A Generalized Approach to Complex Networks

TL;DR

This paper introduces a generalized mathematical framework for analyzing complex networks, extending traditional measures to subnetwork and whole-graph levels, enhancing network characterization and comparison capabilities.

Contribution

It formalizes complex network concepts using discrete mathematics and morphology, enabling new measurements and models for network topology and growth.

Findings

Extended node degree and clustering coefficient for subnetworks

Signature-based network characterization across scales

Improved node correspondence identification in protein networks

Abstract

This work describes how the formalization of complex network concepts in terms of discrete mathematics, especially mathematical morphology, allows a series of generalizations and important results ranging from new measurements of the network topology to new network growth models. First, the concepts of node degree and clustering coefficient are extended in order to characterize not only specific nodes, but any generic subnetwork. Second, the consideration of distance transform and rings are used to further extend those concepts in order to obtain a signature, instead of a single scalar measurement, ranging from the single node to whole graph scales. The enhanced discriminative potential of such extended measurements is illustrated with respect to the identification of correspondence between nodes in two complex networks, namely a protein-protein interaction network and a perturbed version of it. The use of other measurements derived from mathematical morphology are also suggested as a means to characterize complex networks connectivity in a more comprehensive fashion.