arXiv subjects

Sarath Yasodharan

Publications and source records attributed to Sarath Yasodharan.

9 recordsLinked to original sources

arXiv2026

Gibbs conditioning, atypical consensus and splitting Gibbs measures on random regular graphs

Given n independent Bernoulli(p) random variables X_i, i = 1, ..., n, representing the opinions of individuals connected by an underlying random k-regular graph G_n on {1, ..., n}, we show that when conditioned on an atypical empirical consensus, which is the normalized sum of X_i X_j over neighboring vertices i, j, the joint distribution of the random variables converges, as n goes to infinity, to an Ising measure on the infinite k-regular tree T^k with a specific external field that depends only on the bias parameter p, and a temperature that depends on both p and the atypical consensus value. In particular, we show that conditional on the empirical consensus being smaller (respy, larger) than typical, the limit is a translation-invariant splitting (TIS) antiferromagnetic (respy, ferromagnetic) Ising measure on T^k. Moreover, if the bias is zero, then there is a phase transition: when the consensus exceeds k/(k-1), the conditional limits could be either the plus or minus boundary condition Ising measures. Furthermore, when X_i, i = 1, ..., n, are i.i.d. on a finite space, we show that when conditioned on an atypical value of the scaled sum of h(X_i, X_j) over neighboring vertices i and j, for any symmetric edge potential h, the limiting joint distribution of {X_i} lies in the set of (possibly degenerate) TIS Gibbs measures on T^k. The proofs leverage a tractable form of the large deviation rate function for component empirical measures of random regular graphs with i.i.d. marks and Gibbs conditioning principles, and entail careful analyses of associated non-convex constrained optimization problems. As a by-product of our results, we also obtain an (asymptotic) analog of the maximum entropy principle for Gibbs measures on random regular graphs.

math.PR↗

arXiv2023

On the large deviation rate function for marked sparse random graphs

We consider (annealed) large deviation principles for component empirical measures of several families of marked sparse random graphs, including (i) uniform graphs on $n$ vertices with a fixed degree distribution; (ii) uniform graphs on $n$ vertices with a fixed number of edges; (iii) Erdős-Rényi $G(n, c/n)$ random graphs. Assuming that edge and vertex marks are independent, identically distributed, and take values in a finite state space, we show that the large deviation rate function admits a concise representation as a sum of relative entropies that quantify the cost of deviation of a probability measure on marked rooted graphs from certain auxiliary independent and conditionally independent versions. The proof exploits unimodularity, the consequent mass transport principle, and random tree labelings to express certain combinatorial quantities as expectations with respect to size-biased distributions, and to identify unimodular extensions with suitable conditional laws. We also illustrate how this representation can be used to establish Gibbs conditioning principles that provide insight into the structure of marked random graphs conditioned on a rare event. Additional motivation for this work arises from the fact that such a representation is also useful for characterizing the annealed pressure of statistical physics models with general spins, and large deviations of evolving interacting particle systems on sparse random graphs.

math.PR↗

arXiv2023

A sufficient condition for the quasipotential to be the rate function of the invariant measure of countable-state mean-field interacting particle systems

This paper considers the family of invariant measures of Markovian mean-field interacting particle systems on a countably infinite state space and studies its large deviation asymptotics. The Freidlin-Wentzell quasipotential is the usual candidate rate function for the sequence of invariant measures indexed by the number of particles. The paper provides two counterexamples where the quasipotential is not the rate function. The quasipotential arises from finite horizon considerations. However there are certain barriers that cannot be surmounted easily in any finite time horizon, but these barriers can be crossed in the stationary regime. Consequently, the quasipotential is infinite at some points where the rate function is finite. After highlighting this phenomenon, the paper studies some sufficient conditions on a class of interacting particle systems under which one can continue to assert that the Freidlin-Wentzell quasipotential is indeed the rate function.

math.PR↗

arXiv2021

Large deviations of mean-field interacting particle systems in a fast varying environment

