arXiv subjects

Elena Magnanini

Publications and source records attributed to Elena Magnanini.

12 recordsLinked to original sources

arXiv2026

Spatial Coagulation Systems with Mercer Kernels: Replicator Dynamics and Gelation

We study a spatial version of the Smoluchowski coagulation equation in which particles carry both an integer mass and a spatial location, and coagulate according to a mass-space product kernel. After coagulation, the resulting particle inherits the sum of the parent masses and a single spatial location. We provide a recursive formula for the solution and show that, while the first local mass moments are conserved up to gelation time at each location, the second local mass moments evolve according to a closed system of differential equations. When the spatial part of the kernel is of Mercer type, we further show that this system reduces to a replicator equation under a time rescaling and normalisation. We then use the fact that the replicator equation admits monotone functionals to quantify how spatial heterogeneity influences the rate of gelation and derive bounds on the gelation time. We also discuss the implications of these results for several important classes of Mercer kernels, including radial, diffusion, and translation-invariant kernels.

math.AP↗

arXiv2026

Mean-field limits for stochastic particle systems on dense graphs

We study stochastic interacting particle systems whose interaction structure is described by dense weighted directed graphs converging to a graphon. In the thermodynamic limit, we prove a law of large numbers for the empirical measure process and derive a deterministic nonlinear master equation describing the macroscopic evolution. The limiting equation retains the heterogeneous interaction structure of the microscopic system through the limiting graphon, allowing for spatially non-homogeneous behaviors such as localized or community-type interactions.

math.PR↗

arXiv2026

ERGMs on block models

We extend the classical edge-triangle Exponential Random Graph Model (ERGM) to an inhomogeneous setting in which vertices carry types determined by an underlying partition. This leads to a block-structured ERGM where interaction parameters depend on vertex types. We establish a large deviation principle for the associated sequence of measures and derive the corresponding variational formula for the limiting free energy. In the ferromagnetic regime, where the parameters governing triangle densities are nonnegative, we reduce the variational problem to a scalar optimization problem, thereby identifying the natural block counterpart of the replica symmetric regime. Under additional restrictions on the parameters, comparable to the classical Dobrushin's uniqueness region, we prove uniqueness of the maximizer and derive a law of large numbers for the edge density.

math.PR↗

arXiv2025

A standard CLT for triangles in a class of ERGs

We prove a standard Central Limit Theorem for the (normalized) number of triangles in a class of Exponential Random Graphs derived from a slight modification of the edge-triangle model. Our main theorem covers the whole analyticity region of the free energy, and is based on a polynomial representation of the partition function.

math.PR↗

arXiv2024

Convergence of cluster coagulation dynamics

We study hydrodynamic limits of the cluster coagulation model; a coagulation model introduced by Norris [$\textit{Comm. Math. Phys.}$, 209(2):407-435 (2000)]. In this process, pairs of particles $x,y$ in a measure space $E$, merge to form a single new particle $z$ according to a transition kernel $K(x, y, \mathrm{d} z)$, in such a manner that a quantity, one may regard as the total mass of the system, is conserved. This model is general enough to incorporate various inhomogeneities in the evolution of clusters, for example, their shape, or their location in space. We derive sufficient criteria for trajectories associated with this process to concentrate among solutions of a generalisation of the $\textit{Flory equation}$, and, in some special cases, by means of a uniqueness result for solutions of this equation, prove a weak law of large numbers. This multi-type Flory equation is associated with $\textit{conserved quantities}$ associated with the process, which may encode different information to conservation of mass (for example, conservation of centre of mass in spatial models). We also apply criteria for gelation in this process to derive sufficient criteria for this equation to exhibit gelling solutions. When this occurs, this multi-type Flory equation encodes, via the associated conserved property, the interaction between the gel and the finite size sol particles.

math.PR↗

arXiv2024

The Erd\H{o}s-R\'enyi Random Graph Conditioned on Every Component Being a Clique

Motivated by an application in community detection, we consider an \ER random graph conditioned on the rare event that all connected components are fully connected. Such graphs can be considered as partitions of vertices into cliques. Hence, this conditional distribution defines a distribution over partitions. We show that a popular community detection method is equivalent to Bayesian inference with this distribution as prior over the community partitions. Using tools from analytic combinatorics, we prove limit theorems for several graph observables in this conditional distribution: the number of cliques; the number of edges; and the degree distribution. We consider several regimes of the connection probability $p$ as the number of vertices $n$ diverges. For $p=\tfrac{1}{2}$, the conditioning yields the uniform distribution over set partitions, which is well-studied, but has not been studied as a graph distribution before. For $p<\tfrac{1}{2}$, we show that the number of cliques is of the order $n/\sqrt{\log n}$, while for $p>\tfrac{1}{2}$, we prove that the graph consists of a single clique with high probability. This shows that there is a phase transition at $p=\tfrac{1}{2}$. We additionally study the near-critical regime $p_n\downarrow\tfrac{1}{2}$, as well as the sparse regime $p_n\downarrow0$. Finally, we discuss the implications of these results for community detection.

