Related papers: Modularity of Erd\H{o}s-R\'enyi random graphs

We systematically study a natural problem in extremal graph theory, to minimize the number of edges in a graph with a fixed number of vertices, subject to a certain local condition: each vertex must be in a copy of a fixed graph $H$. We…

We consider a time varying analogue of the Erd{\H o}s-R{\' e}nyi graph and study the topological variations of its associated clique complex. The dynamics of the graph are stationary and are determined by the edges, which evolve…

We propose the following model of a random graph on n vertices. Let F be a distribution in R_+^{n(n-1)/2} with a coordinate for every pair i$ with 1 \le i,j \le n. Then G_{F,p} is the distribution on graphs with n vertices obtained by…

The study of network structure is pervasive in sociology, biology, computer science, and many other disciplines. One of the most important areas of network science is the algorithmic detection of cohesive groups of nodes called…

There are good arguments to support the claim that deep neural networks (DNNs) capture better feature representations than the previous hand-crafted feature engineering, which leads to a significant performance improvement. In this paper,…

Given two Erd\H{o}s-R\'enyi graphs with $n$ vertices whose edges are correlated through a latent vertex correspondence, we study complexity lower bounds for the associated correlation detection problem for the class of low-degree polynomial…

We consider the upper tail large deviations of subgraph counts for irregular graphs $\mathrm{H}$ in $\mathbb{G}(n,p)$, the sparse Erd\H{o}s-R\'enyi graph on $n$ vertices with edge connectivity probability $p \in (0,1)$. For $n^{-1/\Delta}…

Modularity, since its introduction, has remained one of the most widely used metrics to assess the quality of community structure in a complex network. However the resolution limit problem associated with modularity limits its applicability…

Let $G_{n,p}$ be the standard Erd\H{o}s-R\'enyi-Gilbert random graph and let $G_{n,n,p}$ be the random bipartite graph on $n+n$ vertices, where each $e\in [n]^2$ appears as an edge independently with probability $p$. For a graph $G=(V,E)$,…

Community detection using both graphs and social networks is the focus of many algorithms. Recent methods aimed at optimizing the so-called modularity function proceed by maximizing relations within communities while minimizing…

We give a simple, computationally efficient, and node-differentially-private algorithm for estimating the parameter of an Erdos-Renyi graph---that is, estimating p in a G(n,p)---with near-optimal accuracy. Our algorithm nearly matches the…

The most widely used techniques for community detection in networks, including methods based on modularity, statistical inference, and information theoretic arguments, all work by optimizing objective functions that measure the quality of…

We consider the quantity $P(G)$ associated with a graph $G$ that is defined as the probability that a randomly chosen subtree of $G$ is spanning. Motivated by conjectures due to Chin, Gordon, MacPhee and Vincent on the behaviour of this…

We study the problem of learning an unknown graph via group queries on node subsets, where each query reports whether at least one edge is present among the queried nodes. In general, learning arbitrary graphs with $n$ nodes and $k$ edges…

In a graph or complex network, communities and anti-communities are node sets whose modularity attains extremely large values, positive and negative, respectively. We consider the simultaneous detection of communities and anti-communities,…

Inspired by the increasing interest in self-organizing social opportunistic networks, we investigate the problem of distributed detection of unknown communities in dynamic random graphs. As a formal framework, we consider the dynamic…

An uncertain graph $\mathcal{G} = (V, E, p : E \rightarrow (0,1])$ can be viewed as a probability space whose outcomes (referred to as \emph{possible worlds}) are subgraphs of $\mathcal{G}$ where any edge $e\in E$ occurs with probability…

Many complex networks exhibit a modular structure of densely connected groups of nodes. Usually, such a modular structure is uncovered by the optimization of some quality function. Although flawed, modularity remains one of the most popular…

We prove asymptotically optimal bounds on the number of edges a graph $G$ must have in order that any $r$-colouring of $E(G)$ has a colour class which contains every $D$-degenerate graph on $n$ vertices with bounded maximum degree. We also…

Community detection is considered as a fundamental task in analyzing social networks. Even though many techniques have been proposed for community detection, most of them are based exclusively on the connectivity structures. However, there…