The Denser the Road Network, the More Resilient It Is?—A Multi-Scale Analytical Framework for Measuring Road Network Resilience

Abstract

A road network is an important spatial carrier for the efficient and reliable operation of urban services and material flows. In recent years, the “high road density, small block size” trend has become a major focus in urban planning practices. However, whether high-density road networks are highly resilient lacks quantitative evidence. This study presents a multi-scale analytical framework for measuring road network resilience from a topological perspective. We abstract 186 ideal orthogonal grid density models from an actual urban road network, quantifying resilience under two disturbance scenarios: random failures and intentional attacks. The results indicate that road network density indeed has a significant impact on resilience, with both scenarios showing a trend where higher densities correlate with greater resilience. However, the increase in resilience value under the intentional attack scenario is significantly higher than that under the random failure scenario. The findings indicate that network density plays a decisive role in determining resilience levels when critical edges fail. This is attributed to the greater presence of loops in denser networks, which helps maintain connectivity even under intentional disruption. In the random failure scenario, network resilience depends on the combined effects of the node degree and density. This study offers quantitative insights into the design of resilient urban forms in the face of disruptive events, establishing reference benchmarks for road network spacing at both meso- and micro-scales. The results provide practical guidance for resilient city planning in both newly developed and existing urban areas, supporting informed decision-making in urban morphology and disaster risk management.

1. Introduction

The dual impact of urbanization and climate change has resulted in an increasingly severe disaster risk faced by modern cities. In response to crises and changes, resilient cities provide a new paradigm and pathway for urban disaster management and sustainable development [1,2]. The construction of resilient cities has become a global consensus. The spatial form of cities serves as the fundamental foundation for building resilient urban environments [3,4], and it is also one of the key areas of spatial intervention carried out by urban planners [5]. When disturbances occur, a well-designed urban spatial form can support and preserve the integrity of urban spaces, ensuring the continued operation of essential functions [4,6].

The road network, as the backbone of urban spatial forms, is not only a critical underlying network that supports the robust functioning of cities but also one of the most essential infrastructures for disaster risk management [7]. Road segments and intersections are the fundamental components of a road network, which, from a graph theory perspective, can be conceptualized as edges and nodes, respectively. The overall topological structure of the road network, the morphological characteristics of road segments (edges), and the design and management of intersections (nodes) are all crucial for ensuring the sustained, efficient, and reliable operation of urban infrastructure [4,8,9,10,11]. Under normal conditions, the road network serves as a primary space for residents’ daily public activities and their perceptions of urban form, and it functions as the main conduit for the flow of materials and services within the city [8]. In the event of a disaster, the road network becomes a vital evacuation route and a key lifeline system for delivering resources and services (such as water, food, and medical care) to those affected [7,10,12,13,14]. However, road networks are often among the first systems to fail during disasters. Earthquakes and floods can damage road segments or intersections, resulting in diminished network connectivity and a decline in public service performance [7]. A well-designed road network can restore connectivity through its remaining segments, maintaining performance similar to its initial capacity, while poorly structured networks may fail to establish new connections due to the failure of localized road sections, resulting in rapid collapse and potentially severe socio-economic consequences. Therefore, understanding the resilience characteristics of road network structures is a crucial prerequisite and foundation for urban planners and decision-makers in designing, implementing, and intervening in resilient urban forms.

Recently, in the context of sustainable development and the return of humanism, the super-block spatial planning model has been widely investigated due to its prioritization of vehicular convenience and extensive land development, while the “high road density, small block size” planning pattern based on the New Urbanism planning concept has been applied in global urban design practices [15,16]. For instance, New Urbanism advocates for road networks with an interval of approximately 100 m. Existing research has demonstrated the positive impacts of smaller blocks and denser road networks in promoting walkability, public transport usage, and efficient land use; reducing carbon emissions; and enhancing urban vitality [16,17,18]. However, within the emerging field of resilient cities, the contribution and impact of density on resilience have received less attention. While the prevailing view in the existing literature is that high-density road networks correlate with greater resilience—due to their increased connectivity and redundancy, which are key resilience features [19,20,21,22]—most conclusions are qualitative. A few quantitative studies suggest that walkability and mixed land uses can enhance social interaction and social capital, thereby improving community resilience [23]. Nonetheless, most research remains rooted in the sustainable development framework, directly translating the social and ecological effects of high-density urban forms into resilience performance. A research study conducted on five major global cities has also suggested that density is not the sole determinant of road network resilience; rather, resilience levels result from the combined effects of multiple factors, including morphological patterns, hierarchical structure, and network density [24]. The question of whether high-density road networks are inherently more resilient requires quantitative evidence that is supported by a clear definition of resilience, its goals, and its underlying components.

Empirical research aimed at clarifying whether density contributes to the resilience of road networks—and to what extent and in which aspects it enhances the resilience level of these networks—can provide valuable support for practical spatial planning. Therefore, we propose the following research questions: (1) How can road network resilience be characterized and measured? (2) How does road network density influence resilience at different scales? (3) What types of planning strategies can be developed from quantitative findings to enhance urban resilience?

To quantitatively explore the relationship between road network density and resilience, it is essential to isolate the effects of other complex real-world factors, such as morphology, structural hierarchy, and spatial scale. This necessitates a comparative analysis of road networks with identical structures but varying densities. However, such networks with consistent structures and systematically varying densities are rarely found in reality. Therefore, we abstract the density characteristics of actual road networks into idealized models with different grid spacing, ensuring the stability of relevant variables for controlled comparisons. The increasing density is represented by systematically reducing grid spacing. Within a multi-scale resilience measurement framework, we simulated how the two different disturbance scenarios affect the global efficiency of road networks, enabling us to measure resilience. By examining how road network resilience changes with increasing density, we provide insights into the design of resilient urban forms. These findings can inform resilience planning strategies for both newly developed areas and existing urban districts.

The paper is structured as follows: Section 2 provides a comprehensive literature review relevant to the research theme. Section 3 details the research methodology, including the density and scale analysis modules of the road network and resilience measurement models. Section 4 presents an in-depth analysis of the research findings. Section 5 examines the impact of density on network resilience, explores corresponding planning enhancement strategies, and critically discusses the model’s limitations and potential future research directions. Finally, Section 6 summarizes the key conclusions of the study.

2. Literature Review

2.1. Measuring the Resilience of Urban Road Networks

The term “resilience” originates from the Latin word “resilio”, meaning the ability of an object to return to its original state after damage. In the 1990s, the concept of resilience gradually extended from ecology to urban studies, initially focusing on urban physical environments and infrastructure responses to disasters. At its core, resilience embodies the fundamental notion that all systems undergo constant change. Existing research typically defined resilience as the capacity of urban systems to absorb and adapt to external disturbances while maintaining their original structure and critical functions. As the understanding of urban complexity evolved, Ostadtaghizadeh et al. [25] categorized resilient urban systems into five dimensions: the physical, economic, natural, institutional, and social dimensions. Later, Ribeiro et al. [26] conceptualized resilience through four pillars: resistance, recovery, adaptation, and transformation. Thus, resilient cities can be understood as multi-system entities capable of the following: absorbing disturbances and mitigating their impacts; restoring system performance through gradual recovery; adapting to new conditions; and transforming challenges into opportunities [26,27,28]. For urban planning disciplines, the primary goal is to enhance spatial resilience through urban design and strategic planning, ensuring the continuous operation of urban systems under disruptive scenarios.

As shown in Figure 1, we describe the process of urban system performance changes under disturbances from multiple dimensions, such as the subject, object, goal, state, and phase of resilience [10,29]. The subject of resilience is the performance of the urban system, which can be summarized as the functioning of the city’s physical spatial system and social system [30]. The object of resilience refers to the various disturbances faced by the city, including threats from external uncertainties such as disasters and internal daily fluctuations within the system. The goal of resilience is to achieve a specific resilience characteristic in different disturbance phases. The resilience concept can be expressed in terms of two core features: robustness (the ability to withstand external shocks) and rapid recovery (the ability to recover quickly) [31,32,33]. Robustness is related to physical space, while recovery is more closely tied to the social system and its management capabilities. When a disturbance occurs, the performance of the urban system is impacted, with the magnitude of the decline depending on the robustness of the physical spatial structure relative to the disturbance. Typically, infrastructure does not require immediate recovery to maintain essential functions after a disturbance [31]. The system’s robustness has a tolerance threshold, and once the disturbance exceeds this threshold, the system’s performance is at risk of collapse. However, such instances are rare. Therefore, Gonçalves and Ribeiro [31] define the robustness of physical characteristics as static resilience, which is related to the intensity of disturbances a system can withstand. In most cases, disturbances cause temporary failure with respect to part of the system’s performance; through the intervention of the social system (e.g., pre-disaster preparation and post-disaster adaptation), the system’s performance can be quickly restored to an acceptable level. Therefore, the speed of recovery is related to the disaster management capacity of a social system [30,31]. Specifically, the disaster management capacity affects how quickly resources can be mobilized to restore the normal functioning of both the material and social systems and how the systems adapt and transform to the disturbance through learning: that is, the system’s adaptation and transformation capacity after performance recovery. The ability of an urban social system to respond to disasters determines the possible recovery states of the system’s performance, and these states include exceeding, equaling, or falling below the original performance level (Figure 1). Gonçalves and Ribeiro [31] define the restorative capacity of social systems as dynamic resilience, which is associated with the recovery time during disturbances. This study focuses solely on the resilience exhibited by the physical space of the road network itself and excludes the social system’s ability to adapt and transform in response to disturbances. We define the resilience of road networks as follows: when shocks or threats occur, the road network’s ability to resist, mitigate, and absorb the impacts of disturbances while maintaining its basic structure and critical performance at an acceptable service level comprises its resilience [31].

