With a core-periphery structure of networks, core nodes are densely interconnected, peripheral nodes are connected to core nodes to different extents, and peripheral nodes are sparsely interconnected. Core-periphery structure composed of a single core and periphery has been identified for various networks. However, analogous to the observation that many empirical networks are composed of densely interconnected groups of nodes, i.e., communities, a network may be better regarded as a collection of multiple cores and peripheries. We propose a scalable algorithm to detect multiple nonoverlapping groups of core-periphery structure in a network. We illustrate our algorithm using synthesized and empirical networks. For example, we find distinct core-periphery pairs with different political leanings in a network of political blogs and separation between international and domestic subnetworks of airports in some single countries in a worldwide airport network.

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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. other types of core-periphery and related structure, such as continuous versions of core-periphery structure [8,10,14,17,25], transport-based core-periphery structure [13,14,17,21,36], k-core [37] and richclubs [38,39].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. If a network has such core-periphery structure, the core block should have more intra-block edges and the periphery block should have fewer intra-block edges than a reference. We argue that the core-periphery structure that Borgatti and Everett proposed (figure 1), which many of the subsequent work is based on, is impossible if we use the configuration model [42] as the null model and there are just one core and one periphery. The configuration model is a common class of random graph models that preserve the degree or its mean value of each node. Therefore, our claim implies that there is no core-periphery structure a la mode de Borgatti and Everett beyond the expectation from the degree of each node (i.e., hubs are core nodes), which is, in fact, consistent with some previous observations [16,28,30].Then, we are led to a question: what is a core-periphery structure? To answer this question, let us look at the status of the configuration model in other measurements of networks. We have a plethora of centrality measures for nodes because the degree is often not a useful measure of the importance of nodes [1]. In other words, different centrality measures provide rank orders of nodes in the given network that ar...

Many empirical networks have community structure, in which nodes are densely interconnected within each community (i.e., a group of nodes) and sparsely across different communities. Like other local and meso-scale structure of networks, communities are generally heterogeneous in various aspects such as the size, density of edges, connectivity to other communities and significance. In the present study, we propose a method to statistically test the significance of individual communities in a given network. Compared to the previous methods, the present algorithm is unique in that it accepts different community-detection algorithms and the corresponding quality function for single communities. The present method requires that a quality of each community can be quantified and that community detection is performed as optimisation of such a quality function summed over the communities. Various community detection algorithms including modularity maximisation and graph partitioning meet this criterion. Our method estimates a distribution of the quality function for randomised networks to calculate a likelihood of each community in the given network. We illustrate our algorithm by synthetic and empirical networks.

Maritime transport accounts for a majority of trades in volume, of which 70% in value is carried by container ships that transit regular routes on fixed schedules in the ocean. In the present paper, we analyse a data set of global liner shipping as a network of ports. In particular, we construct the network of the ports as the one-mode projection of a bipartite network composed of ports and ship routes. Like other transportation networks, global liner shipping networks may have core-periphery structure, where a core and a periphery are groups of densely and sparsely interconnected nodes, respectively. Core-periphery structure may have practical implications for understanding the robustness, efficiency and uneven development of international transportation systems. We develop an algorithm to detect core-periphery pairs in a network, which allows one to find core and peripheral nodes on different scales and uses a configuration model that accounts for the fact that the network is obtained by the one-mode projection of a bipartite network. We also found that most ports are core (as opposed to peripheral) ports and that ports in some countries in Europe, America and Asia belong to a global core-periphery pair across different scales, whereas ports in other countries do not.