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We present a novel approach to study the evolution of the size (i.e. the number of vertices) of the giant component of a random graph process. It is based on the exploration algorithm called simultaneous breadth-first walk, introduced by…
We consider random walks in the form of nearest-neighbor hopping on Erdos-Renyi random graphs of finite fixed mean degree c as the number of vertices N tends to infinity. In this regime, using statistical field theory methods, we develop an…
We introduce the notion of a "random basic walk" on an infinite graph, give numerous examples, list potential applications, and provide detailed comparisons between the random basic walk and existing generalizations of simple random walks.…
This paper introduces a class of objects called decision rules that map infinite sequences of alternatives to a decision space. These objects can be used to model situations where a decision maker encounters alternatives in a sequence such…
This paper presents the Persistent Weisfeiler-Lehman Random walk scheme (abbreviated as PWLR) for graph representations, a novel mathematical framework which produces a collection of explainable low-dimensional representations of graphs…
The extremal process of a branching random walk is the point measure recording the position of particles alive at time $n$, shifted around the expected position of the minimal position. Madaule proved that this point measure converges, as…
Typical performance of approximation algorithms is studied for randomized minimum vertex cover problems. A wide class of random graph ensembles characterized by an arbitrary degree distribution is discussed with some theoretical frameworks.…
Recently, it has been claimed that some complex networks are self-similar under a convenient renormalization procedure. We present a general method to study renormalization flows in graphs. We find that the behavior of some variables under…
We develop a general theory of random walks on hypergraphs which includes, as special cases, the different models that are found in literature. In particular, we introduce and analyze general random walk Laplacians for hypergraphs, and we…
We characterise the form of all simple, finite graphs for which the girth of the graph is equal to the circumference of the graph. We apply this to prove a bound on the number of edges in such a graph.
The transition law of every exchangeable Feller process on the space of countable graphs is determined by a $\sigma$-finite measure on the space of $\{0,1\}\times\{0,1\}$-valued arrays. In discrete-time, this characterization amounts to a…
Paper fragments in free fall constitute a simple yet paradigmatic mechanical system exhibiting remarkably complex motions. Despite a long history of investigation, this system has defied comprehensive first-principles modeling, motivating…
We tackle the problem of attributed graph transformations and propose a new algorithmic approach for defining parallel graph transformations allowing overlaps. We start by introducing some abstract operations over graph structures. Then, we…
We couple projective limits of probability measures to direct limits of their symmetry groups. We show that the direct limit group is the group of symmetries of the projective limit probability measure. If projective systems of probability…
We define a new type of self-similarity for one-parameter families of stochastic processes, which applies to a number of important families of processes that are not self-similar in the conventional sense. This includes a new class of…
We develop a formalism to address statistical pattern recognition of graph valued data. Of particular interest is the case of all graphs having the same number of uniquely labeled vertices. When the vertex labels are latent, such graphs are…
We study the Parallel Replica Dynamics in a general setting. We introduce a trajectory fragment framework that can be used to design and prove consistency of Parallel Replica algorithms for generic Markov processes. We use our framework to…
Yin, Rinaldo, and Fadnavis classified the extremal behavior of the edge-triangle exponential random graph model by first taking the network size to infinity, then the parameters diverging to infinity along straight lines. Lubetzky and Zhao…
We consider asynchronous message-passing systems in which some links are timely and processes may crash. Each run defines a timeliness graph among correct processes: (p; q) is an edge of the timeliness graph if the link from p to q is…
In this work, we study the properties of sampling sets on families of large graphs by leveraging the theory of graphons and graph limits. To this end, we extend to graphon signals the notion of removable and uniqueness sets, which was…