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Intermediate-scale (or `meso-scale) structures in networks have received considerable attention, as the algorithmic detection of such structures makes it possible to discover network features that are not apparent either at the local scale of nodes and edges or at the global scale of summary statistics. Numerous types of meso-scale structures can occur in networks, but investigations of such features have focused predominantly on the identification and study of community structure. In this paper, we develop a new method to investigate the meso-scale feature known as core-periphery structure, which entails identifying densely-connected core nodes and sparsely-connected periphery nodes. In contrast to communities, the nodes in a core are also reasonably well-connected to those in the periphery. Our new method of computing core-periphery structure can identify multiple cores in a network and takes different possible cores into account. We illustrate the differences between our method and several existing methods for identifying which nodes belong to a core, and we use our technique to examine core-periphery structure in examples of friendship, collaboration, transportation, and voting networks.
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Core-periphery structure, the arrangement of a network into a dense core and sparse periphery, is a versatile descriptor of various social, biological, and technological networks. In practice, different core-periphery algorithms are often applied interchangeably, despite the fact that they can yield inconsistent descriptions of core-periphery structure. For example, two of the most widely used algorithms, the k-cores decomposition and the classic two-block model of Borgatti and Everett, extract fundamentally different structures: the latter partitions a network into a binary hub-and-spoke layout, while the former divides it into a layered hierarchy. We introduce a core-periphery typology to clarify these differences, along with Bayesian stochastic block modeling techniques to classify networks in accordance with this typology. Empirically, we find a rich diversity of core-periphery structure among networks. Through a detailed case study, we demonstrate the importance of acknowledging this diversity and situating networks within the core-periphery typology when conducting domain-specific analyses.
A growing number of systems are represented as networks whose architecture conveys significant information and determines many of their properties. Examples of network architecture include modular, bipartite, and core-periphery structures. However inferring the network structure is a non trivial task and can depend sometimes on the chosen null model. Here we propose a method for classifying network structures and ranking its nodes in a statistically well-grounded fashion. The method is based on the use of Belief Propagation for learning through Entropy Maximization on both the Stochastic Block Model (SBM) and the degree-corrected Stochastic Block Model (dcSBM). As a specific application we show how the combined use of the two ensembles -SBM and dcSBM- allows to disentangle the bipartite and the core-periphery structure in the case of the e-MID interbank network. Specifically we find that, taking into account the degree, this interbank network is better described by a bipartite structure, while using the SBM the core-periphery structure emerges only when data are aggregated for more than a week.
Core-periphery networks are structures that present a set of central and densely connected nodes, namely the core, and a set of non-central and sparsely connected nodes, namely the periphery. The rich-club refers to a set in which the highest degree nodes show a high density of connections. Thus, a network that displays a rich-club can be interpreted as a core-periphery network in which the core is made up by a number of hubs. In this paper, we test the resilience of networks showing a progressively denser rich-club and we observe how this structure is able to affect the network measures in terms of both cohesion and efficiency in information flow. Additionally, we consider the case in which, instead of making the core denser, we add links to the periphery. These two procedures of core and periphery thickening delineate a decision process in the placement of new links and allow us to conduct a scenario analysis that can be helpful in the comprehension and supervision of complex networks under the resilience perspective. The advantages of the two procedures, as well as their implications, are discussed in relation to both network effciency and node heterogeneity.
Consider the following process on a network: Each agent initially holds either opinion blue or red; then, in each round, each agent looks at two random neighbors and, if the two have the same opinion, the agent adopts it. This process is known as the 2-Choices dynamics and is arguably the most basic non-trivial opinion dynamics modeling voting behavior on social networks. Despite its apparent simplicity, 2-Choices has been analytically characterized only on restricted network classes---under assumptions on the initial configuration that establish it as a fast majority consensus protocol. In this work, we aim at contributing to the understanding of the 2-Choices dynamics by considering its behavior on a class of networks with core-periphery structure, a well-known topological assumption in social networks. In a nutshell, assume that a densely-connected subset of agents, the core, holds a different opinion from the rest of the network, the periphery. Then, depending on the strength of the cut between the core and the periphery, a phase-transition phenomenon occurs: Either the cores opinion rapidly spreads among the rest of the network, or a metastability phase takes place, in which both opinions coexist in the network for superpolynomial time. The interest of our result is twofold. On the one hand, by looking at the 2-Choices dynamics as a simplistic model of competition among opinions in social networks, our theorem sheds light on the influence of the core on the rest of the network, as a function of the cores connectivity toward the latter. On the other hand, we provide one of the first analytical results which shows a heterogeneous behavior of a simple dynamics as a function of structural parameters of the network. Finally, we validate our theoretical predictions with extensive experiments on real networks.
Temporal communities result from a consistent partitioning of nodes across multiple snapshots of an evolving complex network that can help uncover how dense clusters in a network emerge, combine, split and decay with time. Current methods for finding communities in a single snapshot are not straightforwardly generalizable to finding temporal communities since the quality functions used for finding static communities have highly degenerate landscapes, and the eventual partition chosen among the many partitions of similar quality is highly sensitive to small changes in the network. To reliably detect temporal communities we need not only to find a good community partition in a given snapshot but also ensure that it bears some similarity to the partition(s) found in immediately preceding snapshots. We present a new measure of partition distance called estrangement motivated by the inertia of inter-node relationships which, when incorporated into the measurement of partition quality, facilitates the detection of meaningful temporal communities. Specifically, we propose the estrangement confinement method, which postulates that neighboring nodes in a community prefer to continue to share community affiliation as the network evolves. Constraining estrangement enables us to find meaningful temporal communities at various degrees of temporal smoothness in diverse real-world datasets. Specifically, we study the evolution of voting behavior of senators in the United States Congress, the evolution of proximity in human mobility datasets, and the detection of evolving communities in synthetic networks that are otherwise hard to find. Estrangement confinement thus provides a principled approach to uncovering temporal communities in evolving networks.
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