Related papers: Eigenvector localization in real networks and its …
It is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing…
As in random matrix theories, eigenvector/value distributions are important quantities of random tensors in their applications. Recently, real eigenvector/value distributions of Gaussian random tensors have been explicitly computed by…
How does coarsening affect the spectrum of a general graph? We provide conditions such that the principal eigenvalues and eigenspaces of a coarsened and original graph Laplacian matrices are close. The achieved approximation is shown to…
In this paper new results on personalized PageRank are shown. We consider directed graphs that may contain dangling nodes. The main result presented gives an analytical characterization of all the possible values of the personalized…
The computational demands of community detection algorithms such as Louvain and spectral optimization can be prohibitive for large networks. Eigenvector centrality and Katz centrality are two network statistics commonly used to describe the…
It is a longstanding debate on the absence of threshold for susceptible-infected-susceptible (SIS) model on networks with finite second order moment of degree distribution. The eigenvector localization of the adjacency matrix for a network…
We analyse the eigenvectors of the adjacency matrix of a critical Erd\H{o}s-R\'enyi graph $\mathbb G(N,d/N)$, where $d$ is of order $\log N$. We show that its spectrum splits into two phases: a delocalized phase in the middle of the…
Consider a random regular graph of fixed degree $d$ with $n$ vertices. We study spectral properties of the adjacency matrix and of random Schr\"odinger operators on such a graph as $n$ tends to infinity. We prove that the integrated density…
Centrality is a key property of complex networks that influences the behavior of dynamical processes, like synchronization and epidemic spreading, and can bring important information about the organization of complex systems, like our brain…
The largest eigenvalue of the matrix describing a network's contact structure is often important in predicting the behavior of dynamical processes. We extend this notion to hypergraphs and motivate the importance of an analogous eigenvalue,…
This review presents an account of the major works done on spectra of adjacency matrices drawn on networks and the basic understanding attained so far. We have divided the review under three sections: (a) extremal eigenvalues, (b) bulk part…
Eigenvectors of large matrices (and graphs) play an essential role in combinatorics and theoretical computer science. The goal of this survey is to provide an up-to-date account on properties of eigenvectors when the matrix (or graph) is…
The distribution of the ratios of consecutive eigenvalue spacings of random matrices has emerged as an important tool to study spectral properties of many-body systems. This article numerically investigates the eigenvalue ratios…
Recently developed techniques to acquire high-quality human mobility data allow large-scale simulations of the spread of infectious diseases with high spatial and temporal resolution.Analysis of such data has revealed the oversimplification…
Eigenvector-based centrality measures are among the most popular centrality measures in network science. The underlying idea is intuitive and the mathematical description is extremely simple in the framework of standard, mono-layer…
Many natural and social systems develop complex networks, that are usually modelled as random graphs. The eigenvalue spectrum of these graphs provides information about their structural properties. While the semi-circle law is known to…
We investigate delocalization phenomena for eigenvectors of real random matrices that are invariant by orthogonal transformations. A specific phenomenon with these ensembles is that an eigenvector is typically more localized when its…
The largest eigenvalue of the adjacency matrix of a network (referred to as the spectral radius) is an important metric in its own right. Further, for several models of epidemic spread on networks (e.g., the `flu-like' SIS model), it has…
Most real-world networks are not isolated. In order to function fully, they are interconnected with other networks, and this interconnection influences their dynamic processes. For example, when the spread of a disease involves two species,…
We consider random matrices of the form $H = W + \lambda V$, $\lambda\in\mathbb{R}^+$, where $W$ is a real symmetric or complex Hermitian Wigner matrix of size $N$ and $V$ is a real bounded diagonal random matrix of size $N$ with i.i.d.\…