Improving robustness of spatial networks via reinforced nodes

TL;DR

This paper investigates how reinforced nodes enhance the resilience of spatial networks, revealing that the effectiveness depends on the network's spatial embedding and the stage of failure, with reinforced nodes acting like an external field in phase transitions.

Contribution

It introduces a spatial network model with reinforced nodes, analyzes different reinforcement strategies, and links reinforced nodes to phase transition phenomena in percolation theory.

Findings

Reinforced nodes significantly improve network resilience.

The optimal reinforcement strategy depends on the failure stage.

Reinforced nodes exhibit critical behavior similar to an external field.

Abstract

Many real-world networks are embedded in space, and their resilience in the presence of reinforced nodes has not been studied. Here we model such networks using a spatial network model that have an exponential distribution of link length $r$ having a characteristic length $\zeta$. We find that reinforced nodes can significantly increase the resilience of the networks which varies with strength of spatial embedding. We also study different reinforced node distribution strategies for improving the network resilience. Interestingly, we find that the best strategy is highly dependent on the stage of the percolation process, i.e., the expected fraction of failures. Finally, we show that the reinforced nodes are analogous to an external field in percolation phase transition i.e., having the same critical exponents and that the critical exponents satisfy Widom's relation.