The inhomogeneous evolution of subgraphs and cycles in complex networks

TL;DR

This paper investigates how subgraph and cycle structures in complex networks evolve over time, revealing that some subgraphs remain stable while others grow at rates influenced by network properties, causing shifts in network topology.

Contribution

It introduces the concept of inhomogeneous subgraph evolution in real networks, linking growth patterns to degree distribution and clustering, supported by empirical measurements.

Findings

Certain subgraphs maintain constant density during network growth.

Other subgraphs increase in density at rates determined by network properties.

Network evolution causes systematic shifts in subgraph spectra and overrepresentation of specific structures.

Abstract

Subgraphs and cycles are often used to characterize the local properties of complex networks. Here we show that the subgraph structure of real networks is highly time dependent: as the network grows, the density of some subgraphs remains unchanged, while the density of others increase at a rate that is determined by the network's degree distribution and clustering properties. This inhomogeneous evolution process, supported by direct measurements on several real networks, leads to systematic shifts in the overall subgraph spectrum and to an inevitable overrepresentation of some subgraphs and cycles.