A stochastic computational framework for the joint transportation network fragility analysis and traffic flow distribution under extreme events

The study addresses how spatial correlation of bridge damage under extreme events influences transportation network performance. While performance-based assessment requires component-level damage estimation (e.g., bridge fragility under earthquakes), socio-economic impacts emerge at the network level through traffic disruption and re-routing. The paper proposes a stochastic, computationally efficient framework that integrates structural fragility analyses with traffic distribution/assignment to quantify how correlation distance in bridge damage affects connectivity and total travel time. The key research question is: to what extent does the correlation distance of bridge damage alter network performance metrics, and how sensitive are results to this spatial dependence? The purpose is to enable realistic performance assessment and to avoid unconservative assumptions of independent bridge damages.

The work builds on and contributes to several streams: (i) probabilistic network performance and life-cycle management of infrastructure networks, including bridge system reliability and resilience; (ii) concepts of community resilience and recovery for interdependent lifelines; (iii) system reliability of networks via matrix-based methods and surveys; (iv) network interactions and systems-of-networks perspectives; (v) traffic assignment and user-equilibrium principles for transportation networks. Prior studies often neglect or simplify spatial dependency of component damage, potentially biasing network-level metrics. This paper introduces a practical approach to estimate and incorporate spatial correlation based on fragility analyses and random field modeling, extending existing probabilistic frameworks for network performance under hazards.

The framework comprises five main steps (Fig. 1): 1) Structural fragility analysis: Select the region, highway segments, and bridges. Using HAZUS-MH, compute bridge-specific fragility by combining ground shaking and ground failure for each bridge class and updating for bridge properties (e.g., span, skew). For each scenario, compute a scalar average damage index for bridge b in [0,4] by combining probabilities of reaching damage states (no, minor, moderate, major, collapse) via a weighted sum. 2) Spatial correlation estimation: Map bridge lat/long to planar coordinates. Using multiple hazard scenarios (e.g., different epicenters, magnitudes), estimate empirical correlation of bridge damage versus inter-bridge distance. Fit an exponential-type correlation model ρ(ξ) = A·exp(−ξ²/λ²) + K by least squares or maximum likelihood to obtain correlation length λ (and A, K), along with goodness-of-fit. 3) Random field simulation: Assume a neutral marginal for damage, here Uniform[0,4] (mean ≈ 2, variance ≈ 1.33) to isolate correlation effects. Use the fitted correlation structure to define the autocorrelation R and spectral density S of a homogeneous, isotropic 2D field. Generate 2D random field samples of damage over the network area for a range of λ values. Retrieve damage values at bridge locations from the field samples. The field is simulated once per λ and reused across analyses. 4) Network analysis: Build a directed graph where edges represent directed highway segments and nodes represent junctions (and, in this study, trip origins/destinations). For each edge, compute free-flow travel time from segment length and speed, and practical capacity from speed, lane count, and spacing assumptions. Post-event updating uses a continuous residual capacity model: an edge’s practical capacity is reduced according to the maximum damage among its bridges (dominant assumption), with parameters (e.g., αc = 0.03, βc = 4.0). Trip generation/attraction at nodes (O0, D0) are held at pre-event levels for sensitivity analysis (no exogenous demand reduction). Solve a combined traffic distribution and assignment via an iterative procedure: initialize shortest paths using free-flow times, construct an origin–destination (OD) matrix via a gravity model with balancing factors, assign flows to shortest paths, then iterate to update edge times using a congestion function (e.g., Bureau of Public Roads form with α = 0.15, β = 4.0), recompute shortest paths, update the OD via nested iteration, and perform a line search over step size to minimize a convex objective that combines total travel time (user equilibrium) with the gravity model consistency. Convergence yields OD, link flows, and edge travel times. 5) Statistical result analysis: For each λ, compute performance metrics over multiple field samples: (i) Fully Connected Ratio (FCR), the percentage of samples where all nodes are mutually reachable; (ii) Total travel time P, the sum over all flows times edge travel times. Samples with disconnections lead to divergent P and are excluded from P statistics to avoid bias. For fair comparison across λ, the analysis selects the same number of best (lowest) P values equal to the minimum number of connected samples across λ (based on the minimum FCR).