Based on the process analysis of the resilience concept, we can divide the resilience measurement objectives (resilience for what) into two types: robustness measurement based on disaster intensity sequences and recovery measurement based on time sequences. Recovery measurement methods typically quantify resilience by evaluating the extent of system performance recovery per unit of time after a disturbance [34], with disturbances simulated as specific disasters, such as earthquakes or floods of varying intensities. The “resilience triangle” proposed by Bruneau and Reinhorn [35] is a representative method for evaluating resilience, and it involves using an area enclosed by performance trend curves and time as the quantitative indicator of resilience [35,36,37,38]. However, due to the difficulty in obtaining post-disaster data, few studies focus on the recovery process [14]. Some studies also use hypothetical scenarios of different types of social actions to measure recovery [8].

The majority of research concentrates on robustness measurements. Before the concept of resilience was introduced, mature methods for measuring the robustness of complex networks already existed, and previous studies often conflated the concepts of resilience, robustness, and vulnerability. In the field of computer science, network robustness is typically tested through vulnerability attack experiments, in which a certain proportion of nodes or edges are sequentially removed, and the performance degradation of the remaining network is observed [14,39,40]. The disturbance scenarios of random failures and intentional damage, proposed by Albert and Barabási [41], have been widely applied. A network’s performance is usually represented by the shortest paths or its global efficiency, with the importance of nodes or edges generally measured via the node degree [41] or betweenness centrality [42]. In recent years, some studies have used flood risk levels as a scenario for the failure of network nodes or edges [12,43]. Later, scholars introduced percolation theory into network robustness research. Percolation theory suggests that the size of the largest connected component can be used to measure the robustness of a network, as it reflects the proportion of the network that is functioning [44]. Percolation theory also posits that a transition point will appear during network disruption experiments, where the overall connectivity of the network experiences a sharp decline, representing the network performance collapse threshold. However, Dong et al. [7]—in their study of the robustness of road network topologies—pointed out that a single transition threshold cannot fully capture the robustness of a network. There may be contradictions between the threshold point and the largest connected subgraph. Therefore, the direct use of the threshold as a measure of network resilience can result in overlooking the overall trends in system performance changes.

2.2. Road Network Density and Resilience

Street network density reflects the compactness of the urban fabric and is typically measured via the number of specific elements within a given area, such as intersections, road segments, segment lengths, and cyclomatic numbers. Density metrics are closely linked to road network connectivity [19]. Increasing the number of intersections and road segments enhances both connectivity and permeability, while cyclomatic numbers indicate route redundancy [19]. A higher number of intersections leads to shorter road segments and closer junctions, producing a finer grid with multiple routes and frequent shortcuts [20,45]. This redundancy in connectivity and accessibility is crucial for effective emergency response [10], ensuring that, even if one route fails, multiple alternative paths remain available for quickly restoring connectivity [4,19,21,46,47]. Consequently, high-density road networks are often considered highly resilient. However, an excessive number of intersections can negatively impact vehicular traffic efficiency; compromise pedestrian safety; and contribute to other urban challenges, such as reducing green spaces and public areas—both of which are critical for post-disaster evacuation [19]. Additionally, such networks may exacerbate the rapid spread of infectious diseases [9]. Thus, while high-density road networks offer resilience benefits, they also pose significant trade-offs.

Salat [21,22] argues that large superblock models, such as those proposed in Le Corbusier’s modernist “Radiant City”, exhibit low levels of resilience due to their lack of complexity and scale hierarchy. These superblocks, which often measure 400 m per side and are dominated by isolated mega-buildings, are also common in many Chinese cities. Salat [22] suggests that resilient urban forms should follow a power-law distribution in terms of scale hierarchy, with small, medium, and large numbers of high-density, moderate-density, and low-density nodes and links, respectively. The superblock model lacks smaller-scale sub-layers and the connections between them, resulting in lower resilience. In contrast, fine-grained urban fabrics, such as those in Tokyo and New York, with average road segment lengths of 50–100 m, demonstrate higher resilience. High-density road networks result in smaller land parcels, which can adapt more quickly and at lower costs relative to changes and disturbances [5,20,22]. Moreover, historically, fine-grained cities have continuously adjusted their road network structure and function through cycles of destruction and rebuilding across different spatial and temporal scales [22], allowing them to meet the evolving needs of different historical periods [21]. This structural and functional complexity reflects a city’s capacity to adapt to change.

Significant progress has been made in the construction of resilient city theory and the development of road network resilience measurement methods, primarily focusing on the quantitative assessment of two core characteristics: robustness and recovery. This is exemplified by typical methods such as interruption simulation experiments in complex networks and the resilience triangle measurement approach. However, due to limitations in acquiring full-cycle disaster data, current research on infrastructure resilience remains dominated by robustness measurements. Existing methods generally rely on single indicators, such as threshold values or the size of the largest connected subgraph, to represent system resilience. Such simplifications fail to capture the dynamic evolution of system performance, presenting clear theoretical limitations. Particularly in the field of road network density and resilience relationships, existing studies are often confined to qualitative analysis or historical experience, with no established quantitative correlation between density features (such as redundancy and connectivity) and resilience levels. Moreover, there is a lack of systematic definitions of resilience subjects, objects, goals, and phases. As a result, critical scientific questions, such as whether the ‘small block, dense road network’ urban design paradigm exhibits high resilience characteristics, and how road network density influences resilience through topological structure, still lack quantitative evidence. To fill this research gap, we combine threshold identification and function fitting methods to construct a resilience measurement framework for road networks. Drawing on the resilience framework proposed by Sharifi [10], we address the following research questions: What is the subject of the resilience measure? (subject)—the efficiency of the road network; what are these networks resilient to (object)—random failures and intentional attacks; in what manner does resilience manifest itself (goal)—in the robustness of the road network; at what stage is resilience observed (phase)—pre-disaster preparedness phase; at what scale does resilience occur (scale)—the meso-scale (Figure 1). The research findings assist planners and decision-makers in understanding the role of road network density in reducing urban risks and enhancing adaptability, and they provide a quantitative, evidence-driven framework for resilience planning practices, such as urban form design, disaster spatial management, and community resilience building.

3. Methods

Figure 2 shows a multi-scale analytical framework for resilience measurements in order to disentangle relations between road network density and resilience. In the first step, we created ideal density models of road networks. The ideal models include 19 density units and 11 scale units, and a total of 186 ideal models were selected for this study. In the second step, we used graph theory to abstract the ideal model into a “node-edge” topology. Two perturbation scenarios are considered: random failures and intentional attacks, removing 1% of edges in a sequence according to the principle of randomness or the importance degree of edges. In the third step, we characterized the performance of road networks using cluster sizes and global efficiencies under perturbations. Therefore, we used two elements to quantify network resilience: One is the threshold point of the network’s collapse, which is identified by the size of the connected subgraph. The second is the decline in global efficiency, which is expressed by the efficiency decline curve. Finally, based on the resilience measurement results, the influencing characteristics of density with respect to resilience are identified, and corresponding planning strategies are proposed to improve the resilience level of real cities.

3.1. Designing the Analysis Unit for the Road Network

First, a grid-based road network is employed to construct the idealized model, providing a simplified yet structured framework for controlled analysis. The ideal model is abstracted from the actual road patterns of global cities. First, the ideal model adopts a structured pattern of orthogonal grids. Constrained by geographical space, the topological structure of road networks exhibits distinct characteristics. Typically, road intersections connect 3 to 5 segments depending on the morphological characteristics of the network [7,8]. The grid structure is a highly connected and representative form of road network configurations [48,49]. We use the grid interval to represent the density of the road network, with a reduction in the grid interval in the idealized model indicating an increase in density. On the other hand, urban design principles represented by New Urbanism advocate for high-density orthogonal grid networks. This has become a dominant trend in recent urban planning practices and is widely adopted in the development of new urban districts in China. Therefore, studying orthogonal grid-based road network patterns holds significant practical relevance.

Second, the design of density analysis units was considered. The grid interval in the idealized model was determined based on the average road segment lengths observed in real cities. We selected representative global cities commonly used in road network density analyses—such as Shanghai, Tokyo, New York, and London [21]—and measured the average road spacing within their typical urban blocks. The results show that road segment intervals in real cities mostly fall within the range of 50 to 500 m. In particular, Tokyo’s central districts feature intervals of 50–80 m, with 50 m representing the minimum spacing observed in the high-density city. In London’s historic core, the average road spacing typically ranges from 80 to 100 m. In Manhattan, New York, the typical block size falls between 120 and 200 m. A spacing of 200–300 m corresponds to the road network layout of central Shanghai, while suburban areas of Shanghai often exhibit intervals of 300–400 m. A spacing of 500 m reflects the large block sizes commonly found in many Chinese cities. As shown in Figure 3, the maximum grid interval in the ideal model is set to 500 m, and the minimum is set to 50 m, with a decrement of 25 m between each, resulting in 19 density analysis units with different grid intervals. A decrement step of 25 m ensures that the full range of observed real-world road network intervals falls within the parameter space of our model, thereby validating the practical relevance of the selected parameters.

