Spectral density of equitable core-periphery graphs

Using methods from random matrix theory researchers have recently calculated the full spectra of random networks with arbitrary degrees and with community structure. Both reveal interesting spectral features, including deviations from the Wigner semicircle distribution and phase transitions in the spectra of community structured networks. In this paper we generalize both calculations, giving a prescription for calculating the spectrum of a network with both community structur...

We investigate connections between the symmetries (automorphisms) of a graph and its spectral properties. Whenever a graph has a symmetry, i.e. a nontrivial automorphism $\phi$, it is possible to use $\phi$ to decompose any matrix $M\in\mathbb{C}^{n \times n}$ appropriately associated with the graph. The result of this decomposition is a number of strictly smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix $M$. Some of the mat...

We analyse the density of states of the random graph Laplacian in the percolating regime. A symmetry argument and knowledge of the density of states in the nonpercolating regime allows us to isolate the density of states of the percolating cluster (DSPC) alone, thereby eliminating trivially localised states due to finite subgraphs. We derive a nonlinear integral equation for the integrated DSPC and solve it with a population dynamics algorithm. We discuss the possible existen...

In this article we give an in depth overview of the recent advances in the field of equilibrium networks. After outlining this topic, we provide a novel way of defining equilibrium graph (network) ensembles. We illustrate this concept on the classical random graph model and then survey a large variety of recently studied network models. Next, we analyze the structural properties of the graphs in these ensembles in terms of both local and global characteristics, such as degree...

The largest eigenvalue of a network's adjacency matrix and its associated principal eigenvector are key elements for determining the topological structure and the properties of dynamical processes mediated by it. We present a physically grounded expression relating the value of the largest eigenvalue of a given network to the largest eigenvalue of two network subgraphs, considered as isolated: The hub with its immediate neighbors and the densely connected set of nodes with ma...

Various methods have been proposed in the literature to determine an optimal partitioning of the set of actors in a network into core and periphery subsets. However, these methods either work only for relatively small input sizes, or do not guarantee an optimal answer. In this paper, we propose a new algorithm to solve this problem. This algorithm is efficient and exact, allowing the optimal partitioning for networks of several thousand actors to be computed in under a second...

In order to clarify the statistical features of complex networks, the spectral density of adjacency matrices has often been investigated. Adopting a static model introduced by Goh, Kahng and Kim, we analyse the spectral density of complex scale free networks. For that purpose, we utilize the replica method and effective medium approximation (EMA) in statistical mechanics. As a result, we identify a new integral equation which determines the asymptotic spectral density of scal...

Community and core-periphery are two widely studied graph structures, with their coexistence observed in real-world graphs (Rombach, Porter, Fowler \& Mucha [SIAM J. App. Math. 2014, SIAM Review 2017]). However, the nature of this coexistence is not well understood and has been pointed out as an open problem (Yanchenko \& Sengupta [Statistics Surveys, 2023]). Especially, the impact of inferring the core-periphery structure of a graph on understanding its community structure i...

The extreme eigenvalues of adjacency matrices are important indicators on the influences of topological structures to collective dynamical behavior of complex networks. Recent findings on the ensemble averageability of the extreme eigenvalue further authenticate its sensibility in the study of network dynamics. Here we determine the ensemble average of the extreme eigenvalue and characterize the deviation across the ensemble through the discrete form of random scale-free netw...

The study of complex networks has been one of the most active fields in science in recent decades. Spectral properties of networks (or graphs that represent them) are of fundamental importance. Researchers have been investigating these properties for many years, and, based on numerical data, have raised a number of questions about the distribution of the eigenvalues and eigenvectors. In this paper, we give the solution to some of these questions. In particular, we determine t...