Related papers: Localization of nonbacktracking centrality on dens…
The spectrum of the non-backtracking matrix plays a crucial role in determining various structural and dynamical properties of networked systems, ranging from the threshold in bond percolation and non-recurrent epidemic processes, to…
The spectrum of the nonbacktracking matrix associated to a network is known to contain fundamental information regarding percolation properties of the network. Indeed, the inverse of its leading eigenvalue is often used as an estimate for…
Message-passing theories have proved to be invaluable tools in studying percolation, non-recurrent epidemics and similar dynamical processes on real-world networks. At the heart of the message-passing method is the nonbacktracking matrix…
Eigenvector centrality is a common measure of the importance of nodes in a network. Here we show that under common conditions the eigenvector centrality displays a localization transition that causes most of the weight of the centrality to…
The non-backtracking matrix and its eigenvalues have many applications in network science and graph mining, such as node and edge centrality, community detection, length spectrum theory, graph distance, and epidemic and percolation…
The spectral properties of the adjacency matrix provide a trove of information about the structure and function of complex networks. In particular, the largest eigenvalue and its associated principal eigenvector are crucial in the…
The spectral properties of the adjacency matrix, in particular its largest eigenvalue and the associated principal eigenvector, dominate many structural and dynamical properties of complex networks. Here we focus on the localization…
Message passing equations yield a sharp percolation transition in finite graphs, as an artifact of the locally treelike approximation. For an arbitrary finite, connected, undirected graph we construct an infinite tree having the same local…
We explore the spectra and localization properties of the N-site banded one-dimensional non-Hermitian random matrices that arise naturally in sparse neural networks. Approximately equal numbers of random excitatory and inhibitory…
In case of sparse graphs, relation between the real eigenvalues of the non-backtracking matrix and those of the non-backtracking transition probability matrix is considered with respect to vertex clustering. For this purpose, the random…
The eigendecomposition of the coupling matrix of large biological networks is central to the study of the dynamics of these networks. For neural networks, this matrix should reflect the topology of the network and conform with Dale's law…
Percolation threshold of a network is the critical value such that when nodes or edges are randomly selected with probability below the value, the network is fragmented but when the probability is above the value, a giant component…
We study structure, eigenvalue spectra and diffusion dynamics in a wide class of networks with subgraphs (modules) at mesoscopic scale. The networks are grown within the model with three parameters controlling the number of modules, their…
We study the stochastic susceptible-infected-susceptible model of epidemic processes on finite directed and weighted networks with arbitrary structure. We present a new lower bound on the exponential rate at which the probabilities of nodes…
The non-backtracking operator of a graph is a powerful tool in spectral graph theory and random matrix theory. Most existing results for the non-backtracking operator of a random graph concern only eigenvalues or top eigenvectors. In this…
Random walks are a fundamental tool for analyzing realistic complex networked systems and implementing randomized algorithms to solve diverse problems such as searching and sampling. For many real applications, their actual effect and…
We present a unified theoretical framework connecting the first property of Deep Neural Collapse (DNC1) to the emergence of implicit low-rank bias in nonlinear networks trained with $L^2$ weight decay regularization. Our main contributions…
We examine the heterogeneous responses of individual nodes in sparse networks to the random removal of a fraction of edges. Using the message-passing formulation of percolation, we discover considerable variation across the network in the…
In this survey paper it is illustrated how spectral clustering methods for unweighted graphs are adapted to the dense and sparse regimes. Whereas Laplacian and modularity based spectral clustering is apt to dense graphs, recent results show…
Many models for undirected graphs are based on factorizing the graph's adjacency matrix; these models find a vector representation of each node such that the predicted probability of a link between two nodes increases with the similarity…