Titles and Abstracts

Rumours, Epidemics and Consensus on Networks

The course will develop techniques for studying the long-term behaviour of random processes on networks. A large part of the motivation comes from computer science, where one is interested in designing decentralised algorithms for large networked systems such as the Internet or P2P networks or blockchain systems. Decentralised mechanisms for disseminating information or achieving consensus are typically a basic building block of such algorithms. Randomness can help create mechanisms which are exceptionally simple as well as highly robust, both to random faults and to malicious agents. Other motivations for the material studied in this course come from ecology, social science and economics, where one is interested in collective decision-making in humans and other animals, and from infectious disease epidemiology. The course will develop probabilistic tools for analysing these random processes. These mathematical methods have considerable elegance of their own.

References: 

• M. Draief and L. Massoulie, Epidemics and Rumours in Complex Networks, Cambridge University Press, 2009.

• A. J. Ganesh,  Complex Networks,  Course webpage.

Pre-requisites:  Knowledge of discrete probability is essential. Familiarity with Markov chains - at least discrete time and finite state space; for example see Chapter 1 of the 'Complex Networks' course listed above. Familiarity with Poisson processes will be helpful.

Statistical Inference for Combinatorial Data:
Graphical Models, Networks, and Rankings

The course will focus on statistical inference for combinatorial data, organized around three interrelated themes: graphical models, networks, and rankings. We will first discuss methods of estimation for dependent combinatorial data, specifically through the lens of the Ising model. Next, we will study estimation in inhomogeneous random graph models, graphons, and exponential random graph models. Finally, we will consider estimation problems for ranking models, such as the Bradley-Terry and Mallows models, and their applications to social choice theory.

References: 

• S. Janson, T. Łuczak, and A. Ruciński, Random Graphs, 2000.

• L. Lovász, Large Networks and Graph Limits, 2012. 

• S. Chatterjee, Large Deviations for Random Graphs, Lecture Notes in Mathematics, 2017

Prerequisites: Some background in discrete probability (first 2 chapters of the Janson et. al. book can be useful) and basic statistical inference (at the level of  G. Casella and R. L. Berger, Statistical Inference, 2002) would be helpful.  

Bhaswar Bhattacharya, University of Pennsylvania

Higher-Order Asymptotics of Random Combinatorial Structures and Applications

We will survey some recent results on higher-order fluctuations of random graphs, tournaments, and graphical model, and their applications in the analysis of dependent combinatorial data. 

Mona Azadkia, London School of Economics and Political Science

Coverage correlation: detecting singular dependencies between random variables

We introduce the coverage correlation coefficient, a novel nonparametric measure of statistical association designed to quantify the extent to which two random variables have a joint distribution concentrated on a singular subset with respect to the product of the marginals. Our correlation statistic consistently estimates an f-divergence between the joint distribution and the product of the marginals, which is 0 if and only if the variables are independent and 1 if and only if the copula is singular. Using Monge--Kantorovich ranks, the coverage correlation naturally extends to measure association between random vectors. It is distribution-free, admits an analytically tractable asymptotic null distribution, and can be computed efficiently, making it well-suited for detecting complex, potentially nonlinear associations in large-scale pairwise testing.

Ayalvadi Ganesh, University of Bristol

Steiner trees in the stochastic mean-field model of distance

Consider the complete graph on n vertices, with edge weights drawn independently from the exponential distribution with unit mean. Janson showed that the typical distance between two vertices scales as logn/n, whereas the diameter (maximum distance between any two vertices) scales as 3logn/n. Bollob\'{a}s et al. showed that, for any fixed k, the weight of the Steiner tree connecting k typical vertices scales as (k−1)logn/n, which recovers Janson's result for k=2. We extend this result to show that the worst case k-Steiner tree, over all choices of k vertices, has weight scaling as (2k−1)logn/n. We also study fluctuations of the weight of a Steiner tree connecting k typical vertices.

