# Measuring Multimodal Transportation Network Resilience Using Curvature-Core Decomposition and Flow Inequality Metrics

Guillaume Bouleux, Giacomo Kahn, Aurelie Charles

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Guillaume Bouleux, Giacomo Kahn, Aurelie Charles. Measuring Multimodal Transportation Network Resilience Using Curvature-Core Decomposition and Flow Inequality Metrics. 11th International Conference on Control, Decision and Information Technologies (CoDIT 2025), Jul 2025, Split, Croatia. hal-05059128

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# Measuring Multimodal Transportation Network Resilience Using Curvature-Core Decomposition and Flow Inequality Metrics

1st Guillaume Bouleux

INSA Lyon, Université Lumière Lyon 2,
Université Claude Bernard Lyon 1,
Université Jean Monnet Saint-Etienne,
DISP UR4570

Villeurbanne, 69621, France
guillaume.bouleux@insa-lyon.fr

2nd Giacomo Kahn

Université Lumière Lyon 2, INSA-Lyon,
Université Claude Bernard Lyon 1,
Université Jean Monnet Saint-Etienne,
DISP UR4570,

Villeurbanne, 69621, France
giacomo.kahn@univ-lyon2.fr

3rd Aurélie Charles

Université Lumière Lyon 2, INSA-Lyon,
Université Claude Bernard Lyon 1,
Université Jean Monnet Saint-Etienne,
DISP UR4570,

Villeurbanne, 69621, France
a charles@univ-lyon2.fr

***Abstract***—Production, health, transport and goods systems have been severely affected by recent health crises, and continue to be so in the face of current climatic and geopolitical disruptions. In this unstable context, it is essential not only to ensure their robustness, but also to reinforce their resilience. Many works in the literature deal with the robustness of systems, but very few focus on proposals related to resilience. We therefore propose to approach this concept within the framework of a multimodal transport network modeled by a multilayer network. We develop a resilience strategy based on the curvature-core decomposition of the network and compare it with a random decision strategy and a decision strategy based on maximum transport capacities. The results show that the strategy based on the curvature-core of the multilayer network enables a better decision to be made, avoiding congestion and link saturation during reconstruction, and finally enabling a balance in the distribution of flows.

***Index Terms***—Transport Network, Multimodal network, Resilience, Ollivier-Ricci curvature, k-core decomposition, multilayer

## I. INTRODUCTION

Production, healthcare, transport and goods systems as a whole have been hard hit by recent health events, but they are also challenged by the climatic and geo-political changes of our time [4]. In such a context, it is vital to guarantee the robustness of these systems, as well as their resilience. These two notions, which often go hand in hand [7], nevertheless call for different analyses and strategies. Robustness is understood as the ability of a system to withstand the disturbances to which it is subjected, without requiring structural modification. A growing body of literature is proposing robustness indicators based on percolation and phase transition principles [3], or on efficiency [13] or flow principles [1]. Resilience is the phase directly following the damage, during which the actions taken should restore the system to an acceptable level of

functionality as quickly as possible [9]. To be highly resilient, then, is to have the ability to quickly return to a functional state. In this work, we study and measure the resilience of a multimodal transportation network modeled as a complex multilayer network [12].

The notion of resilience can be approached from a dynamic point of view [6]. The complete topology of the network is used to infer a resilience function that distinguishes zones where the dynamics of the complex network would allow a more or less easy return to a stable state (its functionality). This is in direct line with dynamic systems approaches, in which the system's critical points are identified and the dynamic trajectories leading to them characterized. Based on the state of the network at a given moment, it is possible to identify the trajectory to which it belongs, and thus propose evolutionary measures to return it to its original functionality. Another interesting theoretical theme concerns the concept of "rewiring" [12] or random "rewiring". This method enables the dynamic construction of a network so that information, a flow, can circulate as efficiently as possible [14]. In the same vein, the concept of "message passing" is used to improve the network's ability to distribute a flow more or less rapidly. In addition, recent work has focused on the geometric representation of the network by calculating its curvature [5]. Ricci curvature seems to be the emerging indicator [8] and is used to characterize network locations that clog up the flow. This is the concept of "over-squashing" introduced in [11]. In this work, we propose to approach resilience in terms of the re-wiring and curvature aspects adapted to a multimodal transport network, and then to briefly compare different resilience strategies using indicators based on information about the distribution of flows in the network.

## II. MAIN CONTRIBUTIONS

Firstly, we address the concept of resilience in the context of multimodal transportation networks, which we model as multilayer networks. While this modeling approach is gaining increasing attention and interest within the scientific community, it remains relatively underexplored to date.

