Hierarchical core-periphery structure in networks

We study core-periphery structure in networks using inference methods based on a flexible network model that allows for traditional onion-like cores within cores, but also for hierarchical tree-like structures and more general non-nested types of structure. We propose an efficient Monte Carlo scheme for fitting the model to observed networks and report results for a selection of real-world data sets. Among other things, we observe an empirical distinction between networks showing traditional core-periphery structure with a dense core weakly connected to a sparse periphery, and an alternative structure in which the core is strongly connected both within itself and to the periphery. Networks vary in whether they are better represented by one type of structure or the other. We also observe structures that are a hybrid between core-periphery structure and community structure, in which networks have a set of non-overlapping cores that correspond roughly to communities, surrounded by a single undifferentiated periphery. Computer code implementing our methods is available.Comment: code available: https://github.com/apolanco115/hc

Detection of Core-Periphery Structure in Networks Using Spectral Methods and Geodesic Paths

We introduce several novel and computationally efficient methods for detecting "core--periphery structure" in networks. Core--periphery structure is a type of mesoscale structure that includes densely-connected core vertices and sparsely-connected peripheral vertices. Core vertices tend to be well-connected both among themselves and to peripheral vertices, which tend not to be well-connected to other vertices. Our first method, which is based on transportation in networks, aggregates information from many geodesic paths in a network and yields a score for each vertex that reflects the likelihood that a vertex is a core vertex. Our second method is based on a low-rank approximation of a network's adjacency matrix, which can often be expressed as a tensor-product matrix. Our third approach uses the bottom eigenvector of the random-walk Laplacian to infer a coreness score and a classification into core and peripheral vertices. We also design an objective function to (1) help classify vertices into core or peripheral vertices and (2) provide a goodness-of-fit criterion for classifications into core versus peripheral vertices. To examine the performance of our methods, we apply our algorithms to both synthetically-generated networks and a variety of networks constructed from real-world data sets.Comment: This article is part of EJAM's December 2016 special issue on "Network Analysis and Modelling" (available at https://www.cambridge.org/core/journals/european-journal-of-applied-mathematics/issue/journal-ejm-volume-27-issue-6/D245C89CABF55DBF573BB412F7651ADB

Core-periphery structure in networks (revisited)

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 meso-scale network 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 coreperiphery structure, which consists of an identification of a network's nodes into a densely connected core and a sparsely connected periphery. In contrast to traditional network 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, thereby enabling a detailed description of core-periphery structure. We illustrate the differences between our method and existing methods for identifying which nodes belong to a core, and we use it to classify the most important nodes using examples of friendship, collaboration, transportation, and voting networks

Efficient Network Structures with Separable Heterogeneous Connection Costs

We introduce a heterogeneous connection model for network formation to capture the effect of cost heterogeneity on the structure of efficient networks. In the proposed model, connection costs are assumed to be separable, which means the total connection cost for each agent is uniquely proportional to its degree. For these sets of networks, we provide the analytical solution for the efficient network and discuss stability impli- cations. We show that the efficient network exhibits a core-periphery structure, and for a given density, we find a lower bound for clustering coefficient of the efficient network.Comment: 9 page