Gourab Ghatak, Indian Institute of Technology Delhi

Shortest Paths on Random Line Networks: From Exact Distributions to Multi-Turn Geometry

Random line networks arise naturally in stochastic models of transportation systems, communication infrastructure, and spatial connectivity. When points are distributed along such networks, the relevant notion of distance is often not the Euclidean distance, but the length of the shortest path constrained to the underlying lines. In this talk, we explore shortest-path distances in Poisson line Cox processes, where points are distributed along a random collection of lines. Using classical results from Poisson process theory, we first derive exact nearest-neighbor and contact-distance distributions for paths involving at most one turn. We then examine the mathematical challenges of extending these results to multiple turns, particularly the overlapping reachable regions and geometric dependencies introduced by triangular and bow-tie configurations. Motivated by these difficulties, we explore an alternative approach based on intrinsic metric balls and outline some preliminary results and directions for further investigation. We conclude by discussing approaches that appear to reach mathematical dead ends, along with several open questions.

Vinay Kumar B. R., Indian Institute of Technology Bombay

Variants of the hidden clique problem

The planted clique problem is a central paradigm for studying statistical and computational boundaries in network inference. While classical formulations focus on simple binary graphs, many real-world applications yield weighted network observations or higher-order relational data. In this talk, we examine statistical detection limits and efficient algorithms for recovering planted combinatorial structures across two non-standard graph settings. First, we study the weighted hidden clique problem, where edge weights inside and outside the planted structure follow distinct probability distributions. We highlight fundamental information-theoretic limits for hypothesis testing in this model and show how spectral methods can be designed to exploit differences in edge weight distributions.

Second, we consider hypergraph data observed strictly through a projected weighted adjacency matrix that counts shared hyperedges between node pairs. Although this low-dimensional projection introduces dependent matrix entries and potential information loss, we show that spectral norm testing and eigenvector-based recovery methods remain highly effective across both dense and sparse regimes. Together, these results illustrate how spectral and probabilistic tools adapt to non-standard edge distributions and structural dependencies in complex networks.

Somabha Mukherjee, National University of Singapore

Fluctuations and Optimal Inference for Ising Models on Inhomogeneous Random Graphs

In this talk, I will discuss high-temperature Ising models on sparse and dense inhomogeneous random graphs generated by graphons. I will present scaling limits for the Hamiltonian, linear statistics and the empirical spin field, and show how the underlying fluctuation theory yields sharp inference of the model parameter, based on a single spin configuration. This includes a computationally efficient estimator matching the asymptotic efficiency of maximum likelihood, local asymptotic minimaxity, and limits of experiments that are Gaussian in the sparse regime but generally non-Gaussian in the dense regime. Applications of our results to Bayesian neural networks and causal inference under network interference will also be discussed.

Neeraja Sahasrabudhe, IISER Mohali

Tampered Memory Elephant Random Walk

The Elephant Random Walk (ERW), introduced by Schütz and Trimper in 2004, is a memory-based random walk in which each step depends on the entire past history of the walk. In this talk, we introduce a variation of the classical ERW, called the tampered memory elephant random walk (TMERW), in which the memory is partitioned into two sets. At time n>1, the walk follows the classical ERW dynamics when the uniformly chosen memory index belongs to a prescribed subset $D_{n-1} \subseteq \{1,\ldots,n-1 \}$, while the increment is replaced by an independent innovation when the index belongs to its complement. Our aim is to understand how much memory must be retained for ERW-like behaviour to persist. We discuss limit theorems for this model and show how the asymptotic behaviour of the walk depends on the size and structure of the memory sets. In particular, for deterministic memory partitions, we identify a sharp threshold: the phase transition between diffusive, critical and superdiffusive regimes persists when the proportion of untampered memory exceeds one-half, whereas below this threshold the walk exhibits only diffusive behaviour. This talk is based on joint work with Dr. Debleena Thacker and Vinita Mulay.

Devavrat Shah, Massachusetts Institute of Technology, USA

A Research Program on Time Series: Past and Future

In this talk, I will describe the research program on time series that I have been pursuing with various collaborators over the past decade. I will start by briefly setting the context for the literature on time series. I will discuss representation for time series data through the lens of low-rank matrices and its implications for robust, scalable methods. Finally, I will touch upon ongoing work with the goal of developing efficient time series foundation model.