Third, we explored the definition of scale analysis units. To evaluate resilience across these 19 density units, we divided them into 11 scale analysis units, ranging from the micro-scale to the meso-scale. The smallest unit, a 1.5 × 1.5 km grid, represents a typical walking range, with each successive unit increasing by 500 m up to a 6.5 × 6.5 km district-scale grid. The smallest unit, a 1.5 × 1.5 km grid, corresponds to a typical 15-minute walking catchment area (approximately 1 km walking distance), which closely aligns with the spatial threshold of pedestrian accessibility (800–1000 m) and represents the basic spatial unit in urban community planning. Following the spatial prototype of the Perry neighborhood unit, the intermediate scales encompass 30-minute isochrones for walking (approximately 2.5–3.5 km) and 10-minute cycling zones (radius of 3.0–4.0 km), corresponding to the typical extent of neighborhood units. The largest unit, a 6.5 × 6.5 km grid, captures the primary zones of residents’ daily activities—such as accessing services, procuring supplies, seeking medical care, and taking part in recreation [5]—and aligns with the common travel ranges associated with commuting, schooling, and other routine tasks. This spatial scale also corresponds to the administrative scope of zoning or unit planning within the urban planning system. In this study, a 500-meter interval was selected as the incremental step size based on the following considerations: (1) Empirically, this scale aligns with the widely observed “400-m rule” in various periods of urban design paradigms, representing the typical spacing between arterial roads [50]. (2) The 500-meter increment is compatible with the spatial extent of standard neighborhood units (500–800 m) and the service radius of public transit stops (typically 400–600 m) [45]. (3) It also corresponds to the typical land-use module size (e.g., 500 × 500 m blocks) commonly observed in urban expansion.

Finally, based on the 19 density analysis units and 11 scale analysis units, we constructed 209 ideal grid models of road networks using the NetworkX package in Python 2.8.8 (Figure 3). Since both disturbance scenarios simulate the removal of 1% of the network’s edges each time, models with fewer than 100 edges cannot be simulated. Therefore, 23 grid models with fewer than 100 edges were excluded, and the final sample for the study consisted of 186 models. To minimize potential errors, the edge-shifting algorithm for small-scale networks is detailed as follows: The experiment found that for ideal models with a smaller total number of edges, directly rounding when removing 1% of edges results in significant errors in the results. For example, in an ideal model with a total of 180 edges, 1% of the edges would be 1.8 edges. If this is directly rounded down, one edge is removed each time. After 100 removals, there would still be 80 edges remaining, which results in subgraph and efficiency change curves that do not reflect the outcome after all edges have been removed. Therefore, we modified the edge removal method as follows: the number of edges to be removed in the nth step = total number of edges × n% − sum of edges removed in previous (n – 1) steps. After calculating the required number of edges to remove, the value is rounded down, and the process continues until all edges are removed by the 100th step. As shown in Appendix A 

Table A1. Total number of nodes in the ideal models.

Grid Interval	Scale (km)
	1.5 × 1.5	2.0 × 2.0	2.5 × 2.5	3.0 × 3.0	3.5 × 3.5	4.0 × 4.0	4.5 × 4.5	5.0 × 5.0	5.5 × 5.5	6.0 × 6.0	6.5 × 6.5
500 m	16	25	36	49	64	81	100	121	144	169	196
475 m	25	36	49	64	81	100	121	144	169	196	225
450 m	25	36	49	64	81	100	121	169	169	225	256
425 m	25	36	49	81	100	121	144	169	196	256	289
400 m	25	36	64	81	100	121	169	196	225	256	324
375 m	25	49	64	81	121	144	169	225	237	289	361
350 m	36	49	81	100	121	169	196	256	289	361	400
325 m	36	64	81	121	144	196	225	289	324	400	441
300 m	36	64	100	121	169	225	256	324	400	441	529
275 m	49	81	121	144	196	256	324	400	441	529	625
250 m	49	81	121	169	225	289	361	441	529	625	729
225 m	64	100	169	225	289	361	441	576	676	784	900
200 m	64	121	196	256	361	441	576	676	841	961	1156
175 m	81	144	256	361	441	576	729	900	1089	1296	1521
150 m	121	196	324	441	625	784	961	1225	1444	1681	2025
125 m	169	289	441	625	841	1089	1369	1681	2025	2401	2809
100 m	256	441	676	961	1296	1681	2116	2601	3136	3721	4356
75 m	441	784	1225	1681	2304	3025	3721	4624	5625	6561	7744
50 m	961	1681	2601	3721	5041	6561	8281	10,201	12,321	14,641	17,161

,

Table A2. Total number of edges in the ideal models.

Grid Interval	Scale (km)
	1.5 × 1.5	2.0 × 2.0	2.5 × 2.5	3.0 × 3.0	3.5 × 3.5	4.0 × 4.0	4.5 × 4.5	5.0 × 5.0	5.5 × 5.5	6.0 × 6.0	6.5 × 6.5
500 m	24	40	60	84	112	144	180	220	264	312	364
475 m	40	60	84	112	144	180	220	264	312	364	420
450 m	40	60	84	112	144	180	220	312	364	420	480
425 m	40	60	84	144	180	220	264	312	364	480	544
400 m	40	60	112	144	180	220	312	364	420	480	612
375 m	40	84	112	144	220	264	312	420	454	544	684
350 m	60	84	144	180	220	312	364	480	544	684	760
325 m	60	112	144	220	264	364	420	544	612	760	840
300 m	60	112	180	220	312	420	480	612	760	840	1012
275 m	84	144	220	264	364	480	612	760	840	1012	1200
250 m	84	144	220	312	420	544	684	840	1012	1200	1404
225 m	112	180	312	420	544	684	840	1104	1300	1512	1740
200 m	112	220	364	480	684	840	1104	1300	1624	1860	2244
175 m	144	264	480	648	840	1104	1404	1740	2112	2520	2964
150 m	220	364	612	840	1200	1512	1860	2380	2812	3280	3960
125 m	312	544	840	1200	1624	2112	2664	3280	3960	4704	5512
100 m	480	840	1300	1860	2520	3280	4140	5100	6160	7320	8580
75 m	840	1512	2380	3280	4512	5940	7320	9112	11,100	12,960	15,312
50 m	1860	3280	5100	7320	9940	12,960	16,380	20,200	24,420	29,040	34,060

and

Table A3. Total length of the road network (km).

Grid Interval	Scale (km)
	1.5 × 1.5	2.0 × 2.0	2.5 × 2.5	3.0 × 3.0	3.5 × 3.5	4.0 × 4.0	4.5 × 4.5	5.0 × 5.0	5.5 × 5.5	6.0 × 6.0	6.5 × 6.5
500 m					56	72	90	110	132	156	182
475 m				48	63	80	99	120	143	168	195
450 m				48	63	80	99	130	154	180	208
425 m				54	70	88	108	130	154	192	221
400 m			40	54	70	88	117	140	165	192	234
375 m			40	54	77	96	117	150	176	204	247
350 m			45	60	77	104	126	160	187	228	260
325 m		32	45	66	84	112	135	170	198	240	273
300 m		32	50	66	91	120	144	180	220	252	299
275 m		36	55	72	98	128	162	200	231	276	325
250 m		36	55	78	105	136	171	210	253	300	351
225 m	24	40	65	90	119	152	189	240	286	336	390
200 m	27	44	70	96	133	168	216	260	319	372	442
175 m	30	52	80	114	147	192	243	300	363	432	507
150 m	33	60	90	126	175	224	279	350	418	492	585
125 m	39	68	105	150	203	264	333	410	495	588	689
100 m	48	84	130	186	252	328	414	510	616	732	858
75 m	63	112	175	246	336	440	549	680	825	972	1144
50 m	93	164	255	366	497	648	819	1010	1221	1452	1703

, the total number of nodes and edges and the total length of the road network increase as the grid interval decreases, intuitively reflecting the low-to-high changes in road network density. To validate the statistical significance of the constructed idealized models, one-way ANOVA tests were conducted using scale and grid spacing as the grouping variables. The total number of edges and nodes in each model served as dependent variables. The results show that all p-values were less than 0.05, indicating significant differences in network size across models with varying analytical scales and densities (see Appendix A 

Table A12. One-way analysis of variance of the total number of nodes under different scales.

Source	Sum of Squares	Degrees of Freedom	Mean Square	F	Significance
Between	92,644,663.14	10	9,264,466.314	1.952	0.040
Within	939,599,202.3	198	4,745,450.517
Total	103,224,865	208

,

Table A13. One-way analysis of variance of the total number of nodes under different grid intervals.

Source	Sum of Squares	Degrees of Freedom	Mean Square	F	Significance
Between	637,684,750.7	18	35,426,930.60	17.060	0.000
Within	394,559,114.7	190	2,076,626.920
Total	1,032,243,865	208

,

Table A14. One-way analysis of variance of the total number of edges under different scales.

Source	Sum of Squares	Degrees of Freedom	Mean Square	F	Significance
Between	361,359,539.1	10	36,135,953.91	1.937	0.042
Within	3,694,146,164	198	18,657,303.83
Total	4,055,505,703	208

and

Table A15. One-way analysis of variance of the total number of edges under different grid intervals.

Source	Sum of Squares	Degrees of Freedom	Mean Square	F	Significance
Between	2,500,238,360	10	138,902,131.1	16.969	0.000
Within	1,555,267,343	198	8,185,617.596
Total	4,055,505,703	208

). These results confirm the suitability of the models for comparative analyses.

3.2. Efficiency of the Road Network

Network efficiency is a core characteristic of resilient systems, as it reflects the overall directness of connections between nodes [10]. In existing studies on road networks, global efficiencies are also widely adopted as key performance indicators in evaluating network resilience under disruption scenarios [2,40,51]. We topologized the ideal model, where road segments and intersections are represented as “edges” and “nodes”, respectively, and lengths are used as weights for edges. The primal method retains the network’s physical spatial information by weighting attributes such as street length, width, and design speed [48,52]. It is considered more suitable for simulating real-world scenarios where certain street segments fail during a disaster. The resulting graph (G = N, E) consists of two sets: a set of nodes N and a set of edges E. If there exists an edge between node i and node j, it is denoted as (i, j), indicating that the two nodes are connected or adjacent. In a weighted network based on the edge length, the distance between adjacent nodes i and j is defined by the length of the edge (i, j); if no edge exists between them, their distance is considered to be zero.