This paper studies large deviations of a ``fully coupled" finite state mean-field interacting particle system in a fast varying environment. The empirical measure of the particles evolves in the slow time scale and the random environment evolves in the fast time scale. Our main result is the path-space large deviation principle for the joint law of the empirical measure process of the particles and the occupation measure process of the fast environment. This extends previous results known for two time scale diffusions to two time scale mean-field models with jumps. Our proof is based on the method of stochastic exponentials. We characterise the rate function by studying a certain variational problem associated with an exponential martingale.

math.PR↗

arXiv2021

The Four Levels of Fixed-Points in Mean-Field Models

The fixed-point analysis refers to the study of fixed-points that arise in the context of complex systems with many interacting entities. In this expository paper, we describe four levels of fixed-points in mean-field interacting particle systems. These four levels are (i) the macroscopic observables of the system, (ii) the probability distribution over states of a particle at equilibrium, (iii) the time evolution of the probability distribution over states of a particle, and (iv) the probability distribution over trajectories. We then discuss relationships among the fixed-points at these four levels. Finally, we describe some issues that arise in the fixed-point analysis when the system possesses multiple fixed-points at the level of distribution over states, and how one goes beyond the fixed-point analysis to tackle such issues.

cs.NI↗

arXiv2021

Large Time Behaviour and the Second Eigenvalue Problem for Finite State Mean-Field Interacting Particle Systems

This article examines large time behaviour of finite state mean-field interacting particle systems. Our first main result is a sharp estimate (in the exponential scale) on the time required for convergence of the empirical measure process of the $N$-particle system to its invariant measure; we show that when time is of the order of $\exp\{NΛ\}$ for a suitable constant $Λ\geq 0$, the process has mixed well and it is close to its invariant measure. We then obtain large-$N$ asymptotics of the second largest eigenvalue of the generator associated with the empirical measure process when it is reversible with respect to its invariant measure. We show that its absolute value scales as $\exp\{-NΛ\}$. The main tools used in establishing our results are the large deviation properties of the empirical measure process from its large-$N$ limit. As an application of the study of large time behaviour, we also show convergence of the empirical measure of the system of particles to a global minimum of a certain `entropy' function when particles are added over time in a controlled fashion. The controlled addition of particles is analogous to the cooling schedule associated with the search for a global minimum of a function using the simulated annealing algorithm.

math.PR↗

arXiv2020

COVID-19: Optimal Design of Serosurveys for Disease Burden Estimation

We provide a methodology by which an epidemiologist may arrive at an optimal design for a survey whose goal is to estimate the disease burden in a population. For serosurveys with a given budget of $C$ rupees, a specified set of tests with costs, sensitivities, and specificities, we show the existence of optimal designs in four different contexts, including the well known c-optimal design. Usefulness of the results are illustrated via numerical examples. Our results are applicable to a wide range of epidemiological surveys under the assumptions that the estimate's Fisher-information matrix satisfies a uniform positive definite criterion.

stat.AP↗

arXiv2020

City-Scale Agent-Based Simulators for the Study of Non-Pharmaceutical Interventions in the Context of the COVID-19 Epidemic

We highlight the usefulness of city-scale agent-based simulators in studying various non-pharmaceutical interventions to manage an evolving pandemic. We ground our studies in the context of the COVID-19 pandemic and demonstrate the power of the simulator via several exploratory case studies in two metropolises, Bengaluru and Mumbai. Such tools become common-place in any city administration's tool kit in our march towards digital health.

q-bio.PE↗

arXiv2019

Nonzero-sum Adversarial Hypothesis Testing Games

We study nonzero-sum hypothesis testing games that arise in the context of adversarial classification, in both the Bayesian as well as the Neyman-Pearson frameworks. We first show that these games admit mixed strategy Nash equilibria, and then we examine some interesting concentration phenomena of these equilibria. Our main results are on the exponential rates of convergence of classification errors at equilibrium, which are analogous to the well-known Chernoff-Stein lemma and Chernoff information that describe the error exponents in the classical binary hypothesis testing problem, but with parameters derived from the adversarial model. The results are validated through numerical experiments.

cs.GT↗