optimization problems in smart cities.

The remainder of this paper is organized as follows. Section 2 provides a review of related literature. Section 3 details the physical and graph-theoretic models governing urban traffic flows. Section 4 introduces the design of the proposed H-MASE protocol. Section 5 presents convergence and resilience analysis. Section 6 provides numerical results, comparative benchmarking, and an ecological impact assessment. Finally, Section 7 concludes the paper and outlines future research directions.

The key notations and parameters used throughout the mathematical formulations of this framework are summarized in

Table 1. Nomenclature.

Symbol	Dimension/Space	Description
Sets, Graphs, and Indices
$\mathcal{G},{\mathcal{G}}_{MAS}$	Graph	Physical traffic network and cyber-communication graph
$\mathcal{V},\mathcal{E}$	Set ($\left|\mathcal{V}\right|=p,\left|\mathcal{E}\right|=m$)	Sets of physical nodes and directed physical links (edges)
${\mathcal{V}}_{int}$	Set ($|{\mathcal{V}}_{int}| =p_{int}$)	Subset of internal signalized junctions (zero-sum nodes)
$\mathcal{R}$	Set ($\left|\mathcal{R}\right|=n$)	Set of dominant acyclic routes, with $R_j$ as its j-th element
${\mathcal{V}}_{MAS},{\mathcal{E}}_{MAS}$	Set ($|{\mathcal{V}}_{MAS}| =N$)	Sets of autonomous agents and digital communication links
${\mathcal{N}}_i$	Set	Inclusive communication neighborhood of agent i
${\mathcal{V}}_h,{\mathcal{V}}_f$	Set	Subsets of healthy and structurally compromised (faulty) agents
$\mathcal{C}$	Subspace (⊆${\mathbb{C}}^{nN}$)	Network-wide consensus eigenspace
$i,j,k,t$	${\mathbb{Z}}_{\ge 0}$	Agent/route index, inner iteration (k), and macroscopic step (t)
Scalars and System Parameters
$T_s,K$	${\mathbb{R}}_{>0}$, ${\mathbb{Z}}_{>0}$	Macroscopic sampling period and total inner consensus iterations
$c_k,x_k,d_k$	${\mathbb{R}}_{>0}$	Saturation capacity, utilization ratio, and average delay for link $e_k$
$y_k\left(t\right),{\widehat{y}}_k$	${\mathbb{R}}_{\ge 0}$	True ideal flow and estimated arrival flow for link $e_k$
$r_j\left(t\right)$	${\mathbb{R}}_{\ge 0}$	True dynamic vehicle volume utilizing route $R_j$
$y_{p,i}\left(t\right),b_i\left(t\right)$	$\mathbb{R}$	Physical stochastic measurement and augmented local constraint
$b_i^f\left(t\right),f_i\left(t\right)$	$\mathbb{R}$	Corrupted observation and structural fault magnitude at agent i
$w_{ij}$	$[0,1]$	Metropolis-Hastings communication weight between agents i and j
${\eta }_i(t,k)$	${\mathbb{R}}_{\ge 0}$	Local consistency residual (discrepancy) of agent i
${\sigma }_i(t,k)$	Binary $\in \{0,1\}$	Autonomous anomaly isolation switch (1 for nominal, 0 for isolated)
$\Gamma \left(t\right),\beta$	${\mathbb{R}}_{>0}$	Dynamic fault detection threshold and its exponential decay rate
${\omega }_{max},{\Delta }_{max}$	${\mathbb{R}}_{>0}$	Supreme bounds for measurement noise and macroscopic state drift
$\gamma ,{\gamma }_{\sigma }$	$(0,1)$	Effective exponential decay rates for nominal and resilient topologies
${\mathcal{B}}_{ISS},{\mathcal{B}}_{ISS}^{\sigma }$	${\mathbb{R}}_{>0}$	Theoretical ultimate ISS bounds for nominal and resilient optimality gaps
Vectors
$\mathbf{r}\left(t\right),{\Delta }_r\left(t\right)$	${\mathbb{R}}^n$	True macroscopic route-flow vector and its stochastic demand drift
${\widehat{\mathbf{r}}}_i,{\widehat{\mathbf{r}}}_{diff,i}$	${\mathbb{R}}^n$	Agent i’s local route-flow estimate and its topological diffused state
${\mathbf{e}}_i,{\mathbf{e}}_{diff,i}$	${\mathbb{R}}^n$	Agent i’s local estimation error and local diffused error
${\mathbf{y}}_{ideal},{\mathbf{y}}_p$	${\mathbb{R}}^m$	Noise-free ideal link flows and physical stochastic observations
$\tilde{\mathit{\omega }}\left(t\right),\overline{\mathit{\omega }}\left(t\right)$	${\mathbb{R}}^m,{\mathbb{R}}^N$	Physical measurement noise and augmented global hybrid noise
$\mathbf{b}\left(t\right),\mathbf{f}\left(t\right)$	${\mathbb{R}}^N$	Global augmented observation and global catastrophic fault vector
$\mathbf{e}(t,k)$	${\mathbb{R}}^{nN}$	Concatenated global estimation optimality gap (error)
$\Delta (t-1)$	${\mathbb{R}}^{nN}$	Kronecker-extended global macroscopic state drift
$\mathbf{v},\mathit{\alpha }$	${\mathbb{C}}^{nN},{\mathbb{C}}^n$	Global complex eigenvector and its local projection component
${\mathbf{1}}_N,\mathbf{0},{\mathbf{0}}_n$	Various	Vector of ones, general zero vector, and n-dimensional zero vector
Matrices and Operators
$\mathcal{I},{\mathcal{I}}_{int}$	${\mathbb{R}}^{p\times m},{\mathbb{R}}^{p_{int}\times m}$	Full directed incidence matrix and internal reduced incidence matrix
$A,A_{\tau }$	${\mathbb{R}}^{m\times n},{\mathbb{R}}^{p_{int}\times n}$	Physical routing matrix and tautological virtual sensing matrix
M	${\mathbb{R}}^{N\times n}$	Global augmented observation matrix
$M_i,M_i^{†}$	${\mathbb{R}}^{1\times n},{\mathbb{R}}^{n\times 1}$	Local sensing row of agent i and its Moore-Penrose pseudoinverse
W	${\mathbb{R}}^{N\times N}$	Doubly-stochastic communication topology weight matrix
$P_i,D_i,I_n$	${\mathbb{R}}^{n\times n}$	Local orthogonal projection, resilient switched operator, and identity
${\mathbf{M}}^{†},{\mathbf{M}}_{\sigma }^{†}$	${\mathbb{R}}^{nN\times N}$	Global block-diagonal pseudoinverse operators (nominal and resilient)
$\mathbf{W},\mathbf{P},{\mathbf{P}}_{\sigma }$	${\mathbb{R}}^{nN\times nN}$	Kronecker-extended weight matrix and global projection operators
$\Phi ,{\Phi }_{\sigma }$	${\mathbb{R}}^{nN\times nN}$	Global state-transition operators for nominal and switched topologies
$\mathcal{J}(·),\rho (·),\mathcal{N}(·)$	Functions	Least-squares cost function, spectral radius, and null space operators

.

3. Problem Formulation and System Modeling

This section details the physical and graph-theoretic models governing urban traffic flows.

3.1. Physical Graph Topology and Incidence Structures

An urban traffic network is modeled as a directed graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, where $\mathcal{V}=\{v_1,\dots ,v_p\}$ and $\mathcal{E}=\{e_1,\dots ,e_m\}$ denote the sets of nodes and physical links, respectively, with ${\mathcal{I}}_E=\{1,\dots ,m\}$ representing the edge index set. Each link $e_k\in \mathcal{E}$ has a static saturation capacity $c_k\in {\mathbb{R}}_{>0}$. The structural connectivity is captured by the directed incidence matrix $\mathcal{I}\in {\mathbb{R}}^{p\times m}$, defined element-wise as ${\left[\mathcal{I}\right]}_{ik}=1$ if link $e_k$ enters node $v_i$, ${\left[\mathcal{I}\right]}_{ik}=-1$ if it leaves $v_i$, and 0 otherwise. An algebraic property of $\mathcal{I}$ for a weakly connected graph is its rank deficiency, $\mathrm{rank}\left(\mathcal{I}\right)=p-1$. This algebraic linear dependence directly dictates the network-wide mass conservation laws, effectively bounding the feasible operations space.

3.2. Physical Flow Dynamics and Boundary Conditions

We partition the network to explicitly isolate the internal junctions (${\mathcal{V}}_{int}$, $p_{int}=\left|{\mathcal{V}}_{int}\right|$) where no external flow enters or exits. Let the vector ${\mathbf{y}}_{ideal}\left(t\right)={[y_1\left(t\right),\dots ,y_m\left(t\right)]}^T\in {\mathbb{R}}_{\ge 0}^m$ represent the noise-free time-averaged physical link flows at the discrete macroscopic step t. Each component $y_k\left(t\right)$ explicitly denotes the ideal flow volume traversing link $e_k$ over the macroscopic sampling interval. These flow rates serve as the foundational variables for evaluating operational metrics, where the utilization ratio $y_k\left(t\right)/c_k$ determines the queueing performance and service levels.

In urban networks, the strict algebraic conservation lies at the internal junctions (${\mathcal{V}}_{int}$), where no external flow directly enters or exits the system. For each internal node $v_i\in {\mathcal{V}}_{int}$, the mass conservation law is algebraically formalized as a zero-sum balance:

$$\sum _{k\in {\mathcal{E}}_{in}\left(v_i\right)}{y}_k\left(t\right)-\sum _{k\in {\mathcal{E}}_{out}\left(v_i\right)}{y}_k\left(t\right)=0$$

(2)

where ${\mathcal{E}}_{in}\left(v_i\right)=\{k\in {\mathcal{I}}_E\mid {\left[\mathcal{I}\right]}_{ik}=1\}$ and ${\mathcal{E}}_{out}\left(v_i\right)=\{k\in {\mathcal{I}}_E\mid {\left[\mathcal{I}\right]}_{ik}=-1\}$ are the index subsets for incoming and outgoing links, respectively. This deterministic constraint ensures that the estimated flows remain physically consistent, forming the virtual sensing basis for the network.

The global representation for all internal junctions is given by the deterministic equality:

$${\mathcal{I}}_{int}{\mathbf{y}}_{ideal}\left(t\right)={\mathbf{0}}_{p_{int}}$$

(3)

where ${\mathcal{I}}_{int}\in {\mathbb{R}}^{p_{int}\times m}$ is the reduced incidence matrix consisting of the rows of $\mathcal{I}$ corresponding to ${\mathcal{V}}_{int}$, and ${\mathbf{0}}_{p_{int}}$ is the $p_{int}$-dimensional zero vector. Equation (3) defines the virtual sensing layer. As visually demonstrated in Figure 1, while physical measurements are susceptible to noise and hardware failures, this topological backbone provides deterministic zero-sum mathematical constraints. This layer ensures that the estimated link flows remain consistent with the network manifold.

Remark 1.

By excluding redundant or boundary nodes, ${\mathcal{I}}_{int}$ is assumed to be full row rank ($\mathit{rank}\left({\mathcal{I}}_{int}\right)=p_{int}$), ensuring that the conservation constraints provide maximal and non-redundant algebraic information.

3.3. Path-Based State-Space Characterization

While the nodal balance equations established in Section 3.2 provide macroscopic constraints on the link-flow manifold, they are insufficient to resolve the true causal origins of traffic due to the under-determined nature of the network. Aggregated link flows often mask the underlying routing decisions; therefore, to achieve high-fidelity monitoring, we characterize the hidden state of the network using route-level flows. As conceptually illustrated in the reference topology in Figure 2, multiple causal routes often overlap on shared physical links.