Global efficiency is a metric based on the shortest path length between pairs of nodes. For any two nodes, there may be multiple paths connecting them, and the shortest path is defined as the one with the minimal cumulative length. The global shortest path efficiency of a network is calculated as the average of the reciprocal shortest path lengths over all node pairs [52], and it is provided by the following formula:

$$\mathrm{L}={\displaystyle \frac{1}{\mathrm{N}\left(\mathrm{N}-1\right)}}\sum _{i,j\in N;i\ne j}{d}_{ij}$$

(1)

Here, $d_{ij}$ denotes the shortest distance between a pair of nodes. In unweighted networks, $d_{ij}$refers to the minimum number of edges (or steps) required to travel from node i to node j. In weighted networks where edge lengths are assigned, $d_{ij}$ represents the sum of edge lengths along the shortest path from node i to node j.

Although the shortest path length is an effective measure of network connectivity, problems arise when disconnected node pairs exist, as the shortest path distance between them is infinite $(d_{ij}$ = ∞). To overcome this limitation, global efficiency is typically defined using the reciprocal of the shortest path length ($1/d_{ij}$).When no path exists between two nodes, their efficiency is considered zero. The global efficiency of a network is therefore computed as follows:

$$E^{Glob}={\displaystyle \frac{1}{\mathrm{N}\left(\mathrm{N}-1\right)}}\sum _{i,j\in N;i\ne j}{\displaystyle \frac{1}{d_{ij}}}$$

(2)

Global efficiency ranges between 0 and 1. In an unweighted network, the global efficiency $E^{Glob}=1$ when the network is fully connected—that is, every pair of nodes is directly linked by an edge.

Global efficiency ($E^{glob\left(l\right)}$) is the average of the efficiencies of all pairs of nodes (a pair of nodes) within the network [53], i.e., the average of the inverse of the shortest path lengths between all pairs of nodes. We normalized the efficiency between the pairs of nodes as the straight-line distance between two nodes ($d_{ij}^{Eucl}$) and the actual shortest distance ($d_{ij}^l$) ratio, in which the actual shortest distance ($d_{ij}^l$) refers to the sum of the road segment lengths of the shortest path between pairs of connected nodes. In a valued graph, the shortest path length between i and j is defined as the smallest sum of the edge lengths throughout all possible paths in the graph from i to j. N(N − 1) refers to the number of node pairs in the network. In distance-weighted networks, global efficiency serves as a proxy for evaluating how effectively the road network facilitates spatial connectivity across geographic space, and the calculation is shown in Equation (3).

$$E^{glob\left(l\right)}={\displaystyle \frac{1}{N\left(N-1\right)}}\sum _{i,j\in G;i\ne j}{\displaystyle \frac{d_{ij}^{Eucl}}{d_{ij}^l}}$$

(3)

3.3. Designing Two Simulation Scenarios

Overall, the causes of road disruptions can be broadly classified into two categories: segment failure due to uncertain factors and road interruptions caused by intentional human attacks. These two types of factors can be subdivided into distinguished disturbances, which are further subdivided into internal threats and external threats [14] (Figure 4). Internal threats originate from mistakes by staff members or users that result in accidental events. Uncertain events caused by external factors come from extreme weather and natural disasters triggered by climate change. Intentional attacks caused by external factors include antagonistic actions; for example, public transport stations with high-density crowds have all been targets of several terrorist actions, and usually, these stops are located on roads with high connectivity. The direct consequence of these disturbances is that segments of roads may be blocked, which in turn increases the travelling distance and time of citizens. In more serious cases, some trips have to stop. Indirect effects result in social or economic costs.

In response to the above scenario with respect to the possibility of real disturbances, we adopted two disturbance scenarios for the resilience analysis of road networks: random failures and intentional attacks. The random scenario comprises simulated adverse events by deactivating the road network edges at random and step-by-step relative to increments of 1% until all edges are erased. In order to avoid the possible uncertainty caused by randomness, the interrupting experiment of each ideal model is independently performed 100 times. Existing studies have shown that under varying intensities of urban pluvial flooding events (e.g., 10-year, 20-year, 50-year, 100-year, and 500-year return periods), changes in road network global efficiency exhibit patterns similar to those observed under random disturbance scenarios [54]. The key distinction lies in the fact that pluvial flooding is influenced by topography, resulting in a spatially clustered pattern comprising road segment failures. Therefore, the random scenarios simulated in this study are more representative of flooding impacts in cities located on flat terrains.

The intentional attacks are simulated by successively eliminating 1% of edges with high betweenness centrality ($C_i^B$), and the betweenness centrality is recalculated for the next removal. The betweenness centrality refers to the proportion of the number of shortest paths between nodes that contain a particular node or edge relative to the total number of shortest paths in the network [51]. The higher the betweenness centrality of a road segment, the more the shortest paths pass through that segment [51,53]; this means that its importance in determining the shortest distance between all pairs of nodes in the network is greater. The formula is calculated as follows (Equation (4)):

$$C_i^B={\displaystyle \frac{1}{\left(n-1\right)\left(n-2\right)}}\sum _{i,j\in G,j\ne k\ne i}{\displaystyle \frac{n_{ij}\left(k\right)}{n_{ij}}}$$

(4)

where $n_{ij}$ is the number of shortest paths between i and j, and $n_{ij}$(k) is the number of shortest paths between i and j that contain edge k.

3.4. Measuring the Resilience Level of the Road Network

Continuously removing a certain proportion of edges results in the gradual fragmentation of the network structure, where the initially complete network “graph” splits into several smaller network subgraphs. We use the size of the connected subgraphs to represent the degree of fragmentation in the network. The network subgraph with the most nodes is referred to as the largest connected component (largest cluster), and the subgraph with the second-largest number of nodes is the second-largest connected component (second-largest cluster). During this process, the network structure experiences two critical transition points: The first transition point occurs when the initially complete network begins to fragment into several smaller subgraphs; the second transition point marks the threshold of the network’s collapse—where the largest subgraph rapidly shrinks—and the network eventually splits into several smaller, unconnected subgraphs, signaling the complete fragmentation of the network [51]. We define the peak size of the second-largest cluster as the network collapse threshold $\left(f_{max}\right)$ [40,55,56,57,58]. As shown in Figure 5a, when the size of the second-largest cluster reaches its maximum, the size of the largest cluster is similar to that of the second-largest cluster. At this point, the network has fragmented into several small connected subgraphs, with no cross-connections between them, indicating the occurrence of network fracture points.

As shown in Figure 5b, as road segments progressively fail, the length of routes between intersections increases or disappears, resulting in a continuous decline in global efficiency. However, the efficiency curves of the road network at the same threshold point show differences, meaning that using the threshold alone to represent network resilience would overlook the process of global efficiency decline. We measure the resilience value (R) of the road network by calculating the area enclosed by the global efficiency decline curve and the tolerable fluctuation area before reaching the network collapse threshold.

The X-axis represents the proportion of edges removed (f), and the Y-axis represents the rate of change in global road network efficiency (Q(f)), expressed as the ratio of post-removal efficiency ($E_f$) to the initial efficiency ($E_{pre}$). The starting point of the performance curve is 1, and the endpoint is the system performance change rate Q($f_{\left(max\right)}$) at the network collapse threshold, corresponding to the maximum tolerable proportion of failed edges ($f_{\left(max\right)}$). The resilience level of the road network can be calculated using Equation (5).

$$R={\int }_0^{f_{max}}Q\left(f\right)df={\int }_0^{f_{max}}\left({\displaystyle \frac{E_f}{E_{pre}}}\right)df$$

(5)

The resilience index reflects the ability of a road network to maintain its efficiency, and it is determined by two factors: (1) the threshold of the network’s collapse and (2) the magnitude of efficiency decline. If the network threshold is high and the efficiency decline is low, the network’s resilience value will be higher. Specifically, the larger the area enclosed by the global efficiency decline curve and the disturbance intensity, the higher the network’s resilience. As shown in Figure 5, different networks may collapse at the same threshold point, but the efficiency decline process varies (Figure 5b–d). In Figure 5c, the efficiency decline is the lowest; thus, the area enclosed by the efficiency decline curve and the X-axis (representing the proportion of removed segments) is the largest, indicating a higher network resilience level.

4. Results

4.1. Resilience Characteristics of Road Networks Under Random Failure Scenarios

4.1.1. The Overall Trend of Resilience Under Random Failure Scenarios

We ran random disruptions 100 times for 186 ideal models, using the connected subgraph curves to identify the threshold and the efficiency curves to calculate the resilience values that are averages of the 100 experiments. The results in

Table 1. The variation in road network resilience with respect to density in random failure scenarios.