math.PR↗

arXiv2024

Statistics for the triangle density in ERGM and its mean-field approximation

We consider the edge-triangle model (or Strauss model), and focus on the asymptotic behavior of the triangle density when the size of the graph increases to infinity. This random graph belongs to the class of exponential random graphs, which follows the statistical mechanics approach of introducing a Hamiltonian to weigh the probability measure on the state space of graphs. In the analyticity region of the free energy, we prove a law of large numbers for the triangle density. Along the critical curve, where analyticity breaks down, we show that the triangle density concentrates with high probability in a neighborhood of its typical value. A predominant part of our work is devoted to the study of a mean-field approximation of the edge-triangle model, where explicit computations are possible. In this setting we can go further, and additionally prove a standard and non-standard central limit theorem at the critical point, together with many concentration results obtained via large deviations and statistical mechanics techniques. Despite a rigorous comparison between these two models is still lacking, we believe that they are asymptotically equivalent in many respects, therefore we formulate conjectures on the edge-triangle model, partially supported by simulations, based on the mean-field investigation.

math.PR↗

arXiv2023

Gelation in cluster coagulation processes

We consider the problem of gelation in the cluster coagulation model introduced by Norris [\textit{Comm. Math. Phys.}, 209(2):407-435 (2000)], where pairs of clusters of types $(x,y)$ taking values in a measure space $E$, merge to form a new particle of type $z\in E$ according to a transition kernel $K(x,y, \mathrm{d} z)$. This model possesses enough generality to accommodate inhomogeneities in the evolution of clusters, including variations in their shape or spatial distribution. We derive general, sufficient criteria for stochastic gelation in this model. As particular cases, we extend results related to the classical Marcus--Lushnikov coagulation process, showing that reasonable `homogenous' coagulation processes with exponent $\gamma>1$ yield gelation; and also, coagulation processes with kernel $\bar{K}(m,n)~\geq~(m \wedge n) \log{(m \wedge n)}^{3 +\epsilon}$ for $\epsilon>0$.

math.PR↗

arXiv2023

Limit theorems for exponential random graphs

We consider the edge-triangle model, a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. In the so-called replica symmetric regime, the limiting free energy exists together with a complete characterization of the phase diagram of the model. We borrow tools from statistical mechanics to obtain limit theorems for the edge density. First, we investigate the asymptotic distribution of this quantity, as the graph size tends to infinity, in the various phases, and we complement this analysis with a study of the speed of convergence of the average edge density toward its limiting value. Then, we study the fluctuations of the edge density around its average value off the critical curve and formulate conjectures about the behavior at criticality based on the analysis of a mean-field approximation of the model. Some of our results can be extended with no substantial changes to more general classes of exponential random graphs.

math.PR↗

arXiv2021

The GHS and other correlation inequalities for the two-star model

We consider the two-star model, a family of exponential random graphs indexed by two real parameters, $h$ and $α$, that rule respectively the total number of edges and the mutual dependence between them. Borrowing tools from statistical mechanics, we study different classes of correlation inequalities for edges, that naturally emerge while taking the partial derivatives of the (finite size) free energy. In particular, if $α, h\ge 0$, we derive first and second order correlation inequalities and then prove the so-called GHS inequality. As a consequence, under the above conditions on the parameters, the average edge density turns out to be an increasing and concave function of the parameter $h$, at any fixed size of the graph. Some of our results can be extended to more general classes of exponential random graphs.

math.PR↗

arXiv2021

Limit theorems for Lévy flights on a 1D Lévy random medium

We study a random walk on a point process given by an ordered array of points $(ω_k, \, k \in \mathbb{Z})$ on the real line. The distances $ω_{k+1} - ω_k$ are i.i.d. random variables in the domain of attraction of a $β$-stable law, with $β\in (0,1) \cup (1,2)$. The random walk has i.i.d. jumps such that the transition probabilities between $ω_k$ and $ω_\ell$ depend on $\ell-k$ and are given by the distribution of a $\mathbb{Z}$-valued random variable in the domain of attraction of an $α$-stable law, with $α\in (0,1) \cup (1,2)$. Since the defining variables, for both the random walk and the point process, are heavy-tailed, we speak of a Lévy flight on a Lévy random medium. For all combinations of the parameters $α$ and $β$, we prove the annealed functional limit theorem for the suitably rescaled process, relative to the optimal Skorokhod topology in each case. When the limit process is not càdlàg, we prove convergence of the finite-dimensional distributions. When the limit process is deterministic, we also prove a limit theorem for the fluctuations, again relative to the optimal Skorokhod topology.

math.PR↗

arXiv2021

Approximating the cumulant generating function of triangles in the Erdös-Rényi random graph

We study the pressure of the "edge-triangle model", which is equivalent to the cumulant generating function of triangles in the Erdös-Rényi random graph. By analyzing finite graphs of increasing volume, as well as the graphon variational problem in the infinite volume limit, we locate a curve in the parameter space where a one-step replica symmetry breaking transition occurs. Sampling a large graph in the broken symmetry phase is well described by a graphon with a structure very close to the one of an equi-bipartite graph.

cond-mat.stat-mech↗