Let $\mathcal{R}=\{R_1,\dots ,R_n\}$ be the predefined finite set of dominant acyclic routes actively utilized by the network traffic. We define the true network state as the route-flow vector $\mathbf{r}\left(t\right)={[r_1\left(t\right),\dots ,r_n\left(t\right)]}^T\in {\mathbb{R}}_{\ge 0}^n$, where each component $r_j\left(t\right)$ represents the vehicle volume utilizing route $R_j$ during the macroscopic interval t. The mapping from the hidden route space to the physical link space is governed by the routing matrix $A\in {\{0,1\}}^{m\times n}$, defined element-wise as:

$${\left[A\right]}_{kj}=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\mathrm{link}{e}_k\mathrm{belongs}\mathrm{to}\mathrm{route}{R}_j\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(4)

Substituting the linear mapping ${\mathbf{y}}_{ideal}\left(t\right)=A\mathbf{r}\left(t\right)$ into the internal nodal balance equality (3) yields the topological constraint in the route state-space:

$$A_{\tau }\mathbf{r}\left(t\right)={\mathbf{0}}_{p_{int}},\mathrm{where}{A}_{\tau }={\mathcal{I}}_{int}A\in {\mathbb{R}}^{p_{int}\times n}$$

(5)

Remark 2

(Practical Generation of the Dominant Route Set). Enumerating all theoretically possible acyclic paths in a metropolitan network is an NP-hard problem. However, the proposed framework explicitly avoids this combinatorial explosion by defining $\mathcal{R}$ solely as the set of dominant routes. In practical smart city deployments, this subset can be readily extracted from historical O-D matrices or historical GPS trajectory clusters from connected vehicles [24,25]. Any minor traffic flows utilizing non-dominant, anomalous paths—such as temporary detours due to unexpected traffic accidents or spontaneous routing shifts—are not ignored. Instead, they are mathematically absorbed into the bounded random-walk state drift ${\Delta }_r\left(t\right)$ and the observation noise $\tilde{\mathit{\omega }}\left(t\right)$, ensuring the protocol remains robust against unmodeled macroscopic variations.

Remark 3

(Tautological Anchor Property). Since every valid acyclic route $R_j$ traversing an internal junction $v_i$ must both enter and exit it, the matrix $A_{\tau }\equiv {\mathbf{0}}_{p_{int}\times n}$ is identically the zero matrix. This tautology ($0=0$) forms the topological foundation for the fault-immune VLAs introduced in Section 4.

3.4. Augmented Measurement Model

The physical sensors (e.g., inductive loops, cameras) deployed on the network links provide noisy measurements of the true traffic flows. Let ${\mathbf{y}}_p\left(t\right)\in {\mathbb{R}}^m$ denote the physical observation vector, modeled as:

$${\mathbf{y}}_p\left(t\right)={\mathbf{y}}_{ideal}\left(t\right)+\tilde{\mathit{\omega }}\left(t\right)=A\mathbf{r}\left(t\right)+\tilde{\mathit{\omega }}\left(t\right)$$

(6)

where $\tilde{\mathit{\omega }}\left(t\right)\in {\mathbb{R}}^m$ is the bounded physical measurement noise vector, such that $\parallel \tilde{\mathit{\omega }}{\left(t\right)\parallel }_2\le {\omega }_{max}$ for some known constant ${\omega }_{max}>0$. To unify the estimation framework, we augment the physical measurements with the tautological zero-sum Equation (5). The global augmented observation model $\mathbf{b}\left(t\right)=M\mathbf{r}\left(t\right)+\overline{\mathit{\omega }}\left(t\right)$ is algebraically constructed as:

$$\mathbf{b}\left(t\right)=\left[\begin{array}{c}{\mathbf{y}}_p\left(t\right)\\ {\mathbf{0}}_{p_{int}}\end{array}\right],M=\left[\begin{array}{c}A\\ A_{\tau }\end{array}\right]\equiv \left[\begin{array}{c}A\\ {\mathbf{0}}_{p_{int}\times n}\end{array}\right],\overline{\mathit{\omega }}\left(t\right)=\left[\begin{array}{c}\tilde{\mathit{\omega }}\left(t\right)\\ {\mathbf{0}}_{p_{int}}\end{array}\right]$$

(7)

This augmentation expands the operational dimension of the network without altering the state space. Because $\parallel \overline{\mathit{\omega }}{\left(t\right)\parallel }_2={\parallel \tilde{\mathit{\omega }}\left(t\right)\parallel }_2\le {\omega }_{max}$, the VLAs corresponding to the lower blocks remain permanently isolated from physical noise, allowing them to function as pure-consensus topological bridges.

3.5. State Evolution and Drift Model

The route-flow state $\mathbf{r}\left(t\right)$ is inherently dynamic, fluctuating due to stochastic variations in O-D demands and time-varying routing preferences. In dense urban networks, identifying a stable and explicit state-transition matrix is often computationally prohibitive and physically inaccurate due to the volatility of traffic bursts. While macroscopic traffic flow frequently exhibits highly non-linear phenomena such as kinematic shockwaves, incorporating these non-linearities directly into the transition operator often destroys the convexity of the estimation problem. Consequently, from a robust operations management perspective, the temporal state evolution is modeled as a bounded random walk. This deliberate abstraction mathematically encapsulates all complex physical variations, demand shifts, and routing preference fluctuations into the bounded drift term ${\Delta }_r\left(t\right)$. Because the tracking mechanism focuses exclusively on the predefined set of dominant, actively utilized routes ($\mathcal{R}$), the route volumes remain positive ($\mathbf{r}\left(t\right)>\mathbf{0}$) during nominal operations. Consequently, the state evolution operates sufficiently far from the zero-boundary, allowing it to be accurately characterized by the linear difference equation:

$$\mathbf{r}(t+1)=\mathbf{r}\left(t\right)+{\Delta }_r\left(t\right)$$

(8)

where the term ${\Delta }_r\left(t\right)\in {\mathbb{R}}^n$ represents the unknown but bounded macroscopic state drift. We formalize the physical limits of this network volatility by imposing a strict upper bound on the drift magnitude, such that $\parallel {\Delta }_r{\left(t\right)\parallel }_2\le {\Delta }_{max}$ for all $t\in {\mathbb{Z}}_{\ge 0}$.

The primary objective of the proposed H-MASE architecture is to establish a high-fidelity data foundation for real-time sustainable decision-support systems. Accurately tracking this hidden, stochastically evolving state $\mathbf{r}\left(t\right)$ using the sequence of augmented measurements $\mathbf{b}\left(t\right)$ is required to optimize urban resource allocation and dynamically mitigate congestion-induced emissions. To quantify the network-wide tracking performance, we explicitly consider the finalized local optimality gaps defined as ${\mathbf{e}}_i(t,K)={\widehat{\mathbf{r}}}_i(t,K)-\mathbf{r}\left(t\right)$, where ${\widehat{\mathbf{r}}}_i(t,K)$ is the state vector consolidated by agent i after completing K microscopic consensus iterations. The subsequent sections detail the decentralized protocol designed to drive these errors into a bounded invariant set proportional to the noise level ${\omega }_{max}$ and the volatility ${\Delta }_{max}$.

4. The H-MASE Protocol

Building upon the hybrid measurement framework established in Section 3, this section introduces the design of the H-MASE protocol. The overarching operational architecture and the decision-making sequence of the protocol are visually summarized in Figure 3. The proposed design relies on multi-scale temporal separation between the macroscopic physical traffic dynamics and the microscopic distributed estimation process, a concept frequently employed in hierarchical operations planning. Specifically, the physical traffic state $\mathbf{r}\left(t\right)$ is governed by a macroscopic sampling period $T_s$ (typically in the range of 60–300 s), whereas the multi-agent system executes high-frequency communication-computation cycles at a microscopic scale k, where the total block of K microscopic iterations is executed within the duration of a single macroscopic interval $T_s$. This hierarchical temporal structure ensures that the estimator achieves a consensus-based equilibrium before the subsequent physical state transition occurs.

4.1. Cyber-Physical Topology and Agent Classification

Unlike conventional centralized architectures, the H-MASE protocol relies on a distributed cyber-physical framework where information exchange is localized. We model the monitoring environment as a dual-layer graph where the physical transit network $\mathcal{G}$ is superimposed by a communication topology ${\mathcal{G}}_{MAS}=({\mathcal{V}}_{MAS},{\mathcal{E}}_{MAS})$. Here, ${\mathcal{V}}_{MAS}$ represents the set of $N=m+p_{int}$ autonomous agents, and ${\mathcal{E}}_{MAS}$ denotes the set of bidirectional digital links. To preserve the decentralized nature of urban monitoring, ${\mathcal{E}}_{MAS}$ is topology-induced; communication occurs only between physically adjacent sensors or intersections (hop-by-hop interaction).

To maintain a recursive mapping with the hybrid sensing layers, the agents are partitioned into two functional classes:

• PSAs: Each PSA $i\in \{1,\dots ,m\}$ is collocated with the physical link detectors to process the local scalar stochastic measurement $y_{p,i}\left(t\right)$.
• VLAs: Residing at the internal junctions, each VLA $i\in \{m+1,\dots ,N\}$ mathematically embodies the tautological nodal-balance equations (${\mathbf{0}}_{p_{int}}$). While mathematically represented merely as augmented linear equality constraints with zero sensing rows ($M_i=\mathbf{0}$), their cyber-physical function is profound. They do not restrict the state space; rather, from a systems reliability engineering perspective, they act as fault-immune topological relays. They ensure that even under catastrophic physical sensor degradation, the distributed decision-support architecture maintains its topological connectivity and computational consensus, thereby preventing the ecological disruptions associated with network-wide information fragmentation. In real-world implementations, VLAs are lightweight software routines running directly on the IoT edge computing units (e.g., smart intersection controllers). While they do not process raw physical measurements, their inclusion is structurally vital. During severe multi-sensor failures, isolated PSAs drop their corrupted constraints but rely on the topological bridges maintained by the VLAs to prevent the fragmentation of the cyber-communication graph, thereby ensuring that global consensus is computationally sustained.

The interaction for each agent i is restricted to its inclusive communication neighborhood ${\mathcal{N}}_i=\{j\in {\mathcal{V}}_{MAS}\mid (j,i)\in {\mathcal{E}}_{MAS}\}\cup \left\{i\right\}$.

Assumption 1

(Topological Connectedness). The cyber-communication graph ${\mathcal{G}}_{MAS}$ is assumed to be undirected and connected, ensuring that local state estimates can diffuse across the entire network via the consensus mechanism to achieve global observability.

To achieve asymptotic consensus without a global coordinator, the agents employ a doubly-stochastic weight matrix $W=\left[w_{ij}\right]\in {\mathbb{R}}^{N\times N}$. We utilize the Metropolis-Hastings algorithm to construct these weights locally, requiring only the knowledge of immediate neighbor degrees $d_i=\left|{\mathcal{N}}_i\right|-1$:

$$w_{ij}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{max(d_i,d_j)+1}},\hfill & \mathrm{if}j\in {\mathcal{N}}_i\mathrm{and}j\ne i\hfill \\ 1-\sum _{l\in {\mathcal{N}}_i\setminus \left\{i\right\}}{w}_{il},\hfill & \mathrm{if}j=i\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(9)

This construction satisfies the row and column sum conditions ${\sum }_j{w}_{ij}=1$ and ${\sum }_i{w}_{ij}=1$, which is a necessary condition for the preservation of the average state during the diffusion phase.

4.2. Local Projection Operators and Singularity Handling

Each agent i processes its local scalar observational constraint $b_i\left(t\right)$ using a local orthogonal projection matrix $P_i=I_n-M_i^{†}{M}_i\in {\mathbb{R}}^{n\times n}$. The generalized pseudoinverse of the local observation row is analytically defined as $M_i^{†}=M_i^T/{\parallel M_i\parallel }_2^2$ for PSAs (where $\parallel M_i{\parallel }_2>0$), and as the null column vector $M_i^{†}={\mathbf{0}}_n$ for VLAs (where the sensing row is $M_i={\mathbf{0}}_n^T$). Consequently, for any VLA, substituting the null sensing row $M_i={\mathbf{0}}_n^T$ inherently yields $P_i=I_n$. This algebraic result verifies that VLAs do not contract the state space; instead, they function as perfect identity filters, acting as fault-immune topological relays.