Grid Interval	Scale (km)
	1.5 × 1.5	2.0 × 2.0	2.5 × 2.5	3.0 × 3.0	3.5 × 3.5	4.0 × 4.0	4.5 × 4.5	5.0 × 5.0	5.5 × 5.5	6.0 × 6.0	6.5 × 6.5
500 m					0.3853	0.3781	0.3881	0.3882	0.3850	0.3868	0.3911
475 m				0.3810	0.3941	0.3817	0.3925	0.3884	0.3868	0.3831	0.3878
450 m				0.3864	0.3825	0.3856	0.3814	0.3901	0.3903	0.3896	0.3872
425 m				0.3951	0.3866	0.3920	0.3898	0.3908	0.3904	0.3866	0.3918
400 m			0.3828	0.3822	0.3863	0.3913	0.3889	0.3829	0.3878	0.3836	0.3886
375 m			0.3846	0.3875	0.3878	0.3846	0.3836	0.3881	0.3894	0.3952	0.3929
350 m			0.3876	0.3847	0.3876	0.3876	0.3892	0.3915	0.3897	0.3892	0.3929
325 m		0.3896	0.3837	0.3863	0.3895	0.3886	0.3895	0.3923	0.3864	0.3941	0.3932
300 m		0.3894	0.3820	0.3879	0.3907	0.3894	0.3904	0.3952	0.3910	0.3904	0.3932
275 m		0.3917	0.3881	0.3771	0.3891	0.3886	0.3956	0.3963	0.3915	0.3959	0.3940
250 m		0.3915	0.3906	0.3857	0.3825	0.3921	0.3909	0.3927	0.3934	0.3953	0.3972
225 m	0.3801	0.388	0.3904	0.3947	0.3893	0.3920	0.3925	0.3956	0.3966	0.3952	0.3969
200 m	0.3906	0.3875	0.3853	0.3883	0.3944	0.3937	0.3964	0.3956	0.3972	0.3970	0.4024
175 m	0.3836	0.3891	0.3849	0.3943	0.3931	0.3941	0.4002	0.3982	0.3996	0.4045	0.4042
150 m	0.3812	0.3862	0.3939	0.3928	0.3967	0.3993	0.4001	0.4019	0.4011	0.4045	0.4047
125 m	0.3834	0.3867	0.3915	0.3918	0.3978	0.3994	0.4012	0.4026	0.4067	0.4075	0.4092
100 m	0.3894	0.3913	0.3948	0.4001	0.4015	0.4037	0.4069	0.4082	0.4092	0.4099	0.4111
75 m	0.3955	0.4010	0.4012	0.4043	0.4074	0.4080	0.4092	0.4111	0.4123	0.4156	0.4153
50 m	0.3968	0.4048	0.4070	0.4095	0.4133	0.4141	0.4156	0.4172	0.4197	0.4204	0.4219

Note: One-way ANOVA was conducted using SPSS version 26.0 to test for significant differences in resilience values across different scale groups. The results show a significance level of p < 0.05, indicating that the variations in resilience across scales are statistically significant (Appendix A Table A16). In the table, a darker shade of orange indicates a higher resilience value, while a darker shade of blue represents a lower resilience value.

show that, across different analysis scales, the resilience level of the road network increases as the grid interval decreases. The minimum resilience value is 0.3771, while the maximum value is 0.4219. In different scale analysis units, as the grid interval decreases from the maximum to the minimum, the resilience value increases by up to 9.5%. This result indicates that the density of the road network has a positive effect on the resilience value in random disruption scenarios. Figure 6, taking the 4.0 × 4.0 km scale as an example, illustrates the process of calculating the road network resilience value for different grid intervals.

4.1.2. The Trends in Connected Subgraphs Under Random Failure Scenarios

As shown in Figure 6, after continuously removing a certain proportion of edges, the network gradually splits into several smaller connected subgraphs. By observing the changes in the size of the largest connected subgraph and the second-largest connected subgraph, we can track the process of network fragmentation. We divide the fragmentation process into two stages.

The first stage is characterized by the largest connected subgraph remaining at a high level, while the second-largest subgraph remains at a lower level. This indicates that the network is predominantly supported by the largest connected subgraph, and most nodes remain connected, without splitting into multiple subgraphs. This is because the grid structure connects each node to four edges, and after the removal of a small proportion of edges, there are still many redundant routes between node pairs, meaning that only a small number of nodes are isolated. The largest connected subgraph ensures that most of the nodes in the network remain connected.

The second stage involves the gradual splitting of the network into multiple subgraphs and its rapid collapse. As more edges are removed, the total number of nodes in the largest connected subgraph decreases sharply, while the number of nodes in the second-largest subgraph increases and quickly reaches its peak. This indicates that the network is splitting into multiple subgraphs, with the largest subgraph continuously shrinking, while the size of the split subgraphs increases. At the peak of the second-largest subgraph, the size of the largest subgraph is similar to that of the second-largest subgraph, indicating that the network has fragmented into multiple subgraphs of similar sizes, with no connections between them, representing a complete network collapse.

From Figure 6a–f, it can be observed that as the grid interval decreases, the proportion of edge removal at the network’s fragmentation point increases. For instance, when the grid interval is 500 m, the fragmentation point occurs as approximately 35% of the edges removed (Figure 6a), while with a 50 m grid interval, the fragmentation point occurs at about 45% (Figure 6f). This suggests that as the total number of nodes and edges per unit area increases, the network is less likely to fragment into multiple subgraphs after losing a portion of its connections. However, the threshold for network collapse remains relatively similar, with the results showing that the random thresholds of all ideal models are distributed around 50% (Appendix A 

Table A5. The threshold of the road network in random scenarios.

Grid Interval	Scale (km)
	1.5 × 1.5	2.0 × 2.0	2.5 × 2.5	3.0 × 3.0	3.5 × 3.5	4.0 × 4.0	4.5 × 4.5	5.0 × 5.0	5.5 × 5.5	6.0 × 6.0	6.5 × 6.5
500 m					52%	49%	53%	52%	51%	51%	53%
475 m				50%	55%	50%	53%	52%	51%	49%	52%
450 m				52%	50%	52%	50%	52%	51%	52%	51%
425 m				54%	51%	55%	53%	53%	52%	50%	52%
400 m			51%	50%	52%	54%	51%	49%	51%	49%	50%
375 m			52%	53%	51%	50%	50%	51%	51%	53%	52%
350 m			51%	51%	52%	51%	52%	52%	51%	50%	51%
325 m		52%	51%	51%	52%	52%	52%	52%	49%	52%	51%
300 m		52%	50%	52%	52%	52%	51%	53%	51%	50%	51%
275 m		53%	52%	48%	52%	52%	52%	53%	51%	52%	50%
250 m		55%	53%	51%	49%	53%	50%	51%	50%	52%	52%
225 m	50%	53%	52%	54%	51%	51%	50%	51%	51%	49%	50%
200 m	53%	53%	50%	51%	52%	51%	52%	51%	50%	50%	51%
175 m	50%	51%	49%	52%	52%	50%	53%	51%	50%	51%	51%
150 m	50%	50%	52%	50%	52%	52%	51%	51%	50%	51%	50%
125 m	50%	50%	51%	49%	51%	50%	50%	50%	51%	51%	51%
100 m	51%	51%	50%	52%	50%	50%	52%	51%	51%	50%	50%
75 m	52%	52%	50%	51%	51%	50%	50%	50%	50%	51%	50%
50 m	50%	51%	50%	51%	51%	50%	51%	50%	50%	50%	50%

). This is consistent with previous studies on percolation theory regarding the percolation threshold or critical point of a two-dimensional grid lattice [56,59,60]. This result aligns with the structure of the ideal models, all of which are orthogonal grids, meaning that each intersection is connected to four road segments. Therefore, when roads are randomly removed, the probability of connections between the remaining roads is relatively consistent.

4.1.3. The Trends in Global Efficiency Under Random Failure Scenarios

As edges connecting nodes are progressively removed, network efficiency gradually declines. Overall, the trends in the network’s efficiency decline closely resemble trends in the largest connected subgraph’s decline, as the global efficiency of the network is determined by the size of the largest connected subgraph. In the first stage, after removing a proportion of edges, the size of the largest connected subgraph remains large. This also means that most node pairs in the network are still connected. While local disruptions may lengthen the shortest path, only a few small isolated subgraphs remain disconnected from the largest subgraph. Therefore, only a small proportion of node pairs have an infinite shortest distance (i.e., zero efficiency, contributing nothing to global efficiency). Consequently, the overall decline in global efficiency is relatively minor.

In the second stage, when the network begins to split into multiple subgraphs, the size of the largest connected subgraph decreases sharply, while the size of the second-largest subgraph rapidly increases to its peak. This indicates that as the largest connected subgraph shrinks, the global efficiency also decreases rapidly; this is because when there are more disconnected subgraphs, there is an increase in node pairs that lack a connection, which lowers the network’s global efficiency. When the size of the second-largest subgraph approaches the size of the largest subgraph, this indicates that the network has fragmented into several similarly sized subgraphs with no connections between them, meaning the network has completely collapsed and its efficiency has dropped to its lowest point.

Figure 7 compares the efficiency decline trends of six networks with different grid intervals. As the grid interval decreases, the curvature of the efficiency decline curve increases, indicating that the rate of efficiency decline decreases. For example, when the grid interval is 500 m, efficiency declines more quickly, and the network collapses when the efficiency rate reaches 0.28 (Figure 6a and Figure 7). When the grid interval is 50 m, the efficiency decline is slower and remains at a higher level, only rapidly decreasing near the threshold until the network completely breaks down at 0.11 of its original efficiency (Figure 6f and Figure 7). In other words, when the grid interval is smaller, the network maintains its original efficiency more effectively.

This result shows that the threshold is not the decisive factor in random scenarios. The efficiency change curve plays a major role in determining resilience, strongly supporting the conclusion that variations in efficiency are the primary factors affecting resilience values.

4.2. Resilience Characteristics of Road Networks Under Intentional Attack Scenarios

4.2.1. The Overall Trend of Resilience Under Intentional Attack Scenarios

In intentional attack scenarios, we continuously remove important “bridges” that shorten the shortest path between nodes and observe how network resilience changes with density. The results in

Table 2. The variation in road network resilience with respect to density in intentional attack scenarios.

