ICTS

The elephant random walk (ERW) is a discrete-time stochastic process with a long-time memory of its whole history. In this talk, we introduce a variation of the ERW whose steps are polynomially decaying. We show that it admits a phase transition from divergence to convergence (localization) at a critical point, and a sufficiently large memory can shift the critical point for localization. We also discuss precise estimates the Takagi-van der Waerden functions, a well-known class of continuous but nowhere differentiable functions, via variations of ERW remembering the very recent past.  The talk is partly based on the joint work with Masato Takei (Yokohama National University).

The Brownian loop soup (BLS) is a conformally invariant Poisson point process of unrooted loops in the plane. In this talk, we consider a functional of the BLS, called the layering field, which counts the number of loops from the soup covering each point of the domain. We shall discuss the convergence of the layering field to the Gaussian Multiplicative Chaos measures. This talk will be based on the work arxiv:2510.22165.

We study the survival of a random walker moving among mobile traps on $\mathbb{Z}^d$ lattice. Initially, each site contains a Poisson($\nu_y$) number of traps, and both the walker and the traps evolve as independent simple symmetric continuous-time random walks. The walker is killed at rate $\gamma>0$ upon encountering a trap. While the homogeneous case ($\nu_y = \nu >0$) is well understood, including precise asymptotics for the annealed survival probability, and the existence of annealed and quenched Lyapunov exponents, less is known for inhomogeneous trap configurations. In this talk, I will focus on the asymptotic behaviour of the annealed survival probability of the random walker in inhomogeneous trap configurations.

We consider a class of inhomogeneous random graphs Gn(α, ε) where n vertices carry i.i.d. Pareto  weights (Wi)i∈[n] with tail index α > 0. Conditionally on the weights, edges are drawn inde-pendently with probability pij = min(εWiWj , 1), where ε = εn controls sparsity.  The behaviour of the model is driven by the tail index α, with a sharp structural change at the boundary α = 1. The infinite-mean and finite-mean regimes lead to fundamentally different emerging landscapes.
Building on recent work of L. Avena, D. Garlaschelli, R.S. Hazra and M. Lalli (Journal of Applied Probability 2025), we analyze the degree asymptotics across the full range α > 0 and identify the relevant scalings of εn in each regime for the convergence in distribution of the typical degree.

We then characterize the connectivity threshold. In the infinite-mean case α ≤ 1, connectivity is hub-driven and forces a collapse of the diameter to at most two. In the finite-mean regime α > 1, connectivity emerges through a collective mechanism at a density scale distinct from that of ultra-small-world behaviour. A key feature of the infinite-mean regime is that macroscopic connectivity remains intrinsically random: under the relevant scaling εα = λ/n, the size of the largest component, rescaled by n, converges in distribution to a non-degenerate random variable. This reflects the dominant role of extreme weights, which induce persistent fluctuations and highlight a fundamental departure from the classical finite-mean theory, where the giant’s density is typically deterministic. This is joint ongoing work with Luisa Andreis, Luca Avena and Rajat Hazra.

We investigate the problem of identifying planted cliques in random geometric graphs, focusing on two distinct algorithmic approaches: the first based on vertex degrees (VD) and the other on common neighbors (CN). We analyze the performance of these methods under varying regimes of key parameters, namely the average degree of the graph and the size of the planted clique. We demonstrate that exact recovery is achieved with high probability as the graph size increases, in a specific set of parameters. Notably, our results reveal that the CN-algorithm significantly outperforms the VD-algorithm. In particular, in the connectivity regime, tiny planted cliques (even edges) are correctly identified by the CN-algorithm, yielding a significant impact on anomaly detection. Finally, our results are confirmed by a series of numerical experiments, showing that the devised algorithms are effective in practice.

Complex networks, such as social, financial, e-commerce, and criminal networks, provide a powerful framework for representing real-world systems by capturing intricate structural patterns and interactions, consisting of nodes (entities) and edges (connections or interactions). For example, in social networks, nodes represent individuals, and edges denote social connections, while in banking transaction networks, nodes correspond to bank accounts, and edges represent financial transactions. Complex networks are analyzed to understand individual and group behavior at a large scale and solve critical research problems, such as fraud detection, link prediction, social media surveillance, and resource allocation. However, these networks often encode structural inequalities related to gender, ethnicity, race, or socioeconomic status. Moreover, groups’ distribution may be inherently imbalanced, with certain groups being underrepresented or more sparsely connected. If such structural inequalities are not considered while designing network analysis algorithms, the outcome might be unfair, particularly disadvantaging minorities or underrepresented groups.

In this talk, I will highlight how the structural inequalities of complex networks impact the fairness of different network analysis methods using a case study of link prediction. I will first discuss link prediction methods and the impact of structural inequalities on the fairness of link prediction. Next, I will discuss a few approaches in-depth to encounter structural biases for fair and diverse link prediction. Finally, I will conclude my talk with fairness aspect in other network science problems and open future directions.

