Quantifying randomness in real networks.

Chiara Orsini,Marija M Dankulov,Pol Colomer-De-Simón,Amin Vahdat,Priya Mahadevan,Marián Boguñá,Kevin E Bassler,Zoltán Toroczkai,Santo Fortunato,Dmitri Krioukov,Almerima Jamakovic,Guido Caldarelli

doi:10.1038/ncomms9627

Abstract

Represented as graphs, real networks are intricate combinations of order and disorder. Fixing some of the structural properties of network models to their values observed in real networks, many other properties appear as statistical consequences of these fixed observables, plus randomness in other respects. Here we employ the dk-series, a complete set of basic characteristics of the network structure, to study the statistical dependencies between different network properties. We consider six real networks—the Internet, US airport network, human protein interactions, technosocial web of trust, English word network, and an fMRI map of the human brain—and find that many important local and global structural properties of these networks are closely reproduced by dk-random graphs whose degree distributions, degree correlations and clustering are as in the corresponding real network. We discuss important conceptual, methodological, and practical implications of this evaluation of network randomness, and release software to generate dk-random graphs.

Highlights

Represented as graphs, real networks are intricate combinations of order and disorder
A common belief is that a selforganizing system should evolve to a network structure that makes these dynamical processes, or network functions, efficient[3,4,5]
Suppose that the structure of some real network has property X—some statistically over- or under-represented subgraph, or motif[11], for example—that we believe is related to a particular network function

Summary

Introduction

Represented as graphs, real networks are intricate combinations of order and disorder. The dk-series is a converging series of basic interdependent degree- and subgraph-based properties that characterize the local network structure at an increasing level of detail, and define a corresponding series of null models or random graph ensembles. These random graphs have the same distribution of differently sized subgraphs as in a given real network.

Results

Conclusion