Vulnerability and Hierarchy of Complex Networks

TL;DR

This paper introduces a novel approach to quantify the hidden hierarchy in complex networks by analyzing their vulnerability, revealing how symmetry breaking influences hierarchical properties, especially in random networks.

Contribution

It proposes two measures for network hierarchy based on vulnerability analysis, highlighting the role of symmetry breaking in the emergence of hierarchy in complex networks.

Findings

System vulnerability V is suitable for regular networks but not for large random networks.

Relative variance h of vulnerability characterizes hierarchy in random networks.

Hierarchical properties depend on the ratio of nodes to edges and symmetry breaking.

Abstract

We suggest an approach to study hierarchy, especially hidden one, of complex networks based on the analysis of their vulnerability. Two quantities are proposed as a measure of network hierarchy. The first one is the system vulnerability V. We show that being quite suitable for regular networks this characteristic does not allow one to estimate the hierarchy of large random networks. The second quantity is a relative variance h of the system vulnerability that allows us to characterize a "natural" hierarchy level of random networks. We find that hierarchical properties of random networks depend crucially on a ratio between the number of nodes and the number of edges. We note that any graph with a transitive isometry group action (i.e. an absolutely symmetric graph) is not hierarchical. Breaking such a symmetry leads to appearance of hierarchy.