arXiv subjects

Souvik Ray

Publications and source records attributed to Souvik Ray.

7 recordsLinked to original sources

arXiv2025

Large Deviations for Markovian Graphon Processes and Associated Dynamical Systems on Networks

We consider temporal models of rapidly evolving Markovian networks whose edge-formation and dissolution rates are determined by time-dependent spatial kernels. Equivalently, these may be viewed as Markovian networks with $O(1)$ jump rates observed over long time horizons. In this regime, paths of graphon-valued processes obtained by averaging over suitable moving time windows provide natural state descriptors. Under appropriate conditions on the jump-rate kernels, we establish laws of large numbers and large deviation principles for these window-averaged paths, both in the weak topology and in the cut metric. We also show that, without such local averaging, the rapidly oscillating graphon process does not satisfy a nontrivial path-space LDP. The resulting rate functions admit explicit and tractable representations, distinct from those arising in static random graph models and finite-horizon dynamic graph models. We further analyze the associated variational problems in several examples and apply the graphon LDP to node-valent dynamical systems driven by the evolving network.

math.PR↗

arXiv2024

On the stability of solutions to random optimization problems under small perturbations

Consider the Euclidean traveling salesman problem with $n$ random points on the plane. Suppose that one of the points is shifted to a new random location. This gives us a new optimal path. Consider such shifts for each of the $n$ points. Do we get $n$ very different optimal paths? In this article, we show that this is not the case - in fact, the number of truly different paths can be at most $\mathcal{O}(1)$ as $n\to \infty$. The proof is based on a general argument which allows us to prove similar stability results in a number of other settings, such as branching random walk, the Sherrington-Kirkpatrick model of mean-field spin glasses, the Edwards-Anderson model of short-range spin glasses, and the Wigner ensemble of random matrices.

math.PR↗

arXiv2024

Component Spectrum of Large Sparse Uniformly Random Magical Squares

In this paper, we shall try to deduce asymptotic behaviour of component spectrum of random $n \times n$ magical squares with line sum $r \in \mathbb{N}$, which can also be identified as $r$-regular bipartite graphs on $2n$ vertices, chosen uniformly from the set of all possible such squares as the dimension $n$ grows large keeping $r$ fixed. We shall focus on limits (after appropriate centering and scaling) of various statistic depending upon the component structure, e.g., number of small components, size of the smallest and largest components, total number of components etc. We shall observe that for the case $r=2$, this analysis falls into the domain of Logarithmic combinatorial structures, although we shall present a new approach for this case relying only on the asymptotic results for random permutations which also helps us to demonstrate an importance sampling algorithm to estimate parameters defined in terms of uniform distribution on magical squares. The case $r \geq 3$ although does not fall in the domain of the Logarithmic combinatorial structures, we shall establish that its component structure is rather trivial, using techniques based on a power series approach.

math.PR↗

arXiv2024

Random walk in slowly changing environments

A Random Walk in Changing Environment (RWCE) is a weighted random walk on a locally finite, connected graph $G$ with random, time-dependent edge-weights. This includes self-interacting random walks, where the edge-weights depend on the history of the process. In general, even the basic question of recurrence or transience for RWCEs is difficult, especially when the underlying graph contains cycles. In this note, we derive a condition for recurrence or transience that is too restrictive for classical RWCEs but instead works for any graph $G.$ Namely, we show that any bounded RWCE on $G$ with "slowly" changing edge-weights inherits the recurrence or transience of the initial weighted graph.

math.PR↗

arXiv2021

Characterizing the Functional Density Power Divergence Class

Divergence measures have a long association with statistical inference, machine learning and information theory. The density power divergence and related measures have produced many useful (and popular) statistical procedures, which provide a good balance between model efficiency on one hand and outlier stability or robustness on the other. The logarithmic density power divergence, a particular logarithmic transform of the density power divergence, has also been very successful in producing efficient and stable inference procedures; in addition it has also led to significant demonstrated applications in information theory. The success of the minimum divergence procedures based on the density power divergence and the logarithmic density power divergence (which also go by the names $\beta$-divergence and $\gamma$-divergence, respectively) make it imperative and meaningful to look for other, similar divergences which may be obtained as transforms of the density power divergence in the same spirit. With this motivation we search for such transforms of the density power divergence, referred to herein as the functional density power divergence class. The present article characterizes this functional density power divergence class, and thus identifies the available divergence measures within this construct that may be explored further for possible applications in statistical inference, machine learning and information theory.

math.ST↗

arXiv2021

Characterizing Logarithmic Bregman Functions

Minimum divergence procedures based on the density power divergence and the logarithmic density power divergence have been extremely popular and successful in generating inference procedures which combine a high degree of model efficiency with strong outlier stability. Such procedures are always preferable in practical situations over procedures which achieve their robustness at a major cost of efficiency or are highly efficient but have poor robustness properties. The density power divergence (DPD) family of Basu et al.(1998) and the logarithmic density power divergence (LDPD) family of Jones et al.(2001) provide flexible classes of divergences where the adjustment between efficiency and robustness is controlled by a single, real, non-negative parameter. The usefulness of these two families of divergences in statistical inference makes it meaningful to search for other related families of divergences in the same spirit. The DPD family is a member of the class of Bregman divergences, and the LDPD family is obtained by log transformations of the different segments of the divergences within the DPD family. Both the DPD and LDPD families lead to the Kullback-Leibler divergence in the limiting case as the tuning parameter $\alpha \rightarrow 0$. In this paper we study this relation in detail, and demonstrate that such log transformations can only be meaningful in the context of the DPD (or the convex generating function of the DPD) within the general fold of Bregman divergences, giving us a limit to the extent to which the search for useful divergences could be successful.

math.ST↗

arXiv2019

Branching random walk with infinite progeny mean: a tale of two tails

We study the extremes of branching random walks under the assumption that the underlying Galton-Watson tree has infinite progeny mean. It is assumed that the displacements are either regularly varying or they have lighter tails. In the regularly varying case, it is shown that the point process sequence of normalized extremes converges to a Poisson random measure. We study the asymptotics of the scaled position of the rightmost particle in the $n$-th generation when the tail of the displacement behaves like $\exp(-K(x))$, where either $K$ is a regularly varying function of index $r> 0$, or $K$ has an exponential growth. We identify the exact scaling of the maxima in all cases and show the existence of a non-trivial limit when $r> 1$.

math.PR↗