Grid Interval	Scale (km)
	1.5 × 1.5	2.0 × 2.0	2.5 × 2.5	3.0 × 3.0	3.5 × 3.5	4.0 × 4.0	4.5 × 4.5	5.0 × 5.0	5.5 × 5.5	6.0 × 6.0	6.5 × 6.5
500 m					0.0733	0.0637	0.0545	0.0554	0.0551	0.0842	0.0642
475 m				0.0738	0.0639	0.0546	0.0555	0.0552	0.0836	0.0643	0.0715
450 m				0.0735	0.0638	0.0545	0.0554	0.0829	0.0643	0.0716	0.0696
425 m				0.0641	0.0547	0.0555	0.0552	0.0839	0.0642	0.0697	0.0810
400 m			0.0739	0.0639	0.0545	0.0554	0.0831	0.0643	0.0715	0.0695	0.0780
375 m			0.0735	0.0637	0.0555	0.0552	0.0842	0.0716	0.0696	0.0808	0.0797
350 m			0.0640	0.0546	0.0554	0.0834	0.0642	0.0697	0.0809	0.0797	0.0868
325 m		0.0740	0.0638	0.0555	0.0551	0.0643	0.0715	0.0809	0.0779	0.0868	0.0887
300 m		0.0735	0.0547	0.0554	0.0838	0.0716	0.0695	0.0779	0.0868	0.0887	0.0954
275 m		0.0640	0.0555	0.0551	0.0643	0.0696	0.0780	0.0869	0.0887	0.0954	0.1039
250 m		0.0637	0.0554	0.0842	0.0715	0.0808	0.0796	0.0887	0.0953	0.1038	0.1122
225 m	0.0735	0.0545	0.0829	0.0716	0.0809	0.0796	0.0887	0.1027	0.1042	0.1063	0.1213
200 m	0.0639	0.0554	0.0643	0.0695	0.0797	0.0887	0.1027	0.1041	0.1208	0.1217	0.1371
175 m	0.0546	0.0834	0.0697	0.0797	0.0887	0.1026	0.1123	0.1213	0.1297	0.1387	0.1477
150 m	0.0554	0.0716	0.0779	0.0887	0.1040	0.1063	0.1217	0.1367	0.1466	0.1542	0.1652
125 m	0.0842	0.0808	0.0887	0.1038	0.1208	0.1296	0.1253	0.1542	0.1652	0.1624	0.1731
100 m	0.0695	0.0887	0.1041	0.1217	0.1387	0.1542	0.1616	0.1706	0.1890	0.1987	0.2028
75 m	0.0887	0.1063	0.1367	0.1542	0.1566	0.1790	0.1987	0.2110	0.2351	0.2456	0.2578
50 m	0.1217	0.1542	0.1706	0.1987	0.2192	0.2456	0.2571	0.2718	0.289	0.3131	0.3194

Note: One-way ANOVA was conducted using SPSS version 26.0 to test for significant differences in resilience values across different scale groups. The results show a significance level of p < 0.05, indicating that the variations in resilience across scales are statistically significant (Appendix A Table A17). In the table, a darker shade of orange indicates a higher resilience value, while a darker shade of blue represents a lower resilience value.

show that, across different analysis scales, the resilience of the road network increases as the grid interval decreases. The resilience ranges from a minimum value of 0.0551 to a maximum value of 0.3194. Across the different scale analysis units, the resilience value increases by up to 424%, which is 45 times the increase observed in random scenarios. This result indicates that the grid interval has a decisive impact on resilience in intentional attack scenarios.

Contrary to our expectations, the resilience value in intentional attack scenarios does not uniformly increase as the grid interval decreases. For larger grid intervals, both the threshold and resilience values remain relatively low, with a significant fluctuation occurring. The increase in the threshold is 116% (i.e., from 6% to 13%), and the resilience value increases by 51% (i.e., from 0.0552 to 0.0834), a pattern observed across all different analysis scales. Further analysis reveals that these networks are ideal models with the same number of nodes and edges (Appendix A Table A1 and Table A2; E = 312, N = 169). Therefore, we can conclude that in intentional attack scenarios, when the network structure, in addition to the number of nodes and edges, is identical, both the threshold and resilience value remain consistent.

4.2.2. The Trends in Connected Subgraphs Under Intentional Attack Scenarios

Figure 8, using a 4.0 × 4.0 km scale as an example, illustrates the changes in subgraphs and efficiencies under intentional attack scenarios. Unlike random scenarios, the two stages of subgraph changes in intentional attack scenarios are substantially shorter. Specifically, in the first stage, the maximum connected subgraph remains at a higher level for a short period, and the network quickly splits into multiple subgraphs. In the second stage, the network does not gradually fragment; instead, it collapses rapidly. As a result, the threshold in intentional attack scenarios is generally lower because the removed road segments are those with higher connectivity, resulting in faster network collapse. The lowest threshold point in intentional attack scenarios is 6%, while the highest is 41% (Appendix A 

Table A6. The threshold of the road network in intentional attack scenarios.

Grid Interval	Scale (km)
	1.5 × 1.5	2.0 × 2.0	2.5 × 2.5	3.0 × 3.0	3.5 × 3.5	4.0 × 4.0	4.5 × 4.5	5.0 × 5.0	5.5 × 5.5	6.0 × 6.0	6.5 × 6.5
500 m					8%	7%	6%	6%	6%	13%	7%
475 m				8%	7%	6%	6%	6%	13%	7%	8%
450 m				8%	7%	6%	6%	13%	7%	8%	8%
425 m				7%	6%	6%	6%	13%	7%	8%	9%
400 m			8%	7%	6%	6%	13%	7%	8%	8%	9%
375 m			8%	7%	6%	6%	13%	8%	8%	9%	9%
350 m			7%	6%	6%	13%	7%	8%	9%	9%	10%
325 m		8%	7%	6%	6%	7%	8%	9%	9%	10%	10%
300 m		8%	6%	6%	13%	8%	8%	9%	10%	10%	11%
275 m		7%	6%	6%	7%	8%	9%	10%	10%	11%	12%
250 m		7%	6%	13%	8%	9%	9%	10%	11%	12%	13%
225 m	8%	6%	13%	8%	9%	9%	10%	12%	12%	12%	14%
200 m	7%	6%	7%	8%	9%	10%	12%	12%	14%	14%	16%
175 m	6%	13%	8%	9%	10%	12%	13%	14%	15%	16%	17%
150 m	6%	8%	9%	10%	12%	12%	14%	16%	17%	18%	19%
125 m	13%	9%	10%	12%	14%	15%	14%	18%	19%	19%	20%
100 m	8%	10%	12%	14%	16%	18%	19%	20%	22%	23%	24%
75 m	10%	12%	16%	18%	18%	21%	23%	25%	28%	29%	31%
50 m	14%	18%	20%	23%	26%	29%	31%	33%	36%	39%	41%

), with a difference of 37%.

For larger grid intervals, the maximum subgraph follows a stepwise decline trend (Figure 8a–c), whereas the maximum subgraph for smaller grid intervals first remains stable and then drops sharply at the threshold point (Figure 8d–f). This suggests that as more critical nodes in the network fail, the higher the network density and the stronger the ability to maintain the maximum subgraph.

4.2.3. The Trends in Global Efficiency Under Intentional Attack Scenarios

Similarly to the random scenario, the efficiency decline trend in the intentional attack scenario changes along with variations in the largest subgraph. The difference lies in the fact that in the random scenario, the efficiency decline is smaller, while in the intentional attack scenario, the efficiency decline is more significant. As shown in Figure 8, during the first stage, when the largest subgraph remains at a high level, the efficiency experiences a greater decline. In the second stage, after the subgraph fragments, the efficiency declines sharply. From the efficiency decline curve, it can be observed that before the threshold point, the efficiency decline curve is closer to a straight line, and as it approaches the network collapse threshold point, the network’s efficiency decreases almost linearly. As shown in Figure 8a–d, when the grid interval shrinks from 500 m to 50 m, the efficiency relative to the network collapse remains at 51% and 40% of the original efficiency, respectively. This result indicates that, in the intentional attack scenario, the cumulative area under the efficiency decline curve is jointly affected by both the threshold and the degree of efficiency decline. From the comparison in Figure 9, it can be observed that as the grid interval becomes smaller, the influence of the threshold becomes more significant than the extent of the efficiency decline.

4.3. Comparative Analysis of Two Scenarios

4.3.1. Comparative Analysis of Density Distribution Characteristics of Road Network Resilience

The results of the study show that in both scenarios, smaller grid intervals are associated with higher resilience. As shown in Figure 10, in both scenarios, the grid interval range with higher resilience values is between 50 m and 150 m, the range with moderate resilience values is between 150 m and 300 m, and the range with the lowest resilience values is between 300 m and 500 m. However, the resilience values did not increase uniformly with respect to decreasing grid intervals as we had expected. In the larger grid interval range, the resilience values are lower and fluctuate slightly, whereas in the smaller grid interval range, the resilience values are higher, with a clear resilience increase trend as the grid interval decreases. Compared to the random scenario, the increase in resilience in the intentional attack scenario is more significant. For example, at the 6.5 × 6.5 km analysis scale, when the grid interval decreases from 500 m to 300 m, resilience increases by 0.5% in the random scenario and 38% in the intentional attack scenario; when the grid interval decreases from 300 m to 150 m, resilience increases by 3% in the random scenario and 86% in the intentional attack scenario; when the grid interval decreases from 150 m to 50 m, resilience increases by 4% in the random scenario and 93% in the intentional attack scenario (Figure 10a,b; Table 1 and Table 2). This result suggests that density has a stronger effect on resilience in the intentional attack scenario than in the random scenario. In the orthogonal grid structure, when critical connectivity roads fail, network density plays a decisive role in resilience. In contrast, when roads are randomly removed, the positive impact of network density on resilience is smaller in magnitude.

4.3.2. Comparative Analysis of the Dynamic Evolution Process of Network Structure

Figure 11 illustrates several patterns of dynamic network structural evolution under two types of disturbance scenarios. A comparison of network fragmentation processes reveals that in the random failure scenario, the breakdown is gradual (Figure 11a; Appendix A 

Table A7. Dynamic evolution process of the network structure under random failure scenarios. (4.0 × 4.0 km scale-350 m grid interval).

Network Structure 1	Network Structure 2	Network Structure 3	Network Structure 4	Network Structure 5
f = 5%	f = 10%	f = 15%	f = 20%	f = 25%
Network Structure 6	Network Structure 7	Network Structure 8	Network Structure 9	Network Structure 10
f = 30%	f = 35%	f = 40%	f = 45%	Threshold f = 50%

Note: f is the fraction of edge removal.

and

Table A8. Dynamic evolution process of the network structure in random failure scenarios. (4.0 × 4.0 km scale-125 m grid interval).