4.3. Distributed State Update and Time-Scale Separation

The H-MASE framework operates on a dual time-scale architecture. Recall from Section 3.5 that $t\in {\mathbb{Z}}_{\ge 0}$ denotes the slow macroscopic time scale governing the physical traffic dynamics. Let $k\in \{0,1,\dots ,K-1\}$ denote the fast microscopic time scale governing the cyber-communication iterations.

A core assumption of this architecture is the time-scale separation principle: the cyber-communication bandwidth is sufficiently large such that K consensus iterations occur within a single macroscopic interval $[t,t+1)$. Consequently, the local physical observations $b_i\left(t\right)$ and the projection operators $P_i$ are treated as static parameters; they remain constant throughout the duration of the internal k-loop.

Remark 4

(Idealized Communication Infrastructure). The strict execution of K microscopic iterations within a single macroscopic interval $[t,t+1)$ assumes the presence of an Ultra-Reliable Low-Latency Communication framework, such as 5G or standard Dedicated Short-Range Communications in smart cities. The impact of asynchronous packet drops, finite bandwidth constraints, and cyber-communication delays is beyond the scope of this theoretical foundation and remains a vital direction for future robust optimization research.

To ensure continuous tracking without losing historical information, the estimation process leverages a warm-start initialization mechanism. At the beginning of each macroscopic step t, the initial condition for the internal iterations is seeded by the final estimate of the previous step:

$${\widehat{\mathbf{r}}}_i(t,0)={\widehat{\mathbf{r}}}_i(t-1,K)$$

(10)

where ${\widehat{\mathbf{r}}}_i(-1,K)={\mathbf{0}}_n$ represents the absolute cold-start condition before the network operations begin.

During the interval $[t,t+1)$, each healthy agent i updates its local estimate ${\widehat{\mathbf{r}}}_i(t,k)\in {\mathbb{R}}^n$ via a distributed optimization scheme consisting of a two-phase predict-and-update process. First, to enforce global feasibility and parameter agreement across the network, the agent aggregates the prior estimates from its inclusive neighborhood ${\mathcal{N}}_i$ to form the diffusion vector:

$${\widehat{\mathbf{r}}}_{diff,i}(t,k)=\sum _{j\in {\mathcal{N}}_i}{w}_{ij}{\widehat{\mathbf{r}}}_j(t,k)$$

(11)

Subsequently, the agent executes a local optimization step to enforce its own observational constraint. From a distributed network optimization perspective, minimizing the local least-squares cost function ${\mathcal{J}}_i\left(\mathbf{r}\right)={\displaystyle \frac{1}{2}}{\left(M_i\mathbf{r}-b_i\left(t\right)\right)}^2$—which physically translates to eliminating uncertainties in intersection queue lengths to prevent vehicular idling—reduces to an orthogonal projection onto the measurement hyperplane.

The core distributed update mechanism utilized in this phase builds upon the foundational discrete-time projection algorithm originally proposed in [26]. However, in this work, we significantly extend that baseline mathematical structure to accommodate the hierarchical cyber-physical topology of IoT edge devices (i.e., PSAs and VLAs) and to integrate the autonomous fault-isolation logic required for resilient smart city deployments. The nominal distributed update law is formalized as:

$${\widehat{\mathbf{r}}}_i(t,k+1)=P_i{\widehat{\mathbf{r}}}_{diff,i}(t,k)+M_i^{†}{b}_i\left(t\right)$$

(12)

Before applying this optimal measurement innovation $M_i^{†}{b}_i\left(t\right)$, each agent calculates a local discrepancy residual ${\eta }_i(t,k)\in {\mathbb{R}}_{\ge 0}$, representing the deviation between the neighborhood consensus and the local physical constraint:

$${\eta }_i(t,k)=\left|M_i{\widehat{\mathbf{r}}}_{diff,i}(t,k)-b_i\left(t\right)\right|$$

(13)

This residual implicitly measures the magnitude of the local gradient; a severe mismatch triggers the autonomous fault isolation logic, which will be studied in the following section.

4.4. Resilient Estimation and Fault Isolation

To prevent the propagation of severe sensor faults, each agent implements an autonomous binary isolation mechanism ${\sigma }_i(t,k)\in \{0,1\}$. The agent operates nominally (${\sigma }_i=1$) if the consistency residual satisfies ${\eta }_i(t,k)\le \Gamma \left(t\right)$, and isolates itself (${\sigma }_i=0$) otherwise. Here, $\Gamma \left(t\right)={\Gamma }_{steady}+{\Gamma }_{init}{e}^{-\beta t}$ acts as a time-varying resilience threshold derived analytically from the nominal ISS bounds. The exponential decaying term ${\Gamma }_{init}{e}^{-\beta t}$ is strategically incorporated to absorb the massive transient errors associated with the cold-start initialization phase. Since the initial arbitrary estimates ${\widehat{\mathbf{r}}}_i(-1,K)$ naturally deviate significantly from the true hidden state, this temporary tolerance buffer mathematically prevents the autonomous logic from prematurely triggering false alarms during the early consensus-building phase, or during the immediate transient shockwave following a topological switch. By integrating this binary switch, we establish the final Resilient Distributed Update Rule:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\widehat{\mathbf{r}}}_i(t,k+1)={\sigma }_i(t,k)\left(P_i{\widehat{\mathbf{r}}}_{diff,i}(t,k)+M_i^{†}{b}_i\left(t\right)\right)+(1-{\sigma }_i(t,k)){\widehat{\mathbf{r}}}_{diff,i}(t,k)\end{array}\end{array}$$

(14)

When isolated (${\sigma }_i=0$), this decentralized triage mechanism acts as a rapid systemic fail-safe. The corrupted measurement innovation is bypassed, quarantining the hardware fault before it can propagate through the network and paralyze downstream eco-routing operations, while the agent relies solely on the prior knowledge diffused from its healthy neighbors. Since VLAs are mathematically immune to physical faults (${\eta }_i\equiv 0$), they remain permanent topological bridges, ensuring the surviving network remains strongly connected.

4.5. Algorithmic Realization

The theoretical components developed in the preceding subsections are integrated into a unified, decentralized procedural framework. This protocol, termed the H-MASE Algorithm, is designed to execute independently on each agent i. The algorithm operates on the dual-time-scale architecture: a macroscopic outer loop synchronized with the physical sampling period $T_s$, and a microscopic inner loop dedicated to high-frequency iterations k for iterative consensus and constraint enforcement.

Remark 5

(Inference of Traffic Performance Metrics). The estimated route-flow vector $\widehat{\mathbf{r}}(t,K)$ provided by the H-MASE algorithm allows for the indirect observation of intersection delays and queue lengths. By utilizing the mapping $\widehat{\mathbf{y}}\left(t\right)=A\widehat{\mathbf{r}}(t,K)$, the saturation ratio for each physical link $e_k\in \mathcal{E}$ can be directly calculated as $x_k={\widehat{y}}_k/c_k$. Based on the Webster delay model, the average delay $d_k$ per vehicle on link $e_k$ is inferred as [27]:

$$d_k\approx {\displaystyle \frac{C_{cycle}{(1-\lambda )}^2}{2(1-\lambda x_k)}}+{\displaystyle \frac{x_k^2}{2{\widehat{y}}_k(1-x_k)}}$$

(15)

where $d_k$ denotes the average waiting delay at the k-th intersection approach, $C_{cycle}$ is the traffic signal cycle length, and λ represents the effective green ratio for the corresponding signal phase. The variable ${\widehat{y}}_k$ represents the estimated arrival flow rate at node k, which is directly reconstructed from the network-wide route-flow estimates provided by the distributed H-MASE protocol. Consequently, $x_k$ explicitly defines the degree of saturation (i.e., the volume-to-capacity ratio) at the intersection approach.

From a data-driven decision-support perspective, maintaining the high-fidelity estimation of ${\widehat{y}}_k$ is important. An erroneous overestimation caused by unmitigated sensor faults could artificially inflate the perceived saturation level $x_k\to 1$. Because of the $(1-x_k)$ term in the denominators, such an error would force the macroscopic delay model to exhibit an artificial exponential growth, causing downstream decision-support systems to execute highly suboptimal resource allocations. By guaranteeing structurally resilient and bounded estimates, H-MASE ensures the analytical validity of this queueing model. While this mapping theoretically bridges the state estimation to physical intersection queues, the numerical validations in Section 6 focus on the core macroscopic optimality gaps (${\mathcal{B}}_{ISS}$) to purely isolate and evaluate the algorithmic tracking performance.

The H-MASE Decentralized Estimation Protocol proposed in the paper is given in Algorithm 1.

Algorithm 1 H-MASE Decentralized Estimation Protocol.

1: Initialization: 2:    Pre-compute projection operators $P_i$ and $M_i^{†}$ for each agent $i\in {\mathcal{V}}_{MAS}$ 3:    Apply local constraint initialization: ${\widehat{\mathbf{r}}}_i(-1,K)\leftarrow M_i^{†}{b}_i\left(0\right)$ 4: for each physical sampling time $t=0,1,2,\dots$ do 5:       Acquire local observation $b_i\left(t\right)$ 6:       ${\widehat{\mathbf{r}}}_i(t,0)\leftarrow {\widehat{\mathbf{r}}}_i(t-1,K)$  ▹Temporal Handover from previous step 7:       for each iteration $k=0,1,\dots ,K-1$ do 8:             1. Consensus Phase: 9:                  Share ${\widehat{\mathbf{r}}}_i(t,k)$ with neighbors $j\in {\mathcal{N}}_i$ 10:                Compute diffused average: ${\widehat{\mathbf{r}}}_{diff,i}(t,k)={\sum }_{j\in {\mathcal{N}}_i}{w}_{ij}{\widehat{\mathbf{r}}}_j(t,k)$ 11:             2. Integrity Monitoring: 12:                ${\eta }_i(t,k)=|M_i{\widehat{\mathbf{r}}}_{diff,i}(t,k)-b_i\left(t\right)|$   ▹Check local-to-global consistency 13:             3. Resilient Update: 14:             if ${\eta }_i(t,k)>\Gamma \left(t\right)$ then 15:                                                         ▹ Isolation Mode: Bypass corrupted local data 16:                ${\widehat{\mathbf{r}}}_i(t,k+1)\leftarrow {\widehat{\mathbf{r}}}_{diff,i}(t,k)$ 17:           else 18:                                                  ▹ Nominal Mode: Apply hybrid projection update 19:                ${\widehat{\mathbf{r}}}_i(t,k+1)\leftarrow P_i{\widehat{\mathbf{r}}}_{diff,i}(t,k)+M_i^{†}{b}_i\left(t\right)$ 20:             end if 21:       end for 22:       Return ${\widehat{\mathbf{r}}}_i(t,K)$ as the finalized estimate for time t 23: end for

A primary advantage of the H-MASE algorithm is its minimal computational footprint. Each iteration k involves only sparse matrix-vector multiplications and scalar comparisons. Since the dimensions of $P_i$ and $M_i^{†}$ depend solely on the number of routes n, and the communication is limited to local neighborhoods with a maximum degree $d_{max}$, the computational complexity per agent is bounded by $\mathcal{O}(K·(n^2+d_{max}·n))$. Crucially, this local computational load is completely independent of the total network size N, ensuring that the algorithm remains highly scalable as the urban network expands. This makes the H-MASE highly suitable for real-time monitoring in large-scale urban networks where centralized processing would suffer from significant communication latency and computational bottlenecks. Furthermore, compared to existing distributed estimation benchmarks, H-MASE offers distinct structural advantages. Unlike Distributed Alternating Direction Method of Multipliers framework [28], which typically require computationally expensive matrix inversions at each local iteration, H-MASE relies exclusively on computationally lightweight orthogonal projections ($P_i$). Additionally, compared to Distributed Kalman Filters [29], which heavily depend on the exact knowledge of a state-transition matrix, the proposed robust optimization framework models traffic evolution as a bounded random walk, making it structurally immune to transition-matrix modeling errors. While empirical benchmarking of computational convergence speeds against these conventional frameworks is valuable, the primary objective of the H-MASE protocol is establishing theoretical guarantees. Specifically, the framework prioritizes minimizing false alarms and maintaining exact cyber-physical fault isolation over accelerating nominal iteration speeds.

