相关论文: Sharp thresholds in inference of planted subgraphs

The classical Erd\H os-Ko-Rado (EKR) Theorem states that if we choose a family of subsets, each of size (k), from a fixed set of size (n (n > 2k)), then the largest possible pairwise intersecting family has size (t ={n-1\choose k-1}). We…

We give an algorithm for finding the arboricity of a weighted, undirected graph, defined as the minimum number of spanning forests that cover all edges of the graph, in $\sqrt{n} m^{1+o(1)}$ time. This improves on the previous best bound of…

For a given hypergraph $H$ and a vertex $v\in V(H)$, consider a random matching $M$ chosen uniformly from the set of all matchings in $H.$ In $1995,$ Kahn conjectured that if $H$ is a $d$-regular linear $k$-uniform hypergraph, the…

We consider the sensitivity of algorithms for the maximum matching problem against edge and vertex modifications. Algorithms with low sensitivity are desirable because they are robust to edge failure or attack. In this work, we show a…

A wide variety of complex networks (social, biological, information etc.) exhibit local clustering with substantial variation in the clustering coefficient (the probability of neighbors being connected). Existing models of large graphs…

Graph Neural Networks (GNNs) have achieved impressive performance in graph-related tasks. However, they suffer from poor generalization on out-of-distribution (OOD) data, as they tend to learn spurious correlations. Such correlations…

Detection of a planted dense subgraph in a random graph is a fundamental statistical and computational problem that has been extensively studied in recent years. We study a hypergraph version of the problem. Let $G^r(n,p)$ denote the…

A graph is said to be $\mathcal{H}(n, \Delta)$-universal if it contains every graph on $n$ vertices with maximum degree at most $\Delta$. Using a `matching-based' embedding technique introduced by Alon and F\"uredi, Dellamonica, Kohayakawa,…

We aim to understand the extent to which the noise distribution in a planted signal-plus-noise problem impacts its computational complexity. To that end, we consider the planted clique and planted dense subgraph problems, but in a different…

During the past 15 years, several extensions of the concept of noise sensitivity first coined in~\cite{schramm2000}, has been studied. One such extension was studied in ??, where the definition of noise sensitivity was extended to noise…

In neuroscience, identifying distinct patterns linked to neurological disorders, such as Alzheimer's and Autism, is critical for early diagnosis and effective intervention. Graph Neural Networks (GNNs) have shown promising in analyzing…

In their seminal paper introducing the theory of random graphs, Erd\H{o}s and R\'{e}nyi considered the evolution of the structure of a random subgraph of $K_n$ as the density increases from $0$ to $1$, identifying two key points in this…

Randomized parallel algorithms for many fundamental problems achieve optimal linear work in expectation, but upgrading this guarantee to hold with high probability (whp) remains a recurring theoretical challenge. In this paper, we address…

We study a well known noisy model of the graph isomorphism problem. In this model, the goal is to perfectly recover the vertex correspondence between two edge-correlated Erd\H{o}s-R\'{e}nyi random graphs, with an initial seed set of…

Consider the upper tail probability that the homomorphism count of a fixed graph $H$ within a large sparse random graph $G_n$ exceeds its expected value by a fixed factor $1+\delta$. Going beyond the Erd\H{o}s-R\'enyi model, we establish…

When each vertex is assigned a set, the intersection graph generated by the sets is the graph in which two distinct vertices are joined by an edge if and only if their assigned sets have a nonempty intersection. An interval graph is an…

The celebrated canonical Ramsey theorem of Erd\H{o}s and Rado implies that for a given $k$-uniform hypergraph (or $k$-graph) $H$, if $n$ is sufficiently large then any colouring of the edges of the complete $k$-graph $K^{(k)}_n$ gives rise…

The Lov\'{a}sz Local Lemma (LLL) says that, given a set of bad events that depend on the values of some random variables and where each event happens with probability at most $p$ and depends on at most $d$ other events, there is an…

Let $d,n\in \mathbb{N}$ be such that $d=\omega(1)$, and $d\le n^{1-a}$ for some constant $a>0$. Consider a $d$-regular graph $G=(V, E)$ and the random graph process that starts with the empty graph $G(0)$ and at each step $G(i)$ is obtained…

A central problem in extremal graph theory is to estimate, for a given graph $H$, the number of $H$-free graphs on a given set of $n$ vertices. In the case when $H$ is not bipartite, fairly precise estimates on this number are known. In…