Network Structure 1	Network Structure 2	Network Structure 3	Network Structure 4	Network Structure 5
f = 5%	f = 10%	f = 15%	f = 20%	f = 25%
Network Structure 6	Network Structure 7	Network Structure 8	Network Structure 9	Network Structure 10
f = 30%	f = 35%	f = 40%	f = 45%	Threshold f = 53%

Note: f is the fraction of edge removal.

). Initially, a few small isolated components emerge, and the network does not split into two major subgraphs until approximately 50% of the edges are removed. This trend remains consistent across different network densities. Even after the random removal of a significant proportion of edges, the presence of loops helps maintain the overall connectivity, rendering complete network fragmentation less likely.

In contrast, under the intentional attack scenario, when edges with high connectivity and importance fail—typically those located in the network’s central region—the network rapidly fragments into larger isolated components (Figure 11b,c; Appendix A 

Table A9. Dynamic evolution process of the network structure in intentional attack scenarios. (4.0 × 4.0 km scale-350 m grid interval).

Network Structure 1	Network Structure 2	Network Structure 3	Network Structure 4	Network Structure 5
f = 1%	f = 2%	f = 3%	f = 4%	f = 5%
Network Structure 6	Network Structure 7	Network Structure 8	Network Structure 9	Network Structure 10
f = 6%	f = 7%	f = 8%	f = 9%	f = 10%
Network Structure 11	Network Structure 12	Network Structure 13	Network Structure 14	Network Structure 15
f = 11%	f = 12%	Threshold f = 13%	f = 14%	f = 15%

Note: f is the fraction of edge removal.

,

Table A10. Dynamic evolution process of the network structure in intentional attack scenarios. (4.0 × 4.0 km scale-325 m grid interval).

Network Structure 1	Network Structure 2	Network Structure 3	Network Structure 4	Network Structure 5
f = 1%	f = 2%	f = 3%	f = 4%	f = 5%
Network Structure 6	Network Structure 7	Network Structure 8	Network Structure 9	Network Structure 10
f = 6%	Threshold f = 7%	f = 8%	f = 9%	f = 10%
Network Structure 11	Network Structure 12	Network Structure 13	Network Structure 14	Network Structure 15
f = 11%	f = 12%	f = 13%	f = 14%	f = 15%

Note: f is the fraction of edge removal.

and

Table A11. Dynamic evolution process of the network structure in intentional attack scenarios. (4.0 × 4.0 km scale-125 m grid interval).

Network Structure 1	Network Structure 2	Network Structure 3	Network Structure 4	Network Structure 5
f = 1%	f = 2%	f = 3%	f = 4%	f = 5%
Network Structure 6	Network Structure 7	Network Structure 8	Network Structure 9	Network Structure 10
f = 6%	f = 7%	f = 8%	f = 9%	f = 10%
Network Structure 11	Network Structure 12	Network Structure 13	Network Structure 14	Network Structure 15
f = 11%	f = 12%	f = 13%	f = 14%	Threshold f = 15%

Note: f is fraction of edge removal.

). According to our measurement model, once two dominant subgraphs emerge at their maximum size, this indicates a complete disconnection between components and total network collapse. The threshold at which this collapse occurs is closely related to network density. The question of whether loop structures can form around failed edges to maintain global connectivity becomes critical. Higher network densities correspond to a greater number of loops, which raises the fragmentation threshold (Appendix A Table A9, Table A10 and Table A11). This explains why network densities have a significantly stronger influence on resilience under the intentional attack scenario than under the random failure scenario.

We further investigated the structural evolution patterns of idealized models that exhibited pronounced fluctuations in resilience under the ranked disruption scenario. Taking the 4.0 × 4.0 km model with a 350 m grid interval as an example (Appendix A Table A1 and Table A2; E = 312, N = 169), we found that the primary cause of resilience fluctuation—compared to other grid-based models—was the unique pattern of subgraph fragmentation (Figure 11b,c). Specifically, at 7% edge removal, the network split into a secondary subgraph that was relatively small (approximately one-quarter of the original graph). At 13% edge removal, the network further fragmented into four subgraphs, with the second-largest subgraph now larger than that observed at the 7% removal stage. Hence, the fragmentation threshold was identified at 13%.

In contrast, the 325 m grid model exhibited a different behavior. After removing 6% of the edges with the highest betweenness centrality, the network split vertically from the center into two large left and right subgraphs, which subsequently fragmented into four smaller components. Therefore, the maximum size of the second-largest subgraph occurred at the 6% removal mark. This fluctuation in fragmentation thresholds may be attributed to the number of edges removed in the initial step. In smaller-scale networks, removing an odd number of edges may induce asymmetric fragmentation (e.g., toward the right or bottom) (Figure 11c), whereas removing an even number of edges tends to produce more symmetric top–bottom splits into evenly sized subgraphs (Figure 11b). As a network’s size increases, the influence of whether the initial removals are odd or even becomes less significant, which explains why such fluctuations are not observed in larger-scale models.

5. Discussion

5.1. Effect of Density on the Resilience of Road Networks

First, based on the analysis of the results in Section 4, we can conclude that density significantly affects the resilience level of road networks, and its impact on resilience in intentional attack scenarios is greater than in random failure scenarios. We use global efficiency as the primary measure of resilience, and with an increase in density, the network’s ability to maintain global efficiency strengthens. Specifically, global efficiency is a metric based on the shortest distance, and the impact of density on road network resilience is primarily manifested: High-density networks provide more redundant routes. This is consistent with existing research studies [19,20,21,22,47,53]. Specifically, when local segments fail at the same analysis scale, as network density increases, there are more redundant connecting routes between two points. As a result, the shortest path may gradually lengthen due to disturbances rather than being completely interrupted, which slows down the overall efficiency decline of the network (Figure 6, Figure 7, Figure 8 and Figure 9). This phenomenon is observed in both scenarios, indicating that road networks experience progressively smaller efficiency losses as density increases. More importantly, our findings reveal that higher-density networks develop greater circuit redundancy, which effectively mitigates cascading failures when critical road segments are sequentially disrupted. As a result, these networks exhibit both higher collapse thresholds and improved resilience metrics (Appendix A Table A9, Table A10 and Table A11).

Second, there is a difference compared to the conclusions drawn in similar studies. Our results show that while density is an important factor influencing network resilience, it is not the only factor. In random scenarios, network resilience depends on both network structure and density. The results of the random scenario show that the threshold between different densities is the same (Appendix A Table A5), indicating that the basic structure of the network (node degree, i.e., how many edges a node is connected to) determines the random threshold of the network. Therefore, the impact of density on resilience is limited, and as network density increases in random disruption scenarios, the increase in resilience is relatively small. The influence of the node degree on network resilience has also been confirmed in other studies [7,40,61]. This conclusion also suggests that a single threshold point cannot be used as a measure of network resilience [7,51,57]. In intentional attack scenarios, we found that when networks have the same number of nodes and edges, their threshold and resilience values are similar. Further observations revealed that an increase in network resilience is consistent with an increase in the number of nodes and edges, suggesting that in grid structures, when key nodes or edges fail, the increase in network resilience is more closely related to the number of its nodes and edges rather than its overall scale.

Third, from Appendix A 

Table A4. The initial efficiency of the road network.

Grid Interval	Scale (km)
	1.5 × 1.5	2.0 × 2.0	2.5 × 2.5	3.0 × 3.0	3.5 × 3.5	4.0 × 4.0	4.5 × 4.5	5.0 × 5.0	5.5 × 5.5	6.0 × 6.0	6.5 × 6.5
500 m					0.8115	0.8086	0.8064	0.8046	0.8032	0.8020	0.8011
475 m				0.8162	0.8119	0.8087	0.8063	0.8045	0.8030	0.8018	0.8008
450 m				0.8128	0.8092	0.8066	0.8046	0.8054	0.8033	0.8018	0.8006
425 m				0.8163	0.8104	0.8067	0.8042	0.8025	0.8012	0.8018	0.8004
400 m			0.8172	0.8108	0.8070	0.8046	0.8045	0.8022	0.8007	0.7996	0.7999
375 m			0.8128	0.8086	0.8072	0.8040	0.8020	0.8018	0.8002	0.7990	0.7991
350 m			0.8148	0.8078	0.8046	0.8036	0.8013	0.8011	0.7994	0.7996	0.7982
325 m		0.8190	0.8096	0.8080	0.8037	0.8029	0.8005	0.8001	0.7986	0.7984	0.7974
300 m		0.8128	0.8094	0.8046	0.8027	0.8018	0.7996	0.7990	0.7986	0.7974	0.7971
275 m		0.8130	0.8094	0.8034	0.8016	0.8004	0.7996	0.7990	0.7974	0.7970	0.7967
250 m		0.8086	0.8046	0.8020	0.8003	0.7990	0.7981	0.7974	0.7968	0.7964	0.7960
225 m	0.8128	0.8066	0.8054	0.8018	0.7997	0.7983	0.7974	0.7974	0.7966	0.7961	0.7957
200 m	0.8115	0.8046	0.8022	0.7996	0.7988	0.7974	0.7970	0.7962	0.7961	0.7955	0.7954
175 m	0.8086	0.8032	0.8011	0.7996	0.7974	0.7967	0.7962	0.7959	0.7956	0.7954	0.7952
150 m	0.8046	0.8011	0.7990	0.7974	0.7970	0.7961	0.7955	0.7954	0.7950	0.7947	0.7947
125 m	0.8020	0.7990	0.7974	0.7964	0.7957	0.7953	0.7949	0.7947	0.7945	0.7944	0.7942
100 m	0.7996	0.7974	0.7962	0.7955	0.7950	0.7947	0.7945	0.7943	0.7942	0.7940	0.7940
75 m	0.7974	0.7961	0.7954	0.7947	0.7945	0.7943	0.7940	0.7940	0.7939	0.7938	0.7938
50 m	0.7955	0.7947	0.7943	0.7940	0.7939	0.7938	0.7937	0.7937	0.7936	0.7936	0.7936

, we can observe that as density increases, the network’s initial efficiency decreases, but resilience increases. Some existing studies have incited debates with respect to the perspective of high connectivity, redundancy, modularity, and diversity as resilience attributes [62,63]. Connectivity, redundancy, and diversity are all metrics related to density. Some scholars argue that higher efficiencies can reduce these four attributes, thus hindering network resilience [20]. This conclusion aligns with our research findings, which indicate that there is indeed a potential conflict between high efficiency, high density, and high resilience. Increasing the density may sacrifice the efficiency of the road network, but this helps enhance its resilience. However, from a resilience perspective, urban planning practices need to consider multiple influencing factors to coordinate the relationships between efficiency, density, and resilience [6] in order to balance urban development, the daily needs of residents, and disaster response demands in disturbance scenarios.