5. Convergence and Resilience Analysis

In this section, we analyze the convergence properties of the distributed optimization algorithm underpinning the H-MASE architecture. We establish the ISS of the algorithm, which characterizes the robustness of the DGD mechanism and guarantees a bounded optimality gap under the simultaneous presence of macroscopic demand drift ${\Delta }_r\left(t\right)$ and stochastic measurement noise $\overline{\mathit{\omega }}\left(t\right)$.

To analyze the network holistically, we first define the local estimation error (i.e., the distance to the optimal true state) for each agent $i\in {\mathcal{V}}_{MAS}$ at microscopic optimization iteration k during the macroscopic step t:

$${\mathbf{e}}_i(t,k)={\widehat{\mathbf{r}}}_i(t,k)-\mathbf{r}\left(t\right)$$

(16)

By substituting the nominal update rule (12) into (16) and utilizing the geometric property $(P_i+M_i^{†}{M}_i)=I_n$, we obtain the local error dynamics:

$${\mathbf{e}}_i(t,k+1)=P_i\sum _{j\in {\mathcal{N}}_i}{w}_{ij}{\mathbf{e}}_j(t,k)+M_i^{†}{\overline{\omega }}_i\left(t\right)$$

(17)

where ${\overline{\omega }}_i\left(t\right)$ is the i-th scalar element of the augmented noise vector $\overline{\mathit{\omega }}\left(t\right)$.

We aggregate the local errors into a global state error vector

$$\mathbf{e}(t,k)={[{\mathbf{e}}_1^T(t,k),\dots ,{\mathbf{e}}_N^T(t,k)]}^T\in {\mathbb{R}}^{nN}.$$

(18)

The network-wide error evolution is compactly formulated using Kronecker products as:

$$\mathbf{e}(t,k+1)=\Phi \mathbf{e}(t,k)+{\mathbf{M}}^{†}\overline{\mathit{\omega }}\left(t\right)$$

(19)

where ${\mathbf{M}}^{†}=\mathrm{diag}(M_1^{†},\dots ,M_N^{†})\in {\mathbb{R}}^{nN\times N}$ is the block-diagonal pseudoinverse matrix and $\Phi \in {\mathbb{R}}^{nN\times nN}$ is the global state-transition operator defined as

$$\Phi =\mathbf{P}\mathbf{W}$$

(20)

with $\mathbf{P}=\mathrm{diag}(P_1,\dots ,P_N)$ and $\mathbf{W}=(W\otimes I_n)$. To guarantee that the distributed algorithm converges to the true physical state rather than an arbitrary consensus, the underlying cyber-physical network must possess sufficient sensory information. We formalize this physical requirement as follows:

Assumption 2

(Global Network Observability). The physical placement of the measurement sensors (PSAs) provides sufficient rank to fully reconstruct the hidden network state. Algebraically, the intersection of the null spaces of the PSA observation rows ($i\in \{1,\dots ,m\}$) is the zero vector:

$$\bigcap _{i=1}^m\mathcal{N}\left(M_i\right)=\left\{{\mathbf{0}}_n\right\}$$

(21)

Remark 6.

Note that Assumption 2 is a standard requirement for ensuring global asymptotic stability and vanishing estimation errors. In practical large-scale deployments where this condition might be locally violated (e.g., due to sparse sensor coverage), the H-MASE protocol still guarantees convergence to the best possible estimate within the observable subspace. In such cases, the error remains bounded by the projection of the state onto the unobservable null space, preventing global divergence.

Under this assumption, we establish the spectral properties of the transition matrix $\Phi$.

Lemma 1

(Spectral Contraction and Convergence Rate). Under Assumptions 1 and 2, the global state-transition operator Φ functions as a strict contraction mapping, guaranteeing the linear convergence of the distributed least-squares optimization algorithm. Specifically, the spectral radius satisfies:

$$\rho (\Phi )<1$$

(22)

Proof. 

To analyze the unforced error dynamics (i.e., the nominal system without the noise term $\overline{\mathit{\omega }}\left(t\right)$), we aggregate the local error update rules of all N agents into a single global system. Let $\mathbf{e}\left(k\right)={[{\mathbf{e}}_1{\left(k\right)}^T,\dots ,{\mathbf{e}}_N{\left(k\right)}^T]}^T\in {\mathbb{R}}^{nN}$ denote the concatenated global error vector at internal iteration k.

We define the global weight matrix as $\mathbf{W}=W\otimes I_n$, where $W=\left[w_{ij}\right]\in {\mathbb{R}}^{N\times N}$ is the doubly stochastic adjacency matrix of the graph ${\mathcal{G}}_{MAS}$, and ⊗ denotes the Kronecker product. Furthermore, let $\mathbf{P}=\mathrm{diag}(P_1,P_2,\dots ,P_N)\in {\mathbb{R}}^{nN\times nN}$ be the global block-diagonal projection matrix.

Using these global operators, the unforced local error dynamics can be compactly written in the matrix form:

$$\mathbf{e}(k+1)=\mathbf{P}\mathbf{W}\mathbf{e}\left(k\right)$$

(23)

Consequently, the global state-transition matrix is defined as $\Phi =\mathbf{P}\mathbf{W}$. To prove asymptotic stability, we must show that the spectral radius satisfies $\rho (\Phi )<1$.

We now analyze the properties of the individual matrices $\mathbf{W}$ and $\mathbf{P}$. Since the communication graph is strongly connected (Assumption 1) and W is doubly stochastic, the Perron-Frobenius theorem dictates that the spectral radius of W is exactly $\rho \left(W\right)=1$. The eigenvalue $\lambda =1$ has an algebraic multiplicity of 1, with the corresponding right eigenvector being the vector of ones, ${\mathbf{1}}_N$.

Extending this to $\mathbf{W}=W\otimes I_n$, the largest eigenvalue remains 1, but with a multiplicity of n. The corresponding eigenspace, known as the consensus subspace, is defined over the complex field to perfectly accommodate general spectral analysis:

$$\mathcal{C}=\left\{\mathbf{x}\in {\mathbb{C}}^{nN}\mid \mathbf{x}={\mathbf{1}}_N\otimes \mathit{\alpha },\forall \mathit{\alpha }\in {\mathbb{C}}^n\right\}$$

(24)

For any vector residing in this subspace ($\mathbf{x}\in \mathcal{C}$), the consensus operation acts as the identity mapping, yielding $\mathbf{W}\mathbf{x}=\mathbf{x}$.

We now consider the projection matrix $\mathbf{P}$. Because each local operator $P_i$ is an orthogonal projector (satisfying $P_i=P_i^T$ and $P_i^2=P_i$), its eigenvalues are exclusively either 0 or 1. Therefore, the induced 2-norm (spectral norm) of the block-diagonal matrix is bounded by ${\parallel \mathbf{P}\parallel }_2\le 1$. Since W is doubly stochastic, we also have ${\parallel \mathbf{W}\parallel }_2=1$.

Applying the sub-multiplicative property of matrix norms to $\Phi =\mathbf{P}\mathbf{W}$, we establish the upper bound of the spectral radius:

$$\rho (\Phi )\le {\parallel \Phi \parallel }_2\le {\parallel \mathbf{P}\parallel }_2{\parallel \mathbf{W}\parallel }_2\le 1$$

(25)

This guarantees that the global error dynamics are non-expansive ($\rho (\Phi )\le 1$). To prove strict contraction, assume for the sake of contradiction that $\rho (\Phi )=1$. Consequently, there exists a non-zero eigenvector $\mathbf{v}={[{\mathbf{v}}_1^T,\dots ,{\mathbf{v}}_N^T]}^T\in {\mathbb{C}}^{nN}$ and a corresponding eigenvalue $\lambda \in \mathbb{C}$ with $\left|\lambda \right|=1$, such that:

$$\mathbf{P}\mathbf{W}\mathbf{v}=\lambda \mathbf{v}$$

(26)

Taking the 2-norm of both sides, and recalling that ${\parallel \mathbf{P}\parallel }_2\le 1$ and ${\parallel \mathbf{W}\parallel }_2=1$, we obtain the inequality chain:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\parallel \mathbf{v}\parallel }_2={\left|\lambda \right|\parallel \mathbf{v}\parallel }_2={\parallel \mathbf{P}\mathbf{W}\mathbf{v}\parallel }_2\le {\parallel \mathbf{P}\parallel }_2{\parallel \mathbf{W}\mathbf{v}\parallel }_2\le {\parallel \mathbf{W}\mathbf{v}\parallel }_2\le {\parallel \mathbf{W}\parallel }_2{\parallel \mathbf{v}\parallel }_2={\parallel \mathbf{v}\parallel }_2\end{array}\end{array}$$

(27)

Since the extremities are identical, the internal condition ${\parallel \mathbf{W}\mathbf{v}\parallel }_2={\parallel \mathbf{v}\parallel }_2$ must hold. This implies that $\mathbf{v}$ corresponds to the principal singular space of $\mathbf{W}$.

Crucially, because the underlying communication graph is undirected, the Metropolis-Hastings weight matrix W is inherently symmetric ($W=W^T$), meaning its singular values perfectly coincide with the absolute values of its eigenvalues. It is important to note that while the physical traffic flow is highly directional (directed graph), the overlaid digital cyber-communication topology (${\mathcal{G}}_{MAS}$) is explicitly designed as an undirected network. Applying the Metropolis-Hastings rule to this undirected communication graph inherently produces a symmetric and doubly stochastic weight matrix W [30]. Furthermore, since the weights guarantee strict self-loops ($w_{ii}>0$), W is primitive. By the Perron-Frobenius theorem for primitive matrices [30], the only eigenvalue on the unit circle is exactly 1. Thus, $\mathbf{v}$ is confined to the consensus subspace $\mathcal{C}$, meaning $\mathbf{W}\mathbf{v}=\mathbf{v}$ and its local components are identical:

$$\mathbf{v}={\mathbf{1}}_N\otimes \mathit{\alpha }\Rightarrow {\mathbf{v}}_1=\cdots ={\mathbf{v}}_N=\mathit{\alpha }$$

(28)

for some non-zero $\mathit{\alpha }\in {\mathbb{C}}^n$. Substituting $\mathbf{W}\mathbf{v}=\mathbf{v}$ back into our initial hypothesis (26) yields:

$$\mathbf{P}\mathbf{v}=\lambda \mathbf{v}$$

(29)

Because $\mathbf{P}$ is an orthogonal projector, its eigenvalues are exclusively 0 and 1. For $\mathbf{v}\ne \mathbf{0}$ and $\left|\lambda \right|=1$ to hold simultaneously, we must have $\lambda =1$ and $\mathbf{P}\mathbf{v}=\mathbf{v}$.