This research was funded in whole or in part by Agence Nationale de la Recherche (ANR) under the project ANR-24-CE39-4306-01 ResHI. A CC-BY public copyright license has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission, in accordance with the grant's open access conditions.
Secondly, we examine the notion of resilience specifically within these multimodal transport networks. As previously mentioned, the concept of resilience — particularly in terms of strategies for maintaining or restoring network performance — has not been extensively developed in this context. Through this work, we aim to bridge part of this gap by offering a comprehensive analysis of resilience mechanisms in such complex systems.

We approach the notion of resilience from an innovative perspective, leveraging the concept of network curvature. This methodology, still seldom explored, provides valuable insights into local robustness features and enables a better understanding of flow distributions throughout the network. More specifically, we adapt curvature-based techniques to design and enhance the structure of what we define as the curvature-core, a backbone substructure of the network. This represents a significant contribution compared to the current state of the art.

Finally, we conduct an analysis on some potential strategies for reconstructing the multimodal transport network. In particular, we focus on the formation and prioritization of the curvature-core layers, while monitoring the evolution of flow distribution across the entire network. This comparative evaluation enables us to assess the relative efficiency of different resilience strategies and their impact on the restoration of network performance.

### III. CLASSICAL K-CORE DECOMPOSITION

The k-core decomposition is a structural analysis method widely used to identify dense substructures in complex graphs [2]. An k-core of a graph is defined as the maximal subgraph in which each vertex has at least k neighbors within the same subgraph. This iterative method eliminates vertices of degree less than k, as well as incidental edges, until all remaining vertices satisfy the minimum degree condition.

The k-core decomposition can thus be used to prioritize the nodes of a network according to their local robustness and connectivity, offering a multi-scale view of the graph structure. However, this approach relies solely on the degree of vertices, and doesn't always accurately capture the overall topology or distribution of flows across the network.

### IV. OLLIVIER-RICCI CURVES FOR GRAPHS

Ollivier-Ricci curvature quantifies the “distance” between small probability balls around two connected nodes, using the optimal transport distance (also known as the Wasserstein distance).

Formally, for an edge $(x, y)$ of a graph, the Ollivier-Ricci curvature is defined by:

$$ \kappa(x, y) = 1 - \frac{W_1(m_x, m_y)}{d(x, y)} \quad (1) $$

where:

* $W_1(m_x, m_y)$ is the Wasserstein distance between the probability distributions $m_x$ and $m_y$ centered on vertices $x$ and $y$ respectively,

<figure><img src="image_2.png" /><figcaption>Fig. 1: Illustration of 3-core decomposition of a network.</figcaption></figure>

* $d(x, y)$ is the distance between vertices $x$ and $y$ (generally equal to 1 for direct edges),

* $m_x$ and $m_y$ are local probability distributions, often defined on the basis of edge weights or the degree of neighboring vertices.

Intuitively, positive curvature indicates that the neighborhoods of $x$ and $y$ are very similar, reflecting strong redundancy and robust connectivity around this edge. Conversely, negative curvature suggests that the edge connects topologically distant or fragile areas of the graph, characteristic of “bridges” or “bottlenecks”.

The interest of Ollivier-Ricci curvature lies in its ability to integrate both local and global network information, overcoming the limitations of simple measures of centrality or degree.

### V. CURVATURE-CORE DECOMPOSITION: A NEW APPROACH

We propose here a natural extension of the k-core decomposition, taking advantage of the geometric properties offered by the Ollivier-Ricci curvature. The k-core (or curvature-core) is defined as the maximal subgraph in which all edges have a curvature greater than or equal to a $\kappa$ threshold. This approach offers a different perspective on structural robustness, based not on vertex degree but on the geometric redundancy of local connections.

To further characterize graph structure during the curvature decomposition process, we introduce the concept of onion layers [10]. Each layer corresponds to the set of edges removed at a given iteration of the Ollivier-Ricci curvature-based pruning process. The idea is to analyze the graph structure by following the progressive evolution of the network as curvature thresholds are applied. At each iteration, we remove edges whose curvature is less than the set threshold, as well as peripheral edges that disconnect from the largest connected component. These edges are then aggregated in the corresponding onion layer. This layered approach enables us to observe the local fragility or robustness of the graph, as well as the dynamics of network fragmentation under curvature constraints. It also provides visual and analytical support for identifying critical edges and areas of the network that support the integrity of
<figure><img src="image_3.png" /><figcaption>Fig. 2: Illustration of the new proposed curvature-core decomposition of network. The initial weights are written as labels on edges whereas edges' color stands for the curvature of the network.</figcaption></figure>

the overall structure. In short, curvature-core decomposition enriched by onion layers is a natural extension of classical k-core decomposition, providing a geometric perspective on the structural robustness of complex networks.

