A Nonlinear Spectral Method for Core--Periphery Detection in Networks

Abstract

We derive and analyse a new iterative algorithm for detecting network core-periphery structure. Using techniques in nonlinear Perron-Frobenius theory, we prove global convergence to the unique solution of a relaxed version of a natural discrete optimization problem. On sparse networks, the cost of each iteration scales linearly with the number of nodes, making the algorithm feasible for large-scale problems. We give an alternative interpretation of the algorithm from the perspective of maximum likelihood reordering of a new logistic core-periphery random graph model. This viewpoint also gives a new basis for quantitatively judging a core-periphery detection algorithm. We illustrate the algorithm on a range of synthetic and real networks, and show that it offers advantages over the current state-of-the-art.1. Motivation. Large, complex networks record pairwise interactions between components in a system. In many circumstances, we wish to summarize this wealth of information by extracting high-level information or visualizing key features. Two of the most important and well-studied tasks are• clustering, also known as community detection, where we attempt to subdivide a network into smaller modules such that nodes within each module share many connections and nodes in distinct modules share few connections, and • determination of centrality or rank, where we assign a nonnegative value to each node such that a larger value indicates a higher level of importance. A distinct but closely related problem is to assign each node to either the core or periphery in such a way that core nodes are strongly connected across the whole network whereas peripheral nodes are strongly connected only to core nodes; hence there are relatively weak periphery-periphery connections. More generally, we may wish to assign a non-negative value to each node, with a larger value indicating greater "coreness." The images in the centre and right of Figure 1 indicate the two-by-two block pattern associated with a core-periphery structure.The core-periphery concept emerged implicitly in the study of economic, social and scientific citation networks, and was formalized in a seminal paper of Borgatti and Everett [3]. A review of recent work on modeling and analyzing core-periphery structure, and related ideas in degree assortativity, rich-clubs and nested/bow-tie/onion networks, can be found in [9]. We focus here on the issue of detection: given a large complex network with nodes appearing in arbitrary order, can we discover, quantify and visualize any inherent core-periphery organization?In the next section, we set up our notation and discuss background material. Many detection algorithms can be motivated from an optimization perspective. In section 3 we use such an approach to define and justify the logistic core-periphery *

Community Detection in Networks via Nonlinear Modularity Eigenvectors

“…Among them, the locally greedy algorithm known as Louvain method [5] is arguably the most popular one. In recent years, and mostly in the context of graph partitioning, nonlinear relaxation approaches have been proposed (see for instance [8,9,29,56]). In the context of community detection, a nonlinear relaxation based on the Ginzburg-Landau functional is considered for instance in [6,32], where it is shown to be Γ-convergent to the discrete modularity optimum.…”

Recent advance on detecting core-periphery structure: a survey

“…Some use heuristic algorithms to find near optimal solutions, such as simulated annealing (Lee et al 2014), label switching (Rossa et al 2013), etc., heuristic algorithm can almost guarantee a near optimal solution, but it is often not satisfactory in terms of computational efficiency; some literature transform the optimization problem into matrix computing domain Jia and Benson (2018), for instance, literature (Boyd et al 2010) combined with MINRES algorithm and SVD. The MINRES/ SVD is proposed to transform the problem into the singular value decomposition problem of the matrix; some literature construct the network feature matrix to transform the problem into the cut graph problem Ma et al (2017); some also from the statistical point of view (Zhang et al 2015), use the EM algorithm to calculate the random most similar to the original image, use nonlinear spectral method (Tudisco and Higham 2019). Researchers have considered using integer programming (Brusco 2011) to find the optimal solution, but this method is limited by the size of the network.…”

Random Spatial Network Models for Core-Periphery Structure

“…Since the belief propagation algorithm is sensitive to the initial conditions, we perform 5 independent runs of the algorithm from different random initial conditions and record the best result. The second model defines the edge probability as a logistic function of the "centrality rank" of the incident vertices (henceforth, logistic-CP) [44]. For this model, we tune the hyperparameters s, t in the logistic function with grid search and record the best result.…”

“…The key difference with our work is that we propose a generative model, whereas prior work performs post hoc identification of core-periphery structure using the network topology. The only other generative models are due to Zhang et al [12] and Tudisco and Higham [44] against which we compared in Section 4.2 (unlike our approach, these methods do not use spatial information). Network centrality captures similar properties to core-periphery structure.…”