Sensitivity of matrix function based network communicability measures: Computational methods and a priori bounds

TL;DR

This paper develops efficient computational methods and bounds for assessing how sensitive matrix function-based network measures are to changes, aiding in understanding network robustness and node importance.

Contribution

It introduces linear-cost algorithms for estimating network sensitivities, extending the concept to subgraph centrality and Estrada index, with new a priori bounds based on decay properties of matrix functions.

Findings

Efficient algorithms for network sensitivity estimation with linear complexity.

Extended sensitivity analysis to subgraph centrality and Estrada index.

A priori bounds accurately predict the qualitative behavior of network indices.

Abstract

When analyzing complex networks, an important task is the identification of those nodes which play a leading role for the overall communicability of the network. In the context of modifying networks (or making them robust against targeted attacks or outages), it is also relevant to know how sensitive the network's communicability reacts to changes in certain nodes or edges. Recently, the concept of total network sensitivity was introduced in [O. De la Cruz Cabrera, J. Jin, S. Noschese, L. Reichel, Communication in complex networks, Appl. Numer. Math., 172, pp. 186-205, 2022], which allows to measure how sensitive the total communicability of a network is to the addition or removal of certain edges. One shortcoming of this concept is that sensitivities are extremely costly to compute when using a straight-forward approach (orders of magnitude more expensive than the corresponding communicability measures). In this work, we present computational procedures for estimating network sensitivity with a cost that is essentially linear in the number of nodes for many real-world complex networks. Additionally, we extend the sensitivity concept such that it also covers sensitivity of subgraph centrality and the Estrada index, and we discuss the case of node removal. We propose a priori bounds for these sensitivities which capture well the qualitative behavior and give insight into the general behavior of matrix function based network indices under perturbations. These bounds are based on decay results for Fr\'echet derivatives of matrix functions with structured, low-rank direction terms which might be of independent interest also for other applications than network analysis.