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We consider three distinct and well studied problems concerning network structure: community detection by modularity maximization, community detection by statistical inference, and normalized-cut graph partitioning. Each of these problems…
We introduce an unsupervised graph embedding that trades off local node similarity and connectivity, and global structure. The embedding is based on a generalized graph Laplacian, whose eigenvectors compactly capture both network structure…
The spectral radius of the adjacency matrix can impact both algorithmic efficiency as well as the stability of solutions to an underlying dynamical process. Although much research has considered the distribution of the spectral radius for…
This work analyzes the problem of community structure in real-world networks based on the synchronization of nonidentical coupled chaotic R\"{o}ssler oscillators each one characterized by a defined natural frequency, and coupled according…
Dynamical properties of complex networks are related to the spectral properties of the Laplacian matrix that describes the pattern of connectivity of the network. In particular we compute the synchronization time for different types of…
We study the many-body localization problem in the non-abelian SU(2)-invariant random antiferromagnetic exchange model in 1D. Exact and sparse matrix diagonalization methods are used to calculate eigenvalues and eigenvectors of the…
Localization phenomena permeate many branches of physics playing a fundamental role on dynamical processes evolving on heterogeneous networks. These localization analyses are frequently grounded, for example, on eigenvectors of adjacency or…
We study, analytically and numerically, a simple $\mathcal{PT}$-symmetric tight-binding ring with an onsite energy $a$ at the gain and loss sites. We show that if $a\neq 0$, the system generically exhibits an unbroken…
Communities are a common and widely studied structure in networks, typically under the assumption that the network is fully and correctly observed. In practice, network data are often collected by querying nodes about their connections. In…
We study complex networks under random matrix theory (RMT) framework. Using nearest-neighbor and next-nearest-neighbor spacing distributions we analyze the eigenvalues of adjacency matrix of various model networks, namely, random,…
Results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems (real, random matrices), time-delay dynamical systems (companion matrices), and a set of mappings known for their properties…
In an attempt to characterize the structure of eigenvectors of random regular graphs, we investigate the correlations between the components of the eigenvectors associated to different vertices. In addition, we provide numerical…
Community structure in networks is observed in many different domains, and unsupervised community detection has received a lot of attention in the literature. Increasingly the focus of network analysis is shifting towards using network…
We study the properties of random graphs where for each vertex a {\it neighbourhood} has been previously defined. The probability of an edge joining two vertices depends on whether the vertices are neighbours or not, as happens in Small…
We analyse a collection of empirical networks in a wide spectrum of disciplines and show that strong non-normality is ubiquitous in network science. Dynamical processes evolving on non-normal networks exhibit a peculiar behaviour, as…
Social networks are organized into communities with dense internal connections, giving rise to high values of the clustering coefficient. In addition, these networks have been observed to be assortative, i.e. highly connected vertices tend…
The Laplacian eigenvalues of a network play an important role in the analysis of many structural and dynamical network problems. In this paper, we study the relationship between the eigenvalue spectrum of the normalized Laplacian matrix and…
We derive exact equations for the spectral density of sparse networks with an arbitrary distribution of the number of single edges and triangles per node. These equations enable a systematic investigation of the effect of clustering on the…
We present and discuss a mathematical procedure for identification of small "communities" or segments within large bipartite networks. The procedure is based on spectral analysis of the matrix encoding network structure. The principal tool…
A widely recognized organizing principle of networks is structural homophily, which suggests that people with more common neighbors are more likely to connect with each other. However, what influence the diverse structures embedded in…