LEARNING TO BOOST RESILIENCE OF COMPLEX NET-WORKS VIA NEURAL EDGE REWIRING

Abstract

The resilience of complex networks, a critical structural characteristic in network science, measures the network's ability to withstand noise corruption and structural changes. Improving resilience typically resorts to minimal modifications of the network structure via degree-preserving edge rewiring-based methods. Despite their effectiveness, existing methods are learning-free, sharing the limitation of transduction: a learned edge rewiring strategy from one graph cannot be generalized to another. Such a limitation cannot be trivially addressed by existing graph neural networks (GNNs)-based approaches since there is no rich initial node features for GNNs to learn meaningful representations. However, neural edge rewiring relies on GNNs for obtaining meaningful representations from pure graph topologies to select edges. We found existing GNNs degenerate remarkably with only pure topologies on the resilience task, leading to the undesired infinite action backtracking. In this work, inspired by persistent homology, we specifically design a variant of GNN called FireGNN for learning inductive edge rewiring strategies. Based on meaningful representations from FireGNN, we develop the first end-toend inductive method, ResiNet, to discover resilient network topologies while balancing network utility. ResiNet reformulates network resilience optimization as a Markov decision process equipped with edge rewiring action space and learns to select correct edges successively. Extensive experiments demonstrate that ResiNet achieves a near-optimal resilience gain on various graphs while balancing the utility and outperforms existing approaches by a large margin.

1. INTRODUCTION

Network systems, such as infrastructure systems and supply chains, are vulnerable to malicious attacks. To provide reliable services when facing natural disasters or targeted attacks, networked systems should continue to function and maintain an acceptable level of utility when the network partially fails. Network resiliencefoot_0 , in the context of network science, is a measurement characterizing the ability of a network system to defend itself from such failures and attacks (Schneider et al., 2011) . Studying the resilience of complex networks has found wide applications in many fields, ranging from ecology (Sole & Montoya, 2001 ), biology (Motter et al., 2008 ), economics (Haldane & May, 2011) to engineering (Albert et al., 2004) . To improve network resilience, many learning-free optimization methods have been proposed, typically falling into the categories of heuristic-based (Schneider et al., 2011; Chan & Akoglu, 2016; Yazıcıoglu et al., 2015; Rong & Liu, 2018) and evolutionary computation (Zhou & Liu, 2014) . These methods improve the resilience of complex networks by minimally modifying graph topologies based on a degree-preserving atomic operation called edge rewiring (Schneider et al., 2011; Chan & Akoglu, 2016; Rong & Liu, 2018) . Concretely, for a given graph G = (V, E) and two existing edges AC and BD, an edge rewiring operation alters the graph structure by removing AC and BD and adding AB and CD, where AC, BD ∈ E and AB, CD, AD, BC / ∈ E. Edge rewiring has some nice properties against simply addition or deletion of edges: 1) it preserves node degree, while addition may violate capacity constraints; 2) it achieves minimal utility degradation in terms of graph Laplacian measurement, while addition/deletion may lead to a large network utility degradation (Jaume et al., 2020; Ma et al., 2021) . Despite their success, learning-free methods share the following limitations: AC-BD AB-CD G t+2k+1 D A B C D A B C G t+2k Agent Figure 1 : Action backtracking in successive edge rewirings since GNNs cannot provide distinguishable edge representations on graphs without rich features. After selecting AC and BD from G t+2k for rewiring at step t + 2k, the agent would select AB and CD at step t + 2k + 1, returning back to G t+2k and forming a cycled action backtracking between G t+2k and G t+2k+1 . • Transduction. Existing methods for selecting edges for rewiring are transductive since they search the robust topology on a particular graph instance. This search procedure is performed for every individual graph without generalization. • Local optimality. It is NP-hard to combinatorially choose two edges to rewire to obtain the globally optimal resilience (Mosk-Aoyama, 2008). Previous studies predominantly adopt greedy-like algorithms, yielding local optimality in practice (Chan & Akoglu, 2016). • Utility Loss. Rewiring operation in network resilience optimization may lead to considerable degradation of the network utility, which may jeopardize the network's functioning. To our knowledge, there is no learning-based inductive network resilience optimization method. A key challenge is that many network science tasks, including resilience optimization, only have pure topologies without rich node features. GNNs-based learning paradigms have proved powerful in solving a large variety of graph tasks with rich features inductively (Li et al., 2018; Joshi et al., 2019; Fu et al., 2020; Khalil et al., 2017; Nazari et al., 2018; Peng et al., 2020; Yu et al., 2019) . However, it still remains opaque how to adapt such approaches to graph tasks with only topological structures available, especially to graph tasks requiring distinguishable node/edge representations to select correct nodes/edges for constructing a solution sequentially. For example, Boffa et al. (2022) showed that the performance of GNNs degenerates largely in solving the traveling salesman problem (TSP) when node coordinate features are missing. Similarly, we empirically found that the popular framework of GNNs and reinforcement learning (RL) fails to optimize network resilience. The RL agent would be stuck into the undesired infinite action backtracking without the meaningful edge representations, as shown in Figure 1 . Readers are referred to Appendix E for a more detailed analysis. Therefore, it is demanding to devise a novel GNN that can be applicable for the network resilience optimization task without rich features. In this work, we overcome the above limitation of GNNs in modeling graphs without rich features and develop the first inductive learning-based method for discovering resilient network topology using successive edge rewiring operations. Inspired by persistent homology and the approximation of the persistence diagram (Edelsbrunner & Harer, 2008; Aktas et al., 2019; Hofer et al., 2020; Horn et al., 2022) , we specially design a purely topology-oriented variant of GNN called Filtration enhanced GNN (FireGNN). FireGNN creates a series of subgraphs (the filtration) by successively removing the node with the highest degree from the graph and then learns to aggregate node representations from each subgraph. FireGNN learns meaningful representations via the proposed filtration process. The main contributions of this paper are summarized as follows: 1) We propose the first learning-based method, ResiNet, to boost network resilience without rich node features inductively in a degree-preserving manner with moderate utility loss. ResiNet forms resilience optimization into a successive sequential decision process of neural edge rewirings. Extensive experiments show that ResiNet achieves near-optimal resilience while balancing network utilities and outperforms existing approaches by a large margin. 2) FireGNN, our technical innovation serving as the graph feature extractor, can learn meaningful representations from pure topological structures. FireGNN provides sufficient training signals to train an RL agent to learn successive edge rewiring operations inductively.



In this paper, network resilience and network resilience are used interchangeably.

