Abstract
A network with core-periphery structure consists of core nodes that are densely interconnected. In contrast to a community structure, which is a different meso-scale structure of networks, core nodes can be connected to peripheral nodes and peripheral nodes are not densely interconnected. Although core-periphery structure sounds reasonable, we argue that it is merely accounted for by heterogeneous degree distributions, if one partitions a network into a single core block and a single periphery block, which the famous Borgatti–Everett algorithm and many succeeding algorithms assume. In other words, there is a strong tendency that high-degree and low-degree nodes are judged to be core and peripheral nodes, respectively. To discuss core-periphery structure beyond the expectation of the node’s degree (as described by the configuration model), we propose that one needs to assume at least one block of nodes apart from the focal core-periphery structure, such as a different core-periphery pair, community or nodes not belonging to any meso-scale structure. We propose a scalable algorithm to detect pairs of core and periphery in networks, controlling for the effect of the node’s degree. We illustrate our algorithm using various empirical networks.
Highlights
Many complex systems, biological, physical or social, can be represented by networks [1, 2]
A core-periphery structure in its simplest form refers to a partition of a network into two groups of nodes called core and periphery, where core nodes are densely interconnected, and peripheral nodes are adjacent to the core nodes but not to other peripheral nodes [8,9,10]
Quality and size of detected core-periphery pairs The circles in figure 5 represent the quality and size of core-periphery pairs detected by KM–config in the 12 empirical networks
Summary
Biological, physical or social, can be represented by networks [1, 2]. Given that block structure of networks, or equivalently, hard partitioning of the nodes into groups, has spurred many studies such as community detection [3, 40] and the inference of stochastic block models (SBM) [6, 41], as well as its appeal to intuition, we focus on the discrete version of core-periphery structure based on edge density in the present paper. To solve the conundrum that one does not discover core-periphery structure using the configuration model as the null model, we propose that one must add at least one different block apart from a core block and the corresponding periphery block for a network to have core-periphery structure that is consistent with figure 1. We use the configuration model as the null model, which is different from our previous algorithm [19]
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