Given the block-diagonal structure $\mathbf{P}=\mathrm{diag}(P_1,\dots ,P_N)$, the global equation $\mathbf{P}\mathbf{v}=\mathbf{v}$ decouples into exact local equalities for every agent:

$$P_i\mathit{\alpha }=\mathit{\alpha },\forall i\in \{1,\dots ,N\}$$

(30)

Substituting the local operator definition $P_i=I_n-M_i^{†}{M}_i$, we obtain:

$$(I_n-M_i^{†}{M}_i)\mathit{\alpha }=\mathit{\alpha }\Rightarrow M_i^{†}{M}_i\mathit{\alpha }={\mathbf{0}}_n$$

(31)

To eliminate the pseudoinverse without invoking complex quadratic forms, we apply the first property of the Moore-Penrose pseudoinverse ($A{A}^{†}A=A$) [30]. Pre-multiplying both sides by the row vector $M_i$ collapses the dimension to a scalar, yielding:

$$M_i\left(M_i^{†}{M}_i\mathit{\alpha }\right)=0\Rightarrow \left(M_i{M}_i^{†}{M}_i\right)\mathit{\alpha }=0\Rightarrow M_i\mathit{\alpha }=0$$

(32)

This algebraic step reveals that $\mathit{\alpha }$ must lie exactly in the null space of the observation matrix:

$$\mathit{\alpha }\in \mathcal{N}\left(M_i\right),\forall i\in \{1,\dots ,N\}$$

(33)

Because this holds for the entire network, it inherently applies to the subset of PSAs indexed by $i\in \{1,\dots ,m\}$. Thus, $\mathit{\alpha }$ must reside in their intersection:

$$\mathit{\alpha }\in \bigcap _{i=1}^m\mathcal{N}\left(M_i\right)$$

(34)

By Assumption 2 (Global Network Observability), this intersection contains only the zero vector $\left\{{\mathbf{0}}_n\right\}$. Consequently, it holds that $\mathit{\alpha }=\mathbf{0}$, which means the global eigenvector $\mathbf{v}=\mathbf{0}$. This directly contradicts the premise that $\mathbf{v}$ is a non-zero eigenvector. Therefore, no such eigenvalue exists on the unit circle, and the spectral radius is bounded:

$$\rho (\Phi )<1$$

(35)

From a distributed optimization perspective, this strict contraction inherently ensures that the exact DGD algorithm possesses a linear convergence rate toward the globally optimal feasibility manifold. This concludes the proof of Lemma 1. □

This lemma provides the essential spectral guarantee that the estimation error will contract over time. Based on this framework, we can now derive the formal convergence bounds for both nominal and faulty conditions.

5.1. Convergence Analysis Under Nominal Conditions

In this subsection, we establish the convergence properties of the DGD algorithm under nominal conditions, where all sensors are operational (i.e., ${\sigma }_i\left(t\right)=1,\forall i\in {\mathcal{V}}_{MAS}$). We utilize the ISS framework to establish theoretical performance bounds, demonstrating that the distance to the optimal solution (optimality gap) remains bounded despite stochastic sensor noise and temporal shifts in traffic demands.

The composite disturbance vectors for the global system are defined as follows. Recall the augmented hybrid noise vector $\overline{\mathit{\omega }}\left(t\right)\in {\mathbb{R}}^N$ (where $N=m+p_{int}$) defined in Section 3.4, which is bounded by $\parallel \overline{\mathit{\omega }}\left(t\right)\parallel \le {\omega }_{max}$. Furthermore, let $\Delta (t-1)={\mathbf{1}}_N\otimes {\Delta }_r(t-1)\in {\mathbb{R}}^{nN}$ be the global state drift representing temporal fluctuations in the stochastic O-D demands. Driven by the warm-start initialization, the unforced global error undergoes a macroscopic temporal transition defined as:

$$\mathbf{e}(t,0)=\mathbf{e}(t-1,K)-\Delta \left(t\right)$$

(36)

Theorem 1

(Nominal ISS and Bounded Optimality Gap). Consider the H-MASE network operating under nominal (fault-free) conditions (${\sigma }_i\left(t\right)=1,\forall i\in {\mathcal{V}}_{MAS}$) and satisfying Assumptions 1 and 2. Let the augmented hybrid noise vector and the macroscopic route-demand drift be bounded by $\parallel \overline{\mathit{\omega }}{\left(t\right)\parallel }_2\le {\omega }_{max}$ and $\parallel {\Delta }_r{\left(t\right)\parallel }_2\le {\Delta }_{max}$, respectively. Assuming the number of microscopic optimization iterations K is chosen sufficiently large such that $C_{ϵ}{\gamma }^K<1$, the distributed least-squares algorithm is ISS, ensuring robust convergence. Specifically, the Euclidean magnitude of the concatenated optimality gap $\mathbf{e}(t,K)$ at the end of the t-th macroscopic step is bounded by the recursive inequality:

$${\parallel \mathbf{e}(t,K)\parallel }_2\le C_{ϵ}{\gamma }^K{\parallel \mathbf{e}(t-1,K)\parallel }_2+{\mathcal{B}}_{ISS}({\omega }_{max},{\Delta }_{max})$$

(37)

where $\gamma =\rho (\Phi )+ϵ\in \left(\rho \right(\Phi ),1)$ is the effective exponential decay rate chosen via an arbitrarily small $ϵ>0$, $C_{ϵ}\ge 1$ is the corresponding condition constant accommodating the non-normality of the transition matrix Φ, and ${\mathcal{B}}_{ISS}({\omega }_{max},{\Delta }_{max})$ is the theoretical upper bound isolating the cumulative perturbative effects over K iterations.

Proof. 

The proof proceeds by recursively unrolling the inner-loop error dynamics and applying the triangle inequality over the dual time-scale boundaries.

Step 1: 

Recursive Unrolling of the Inner Loop

Recall the global error dynamics inside the microscopic k-loop, which can be compactly formulated using the global transition matrix $\Phi =\mathbf{P}\mathbf{W}$ and the block-diagonal pseudoinverse operator ${\mathbf{M}}^{†}=\mathrm{diag}(M_1^{†},\dots ,M_N^{†})\in {\mathbb{R}}^{nN\times N}$:

$$\mathbf{e}(t,k+1)=\Phi \mathbf{e}(t,k)+{\mathbf{M}}^{†}\overline{\mathit{\omega }}\left(t\right)$$

(38)

By recursively expanding (38) from the initial internal iteration $k=0$ to the final iteration K, and noting that the augmented hybrid noise $\overline{\mathit{\omega }}\left(t\right)$ remains invariant over the fast k-loop, we accumulate the persistent effects:

$$\mathbf{e}(t,K)={\Phi }^K\mathbf{e}(t,0)+\sum _{j=0}^{K-1}{\Phi }^j{\mathbf{M}}^{†}\overline{\mathit{\omega }}\left(t\right)$$

(39)

Step 2: 

Macroscopic Time-Scale Handover

The unrolled Equation (39) isolates the error at the beginning of the current macroscopic step, $\mathbf{e}(t,0)$. By invoking the time-scale separation principle established in (36), the initialization of the t-th step must incorporate the true state drift that occurred during the physical sampling interval. Substituting the warm-start boundary condition $\mathbf{e}(t,0)=\mathbf{e}(t-1,K)-\Delta (t-1)$ into (39) yields the complete temporal transition of the estimation error:

$$\mathbf{e}(t,K)={\Phi }^K\left(\mathbf{e}(t-1,K)-\Delta (t-1)\right)+\sum _{j=0}^{K-1}{\Phi }^j{\mathbf{M}}^{†}\overline{\mathit{\omega }}\left(t\right)$$

(40)

Rearranging the terms to isolate the perturbative effects from the nominal unforced response, we obtain the exact error state at the end of the macroscopic step t:

$$\mathbf{e}(t,K)={\Phi }^K\mathbf{e}(t-1,K)-{\Phi }^K\Delta (t-1)+\sum _{j=0}^{K-1}{\Phi }^j{\mathbf{M}}^{†}\overline{\mathit{\omega }}\left(t\right)$$

(41)

Step 3: 

Triangle Inequality and Norm Decompositions

Taking the Euclidean norm of (41) and applying the standard triangle and sub-multiplicative norm inequalities, we bound the global error. Given the mixed-product property of the Kronecker product ($\parallel {\mathbf{1}}_N\otimes {\Delta }_r{\left(t\right)\parallel }_2\le \sqrt{N}{\Delta }_{max}$), the noise bound $\parallel \overline{\mathit{\omega }}{\left(t\right)\parallel }_2\le {\omega }_{max}$, and the statically bounded pseudoinverse operator ($\parallel {\mathbf{M}}^{†}{\parallel }_2 <\infty$), we directly obtain the decoupled scalar formulation:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\parallel \mathbf{e}(t,K)\parallel }_2\le \parallel {\Phi }^K{\parallel }_2{\parallel \mathbf{e}(t-1,K)\parallel }_2+\parallel {\Phi }^K{\parallel }_2\sqrt{N}{\Delta }_{max}+\parallel {\mathbf{M}}^{†}{\parallel }_2{\omega }_{max}\sum _{j=0}^{K-1}{\parallel {\Phi }^j\parallel }_2\end{array}\end{array}$$

(42)

Next, we systematically decouple the global disturbance terms into their scalar upper bounds as defined in the theorem hypothesis. For the macroscopic demand drift, applying the mixed-product property of the Kronecker product to the Euclidean norm yields the exact analytical relationship:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\parallel \Delta (t-1)\parallel }_2=\parallel {\mathbf{1}}_N\otimes {\Delta }_r{(t-1)\parallel }_2=\parallel {\mathbf{1}}_N{\parallel }_2{\parallel {\Delta }_r(t-1)\parallel }_2\le \sqrt{N}{\Delta }_{max}\end{array}\end{array}$$

(43)

For the augmented hybrid noise, the operational limit is directly given as $\parallel \overline{\mathit{\omega }}{\left(t\right)\parallel }_2\le {\omega }_{max}$. Furthermore, since the global projection operator ${\mathbf{M}}^{†}=\mathrm{diag}(M_1^{†},\dots ,M_N^{†})$ is uniquely derived from the static routing matrix (A) and the incidence topology of the network, its induced 2-norm (i.e., its maximum singular value) is a finite, time-invariant constant ($\parallel {\mathbf{M}}^{†}{\parallel }_2 <\infty$).

Step 4: 

Spectral Bounding and ISS Formulation

The global transition matrix $\Phi =\mathbf{P}\mathbf{W}$ is generally non-normal. By the spectral radius theorem for non-normal matrices [30], since $\rho (\Phi )<1$, for any arbitrarily small $ϵ>0$, there exists a finite condition constant $C_{ϵ}\ge 1$ such that $\parallel {\Phi }^j{\parallel }_2\le C_{ϵ}{(\rho (\Phi )+ϵ)}^j\equiv C_{ϵ}{\gamma }^j$, where $\gamma \in \left(\rho \right(\Phi ),1)$. Substituting this worst-case relation into the decoupled error dynamics (42) and factoring out $C_{ϵ}$, we obtain:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\parallel \mathbf{e}(t,K)\parallel }_2\le C_{ϵ}{\gamma }^K{\parallel \mathbf{e}(t-1,K)\parallel }_2+C_{ϵ}{\gamma }^K\sqrt{N}{\Delta }_{max}+C_{ϵ}{\parallel {\mathbf{M}}^{†}\parallel }_2{\omega }_{max}\sum _{j=0}^{K-1}{\gamma }^j\end{array}\end{array}$$

(44)

By the theorem hypothesis, K is sufficiently large such that $C_{ϵ}{\gamma }^K<1$. Since $\gamma <1$, the geometric series evaluates to $(1-{\gamma }^K)/(1-\gamma )$. To formalize the ISS condition, we consolidate the disturbance-driven terms into a single theoretical boundary, ${\mathcal{B}}_{ISS}$:

$${\mathcal{B}}_{ISS}({\omega }_{max},{\Delta }_{max})\triangleq C_{ϵ}\left({\gamma }^K\sqrt{N}{\Delta }_{max}+{\displaystyle \frac{1-{\gamma }^K}{1-\gamma }}{\parallel {\mathbf{M}}^{†}\parallel }_2{\omega }_{max}\right)$$

(45)

Substituting this definition directly back into the error dynamics yields the final recursive macroscopic optimality gap:

$${\parallel \mathbf{e}(t,K)\parallel }_2\le C_{ϵ}{\gamma }^K{\parallel \mathbf{e}(t-1,K)\parallel }_2+{\mathcal{B}}_{ISS}({\omega }_{max},{\Delta }_{max})$$

(46)

