Optimization of robustness based on reinforced nodes in a modular network

TL;DR

This paper models how reinforcing nodes within modules of a network enhances overall robustness, using percolation theory to identify optimal reinforcement strategies near critical thresholds.

Contribution

It introduces a new analytical model combining modularity and reinforced nodes, providing solutions for optimal reinforcement partitioning in complex networks.

Findings

Robustness is highly sensitive to reinforced node distribution near the percolation threshold.

An optimal reinforced node partition maximizes robustness and remains stable at high average degrees.

Analytical solutions are derived for different reinforced node distributions in modular networks.

Abstract

Many systems such as critical infrastructure exhibit a modular structure with many links within the modules and few links between them. One approach to increase the robustness of these systems is to reinforce a fraction of the nodes in each module, so that the reinforced nodes provide additional needed sources for themselves as well as for their nearby neighborhood. Since reinforcing a node can be an expensive task, the efficiency of the decentralization process by reinforced nodes is vital. In our study we analyze a new model which combines both above mentioned features of real complex systems - modularity and reinforced nodes. Using tools from percolation theory, we derived an analytical solution for any partition of reinforced nodes; between nodes which have links that connect them to other modules ("inter-nodes") and nodes which have connections only within their modules ("intra-nodes"). Among our results, we find that near the critical percolation point ($p\approx p_c$) the robustness is greatly affected by the distribution. In particular, we find a partition of reinforced nodes which yields an optimal robustness and we show that the optimal partition remains constant for high average degrees.