This hierarchical decomposition reveals successive layers of the graph according to their geometric cohesion. The innermost layers (high $κ$ values) correspond to the most robust areas of the network, highly interconnected and not very sensitive to disturbances. Conversely, peripheral layers (low $κ$ values) include critical edges linking less densely connected communities or regions.

In the example of Fig 2, we observed that the *curvature-core* decomposition highlights network substructures that escape simple analysis by degree. For example, inter-community bridges or high-flow but structurally fragile corridors are identified by negative curvatures, while dense and redundant clusters show high curvatures. This makes the *curvature-core* a powerful complementary tool for analyzing the resilience and vulnerability of complex networks.

## VI. ILLUSTRATION

We propose here to illustrate the method proposed in this article by taking a multimodal Network composed of three layers, a road layer, a rail layer and an air traffic layer. We have chosen a Multilayer-type model network, to better materialize the data-switching links between the different modes of air transport, road traffic and rail traffic. In this context, each city is linked to all three possible modes of transport. Before talking about reconstruction strategies when the network is damaged or attacked, we propose to analyze

the curvature-core of this multilayer network. Figure 3 below illustrates the test network we have chosen. Figure 3 (a) represents the multilayer network itself, while Figure 3 (b) shows the adjacency or supra-adjacency matrix that represents all the connections in this multilayer network. In order to reflect reality, we have chosen a slightly hierarchical structure for this multilayer network. In fact, air-type interconnections are fewer than rail-type interconnections, which in turn are fewer than road-type interconnections. We think this does indeed represent a hierarchical mesh structure that is close to what we can have in countries, for example. As this is a discussion about the curvature of the network, and even of the multilayer network, we have given different weights to each of the edges of each of the layers. These weights are intended to represent average passenger flows per kilometer and per hour. Thus, the weights assigned to all the connections in the road infrastructure layer have values varying randomly between 10 and 50. The weights assigned to all connections in the rail infrastructure layer have values randomly varying between 125 and 150, and the weights associated with the air infrastructure layer vary between 150 and 300. In fig. 3, layer 0 corresponds to road infrastructure, layer 1 to rail infrastructure and layer 2 to air infrastructure.

Fig. 4 shows the curvature-core associated with this multimodal transport network. Once again, we recall that the very idea of curvature-core is to be able to identify locally all the edges that allow a homogeneous flow to pass through the network. The higher the curvature, the greater the number of
<figure><img src="image_4.png" /><figcaption>Fig. 3: Randomly simulated multilayer and hierarchical transportation network of the 20 biggest cities in France. The weights for each edges are plotted in (b) with the corresponding adjacency matrix. There are more roads than flights and the number of passengers hours kilometers (weights) is an average estimation based on real data.</figcaption></figure>

flows passing locally through the network. In addition, high curvature value means clique-like structure of the network for which associated nodes can completely absorb incoming flows, and also enable flows to be redistributed to various neighbors without saturating them.

To assess the effectiveness of different network reconstruction strategies, we analyze two complementary indicators: the entropy of the distribution of maximum flows and the Gini coefficient applied to this same distribution. Three reconstruction strategies are compared. First the random strategy, where edges are randomly reintegrated into the graph then the weighted strategy, where edges with the highest capacity (high weight) are reinserted first and finally the curvature-core strategy, where reconstruction follows the reverse order of layers identified by our curvature-core decomposition, reintegrating edges from inner (highest onion layer value) to outer layers (weakest onion layer value) first.

### A. Entropy of maximum-flow distribution

Entropy is a measure derived from information theory that quantifies the dispersion or uncertainty of a distribution. Applied here to the distribution of maximum flows on the edges, it enables us to assess the extent to which the flow is uniformly distributed across the network.

For a normalized flow distribution entropy is defined by

$$H = - \sum_{i=1}^{n} f_i \log(f_i) \qquad (2)$$

where $f_i$ is the normalized proportion of maximum flow on edge $i$. High entropy indicates a greater diversification of flows across the network, translating into a more robust structure in the face of local disturbances. Conversely, low entropy reveals a concentration of flow on a small number of critical edges.