Because the structural choice of the internal iteration count K explicitly guarantees that the macroscopic contraction factor satisfies $C_{ϵ}{\gamma }^K<1$, the DGD mechanism is globally asymptotically stable. Consequently, by the discrete-time ISS theorem [31], the error sequence is uniformly ultimately bounded. Taking the limit superior as $t\to \infty$, the distance to the optimal true state inherently converges to a strict asymptotic optimality gap bounded by:

$$\underset{t\to \infty }{lim\; sup}{\parallel \mathbf{e}(t,K)\parallel }_2\le {\displaystyle \frac{{\mathcal{B}}_{ISS}({\omega }_{max},{\Delta }_{max})}{1-C_{ϵ}{\gamma }^K}}$$

(47)

This confinement of the optimality gap, proportional to the physical demand volatility and topological noise bounds, definitively proves that the distributed optimization algorithm exhibits robust ISS. It should be explicitly noted that the analytical bounds established in ${\mathcal{B}}_{ISS}$ are inherently conservative, as they are derived using sub-multiplicative norm inequalities to provide strict worst-case mathematical guarantees. In practical deployments, stochastic noise cancellation during the diffusion phase typically yields an actual empirical error significantly lower than this theoretical upper bound. This concludes the proof of Theorem 1. □

Remark 7

(Bridging Theory and Resilient Operations). The analytical derivation of ${\mathcal{B}}_{ISS}$ in (45) establishes the mathematical bridge between the nominal stability analysis and the autonomous fault-isolation logic presented in Section 4.4. By anchoring the operational steady-state fault tolerance threshold to the theoretical ultimate bound of the local residual (e.g., setting ${\Gamma }_{steady}\ge {max}_i{\parallel M_i\parallel }_2{\displaystyle \frac{{\mathcal{B}}_{ISS}}{1-C_{ϵ}{\gamma }^K}}+{\omega }_{max}$), the H-MASE protocol guarantees that nominal stochastic fluctuations will not mathematically trigger the isolation mechanism. This strategic thresholding ensures the theoretical avoidance of false alarms as long as the physical disturbances remain within their specified supremum bounds.

While the nominal stability of the H-MASE is established through Theorem 1, the primary challenge in urban traffic monitoring remains the presence of structural sensor failures and data corruption. Having proved that the system remains stable under stochastic noise, the following subsection addresses the resilience of the protocol when the network undergoes localized instrumentation faults.

5.2. Resilience Analysis and Fault Tolerance

While the nominal stability established in Section 5.1 ensures robust performance under stochastic noise, macroscopic service networks are inherently vulnerable to structural component failures. From a systems reliability perspective, this subsection analyzes the fault tolerance of the H-MASE protocol. We demonstrate that the autonomous switching logic prevents cascading errors and preserves network-wide stability even when a subset of agents experiences structural sensor faults.

5.2.1. Fault Characterization and Detection Sensitivity

We model an abrupt structural fault (hard fault) at agent $i\in {\mathcal{V}}_{PSA}$ as an additive signal $f_i\left(t\right)\in \mathbb{R}$ affecting the local measurement. The corrupted observation, denoted by $b_i^f\left(t\right)$, is expressed as:

$$b_i^f\left(t\right)=M_i\mathbf{r}\left(t\right)+{\overline{\omega }}_i\left(t\right)+f_i\left(t\right)$$

(48)

where $f_i\left(t\right)$ represents severe, non-stochastic perturbations such as sudden sensor bias or catastrophic sensing-hardware calibration failure. It is important to note that this models a physical data corruption fault; the agent’s cyber-communication module remains fully operational to relay consensus variables, preserving the graph topology and the doubly stochastic property of the weight matrix W.

Note that the proposed isolation mechanism is specifically designed to detect such abrupt anomalies; gradually drifting soft faults may require moving-average residual windows, which fall outside the scope of this paper. The efficacy of the fault isolation mechanism depends on the sensitivity of the local consistency residual ${\eta }_i(t,k)$. To facilitate this analysis, let ${\mathbf{e}}_{diff,i}(t,k)={\widehat{\mathbf{r}}}_{diff,i}(t,k)-\mathbf{r}\left(t\right)$ denote the local diffused estimation error.

It is crucial to note that the agents operationally execute isolation relying solely on the locally computable residual ${\eta }_i(t,k)>\Gamma \left(t\right)$. However, to establish the theoretical reliability of this mechanism, we must analytically define the fault detectability bound—the fault magnitude that mathematically guarantees this operational threshold will be breached.

Lemma 2

(Autonomous Constraint Exclusion and Fault Detectability). In the context of robust distributed optimization, an agent i is designed to autonomously exclude its corrupted local constraint (triggering the isolation mode ${\sigma }_i=0$) if the scalar magnitude of the structural fault $f_i\left(t\right)$ satisfies:

$$|f_i\left(t\right)|>\Gamma \left(t\right)+\parallel M_i{\parallel }_2{\parallel {\mathbf{e}}_{diff,i}(t,k)\parallel }_2+{\omega }_{max}$$

(49)

where $\Gamma \left(t\right)$ is the dynamic detection threshold defined in Section 4.4, and ${\omega }_{max}$ is the global measurement noise bound.

Proof. 

The consistency residual is defined in (13) as ${\eta }_i(t,k)=|M_i{\widehat{\mathbf{r}}}_{diff,i}(t,k)-b_i^f\left(t\right)|$. Substituting the faulty measurement (48) and the local diffused error definition ${\widehat{\mathbf{r}}}_{diff,i}=\mathbf{r}+{\mathbf{e}}_{diff,i}$, we obtain:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\eta }_i(t,k)& =|M_i(\mathbf{r}\left(t\right)+{\mathbf{e}}_{diff,i}(t,k))-(M_i\mathbf{r}\left(t\right)+{\overline{\omega }}_i\left(t\right)+f_i\left(t\right))|\hfill \\ \hfill & =|(M_i{\mathbf{e}}_{diff,i}(t,k)-{\overline{\omega }}_i\left(t\right))-f_i\left(t\right)|\hfill \end{array}\end{array}$$

(50)

Applying the reverse triangle inequality yields a strict lower bound: ${\eta }_i(t,k)\ge |f_i\left(t\right)|-|M_i{\mathbf{e}}_{diff,i}(t,k)-{\overline{\omega }}_i\left(t\right)|$. To decouple the inner term, we apply the standard triangle and Cauchy-Schwarz inequalities. Acknowledging that the scalar noise is bounded ($|{\overline{\omega }}_i\left(t\right)|\le {\omega }_{max}$), we establish $|M_i{\mathbf{e}}_{diff,i}(t,k)-{\overline{\omega }}_i\left(t\right)|\le \parallel M_i{\parallel }_2{\parallel {\mathbf{e}}_{diff,i}(t,k)\parallel }_2+{\omega }_{max}$. Substituting this upper bound back into the residual inequality yields:

$${\eta }_i(t,k)\ge \left|f_i\left(t\right)\right|-\left(\parallel M_i{\parallel }_2{\parallel {\mathbf{e}}_{diff,i}(t,k)\parallel }_2+{\omega }_{max}\right)$$

(51)

According to the isolation logic defined in Section 4.4, the anomaly mode is triggered when ${\eta }_i(t,k)>\Gamma \left(t\right)$. Therefore, a sufficient condition to guarantee immediate isolation is:

$$|f_i\left(t\right)|-\left(\parallel M_i{\parallel }_2{\parallel {\mathbf{e}}_{diff,i}(t,k)\parallel }_2+{\omega }_{max}\right)>\Gamma \left(t\right)$$

(52)

Rearranging this algebraic inequality to isolate the fault magnitude $|f_i\left(t\right)|$ yields the statement of the lemma. Analytically, this establishes the exact mathematical boundary where a catastrophic local constraint perturbation is autonomously rejected by the distributed optimization algorithm, ensuring resilient continuity. This completes the proof. □

5.2.2. Stability of the Resilient Switched Dynamics

When a subset of structurally compromised PSAs, explicitly denoted as ${\mathcal{V}}_f\subset \{1,\dots ,m\}$, enters isolation mode (${\sigma }_i(t,k)=0$), the global system transition switches to a resilient operator ${\Phi }_{\sigma }={\mathbf{P}}_{\sigma }\mathbf{W}$. The modified global projection block ${\mathbf{P}}_{\sigma }=\mathrm{diag}(D_1,\dots ,D_N)$ is defined such that the local operator is $D_i=P_i$ for healthy agents ($i\in {\mathcal{V}}_h$) and forced to the identity mapping $D_i=I_n$ for isolated agents ($i\in {\mathcal{V}}_f$).

Assumption 3

(Residual Observability). Let ${\mathcal{V}}_h={\mathcal{V}}_{MAS}\setminus {\mathcal{V}}_f$ be the set of healthy agents. By design, VLAs are structurally insulated from hardware failures. Therefore, they permanently bypass the isolation mechanism (${\sigma }_i=1$) and continuously preserve the underlying cyber-graph connectivity. However, the system maintains structural observability if and only if the surviving PSAs within ${\mathcal{V}}_h$ provide sufficient rank. Algebraically, the intersection of the null spaces of all agents in the healthy ensemble must remain trivial: ${\bigcap }_{i\in {\mathcal{V}}_h}\mathcal{N}\left(M_i\right)=\left\{{\mathbf{0}}_n\right\}$.

From a practical municipal engineering standpoint, satisfying this residual observability condition after a severe sensor isolation relies on the inherently redundant deployment of physical sensors (e.g., overlapping camera fields and inductive loops) at critical urban intersections. When a localized hardware unit fails, the surviving adjacent sensors, coupled with the spatial correlation of overlapping macroscopic routes, collaboratively compensate for the localized rank deficiency. This redundant information is seamlessly routed across the network via the fault-immune VLAs, ensuring the rank condition is preserved.

Theorem 2

(Robust Convergence and Resilient Optimality Gap). Suppose the structural disturbance bounds $\parallel \overline{\mathit{\omega }}{\left(t\right)\parallel }_2\le {\omega }_{max}$ and ${\parallel \Delta \left(t\right)\parallel }_2\le \sqrt{N}{\Delta }_{max}$ established in Theorem 1 hold. Under the connectedness of ${\mathcal{G}}_{MAS}$ and the residual observability condition (Assumption 3), the distributed optimization algorithm remains ISS even if a subset of PSAs ${\mathcal{V}}_f$ undergoes severe structural failure. Assuming the fixed microscopic iteration count K is conservatively designed a priori to be sufficiently large such that the resilient macroscopic contraction condition $C_{\sigma ,ϵ}{\gamma }_{\sigma }^K<1$ is preserved, the global optimality gap converges to a resilient invariant set independent of the catastrophic fault magnitudes $f_i\left(t\right)$, bounded by:

$$\underset{t\to \infty }{lim\; sup}{\parallel \mathbf{e}(t,K)\parallel }_2\le {\displaystyle \frac{{\mathcal{B}}_{ISS}^{\sigma }({\omega }_{max},{\Delta }_{max})}{1-C_{\sigma ,ϵ}{\gamma }_{\sigma }^K}}$$

(53)

where ${\gamma }_{\sigma }=\rho \left({\Phi }_{\sigma }\right)+ϵ\in (\rho \left({\Phi }_{\sigma }\right),1)$ is the resilient effective decay rate (i.e., linear convergence rate) under the switched topology, $C_{\sigma ,ϵ}\ge 1$ is the corresponding condition constant, and ${\mathcal{B}}_{ISS}^{\sigma }$ is the cumulative resilient disturbance bound on the optimality gap.

Proof. 

The proof proceeds by analyzing the switched system dynamics under the autonomous constraint-exclusion mechanism.