In large neural networks, important theoretical insights arise from scaling limits, in particular the infinite-width regime, where the depth is fixed and the number of neurons per layer tends to infinity. In this setting, network behavior simplifies and Gaussian processes emerge as limiting objects. We focus on fully connected architectures, which can be viewed as finite compositions of complete bipartite graphs with nonlinear activations between layers. We establish a large deviation principle (LDP) for the covariance process of fully connected deep neural networks with Gaussian weights, formulated in a functional setting as a process in the space of continuous functions. As applications, we derive posterior LDPs under Gaussian likelihoods in both the infinite-width and mean-field regimes. Our approach relies on proving an LDP for the covariance process seen as a Markov process taking values in the space of non-negative, symmetric trace-class operators endowed with the trace norm. This is a joint work with Federico Bassetti (Milan) and Christian Hirsch (Aarhus).
 

In this talk we study two-type competing first-passage percolation on random graphs generated by the configuration model with power-law degree distributions of exponent $\tau$. Starting from two uniformly chosen vertices, the infections spread through the graph with random edge passage times, and each vertex becomes permanently occupied by the first type that reaches it.

We begin by introducing the classical nearest-neighbor model and reviewing known results across different degree regimes, highlighting how the structure of the graph affects the outcome of the competition. In the infinite-mean regime when $\tau \in (1,2)$, the dynamics are dominated by high-degree vertices, leading to an extreme “winner-takes-all” phenomenon: with high probability, as the graph size $n$ tends to infinity, one type occupies the entire graph except for the starting vertex of the other type, although both types still have a positive probability of winning.

In this regime, we then extend the model by incorporating long-range jumps, allowing each infected vertex to infect uniformly chosen vertices at a rate $\gamma_n$. This global spreading mechanism competes with the local edge-based dynamics and significantly alters the behavior of the system. In particular, it induces a phase transition: below a critical threshold, the “winner-takes-all” phenomenon persists, while above it, long-range jumps enable macroscopic coexistence between the two types.

These results provide insight into how graph heterogeneity and transmission mechanisms interact, with applications ranging from epidemic spread to competing information or malware propagation in complex networks.

We prove a Sanov-type large deviation principle for the component empirical measure of certain families of sparse random graphs whose vertices are marked with i.i.d. random variables. Specifically, we show that the rate function can be expressed in a fairly tractable form involving suitable relative entropies. We illustrate two applications of this result: (i) we quantify probabilities of rare events in stochastic networks on sparse random graphs, and (ii) we characterize the annealed free energy density of a broad class of probabilistic graphical models.

Joint work with I-Hsun Chen and Kavita Ramanan.

In this work, a generalized Wigner matrix perturbed by a finite-rank deterministic matrix is studied. The goal is to understand the fluctuations of the largest eigenvalues, which emerge outside the bulk of the spectrum, and the corresponding eigenvectors. Under certain assumptions on the perturbation and the matrix structure, we derive the first-order behavior of these eigenvalues and show that they are well separated from the bulk. The fluctuations of these eigenvalues are shown to follow a multivariate Gaussian distribution, and the asymptotic behavior of the associated eigenvectors is also studied. Central limit theorems that describe the asymptotic alignment of these eigenvectors with the perturbation's eigenvectors, as well as their Gaussian fluctuations around the origin for the non-aligned components, are proven. Furthermore, the convergence of the eigenvector process in a Sobolev space framework is discussed.

This is a joint work with Bishakh Bhattacharya and Rajat Subhra Hazra.

I shall talk about some recent and ongoing work on a different kind of networks - overparameterized deep neural networks. The work is not necessarily deep.

This talk is based on joint work with Andreas Klippel and Christian Mönch. We study the extreme value statistics of the zero-average Gaussian free field (GFF) on random r-regular graphs and the Gaussian free field on r-regular trees. For random r-regular graphs of diverging size, for every fixed r≥3, we show that the rescaled extremal point process of the field is asymptotically distributed, in the annealed sense, as a Poisson point process on the line with intensity e−xdx. The same limit behaviour is obeyed by the restriction of the GFF on r-regular trees to finite subsets of vertices. Our approach relies on a direct Gaussian comparison argument and precise Green function estimates.

As our world becomes increasingly interconnected, the informational landscape that drives decision-making is marked by ever-expanding scale and interdependencies. Leveraging graph structure, we develop computationally efficient alternatives to canonical subroutines that underlie inference in modern machine learning and optimization infrastructure. We discuss two key directions: First, we optimize graph algorithms for learning from distributed data sources, addressing a key challenge in decentralized settings- namely, identifying simple probabilistic rules for organizing nodes to balance sparsity with reliable connectivity. Our results resolve several open problems related to the exact analysis of connectivity properties in a class of random graph models known as random k-out graphs, widely appearing as heuristics for network design in settings with limited trust. Second, we discuss computationally efficient alternatives to parameter learning in probabilistic graphical models. We develop methods that retain the statistical advantage of classical maximum likelihood estimation while significantly cutting computational costs in the context of high dimensional exponential family models. Summing, our work sheds new light on how the interplay between graph structure and performance can be leveraged to push the frontiers of efficient and provably reliable algorithms.