5.2. Implications of Planning Strategies for Enhancing Road Network Resilience

First, our analytical framework is particularly relevant for meso-scale and micro-scale urban planning practices. At the macro-scale, critical threshold states that influence functionality rarely occur, as indicated by the threshold points in our measurement approach [21,64]. However, at the meso-scale and micro-scale, it is possible for a neighborhood-level road network to fail entirely. From the perspective of experimental simulations, real cities typically do not experience disturbances exactly as simulated in random or intentional attack scenarios, where a certain percentage of road intersections or road segments is sequentially removed. However, in road networks exposed to high-risk disaster zones, events such as floods can directly inundate high-betweenness-centrality roads and other road segments [12], which is similar to a combination of intentional attack and random attack scenarios. The meso-scale and micro-scale are the main spatial ranges of residents’ daily lives, and they are also crucial scales for understanding the impacts of disasters on urban resilience [5]. For example, during an emergency evacuation or the delivery of basic survival goods, the network’s resilience becomes critical. Simultaneously, density is an essential factor at the meso-scale. Understanding the global efficiency of road networks with different densities in relation to their tolerance for disturbances can help reduce the impact of adverse events on residents’ daily lives by improving urban forms.

Second, this study proposes planning recommendations for resilient road network spacing. According to our findings, the highest resilience values under both disruption scenarios were observed in networks with a grid spacing of 50 m. However, designing an urban road network uniformly with a 50 m interval is neither realistic nor appropriate. Sharifi [10] argues that the optimal structure of a street network depends on its primary purpose and function. Enhancing redundancy—in terms of connectivity—to the highest level is desirable for managing transportation demands and ensuring efficient post-disaster evacuation. However, excessively high road network densities can also impose a range of negative impacts on urban environments [18,65]. It is worth noting that the optimal structure of street networks varies depending on what their main purposes and functions are. Increasing redundancy (in terms of connectivity) to the maximum level could be desirable for transport demand management and smooth evacuation in the aftermath of disasters. On the one hand, a high-density road network implies a greater proportion of land allocated to road infrastructure and a higher number of intersections. An excessive number of intersections may pose safety risks for pedestrians, while the increased presence of traffic signals can result in longer travel times for vehicles, thereby reducing overall urban operational efficiency. Moreover, denser road networks may facilitate the rapid spread and transmission of infectious diseases [9]. Barcelona’s “superblock” initiative is a direct response to the challenges posed by dense street networks and small urban blocks, particularly with regard to vehicular traffic efficiency [66].

On the other hand, a high-density road network also implies smaller urban block sizes, which can constrain the provision of green space and public areas. From the perspective of land development patterns, overly small block sizes often result in full-block building during development—where buildings occupy the entire block—leaving little room for internal courtyards or open space. As a result, some scholars have advocated for the maintenance of road network densities within an optimal range to balance accessibility with livability [2]. The definition of an appropriate density range should be determined through a comprehensive assessment based on specific planning contexts and objectives. Siksna [67] suggested that block sizes of 50–70 m are most conducive to concentrating pedestrian activity, while blocks of 80–110 m strike a balance between walkability and vehicular mobility. Based on these insights, we define a grid spacing of 50 to 100 m as appropriate for pedestrian-oriented urban networks and designate a spacing of 100 to 200 m for the lowest tier of vehicular road networks, such as local streets or internal roads within residential communities. Therefore, setting the pedestrian network at a minimum spacing is both feasible and effective, falling within the high-resilience grid spacing range and aligning with existing research on suitable urban land parcel sizes from other perspectives [20,21,66].

Third, ensuring the safety of highly connected road segments and increasing the density of existing road networks through flexible management measures are essential. According to the results of the intentional attack scenario, when high-betweenness, highly connected segments fail, network density becomes a critical factor in determining resilience. Highly connected road segments should be avoided in disaster-prone areas to minimize the risk of essential road segments failing. Simultaneously, maintaining a high density near highly connected road segments ensures that if a segment fails, the overall network efficiency can remain high. For already formed superblocks, urban management strategies can be used to increase road network density. For example, China’s rapid urbanization has resulted in the construction of numerous enclosed superblocks [68]. In normal conditions, these superblocks can remain as they are, but in the event of a disaster, internal roads can be opened for emergency evacuation, increasing accessibility and facilitating the rapid movement of people or relief materials. Pedestrian pathways can also be converted into vehicle lanes to allow emergency vehicles and supplies to reach affected communities. This approach improves the existing urban form at a low cost, does not disrupt daily life, and ensures that the city’s core functions can continue to operate by maintaining a sufficient network density during disasters, preventing extreme disruptions to the road network’s connectivity.

5.3. Limitations and Future Opportunities

Future research can address several limitations in this study. First, the limitations of the idealized model are as follows. The construction of an ideal model inevitably involves simplifying the complexities of real-world systems. This simplification is guided by the researcher’s conceptual understanding of the system’s importance, allowing for a subjective abstraction of reality that facilitates the clear and objective analysis of the resulting topological relationships and network structure [49]. To clarify the study boundaries and isolate the effects of density on resilience, this study employed an orthogonal grid-based idealized model. However, this abstraction also omits many complexities inherent in real-world road networks, such as heterogeneity, hierarchical organization, elevation variations, physical constraints imposed by rivers, hills, or mountains, and variations in traffic volume. Future research should aim to incorporate additional features into the model to more closely approximate real-world road network characteristics. Second, this study focuses solely on grid-based networks; therefore, the findings on the relationship between density and resilience may not be generalizable to other road network forms. Fortunately, the measurement framework proposed here is adaptable and can be applied to networks with different morphological characteristics. Future studies may include comparative resilience assessments across real-world road networks that represent various structural typologies, thereby clarifying the resilience characteristics of alternative street configurations. Third, there are also limitations in our disturbance scenario simulation. While the disruption scenarios in this study are representative, they do not capture cascading failures between interconnected urban systems or the compound impacts of multiple simultaneous hazards—both of which are common in real disaster events. Future research can extend the simulation framework to explore the multi-system impacts of real-world disasters. Finally, this study focuses exclusively on the road network as an isolated system, overlooking the influence of interactions with other critical urban subsystems, such as land use, travel behavior, and the distribution of service facilities. Future studies could examine composite networks that integrate multiple urban systems, thereby broadening the scope of resilience research and aligning it more closely with the complexity of actual urban functions. Moreover, if full-cycle disaster data become available, resilience evaluations can be expanded to consider both physical infrastructure and social systems, particularly regarding the accessibility of critical services during post-disaster recovery.

6. Conclusions

Guided by the principles of compact urban development, high-density street networks have become a prominent trend in urban planning. This study introduces a multi-scale framework for measuring street network resilience and applies it to 186 idealized orthogonal networks with varying densities, providing quantitative evidence to assess whether higher-density road networks are inherently more resilient. We propose a novel method for measuring resilience, based on a street network’s ability to maintain global efficiency under various disruption scenarios. Unlike traditional approaches that rely solely on a single failure threshold, our method calculates the resilience value using the area under the efficiency–decline curve before system collapse. This approach captures both the magnitude of disruption a system can withstand and its performance trajectory during failure.

The results demonstrate that density significantly influences the efficiency-based resilience of street networks. In both random and intentional disruption scenarios, low-density networks exhibit lower and more unstable resilience, while high-density networks show increased and more stable resilience. In both scenarios, the grid interval range with higher resilience values falls between 50 m and 150 m, with peak resilience observed in the 50 m grid interval models. Moderate resilience values are found in the 150 m to 300 m grid interval models, and the lowest resilience values are in the 300 m to 500 m grid interval models.

Further analysis reveals that network subgraphs and efficiency undergo two distinct phases under both disruption scenarios: network fragmentation and network collapse. In random failure scenarios, these transitions occur more gradually, whereas in intentional attack scenarios, they manifest more abruptly. Across different spatial scales, the maximum resilience improvements with increasing density reach 9.5% for random failures and 424% for intentional attacks. This striking disparity highlights the decisive role of network density in maintaining resilience when critical nodes or edges are compromised. This effect is attributed to the higher prevalence of loops in denser networks, which helps sustain connectivity even under intentional disruption. In the random failure scenario, network resilience is determined by the combined effects of node degree and density.

We propose density intervals for resilient urban road networks. The threshold-focused resilience measurement framework is particularly applicable to urban districts or communities located in disaster-prone areas. Based on our findings, we recommend a grid spacing of 50–100 m for pedestrian networks and 100–200 m for the lowest tier of vehicular networks. This ensures a resilient and highly connected underlying network structure while aligning with urban block development scales identified in prior research. Under normal conditions, safety can be maintained on high-connectivity street segments. During disaster events, flexible management measures—such as opening internal roads—can be implemented to enhance overall road network density, thereby cost-effectively sustaining high network efficiency within already developed urban areas.