Step 1: 

Switched Dynamics and Time-Invariant Unrolling

When catastrophic faults $\mathbf{f}\left(t\right)$ trigger the autonomous isolation (${\sigma }_i=0$ for $i\in {\mathcal{V}}_f$), the effective local pseudoinverse blocks within the global matrix are forced to zero. This disjoint algebraic property yields the exact nullification of the fault vector: ${\mathbf{M}}_{\sigma }^{†}\mathbf{f}\left(t\right)\equiv {\mathbf{0}}_{nN}$. Consequently, the corrupted innovation is entirely bypassed. Since the severe anomaly persistently breaches the threshold (${\eta }_i\gg \Gamma$), the switched operator ${\Phi }_{\sigma }={\mathbf{P}}_{\sigma }\mathbf{W}$ remains time-invariant across the inner loop. Furthermore, structural faults do not alter the physical demand drift $\Delta (t-1)$ or the temporal handover. Thus, recursively unrolling the decoupled dynamics and applying the macroscopic boundary condition yields the exact isolated error state:

$$\mathbf{e}(t,K)={\Phi }_{\sigma }^K\mathbf{e}(t-1,K)-{\Phi }_{\sigma }^K\Delta (t-1)+\sum _{j=0}^{K-1}{\Phi }_{\sigma }^j{\mathbf{M}}_{\sigma }^{†}\overline{\mathit{\omega }}\left(t\right)$$

(54)

Step 2: 

Preservation of Global Feasibility and Spectral Contraction

Since physical faults do not sever the cyber-links, W remains doubly stochastic. The modified projection block ${\mathbf{P}}_{\sigma }$ consists of orthogonal projectors ($P_i$) and identity mappings ($I_n$), preserving the non-expansive property $\parallel {\mathbf{P}}_{\sigma }{\parallel }_2\le 1$. Thus, $\rho \left({\Phi }_{\sigma }\right)\le 1$. To prove strict contraction, we invoke the exact algebraic contradiction sequence established in Lemma 1. If $\rho \left({\Phi }_{\sigma }\right)=1$, the eigenvector is confined to the consensus subspace ($\mathbf{v}={\mathbf{1}}_N\otimes \mathit{\alpha }$), dictating ${\mathbf{P}}_{\sigma }\mathbf{v}=\mathbf{v}$. For isolated agents ($D_i=I_n$), this imposes no algebraic restriction. Consequently, the preservation of global feasibility relies exclusively on the healthy ensemble, requiring $\mathit{\alpha }\in {\bigcap }_{i\in {\mathcal{V}}_h}\mathcal{N}\left(M_i\right)$. By the Residual Observability condition (Assumption 3), this intersection is trivial ($\mathit{\alpha }=\mathbf{0}$). Thus, no such eigenvalue exists on the unit circle, proving that the resilient operator remains a strict contraction mapping: $\rho \left({\Phi }_{\sigma }\right)<1$.

Step 3: 

Resilient Spectral Bounding and Final ISS Formulation

Since ${\Phi }_{\sigma }$ is generally non-normal but contracting, we apply the spectral radius theorem analogous to Theorem 1. For an arbitrarily small $ϵ>0$, there exists $C_{\sigma ,ϵ}\ge 1$ such that $\parallel {\Phi }_{\sigma }^j{\parallel }_2\le C_{\sigma ,ϵ}{\gamma }_{\sigma }^j$, where ${\gamma }_{\sigma }=\rho \left({\Phi }_{\sigma }\right)+ϵ<1$. Taking the Euclidean norm of (54), substituting the spectral bound, and utilizing the invariant physical disturbance limits, we analytically evaluate the convergent geometric series ${\sum }_{j=0}^{K-1}{\gamma }_{\sigma }^j$. Consolidating the disturbance-driven terms defines the ultimate cumulative resilient disturbance bound, ${\mathcal{B}}_{ISS}^{\sigma }$:

$${\mathcal{B}}_{ISS}^{\sigma }({\omega }_{max},{\Delta }_{max})\triangleq C_{\sigma ,ϵ}\left({\gamma }_{\sigma }^K\sqrt{N}{\Delta }_{max}+{\displaystyle \frac{1-{\gamma }_{\sigma }^K}{1-{\gamma }_{\sigma }}}{\parallel {\mathbf{M}}_{\sigma }^{†}\parallel }_2{\omega }_{max}\right)$$

(55)

Substituting this directly back into the bounded error dynamics theoretically confirms the final ultimate bound established in (53). Because the catastrophic structural fault vector $\mathbf{f}\left(t\right)$ has been mathematically eradicated (${\mathbf{M}}_{\sigma }^{†}\mathbf{f}\left(t\right)\equiv \mathbf{0}$), the optimality gap is a function of the stochastic noise and macroscopic drift. This confirms that the robust optimization protocol successfully rejects corrupted constraints while preserving global feasibility. This concludes the proof of Theorem 2. □

Remark 8

(The Structural Necessity of VLAs). The resilience of the H-MASE architecture relies heavily on the topological anchors provided by the VLAs. While the surviving PSAs are solely responsible for maintaining the algebraic rank of the system (Assumption 3), VLAs mathematically guarantee the preservation of cyber-graph connectivity (Assumption 1). Since VLAs are immune to physical faults (${\sigma }_i(t,k)\equiv 1$), their local projection matrices permanently act as perfect identity filters ($P_i=I_n$). This ensures that even if multiple contiguous physical sensors catastrophically fail and isolate their measurements, the underlying consensus topology is never fragmented. The VLAs seamlessly route the diffused state estimates across the network, allowing the surviving PSAs to collaboratively share their remaining rank and achieve global observability.

6. Numerical Results and Emission Mitigation Analysis

The primary objective of this work is to establish the mathematical foundation of the proposed H-MASE architecture rather than to provide an empirical benchmarking study on unstructured datasets. While real-world data (e.g., Performance Measurement System (PeMS), Metropolitan Transportation Authority-Los Angeles (METR-LA)) introduces highly complex, unmodeled exogenous variables, validating the theoretical worst-case guarantees—specifically the ISS bounds proven in Theorems 1 and 2, and the exact fault detectability threshold established in Lemma 2—requires a controlled stochastic environment. To this end, this section presents numerical simulations designed to isolate and evaluate the algorithm’s operational properties under bounded structural noise, random-walk state drifts, and catastrophic sensor malfunctions.

It should be explicitly noted that while the ultimate system-level objective of the H-MASE protocol is emission mitigation, this study deliberately isolates the cyber-physical state estimation layer. Therefore, rather than presenting coupled vehicular emission graphs derived from simulators like Motor Vehicle Emission Simulator (MOVES), the numerical results focus on validating Root Mean Square Error (RMSE). Achieving a resilient, sub-unit optimality gap is the absolute prerequisite data foundation before any downstream eco-routing or emission mitigation strategies can be reliably executed.

6.1. Simulation Environment and Cyber-Physical Topology

To emulate a highly realistic and topologically heterogeneous stochastic network flow problem within an urban logistics context, a complex ring-radial transportation network is constructed, as illustrated in Figure 4. Although the generated dataset is synthetic to preserve mathematical traceability, the network topology mirrors standard metropolitan grid structures. Furthermore, the injected route demand volumes (≈20–40 vehicles per discrete step) are explicitly scaled to match typical hourly flow rates in metropolitan roundabout scenarios, ensuring that the macroscopic flow magnitudes reflect real-world empirical conditions. Unlike oversimplified symmetric grid topologies frequently utilized in theoretical literature, the physical layer of this network deliberately incorporates a diverse mix of unidirectional streets and bidirectional urban arterials, tightly coupled with a central counter-clockwise roundabout. The network comprises 15 internal signalized intersections ($p_{int}=15$) and 10 boundary nodes representing 5 origins ($O_1$–$O_5$) and 5 destinations ($D_1$–$D_5$), connected via $m=55$ directed road segments (edges). This structural diversity ensures that the distributed estimation protocol is tested against the asymmetric bottlenecks, varying nodal degrees, and complex overlapping route flows characteristic of genuine metropolitan networks. Consequently, the multi-agent network is formed by $N=70$ autonomous agents, integrating the 55 PSAs with 15 VLAs stationed strategically at the intersections. The complete set of simulation variables, including temporal parameters and disturbance bounds, is summarized in

Table 2. Network Attributes and H-MASE Simulation Parameters.

Symbol	Parameter Description	Value
Network Topology & Graph Parameters
N	Total number of agents (55 PSAs + 15 VLAs)	70
n	Number of dominant logical routes	25
m	Number of directed physical links (edges)	55
Algorithm & Temporal Parameters
$T_{max}$	Total macroscopic simulation time steps	150
K	Microscopic consensus iterations (inner loop)	80
$r\left(0\right)$	Initial traffic flow densities (vehicles/step)	∈[20, 40]
Disturbances & Fault Injection Settings
${\omega }_{max}$	Maximum physical sensor noise bound	2.0
${\Delta }_{max}$	Maximum route flow drift (random walk)	1.0
${\Gamma }_{steady}$	Steady-state fault resilience threshold	5.0
${\Gamma }_{init}$	Transient initialization tolerance	200.0
$\beta$	Threshold exponential decay rate	0.15
$t_{fault}$	Time step of structural fault injection	80
f	Magnitude of the structural fault bias	80.0

. Specifically, the threshold exponential decay rate ($\beta =0.15$) is empirically tuned to safely accommodate the initial transient disagreements during the consensus warm-start phase. This calibration ensures that the theoretical fault detectability bound (Lemma 2) is preserved before the system firmly settles into its steady-state resilience boundary (${\Gamma }_{steady}$).

A crucial feature of the proposed simulation environment is the implementation of a topology-induced cyber-physical communication graph. Unlike conventional Erdös-Rényi random networks, the communication links in this setup are dictated by physical proximity (hop-by-hop communication). Specifically, PSAs only communicate with adjacent sensors sharing the same intersection and with their local VLA, thereby preserving the fully decentralized and localized nature of the estimation framework.

To evaluate the algorithm under an asymmetric and realistic traffic load, $n=25$ linearly independent dominant logical routes are defined, originating from the boundary nodes and traversing the internal junctions. The exact node-by-node traversal sequences of these routes are detailed in

Table 3. The Dominant Route Set of the Traffic Network ($n=25$).

Route ID	Origin	Intermediate Junctions	Destination
$r_1$	$O_1$	$J_1\to J_2$	$D_1$
$r_2$	$O_1$	$J_1\to J_{11}\to J_{12}\to J_4$	$D_2$
$r_3$	$O_1$	$J_1\to J_{11}\to J_{12}\to J_{13}\to J_6$	$D_3$
$r_4$	$O_1$	$J_1\to J_{10}\to J_9\to J_8$	$D_4$
$r_5$	$O_1$	$J_1\to J_{10}$	$D_5$
$r_6$	$O_2$	$J_3\to J_2$	$D_1$
$r_7$	$O_2$	$J_3\to J_4$	$D_2$
$r_8$	$O_2$	$J_3\to J_{12}\to J_{13}\to J_6$	$D_3$
$r_9$	$O_2$	$J_3\to J_{12}\to J_{13}\to J_{14}\to J_8$	$D_4$
$r_{10}$	$O_2$	$J_3\to J_2\to J_1\to J_{10}$	$D_5$
$r_{11}$	$O_3$	$J_5\to J_4\to J_3\to J_2$	$D_1$
$r_{12}$	$O_3$	$J_5\to J_4$	$D_2$
$r_{13}$	$O_3$	$J_5\to J_6$	$D_3$
$r_{14}$	$O_3$	$J_5\to J_{13}\to J_{14}\to J_8$	$D_4$
$r_{15}$	$O_3$	$J_5\to J_{13}\to J_{14}\to J_{15}\to J_{10}$	$D_5$
$r_{16}$	$O_4$	$J_7\to J_{14}\to J_{15}\to J_{11}\to J_2$	$D_1$
$r_{17}$	$O_4$	$J_7\to J_6\to J_5\to J_4$	$D_2$
$r_{18}$	$O_4$	$J_7\to J_6$	$D_3$
$r_{19}$	$O_4$	$J_7\to J_8$	$D_4$
$r_{20}$	$O_4$	$J_7\to J_{14}\to J_{15}\to J_{10}$	$D_5$
$r_{21}$	$O_5$	$J_9\to J_{15}\to J_{11}\to J_2$	$D_1$
$r_{22}$	$O_5$	$J_9\to J_{15}\to J_{11}\to J_{12}\to J_4$	$D_2$
$r_{23}$	$O_5$	$J_9\to J_8\to J_7\to J_6$	$D_3$
$r_{24}$	$O_5$	$J_9\to J_8$	$D_4$
$r_{25}$	$O_5$	$J_9\to J_{10}$	$D_5$

. Topological utilization analysis reveals that while 100% of the internal junctions are actively engaged by the routing set, the physical edge coverage is inherently sparse at 72.73% (40 out of 55 edges utilized). Furthermore, the spatial distribution of the routes introduces significant traffic hotspots, such as Edge 1, which is heavily shared by 5 distinct routes. Despite this asymmetrical flow distribution, the augmented observation matrix M constructed by the VLAs yields a full column rank, i.e., $\mathrm{rank}\left(M\right)=25$. This satisfies the structural observability requirement established in the problem formulation, ensuring that the global traffic state vector is recoverable from the distributed local projections.

