Generalized theory for node disruption in finite size complex networks

TL;DR

This paper develops a general formula to analyze how node removals affect the degree distribution and stability of finite complex networks, providing new insights into percolation thresholds under various attack strategies.

Contribution

It introduces a simple formula to compute the degree distribution after node removal and derives a stability condition for finite networks under arbitrary attacks.

Findings

Derived a formula for the degree distribution after node removal.

Established a stability condition for finite complex networks.

Calculated the percolation threshold with finite-size corrections.

Abstract

After a failure or attack the structure of a complex network changes due to node removal. Here, we show that the degree distribution of the distorted network, under any node disturbances, can be easily computed through a simple formula. Based on this expression, we derive a general condition for the stability of non-correlated finite complex networks under any arbitrary attack. We apply this formalism to derive an expression for the percolation threshold $f_c$ under a general attack of the form $f_k \sim k^{\gamma}$, where $f_k$ stands for the probability of a node of degree $k$ of being removed during the attack. We show that $f_c$ of a finite network of size $N$ exhibits an additive correction which scales as $N^{-1}$ with respect to the classical result for infinite networks.