Related papers: Contagious Sets in Random Graphs

Let $u_k(G,p)$ be the maximum over all $k$-vertex graphs $F$ of by how much the number of induced copies of $F$ in $G$ differs from its expectation in the binomial random graph with the same number of vertices as $G$ and with edge…

Epidemic spreading is well understood when a disease propagates around a contact graph. In a stochastic susceptible-infected-susceptible setting, spectral conditions characterise whether the disease vanishes. However, modelling human…

We say that a hereditary graph class $\mathcal{G}$ is \emph{clique-sparse} if there is a constant $k=k(\mathcal{G})$ such that for every graph $G\in\mathcal{G}$, every vertex of $G$ belongs to at most $k$ maximal cliques, and any maximal…

In this paper, we develop efficient exact and approximate algorithms for computing a maximum independent set in random graphs. In a random graph $G$, each pair of vertices are joined by an edge with a probability $p$, where $p$ is a…

Given a graph $I=(V, E),$ $\emptyset \ne D \subseteq V,$ and an arbitrary nonempty set $X,$ an injective function $f: V\to 2^X \setminus \{\emptyset\}$ is an interference of $D$ with respect to $I,$ if for every vertex $u\in V\setminus D$…

Majority bootstrap percolation on the random graph $G_{n,p}$ is a process of spread of "activation" on a given realisation of the graph with a given number of initially active nodes. At each step those vertices which have more active…

A classic result of Erd\H{o}s, Gy\'arf\'as and Pyber states that for every coloring of the edges of $K_n$ with $r$ colors, there is a cover of its vertex set by at most $f(r) = O(r^2 \log r)$ vertex-disjoint monochromatic cycles. In…

We study the SIR epidemic model with infections carried by $k$ particles making independent random walks on a random regular graph. Here we assume $k\leq n^{\epsilon}$, where $n$ is the number of vertices in the random graph, and $\epsilon$…

Motivated by an application in community detection, we consider an \ER random graph conditioned on the rare event that all connected components are fully connected. Such graphs can be considered as partitions of vertices into cliques.…

We consider an SIR-type (Susceptible $\to$ Infected $\to$ Recovered) stochastic epidemic process with multiple modes of transmission on a contact network. The network is given by a random graph following a multilayer configuration model…

A graph with a trivial automorphism group is said to be rigid. Wright proved that for $\frac{\log n}{n}+\omega(\frac 1n)\leq p\leq \frac 12$ a random graph $G\in G(n,p)$ is rigid whp. It is not hard to see that this lower bound is sharp and…

We show, through local estimates and simulation, that if one constrains simple graphs by their densities $\varepsilon$ of edges and $\tau$ of triangles, then asymptotically (in the number of vertices) for over $95\%$ of the possible range…

We study the diffusion of influence in random multiplex networks where links can be of $r$ different types, and for a given content (e.g., rumor, product, political view), each link type is associated with a content dependent parameter…

Consider~\(n\) nodes~\(\{X_i\}_{1 \leq i \leq n}\) independently distributed in the unit square~\(S,\) each according to a distribution~\(f.\) Nodes~\(X_i\) and~\(X_j\) are joined by an edge if the Euclidean distance~\(d(X_i,X_j)\) is less…

The $d$-process generates a graph at random by starting with an empty graph with $n$ vertices, then adding edges one at a time uniformly at random among all pairs of vertices which have degrees at most $d-1$ and are not mutually joined. We…

For any set $\Omega$ of non-negative integers such that $\{0,1\}\subseteq \Omega$ and $\{0,1\}\ne \Omega$, we consider a random $\Omega$-$k$-tree ${\sf G}_{n,k}$ that is uniformly selected from all connected $k$-trees of $(n+k)$ vertices…

For any $S\subset [n]$, we compute the probability that the subgraph of $\mathcal{G}_{n,d}$ induced by $S$ is a given graph $H$ on the vertex set $S$. The result holds for any $d=o(n^{1/3})$ and is further extended to $\mathcal{G}_{{\bf…

Given positive integers n and m, and a probability measure P on {0, 1, ..., m} the random intersection graph G(n,m,P) on vertex set V = {1,2, ..., n} and with attribute set W = {w_1, w_2, ..., w_m} is defined as follows. Let S_1, S_2, ...,…

The study of networks leads to a wide range of high dimensional inference problems. In many practical applications, one needs to draw inference from one or few large sparse networks. The present paper studies hypothesis testing of graphs in…

In 1990 Bender, Canfield and McKay gave an asymptotic formula for the number of connected graphs on $[n]=\{1,2,\ldots,n\}$ with $m$ edges, whenever $n\to\infty$ and $n-1\le m=m(n)\le \binom{n}{2}$. We give an asymptotic formula for the…