arXiv subjects

Pierfrancesco Dionigi

Publications and source records attributed to Pierfrancesco Dionigi.

7 recordsLinked to original sources

arXiv2026

Preferential Attachment with Local Flexibility

From the formation of social ties to the budding quantum internet, growing networks often exhibit local flexibility upon new nodes attaching to an existing network. In our proposed model, a new node connects uniformly at random to a node within the proximity of the intended target, including, but not restricted to, the target itself. Through numerical simulations and rigorous stochastic analysis, we find this local flexibility to qualitatively change the global network behavior of nonlinear preferential attachment. Depending on whether the preferential attachment is superlinear or (sub)linear, two distinct classes of complex network architectures emerge. The superlinear phase leads to a layered hierarchy, with no stationary degree distribution. Although there is a stationary degree distribution in the linear and sublinear cases, it decays strictly faster than for the Barabási--Albert model. We interpret our results within a two-dimensional phase diagram of network growth models incorporating redirection, with broad implications.

quant-ph↗

arXiv2026

Colorful Exponential Random Graph Models

In this paper, we initiate the study of colored exponential random graph models (ERGMs), a class of exponential-family models for networks with multiple types of edge relations. Using the framework of probability graphons, we first derive a variational representation for the limiting free energy, whose maximizers determine the asymptotic structure of typical samples from the model. Then we identify several general families of colored ERGMs exhibiting replica symmetry, where the variational problem has constant maximizers and the model asymptotically concentrates on product colorings with independent edges. For general colored ERGMs, we derive Euler-Lagrange fixed-point equations for the variational maximizers, which in turn yield a general high-temperature uniqueness criterion. In the complementary zero-temperature regime, we establish a two-level selection principle: the leading energy term determines the ground states, while the lower-order energy terms, combined with entropy, act as a tie-breaker to determine the asymptotic zero-temperature structure of the model. We illustrate this principle through the induced wedge and rainbow triangle ERGMs. Both models have natural interpretations in multitype networks, and their zero-temperature limits exhibit interesting structures that connect to well-known results in extremal combinatorics. We further establish finite-temperature symmetry breaking for both these models and complement the rigorous results with numerical experiments.

math.PR↗

arXiv2025

Large deviations for probability graphons

We establish a large deviation principle (LDP) for probability graphons, which are symmetric functions from the unit square into the space of probability measures. This notion extends classical graphons and provides a flexible framework for studying the limit behavior of large dense weighted graphs. In particular, our result generalizes the seminal work of Chatterjee and Varadhan (2011), who derived an LDP for Erdős-Rényi random graphs via graphon theory. We move beyond their binary (Bernoulli) setting to encompass arbitrary edge-weight distributions. Specifically, we analyze the distribution on probability graphons induced by random weighted graphs in which edges are sampled independently from a common reference probability measure supported on a compact Polish space. We prove that this distribution satisfies an LDP with a good rate function, expressed as an extension of the Kullback-Leibler divergence between probability graphons and the reference measure. This theorem can also be viewed as a Sanov-type result in the graphon setting. Our work provides a rigorous foundation for analyzing rare events in weighted networks and supports statistical inference in structured random graph models under distributional edge uncertainty.

math.PR↗

arXiv2025

The effects of initial conditions on the accuracy of mean-field approximations of Markov processes on large random graphs

We study the evolution of a general class of stochastic processes (containing, e.g. SIS and SIR models) on large random networks, focusing on a particular general class of random graph models. To approximate the expected dynamics of these processes, we employ a mean-field technique known as the N-Intertwined Mean-Field Approximation (NIMFA). Our primary goal is to quantify the impact of the randomness in the network topology on the accuracy of such approximations. While classical work by Kurtz established strong approximation results for density-dependent Markov chains using mean-field on the complete-graph of order $N$, yielding error bounds of order $\frac{1}{\sqrt{N}}$, we demonstrate that in the context of NIMFA on random graphs, the error admits a refined characterization depending on initial conditions. Specifically, for generic initial conditions, the worst case error is of order $\frac{1}{\sqrt{d}}$, where $d$ is the expected average degree of the graph. This result highlights how network sparsity contributes an additional source of variability not captured by classical mean-field approaches. Furthermore, when the initial conditions are taken to be fairly homogeneous, we show that the error term is of order $\frac{1}{\sqrt{N}}+\frac{1}{d}$, where $N$ is the number of nodes, underscoring the intricate role of the choice of initial conditions in the error bounds. Our analysis connects with recent interest in quantifying the role of graph heterogeneity in epidemic thresholds and dynamics.

math.PR↗

arXiv2023

Largest Eigenvalue of the Configuration Model and Breaking of Ensemble Equivalence

We analyse the largest eigenvalue of the adjacency matrix of the configuration model with large degrees, where the latter are treated as hard constraints. In particular, we compute the expectation of the largest eigenvalue for degrees that diverge as the number of vertices $n$ tends to infinity, uniformly on a scale between $1$ and $\sqrt{n}$, and show that a weak law of large numbers holds. We compare with what was derived in our earlier paper "Central limit theorem for the principal eigenvalue and eigenvector of Chung-Lu random graphs" for the Chung-Lu model, which in the regime considered represents the corresponding configuration model with soft constraints, and show that the expectation is shifted down by $1$ asymptotically. This shift is a signature of breaking of ensemble equivalence between the hard and soft (also known as micro-canonical and canonical) versions of the configuration model. The latter result generalizes a previous finding in "A spectral signature of breaking of ensemble equivalence for constrained random graphs" obtained in the case when all degrees are equal.

math.PR↗

arXiv2022

Central limit theorem for the principal eigenvalue and eigenvector of Chung-Lu random graphs

A Chung-Lu random graph is an inhomogeneous Erdős-Rényi random graph in which vertices are assigned average degrees, and pairs of vertices are connected by an edge with a probability that is proportional to the product of their average degrees, independently for different edges. We derive a central limit theorem for the principal eigenvalue and the components of the principal eigenvector of the adjacency matrix of a Chung-Lu random graph. Our derivation requires certain assumptions on the average degrees that guarantee connectivity, sparsity and bounded inhomogeneity of the graph.

math.PR↗

arXiv2021

A spectral signature of breaking of ensemble equivalence for constrained random graphs

For random systems subject to a constraint, the microcanonical ensemble requires the constraint to be met by every realisation ("hard constraint"), while the canonical ensemble requires the constraint to be met only on average ("soft constraint"). It is known that for random graphs subject to topological constraints breaking of ensemble equivalence may occur when the size of the graph tends to infinity, signalled by a non-vanishing specific relative entropy of the two ensembles. We investigate to what extent breaking of ensemble equivalence is manifested through the largest eigenvalue of the adjacency matrix of the graph. We consider two examples of constraints in the dense regime: (1) fix the degrees of the vertices (= the degree sequence); (2) fix the sum of the degrees of the vertices (= twice the number of edges). Example (1) imposes an extensive number of local constraints and is known to lead to breaking of ensemble equivalence. Example (2) imposes a single global constraint and is known to lead to ensemble equivalence. Our working hypothesis is that breaking of ensemble equivalence corresponds to a non-vanishing difference of the expected values of the largest eigenvalue under the two ensembles. We verify that, in the limit as the size of the graph tends to infinity, the difference between the expected values of the largest eigenvalue in the two ensembles does not vanish for (1) and vanishes for (2). A key tool in our analysis is a transfer method that uses relative entropy to determine whether probabilistic estimates can be carried over from the canonical ensemble to the microcanonical ensemble, and illustrates how breaking of ensemble equivalence may prevent this from being possible.

math.PR↗