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We present a new family of graphs with remarkable properties. They are obtained by connecting the points of a random walk when their distance is smaller than a given scale. Their degree (number of neighbors) does not depend on the graph's…
Diffusion models have established themselves as state-of-the-art generative models across various data modalities, including images and videos, due to their ability to accurately approximate complex data distributions. Unlike traditional…
A wide variety of complex networks (social, biological, information etc.) exhibit local clustering with substantial variation in the clustering coefficient (the probability of neighbors being connected). Existing models of large graphs…
Based on methods of structural convergence we provide a unifying view of local-global convergence, fitting to model theory and analysis. The general approach outlined here provides a possibility to extend the theory of local-global…
We study the random graph obtained by random deletion of vertices or edges from a random graph with given vertex degrees. A simple trick of exploding vertices instead of deleting them, enables us to derive results from known results for…
Random graph models have played a dominant role in the theoretical study of networked systems. The Poisson random graph of Erdos and Renyi, in particular, as well as the so-called configuration model, have served as the starting point for…
We provide a well-posedness theory for a class of nonlocal continuity equations on co-evolving graphs. We describe the connection among vertices through an edge weight function and we let it evolve in time, coupling its dynamics with the…
Given a graph $G$, we form a random subgraph $G_p$ by including each edge of $G$ independently with probability $p$. We provide an asymptotic expansion of the expected number of independent sets in random subgraphs of regular bipartite…
Mean-field frozen percolation is a random graph-valued process, which adjusts the dynamics of the classical Erdos-Renyi process with an additional mechanism to 'freeze' potential giant components before they can form. It is known to exhibit…
Time-evolving graphs arise frequently when modeling complex dynamical systems such as social networks, traffic flow, and biological processes. Developing techniques to identify and analyze communities in these time-varying graph structures…
Two-scale models pose a promising approach in simulating reactive flow and transport in evolving porous media. Classically, homogenized flow and transport equations are solved on the macroscopic scale, while effective parameters are…
We study the evolution of convex complete non-compact graphs by positive powers of Gauss curvature. We show that if the initial complete graph has a local uniform convexity, then the graph evolves by any positive power of Gauss curvature…
In the last decades, many authors have used the susceptible-infected-recovered model to study the impact of the disease spreading on the evolution of the infected individuals. However, few authors focused on the temporal unfolding of the…
Random growth models are fundamental objects in modern probability theory, have given rise to new mathematics, and have numerous applications, including tumor growth and fluid flow in porous media. In this article, we introduce some of the…
Random intersection graphs containing an underlying community structure are a popular choice for modelling real-world networks. Given the group memberships, the classical random intersection graph is obtained by connecting individuals when…
We analyze site percolation on directed and undirected graphs with site-dependent open-site probabilities. We construct upper bounds on cluster susceptibilities, vertex connectivity functions, and the expected number of simple open cycles…
Cascade models based on dynamical complex networks are proposed as models of turbulent energy cascade. Taking a simple shell model as the initial regular lattice with only nearest neighbor interactions, small world network models are…
In this paper we examine the percolation properties of higher-order networks that have non-trivial clustering and subgraph-based assortative mixing (the tendency of vertices to connect to other vertices based on subgraph joint degree). Our…
Around 2008, Schramm conjectured that the critical probabilities for Bernoulli bond percolation satisfy the following continuity property: If $(G_n)_{n\geq 1}$ is a sequence of transitive graphs converging locally to a transitive graph $G$…
We study diffusion-type equations supported on structures that are randomly varying in time. After settling the issue of well-posedness, we focus on the asymptotic behavior of solutions: our main result gives sufficient conditions for…