arXiv subjects

Ankan Ganguly

Publications and source records attributed to Ankan Ganguly.

6 recordsLinked to original sources

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

Multiplexons: Limits of Multiplex Networks

In a multiplex network, a set of nodes is connected by different types of interactions, each represented as a separate layer within the network. Multiplexes have emerged as a key instrument for modeling large-scale complex systems, due to the widespread coexistence of diverse interactions in social, industrial, and biological domains. This motivates the development of a rigorous and readily applicable framework for studying properties of large multiplex networks. In this article, we provide a self-contained introduction to the limit theory of dense multiplex networks, analogous to the theory of graphons (limit theory of dense graphs). As applications, we derive limiting analogues of commonly used multiplex features, such as degree distributions and clustering coefficients. We also present a range of illustrative examples, including correlated versions of Erd\H{o}s-R\'enyi and inhomogeneous random graph models and dynamic networks. Finally, we discuss how multiplex networks fit within the broader framework of decorated graphs, and how the convergence results can be recovered from the limit theory of decorated graphs. Several future directions are outlined for further developing the multiplex limit theory.

math.PR↗

arXiv2025

Mean-Field Analysis of Latent Variable Process Models on Dynamically Evolving Graphs with Feedback Effects

We study the mean-field limit of a generic class of dynamic co-evolving latent space networks motivated by the social and opinion dynamics literature. Such models include $n$ agents, whose opinions are given by latent stochastic processes, and a dynamic network process describing agent interactions. Models in this class incorporate (a) bi-directional feedback between the latent processes and the network process, (b) persistence effects, meaning that the network structure at the current time depends on the value of the latent processes at the current time but also on the network structure at the previous time instance and (c) localized interactions, meaning that individual agents do not have global information. We characterize the distributional limit of a random sample taken from the latent space network as the number of nodes in the network diverges. We describe the rich conditional probabilistic structure of the resulting limiting model which we use to establish the limiting behavior of the following quantities: (i) the empirical measure of the latent process, (ii) a conditional empirical measure relating the latent process to the network process and (iii) the network process graphon. In proving our main results, we derive a general conditional propagation of chaos result, which is of independent interest. Our novel approach to studying the limiting behavior of random samples proves to be a very useful methodology for fully grasping the asymptotic behavior of co-evolving particle systems. Numerical results are included to illustrate the theoretical findings.

math.PR↗

arXiv2024

Hydrodynamic Limits of non-Markovian Interacting Particle Systems on Sparse Graphs

Consider an interacting particle system indexed by the vertices of a (possibly random) locally finite graph whose vertices and edges are equipped with marks representing parameters of the model such as the environment and initial conditions. Each particle takes values in a countable state space and evolves according to a pure jump process whose jump intensities depend on only the states (or histories) and marks of itself and particles and edges in its neighborhood. Under mild conditions, it is shown that if the sequence of (marked) interaction graphs converges locally in probability to a limit (marked) graph that satisfies a certain finite dissociability property, then the corresponding sequence of empirical measures of the particle trajectories converges weakly to the law of the marginal dynamics at the root vertex of the limit graph. The proof of this limit relies on several results of independent interest. First, such interacting particle systems are shown to be well-posed on almost surely finitely dissociable graphs, which include graphs of maximal bounded degree and any Galton-Watson tree whose offspring distribution has a finite first moment. A counterexample is provided to show that well-posedness can fail for dynamics on graphs outside this class. Second, the dynamics on a locally convergent sequence of graphs are shown to converge in the local weak sense to the dynamics on the limit graph when the latter is finitely dissociable. Finally, the dynamics are also shown to exhibit an (annealed) asymptotic correlation decay property. These results complement recent work that establishes hydrodynamic limits of locally interacting probabilistic cellular automata and diffusions on sparse random graphs. However, the analysis of jump processes requires very different techniques, including percolation arguments and notions such as (consistent) spatial localization and causal chains.

math.PR↗

arXiv2023

Rotor-Routing Induces the Only Consistent Sandpile Torsor Structure on Plane Graphs

We make precise and prove a conjecture of Klivans about actions of the sandpile group on spanning trees. More specifically, the conjecture states that there exists a unique ``suitably nice'' sandpile torsor structure on plane graphs which is induced by rotor-routing. First, we rigorously define a sandpile torsor algorithm (on plane graphs) to be a map which associates each plane graph (i.e., planar graph with an appropriate ribbon structure) with a free transitive action of its sandpile group on its spanning trees. Then, we define a notion of consistency, which requires a torsor algorithm to be preserved with respect to a certain class of contractions and deletions. Using these definitions, we show that the rotor-routing sandpile torsor algorithm is consistent. Furthermore, we demonstrate that there are only three other consistent algorithms on plane graphs, which all have the same structure as rotor-routing. We also define sandpile torsor algorithms on regular matroids and suggest a notion of consistency in this context. We conjecture that the Backman-Baker-Yuen algorithm is consistent, and that there are only three other consistent sandpile torsor algorithms on regular matroids, all with the same structure.

math.CO↗

arXiv2022

Interacting Jump Processes Preserve Semi-Global Markov Random Fields on Path Space

Consider a system of interacting particles indexed by the nodes of a graph whose vertices are equipped with marks representing parameters of the model such as the environment or initial data. Each particle takes values in a countable state space and evolves according to a (possibly non-Markovian) continuous-time pure jump process whose jump intensities depend only on its own state (or history) and marks as well as the states (or histories) and marks of particles and edges in its neighborhood in the graph. Under mild conditions on the jump intensities, it is shown that the trajectories of the interacting particle system exhibit a certain local or semi-global Markov random field property whenever the initial condition satisfies the same property. Our results complement recent works that establish the preservation of a local second-order Markov random field property for interacting diffusions. Our proof methodology in the context of jump processes is different, and works directly on infinite graphs, thereby bypassing any limiting arguments. Our results apply to models arising in diverse fields including statistical physics, neuroscience, epidemiology and opinion dynamics, and have direct applications to the study of marginal distributions of interacting particle systems on Cayley trees.

math.PR↗