PSI - Issue 84

N. Kheirkhahan et al. / Procedia Structural Integrity 84 (2026) 33–40 N. Kheirkhahan et al./ Structural Integrity Procedia 00 (2026) 000–000

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4.1. Global efficiency A resilient urban spatial structure can absorb disturbances while maintaining the operational continuity of critical urban systems (Lu et al., 2025). The global efficiency metric, defined as the average inverse shortest path length across all node pairs in the network, is calculated as: 1 1 ( 1) i j ij GE n n d  = −  (2) Where n is the total number of nodes and d ij is the shortest path length between nodes i and j (Latora et al., 2001). 4.2. Optimal buffer radius selection Analysis is conducted across buffer radii r ∈ {10,20,30,40,50,60} km around the disrupted edge to identify the distance at which the global efficiency loss following edge removal stabilizes, capturing its maximum spatial impact. This buffer-based approach is grounded in Tobler’s first law of geography, which emphasizes the stronger influence of nearby features (Miller et al., 2004). For each edge e and radius r , the global efficiency change is computed as the absolute percentage variation in global efficiency, following the equation below:

GE

GE

−

(3)

, original r

, removed r

, e r GE  =

GE

, original r

where GE original,r is the baseline global efficiency within the buffer radius r before edge removal, and GE removal,r the global efficiency after edge removal. This study determines the optimal buffer radius, defined as the radius at which a further increase in the distance from the removed edge has a negligible impact on the global efficiency loss. The optimal buffer radius, r opt , for an edge ( e ) is a parameter that objectively defines the spatial extent of its functional influence. The analysis determines r opt (e) as the first radius r n where the relative change in the global efficiency loss between consecutive radii falls below a percentage stabilization threshold τ=1 %.

|

|

GE GE

   

    

, e r   −

|

(4)

, e r

( ) min =

op r e t

r

n





1

i

n

+

GE



,

e

rn

Analysis of global efficiency loss across buffer radii reveals topological criticality in the network. Median ΔGE rankings show that motorways dominate, with drops over four times greater than those of primary roads, reflecting their central role in network flow and the high systemic cost of their disruption. Motorway links, particularly at interchanges, emerge as key topological bottlenecks. Fig. 3 illustrates the relative global efficiency loss across major highways. Table 3 summarizes global efficiency losses by road type. While secondary and tertiary roads show low median ΔGE , their maximum drops exceed those of motorways. This indicates a dual vulnerability: motorways pose the greatest risk under typical failures, but specific road segments, such as unique bridges or tunnels, can cause catastrophic disruption by acting as critical cut points.

Table 3: Topological criticality statistics (global efficiency loss, ΔGE(e,r) ), summarizing the median (Median ΔGE(e,r) ) and maximum (Max ΔGE(e,r) ) efficiency loss caused by the failure of edges, grouped by highway classification.

Road Type

Motorway 0.000456 0.012623

Motorway Link

Primary 0.000108 0.015279

Secondary 0.000107 0.031603

Tertiary 0.000115 0.028490

Residential

Median ΔGE(e,r)

0.000197 0.009718

0.000056 0.010033

Max ΔGE(e,r)