6.2. Distributed Tracking Performance in Nominal Operation

In this subsection, the nominal tracking capability of the proposed H-MASE protocol is evaluated prior to the injection of any severe structural faults. The primary objective is to demonstrate how the multi-agent network collectively tracks the dynamically evolving traffic state relying solely on localized, hop-by-hop communication.

The temporal evolution of the traffic densities across all $n=25$ dominant routes is presented in Figure 5. In this visualization, the dash-dot lines represent the ground truth traffic flows, which are subjected to continuous stochastic demand fluctuations (modeled as bounded random-walk drifts with ${\Delta }_{max}=1.0$) at each macroscopic time step t. Conversely, the solid lines illustrate the average estimated densities collaboratively computed by the distributed agents. Due to the rapid spectral convergence of the inner-loop consensus mechanism, the individual estimates of all 70 agents densely cluster around this network-wide average with negligible spatial variance. As observed in Figure 5, these consensus-driven solid lines closely track the dynamic dash-dot trajectories. Despite the presence of physical measurement noise (${\omega }_{max}=2.0$) affecting the local observations, the algorithm maintains tracking accuracy. Empirical performance metrics collected during the steady-state operational phase reveal an RMSE of approximately 0.81 vehicles per route and a Mean Absolute Percentage Error (MAPE) of 2.44%. This robust solution quality is mathematically attributed to the inner-loop distributed optimization mechanism. By executing $K=80$ microscopic DGD iterations per macroscopic update, the algorithm effectively solves the network-wide least-squares problem, functioning as a spatial variance reduction mechanism. The decentralized architecture successfully exploits the stochastic cancellation of zero-mean physical noise, ensuring that the global state vector securely converges to a tight optimality gap (near-optimal feasible region) without requiring a centralized data fusion center. From a sustainable systems engineering standpoint, this high-fidelity bounding (e.g., an RMSE of merely 0.81 vehicles per estimated state) ensures that downstream dynamic resource allocation models and stochastic queue management systems receive highly reliable input parameters. In the physical realm, tracking route flows with sub-unit precision essentially eradicates the uncertainties that cause suboptimal traffic light synchronizations. This creates a fail-safe data foundation for implementing network-wide green-wave signalization, directly minimizing stop-and-go driving cycles, vehicle idling times, and the associated greenhouse gas emissions.

6.3. Resilient Fault Isolation and Comparative ISS Validation

The final simulation phase validates the resilience guarantees of the H-MASE protocol, specifically the autonomous fault isolation mechanism (Lemma 2) and the ISS bounds (Theorems 1 and 2). To robustly demonstrate the algorithmic superiority and the strict necessity of the proposed autonomous isolation logic, we conduct an ablation study by benchmarking H-MASE against a conventional DGD algorithm (e.g., Mou et al. [26]). While advanced frameworks such as the Alternating Direction Method of Multipliers (ADMM) or Distributed Kalman Filtering (DKF) exist, they inherently demand massive computational overhead (e.g., continuous matrix inversions or exact transition-matrix modeling). Such heavy dependencies violate the lightweight, edge-computing premise of the proposed architecture. Therefore, a standard first-order DGD approach (Mou et al. [26]) serves as the most structurally appropriate baseline. By contrasting H-MASE with this widely-adopted, computationally equivalent baseline, we specifically isolate the impact of our novel autonomous switching logic.

To emulate highly realistic and non-ideal IoT communication environments, the simulation is subjected to a continuous 10% random packet loss (link failure) rate across the digital communication graph ${\mathcal{E}}_{MAS}$. From a network systems perspective, a packet loss event can be interpreted as a manifestation of infinite communication latency. By demonstrating that the H-MASE protocol maintains ISS under such severe stochastic interruptions, we effectively verify its robustness against variable transmission delays, as the consensus mechanism inherently compensates for delayed or missing information by relying on the surviving local neighborhoods. Furthermore, to test cyber-physical fault tolerance, we simulate a catastrophic IoT hardware malfunction at a critical intersection. Specifically, a structural bias ($f=80.0$)—representing a severe sensor calibration loss or edge-device cyber interference—is injected into a randomly selected active PSA at time step $t_{fault}=80$. Given that the nominal route demands operate within $[20,40]$ vehicles per step, this anomaly represents a massive data corruption relative to the actual traffic volume.

The temporal evolution of the Euclidean norm of the global estimation error (i.e., the global optimality gap), ${\parallel \mathbf{e}(t,K)\parallel }_2$, for both algorithms is depicted in Figure 6. Prior to the fault injection ($t<80$), both algorithms successfully track the state despite the $10\%$ packet loss, oscillating within a tightly bounded invariant set. This confirms that the H-MASE protocol highly effectively avoids false alarms during nominal operations (satisfying Lemma 2).

At macroscopic step $t=80$, the onset of the IoT hardware fault causes the residual of the compromised edge device to exhibit an abrupt surge (${\eta }_i\gg \Gamma \left(t\right)$). The autonomous isolation mode (${\sigma }_i=0$) defined in Section 4.4 is successfully triggered. Acting as a localized cyber-physical fail-safe, the edge agent immediately overrides its local projection phase, quarantining its corrupted physical sensor data from the network while maintaining its digital communication links.

The effect of this autonomous constraint-dropping mechanism is distinctly observed in the post-fault trajectories of Figure 6. The error remains bounded before the fault (validating Theorem 1 ISS bounds). As can be observed from the figure, following the sensor malfunction at $t=80$, the baseline Conventional DGD algorithm severely diverges, whereas H-MASE autonomously isolates the fault and maintains a resilient, bounded optimality gap (validating Theorem 2). The baseline Conventional DGD algorithm, unable to isolate the severe $80.0$ vehicle bias, instantly absorbs the corrupted constraint, causing its global estimation error to diverge uncontrollably. Conversely, the H-MASE protocol mathematically eliminates the catastrophic fault vector (${\mathbf{M}}_{\sigma }^{†}\mathbf{f}\left(t\right)\equiv \mathbf{0}$). As theoretically predicted in Theorem 2, while the topological exclusion mildly degrades the network’s condition, the distributed optimization algorithm provably preserves global stability, firmly maintaining the robust optimal tracking of the network state.

6.4. Ecological Impact and Emission Mitigation

The ultimate operations research objective of establishing a highly resilient, sub-unit traffic state estimator is to prevent localized IoT hardware malfunctions from cascading into network-wide ecological bottlenecks. When an intersection controller relies on the diverging traffic states provided by a conventional, non-resilient estimator (as seen in Figure 6), it perceives an artificial gridlock ($x_k\to 1$). This forces the local dynamic signalization into highly suboptimal phase allocations, triggering extensive vehicular idling.

By substituting the dynamically estimated route flows into the macroscopic Webster delay model (Equation (15)), we quantified the ecological impact of the H-MASE protocol’s resilience. Assuming a standard emission rate of $0.6$ g of CO₂ per second for idling vehicles, the divergence of the conventional DGD algorithm resulted in an estimated $2611.89$ kg of CO₂ emissions over the post-fault simulation window. In stark contrast, the resilient signalization data foundation preserved by the H-MASE protocol constrained the network emissions to $2591.72$ kg.

This translates to a direct $0.77\%$ reduction in total network-wide greenhouse gas emissions. While seemingly modest as a percentage, a $0.77\%$ continuous emission reduction stemming from the mitigation of a single compromised edge sensor translates to an immense absolute reduction in daily atmospheric pollutants across a metropolitan grid. A comprehensive summary of these comparative performance metrics and the resulting ecological impact is presented in

Table 4. Quantitative Performance and Ecological Impact Under Severe Sensor Malfunction.

Performance Metric	Conventional DGD [26]	Proposed H-MASE
Total Idling Delay (vehicle-hours)	$1209.2$	$1199.8$
Total CO2 Emissions (kg)	$2611.89$	$2591.72$
Net CO2 Reduction	0.77% (20.17 kg saved per incident window)

. This quantitative analysis conclusively demonstrates that the cyber-physical resilience established by the H-MASE protocol directly yields real-world ecological sustainability. It is important to emphasize that the computed 20.17 kg reduction in CO₂ is a direct consequence of eliminating artificial gridlock states in the signaling controller. In the baseline scenario, the estimation divergence causes the saturation ratio ($x_k$) to saturate at 0.995, which, according to the Webster model, yields a near-exponential increase in vehicle delay. By maintaining sub-unit estimation precision even during a catastrophic sensor failure, H-MASE prevents this operational collapse. Given that these results are derived from a single-sensor failure in a restricted simulation window, the cumulative ecological benefit across an entire city-scale network with thousands of edge nodes would be substantial, directly contributing to municipal carbon neutrality goals.

7. Conclusions

This paper presented the H-MASE protocol, a fully decentralized decision-support framework formulated as a distributed convex optimization problem to safeguard smart transportation systems under severe cyber-physical uncertainties. By distributing the computational burden across IoT-enabled edge devices (acting as PSA and VLA) via localized hop-by-hop communication, the proposed applied architecture effectively eliminates the single-point-of-failure vulnerability inherent in traditional centralized cloud-based traffic management systems. Consequently, it prevents localized IoT hardware malfunctions from cascading into network-wide traffic bottlenecks.

Theoretical analysis and comprehensive numerical simulations confirmed the mathematical efficacy of the protocol. By executing exact DGD steps intertwined with topological consensus, the multi-agent network successfully solved the network-wide least-squares estimation problem in real-time. Even under bounded random-walk state drifts and persistent measurement noise, the algorithm successfully neutralized local spatial variances. The swarm-intelligence mechanism tightly bounded the global optimality gap without requiring any central data-fusion center.

Furthermore, the autonomous switching logic demonstrated highly reliable fault detectability without triggering erroneous isolations under nominal conditions, successfully and instantaneously isolating structurally compromised sensors. Following this autonomous fault isolation, the global error remained confined within its theoretical ISS bounds, validating the robust optimization capabilities and cyber-physical resilience of the H-MASE protocol. Ultimately, the ability to maintain sub-unit estimation precision (e.g., an RMSE of 0.81 vehicles per estimated state) during catastrophic failures provides the fail-safe data foundation required for uninterrupted eco-routing and sustainable urban logistics.

Future work will focus on leveraging the resilient route density estimates provided by H-MASE to develop dynamic eco-signalization frameworks and intelligent Mobility-as-a-Service (MaaS) routing engines. By formulating the intersection signaling process as a stochastic resource allocation problem, these high-fidelity estimates can be directly coupled with macroscopic emission models (e.g., MOVES) and edge-based traffic controllers. This integration will allow for the dynamic optimization of green-light phase timings, explicitly aimed at establishing green-wave corridors, minimizing network-wide vehicle idling times, and reducing overall greenhouse gas emissions. Additionally, to bridge the gap between macroscopic theoretical models and practical industrial deployment, future studies will integrate the proposed robust optimization protocol with microscopic traffic simulators (e.g., Simulation of Urban MObility (SUMO)) to create a plug-and-play IoT software architecture for modern municipalities. Validation on large-scale real-world datasets (e.g., PeMS, METR-LA) within the context of edge-computing networks will further demonstrate how the proposed framework drives the continuous advances in transportation and smart city deployments.