Computational complexity arising from degree correlations in networks

TL;DR

This paper investigates how degree correlations in networks affect the computational complexity of finding minimal vertex covers, revealing that correlations can significantly increase problem difficulty.

Contribution

It introduces a Bethe-Peierls approach to analyze the impact of degree correlations on the complexity of vertex cover problems in networks.

Findings

Correlated networks exhibit higher computational complexity for vertex cover problems.

Heuristic algorithms struggle with highly correlated networks, unlike uncorrelated ones.

Degree correlations can qualitatively change the solution structure of network optimization problems.

Abstract

We apply a Bethe-Peierls approach to statistical-mechanics models defined on random networks of arbitrary degree distribution and arbitrary correlations between the degrees of neighboring vertices. Using the NP-hard optimization problem of finding minimal vertex covers on these graphs, we show that such correlations may lead to a qualitatively different solution structure as compared to uncorrelated networks. This results in a higher complexity of the network in a computational sense: Simple heuristic algorithms fail to find a minimal vertex cover in the highly correlated case, whereas uncorrelated networks seem to be simple from the point of view of combinatorial optimization.