Core–periphery structure in directed networks

Abstract

Empirical networks often exhibit different meso-scale structures, such as community and core–periphery structures. Core–periphery structure typically consists of a well-connected core and a periphery that is well connected to the core but sparsely connected internally. Most core–periphery studies focus on undirected networks. We propose a generalization of core–periphery structure to directed networks. Our approach yields a family of core–periphery block model formulations in which, contrary to many existing approaches, core and periphery sets are edge-direction dependent. We focus on a particular structure consisting of two core sets and two periphery sets, which we motivate empirically. We propose two measures to assess the statistical significance and quality of our novel structure in empirical data, where one often has no ground truth. To detect core–periphery structure in directed networks, we propose three methods adapted from two approaches in the literature, each with a different trade-off between computational complexity and accuracy. We assess the methods on benchmark networks where our methods match or outperform standard methods from the literature, with a likelihood approach achieving the highest accuracy. Applying our methods to three empirical networks—faculty hiring, a world trade dataset and political blogs—illustrates that our proposed structure provides novel insights in empirical networks.

Sublinear domination and core–periphery networks

“…We fit the IGAM model to networks examined in Ref. 9 . More specifically, we examine the world-trade network from Ref.…”

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Figure 1 Results of fitting an IGAM model to the world-trade, cs-faculty, history-faculty, business-faculty, and airports datasets examined in Ref. 9 , 24 – 27 . The Figure displays the predicted values of b and c for the IGAM model, and the total degree at each level h of the skeleton tree of fanout b .

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“…Generative models for core–periphery networks have also been studied at Ref. 3 , 7 – 9 . The closest model to ours is the stochastic blockmodel of Ref.…”

“… 3 which assumes that core–core nodes are connected with probability , periphery–periphery nodes are connected with probability and core–periphery nodes are connected with probability , with , and its recent extension to directed graphs in Ref. 9 .…”

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“…de Jeude et al [26] developed a p-value based method using multinomial hypergeometric distribution to assign a surprise like score to network partitions and categorize them into cores and peripheries. Three methodologies for find core periphery structures in directed graphs were described in [27]. Two of them are based upon the well know link analysis algorithm called Hyperlink-Induced Topic Search (HITS) [28] and one of them is based upon the concept of likelihood maximization.…”