Multi-scale structure and topological anomaly detection via a new network statistic: The onion decomposition.

Laurent Hébert-Dufresne,Joshua A. Grochow,Antoine Allard

doi:10.1038/srep31708

Laurent Hébert-Dufresne, Joshua A. Grochow + Show 1 more

Open Access

https://doi.org/10.1038/srep31708

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Journal: Scientific Reports	Publication Date: Aug 1, 2016
Citations: 36	License type: CC BY 4.0

Affiliation: Santa Fe Institute, University of Barcelona

Abstract

We introduce a network statistic that measures structural properties at the micro-, meso-, and macroscopic scales, while still being easy to compute and interpretable at a glance. Our statistic, the onion spectrum, is based on the onion decomposition, which refines the k-core decomposition, a standard network fingerprinting method. The onion spectrum is exactly as easy to compute as the k-cores: It is based on the stages at which each vertex gets removed from a graph in the standard algorithm for computing the k-cores. Yet, the onion spectrum reveals much more information about a network, and at multiple scales; for example, it can be used to quantify node heterogeneity, degree correlations, centrality, and tree- or lattice-likeness. Furthermore, unlike the k-core decomposition, the combined degree-onion spectrum immediately gives a clear local picture of the network around each node which allows the detection of interesting subgraphs whose topological structure differs from the global network organization. This local description can also be leveraged to easily generate samples from the ensemble of networks with a given joint degree-onion distribution. We demonstrate the utility of the onion spectrum for understanding both static and dynamic properties on several standard graph models and on many real-world networks.

Highlights

We show how the onion decomposition (OD) naturally corresponds to an ensemble of random networks, unlike other centrality measures. With this new analysis method in hand, we define the onion spectrum of a network as the fraction of all nodes which are found in a given layer of the OD
The onion spectrum can be thought of as a structural spectrum as it assigns every node to a given structural role through our new measure of node centrality
We have introduced the onion decomposition that can be useful to (i) characterize complex networks, (ii) identify interesting subgraphs and (iii) approximate their structure—and the dynamics they support—through the onion network ensemble. This is done by running an algorithm that scales almost linearly with network size (O(|E| log |V|)) to reveal how quickly a network can be peeled by removing peripheral nodes

Summary

Introduction

Modularity (e.g., [4, 5]) and other community structure properties probe the meso-scale organization of networks but are still ill-defined, as evidenced by the lack of a common definition for network communities [6] Macroscopic measures such as betweenness centrality [7], eigenvector centrality [1], and mean shortest path length [3] can characterize the role of a given node in the overall network structure. They demand significant computational effort and rarely map back to a good local understanding of how the network might be constructed. There is a dire need for tools that are easy to compute and that complement existing tools by characterizing networks at multiple scales at a glance

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