Localized recovery of complex networks against failure.

Yilun Shang

doi:10.1038/srep30521

Abstract

Resilience of complex networks to failure has been an important issue in network research for decades, and recent studies have begun to focus on the inverse recovery of network functionality through strategically healing missing nodes or edges. However, the effect of network recovery is far from fully understood, and a general theory is still missing. Here we propose and study a general model of localized recovery, where a group of neighboring nodes are restored in an invasive way from a seed node. We develop a theoretical framework to compare the effect of random recovery (RR) and localized recovery (LR) in complex networks including Erdős-Rényi networks, random regular networks, and scale-free networks. We find detailed phase diagrams for the subnetwork of occupied nodes and the “complement network” of failed nodes under RR and LR. By identifying the two competitive forces behind LR, we present an analytical and numerical approach to guide us in choosing the appropriate recovery strategy and provide estimation on its effect by using the degree distribution of the original network as the only input. Our work therefore provides insight for quantitatively understanding recovery process and its implications in infrastructure protection in various complex systems.

Highlights

Is usually easier and more effective to repair a node in the neighborhood than one far apart as functional nodes become connected after recovery and (b) it is sometimes the only way to repair when an external monitor/controller is not available to guide the recovery or limited resources fail to maintain simultaneous recovery in multiple unrelated areas
By identifying the two competitive forces that influence the effect of localized recovery strategy, we develop an analytical tool that can guide us in choosing the appropriate recovery strategy and provide estimation on its effect by using the degree distribution of the original network as the only input
We mention that the localized recovery strategy hinges on the interconnections in a network, especially the topology of its shells[24], which is defined as the set of nodes that are at some distance from a randomly chosen root node

Summary

Introduction

Is usually easier and more effective to repair a node in the neighborhood than one far apart as functional nodes become connected after recovery and (b) it is sometimes the only way to repair when an external monitor/controller is not available to guide the recovery or limited resources fail to maintain simultaneous recovery in multiple unrelated areas. The analytical framework developed here for studying localized recovery of complex networks allows us to examine some percolation properties such as the critical recovery probability rc, marking the threshold of existence/non-existence of giant component in the network of fail nodes, and the fraction P∞ of failed nodes in the giant component of the network of failed nodes after recovery. Let r0 be a second critical recovery probability signifying the threshold of the network of occupied (i.e., functional or recovered) nodes, at which a giant component of occupied nodes first forms. We mention that the localized recovery strategy hinges on the interconnections in a network, especially the topology of its shells[24], which is defined as the set of nodes that are at some distance from a randomly chosen root node. Shell structure has been explored recently by some researchers to collapse an otherwise connected network[25,26,27,28]

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