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This Letter introduces a generalization of known duplication-divergence models for growing random graphs. This general duplication-divergence model includes a new coupled divergence asymmetry rate, which allows to obtain the structure of…

We study the statistical properties of the generation of random graphs according the configuration model, where one assigns randomly degrees to nodes. This model is often used, e.g., for the scale-free degree distribution ~d^gamma. For the…

Subgraphs such as cliques, loops and stars form crucial connections in the topologies of real-world networks. Random graph models provide estimates for how often certain subgraphs appear, which in turn can be tested against real-world…

In this paper we study the persistent homology associated with topological crackle generated by distributions with an unbounded support. Persistent homology is a topological and algebraic structure that tracks the creation and destruction…

A thorough discussion of the statistical ensemble of scale-free connected random tree graphs is presented. Methods borrowed from field theory are used to define the ensemble and to study analytically its properties. The ensemble is…

We study the growth of random networks under a constraint that the diameter, defined as the average shortest path length between all nodes, remains approximately constant. We show that if the graph maintains the form of its degree…

In 2007 we introduced a general model of sparse random graphs with independence between the edges. The aim of this paper is to present an extension of this model in which the edges are far from independent, and to prove several results…

This work develops a flexible and mathematically sound framework for the design and analysis of graph scattering networks with variable branching ratios and generic functional calculus filters. Spectrally-agnostic stability guarantees for…

We introduce a new random graph model motivated by biological questions relating to speciation. This random graph is defined as the stationary distribution of a Markov chain on the space of graphs on $\{1, \ldots, n\}$. The dynamics of this…

We introduce generalizations of Aldous' Brownian Continuous Random Tree as scaling limits for multicritical models of discrete trees. These discrete models involve trees with fine-tuned vertex-dependent weights ensuring a k-th root…

We find that the fractal scaling in a class of scale-free networks originates from the underlying tree structure called skeleton, a special type of spanning tree based on the edge betweenness centrality. The fractal skeleton has the…

Scaling behavior of scale-free evolving networks arising in communications, citations, collaborations, etc. areas is studied. We derive universal scaling relations describing properties of such networks and indicate limits of their…

Given any regularly varying dislocation measure, we identify a natural self-similar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of…

This paper delves into the stability of the $2$-domination number in simple undirected graphs. The $2$-domination number of a graph $G$, $\gamma_2(G)$, represents the minimum size of a vertex subset where every other vertex in the graph is…

A well-known open problem on the behavior of optimal paths in random graphs in the strong disorder regime, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade [31,32,38,70] is as…

We consider a variant of so called power-law random graph. A sequence of expected degrees corresponds to a power-law degree distribution with finite mean and infinite variance. In previous works the asymptotic picture with number of nodes…

We consider random partitions of the vertex set of a given finite graph that can be sampled by means of loop-erased random walks stopped at a random exponential time of parameter $q>0$. The related random blocks tend to cluster nodes…

We show that the stationary distribution of a finite Markov chain can be expressed as the sum of certain normal distributions. These normal distributions are associated to planar graphs consisting of a straight line with attached loops. The…

In spatial networks vertices are arranged in some space and edges may cross. When arranging vertices in a 1-dimensional lattice edges may cross when drawn above the vertex sequence as it happens in linguistic and biological networks. Here…

Systems as diverse as genetic networks or the world wide web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This…