- Spatial correlation of bridge damage strongly affects network connectivity (FCR) and total travel time (P). - Bridges in parallel example: FCR(λ) displays a pronounced non-monotonic dependence on correlation length. The minimum FCR occurs at λ ≈ 8 miles with FCR ≈ 40%, meaning only about 40% of samples yielded full connectivity at that λ; consequently, the lowest 200 P results (matching that minimum count of connected samples) were retained across all λ for consistent statistics. For small λ (weak correlation), FCR is higher (nearly 60% connected), despite conservative uniform damage marginals that often reduce capacity by more than half. For intermediate λ (approximately 4–14 miles), FCR drops below 50% as co-located bridges near a node tend to share high damage simultaneously, isolating nodes. For large λ (>14 miles), damage is nearly uniform across the network, yielding roughly a 50% chance of total functionality vs. total disruption, and FCR tends to about 50%. - The total travel time P mirrors this behavior: best (lowest) P for small λ, worst (highest) P for intermediate λ, and improving again as λ becomes very large. Variability of P is also highest for intermediate λ (about 4–10 miles), indicating reduced reliability. - Bridges in series example: Including multiple bridges per edge degrades performance, especially for low λ, because it is more likely that at least one bridge on an edge is heavily damaged. Both FCR and P indicate worse performance than the parallel case for low λ. For very large λ (fully correlated damage), the performance converges to that of the parallel case because all bridges share similar damage and their count is no longer influential. - Analytical linkage between uniform damage and residual capacity under the exponential capacity reduction model shows a heavy tail toward low residual capacity, reinforcing the conservative nature of the marginal damage assumption and the observed sensitivity to λ. - Overall, assuming independence among bridge damages can significantly overestimate network performance (unconservative), especially in the presence of spatially correlated hazards.

The findings demonstrate that network-level performance under extreme events depends critically on the spatial correlation of component damage, not solely on marginal damage probabilities. Weak correlation distributes damage heterogeneously, enabling alternative routes and maintaining higher connectivity and lower travel times. Intermediate correlation lengths cluster damage around nodes, frequently isolating them and severely degrading performance and increasing variability. Very strong correlation renders the system either widely functional or widely impaired, with corresponding bifurcation in connectivity and travel time. The framework connects structure-scale fragility (bridges) to system-scale impacts (network flows), enabling realistic assessments that capture user equilibrium and demand redistribution via a combined distribution–assignment model. The stochastic random field approach provides computational efficiency and direct control of correlation length for sensitivity analyses, outperforming scenario-by-scenario fragility-only approaches in scalability. Results caution against the common independence assumption for bridge damages, as it can lead to unconservative estimates in planning, monitoring, and vulnerability assessment. The approach can inform post-event accessibility planning (via FCR) and longer-term recovery and economic impact assessments (via P).

The paper contributes: (1) a computational framework integrating structural fragility analyses, spatial random field simulation of bridge damage, and traffic distribution/assignment for transportation network performance under extreme events; (2) a sensitivity analysis revealing a strong influence of the correlation distance of bridge damage on connectivity and travel time, with worst performance at intermediate correlation lengths; (3) a practical method to estimate correlation distance from limited fragility-based scenarios. The framework highlights that neglecting spatial correlation can significantly overestimate network performance. The methodology is adaptable to other hazards and can be extended to multi-variate fields representing different bridge classes with distinct marginals and cross-correlations. The tools and concepts can serve as a core for life-cycle maintenance optimization, network reliability and availability assessments, loss estimation, and component importance ranking.

- Damage is modeled only for bridges; other network components (e.g., roadways, intersections) are assumed undamaged. - Damage fields are assumed homogeneous, isotropic, and characterized by a single correlation length; anisotropy or non-stationarity are not modeled. - The marginal damage distribution is assumed Uniform[0,4] for neutrality; real-world marginals may be non-uniform and bridge-specific. - Bridge structural diversity is represented in an average sense via the random field; detailed bridge-to-bridge variability is not included unless extended to multi-variate fields. - The dominant assumption reduces an edge’s capacity to that of its most damaged bridge; local detours or partial lane operations may mitigate losses but are not fully modeled (except in tabulated residual-capacity cases). - Elevation differences are neglected in distance calculations; mapping uses a planar approximation. - OD demand (O0, D0) is held constant post-event for sensitivity analysis; real events may reduce or shift demand. - Origins and destinations are assumed to coincide with network nodes for simplicity. - Discrete grid sampling of the random field requires careful resolution selection; extremely fine grids increase computation. - The “bridges in series” case is a limit condition that may be overly conservative due to possible local bypasses not modeled.