Network Quotients: Structural Skeletons of Complex Systems

TL;DR

This paper introduces network quotients, a method to simplify complex networks by collapsing redundant structures using symmetry, preserving essential features while reducing size and computational complexity.

Contribution

The paper systematically explores the properties of network quotients and demonstrates their ability to retain key structural and functional features of the original networks.

Findings

Quotients are significantly smaller than original networks.

Quotients preserve key properties like diameter and heterogeneity.

Using quotients can reduce computational complexity in network analysis.

Abstract

A defining feature of many large empirical networks is their intrinsic complexity. However, many networks also contain a large degree of structural repetition. An immediate question then arises: can we characterize essential network complexity while excluding structural redundancy? In this article we utilize inherent network symmetry to collapse all redundant information from a network, resulting in a coarse-graining which we show to carry the essential structural information of the `parent' network. In the context of algebraic combinatorics, this coarse-graining is known as the \emph{quotient}. We systematically explore the theoretical properties of network quotients and summarize key statistics of a variety of `real-world' quotients with respect to those of their parent networks. In particular, we find that quotients can be substantially smaller than their parent networks yet typically preserve various key functional properties such as complexity (heterogeneity and hubs vertices) and communication (diameter and mean geodesic distance), suggesting that quotients constitute the essential structural skeleton of their parent network. We summarize with a discussion of potential uses of quotients in analysis of biological regulatory networks and ways in which using quotients can reduce the computational complexity of network algorithms.