### B. The Gini coefficient of maximum-flow distribution

The Gini coefficient is a measure of inequality used to assess disparity in the distribution of a variable, here applied to maximum edge flows. Unlike entropy, which measures the diversity of the distribution, the Gini coefficient directly quantifies the degree of inequality. Let $F = \{f_1, f_2, \dots, f_n\}$, where flows are sorted in ascending order $f_1 \le f_2 \le \dots \le f_n$. then, the Gini coefficient is given as:

$$G = \frac{\sum_{i=1}^{n} \sum_{j=1}^{n} |f_i - f_j|}{2n \sum_{i=1}^{n} f_i} \qquad (3)$$

where $f_i$ is the maximum flow on edge $i$, $n$ is the total number of edges considered, $n$ is the maximum flow on edge $i$, $n$ is the total number of edges considered. It is bounded between 0 and 1:

* G=0 indicates perfect equality (all flows are equal),

* G=1 reflects maximum inequality (a single flow concentrates all the load).

In our context, a low Gini is desirable as it reflects a homogeneous distribution of flows over the edges of the network, reducing vulnerability to localized failures.

### C. Strategies Comparison

The joint analysis of entropy and Gini coefficient throughout the reconstruction process enables us to dynamically characterize the resilience of the network: (i) the random strategy serves as a baseline for observing restoration without structural bias; (ii) the weighted strategy aims to rapidly restore network capacity by prioritizing high-capacity edges, at the risk of an initial concentration of flows; and (iii) the curvature-core strategy, exploiting the network geometry revealed by Ollivier-Ricci curvature, enables a more balanced restoration, by reintegrating internal edges first, then peripheral ones.
<figure><img src="image_5.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(b)</figcaption></figure>

<caption>Fig. 4: The multilayer curvature-core decomposition illustration of multilayer network displayed in Fig. 3.</caption>

## VII. DISCUSSION

Analysis of the Fig. 6 results finally shows that the strategy based on curvature-core and reconstruction of inter-layer links first, pays off. Indeed, both in terms of the entropy evolution of the max-flow distribution in the network and in terms of the Gini coefficient of this same distribution, the performance of this strategy is better. In particular, we observe a greater difference in the evolution of the Gini coefficient of the max-flow distribution, showing that the curvature-core strategy offers greater fairness in the distribution of flows. This result is supported by the analysis of Fig. 7 where the strategy based on prioritizing high-capacity links clearly induces bottlenecks very quickly by saturating edges. The strategy that randomly reconstructs the network is more moderate in saturating links, but nevertheless shows a higher concentration of flows than all the other strategies.

<figure><img src="image_7.png" /><figcaption>Fig. 5: Supra-adjacency matrix of the degraded multilayer network of Fig. 3. here, the node (Rennes, layer 1) is the epicenter of the degradation, from which a cascading effect occurs next to destroy/degrade 87 others edges.</figcaption></figure>

## VIII. CONCLUSION

The study of the robustness and resilience of transport networks is a subject that has attracted a great deal of interest for some years now, but only a few strategies are proposed in the literature. The most natural modeling approach to address these problems is that of multilayer networks, since it allows us to consider the specificity of each of the transport modes and their inter-modal links. K-core decompositions are effective for quantifying or improving network robustness, but few works propose such a decomposition for multilayer networks. To exploit the flow through the network, we propose to use the curvature of the edges calculated using the Ollivier-Ricci curvature. We also propose a completely new decomposition that combines the multi-layer representation of the transport network with a k-core decomposition adapted to the flow using Ollivier-Ricci curvatures. We obtain a multi-layer curvature-core decomposition that enables us to highlight the homogeneous backbone of passenger flow transport. The process for obtaining such a decomposition is one of identifying elements by their membership of an onion layer, which we propose to use to guide network reconstruction choices when the network is damaged. We propose a new reconstruction strategy to improve resilience, based on the reconstruction of inter-layer edges (inter-modal links) and then layer edges, all ordered according to their depth in the onion decomposition of the multi-layer curvature-core and their intrinsic curvature. A comparison of this strategy with a purely random strategy and a strategy guided by the reconstruction of the most important edges in term of flow shows the importance of our proposal. Indeed, our strategy shows a better distribution of flows during reconstruction, avoiding congestion on certain links and guaranteeing better end-to-end accessibility.
<figure><img src="image_8.png" /><figcaption>Fig. 6: Entropy and Gini coefficient evolution w.r.t. time-ordered iteration of the reconstruction. High entropy value means diversity repartition and low Gini coefficient value means fair repartition of the flow.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7: Heatmap of edges' flow for each iterations of the reconstruction strategy. From left to right, Curvature-core, weight and random strategies. The x-axis is the time ordered iteration and y-axis the edges of the multilayer network. Yellow color represents a heavy-load flow on the edge whereas purple color is a weak-load flow.</figcaption></figure>

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