Binomial Random Graph Research Articles

On the Distribution of the Maximum k-Degrees of the Binomial Random Graph

For the maximum number Δn of common neighbors of k vertices in the random graph G(n, p), there exist functions an and σn such that $$\frac{\Delta_n - a_n}{\sigma_n}$$ converges in distribution to a random variable having the standard Gumbel distribution.

The robot crawler graph process

Information gathering by crawlers on the web is of practical interest. We consider a simplified model for crawling complex networks such as the web graph, which is a variation of the robot vacuum edge-cleaning process of Messinger and Nowakowski. In our model, a crawler visits nodes via a deterministic walk determined by their weightings which change during the process deterministically. The minimum, maximum, and average time for the robot crawler to visit all the nodes of a graph is considered on various graph classes such as trees, multi-partite graphs, binomial random graphs, and graphs generated by the preferential attachment model.

How unproportional must a graph be?

Let uk(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 probability p. This may be viewed as a measure of how close G is to being p-quasirandom. For a positive integer n and 0<p<1, let D(n,p) be the distance from pn2 to the nearest integer. Our main result is that, for fixed k≥4 and for n large, the minimum of uk(G,p) over n-vertex graphs has order of magnitude Θ(max{D(n,p),p(1−p)}nk−2) provided that p(1−p)n1∕2→∞.

Subhypergraphs in non-uniform random hypergraphs

In this paper we focus on the problem of finding (small) subhypergraphs in a (large) hypergraph. We use this problem to illustrate that reducing hypergraph problems to graph problems by working with the 2-section is not always a reasonable approach. We begin by defining a generalization of the binomial random graph model to hypergraphs and formalizing several definitions of subhypergraph. The bulk of the paper focusses on determining the expected existence of these types of subhypergraph in random hypergraphs. We also touch on the problem of determining whether a given subgraph appearing in the 2-section is likely to have been induced by a certain subhypergraph in the hypergraph. To evaluate the model in relation to real-world data, we compare model prediction to two datasets with respect to (1) the existence of certain small subhypergraphs, and (2) a clustering coefficient.

On the total variation distance between the binomial random graph and the random intersection graph

AbstractWhen 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 intersection graph generated by intervals in the real line. A chordal graph can be considered as an intersection graph generated by subtrees of a tree. In 1999, Karoński, Scheinerman, and Singer‐Cohen introduced a random intersection graph by taking randomly assigned sets. The random intersection graph has n vertices and sets assigned to the vertices are chosen to be i.i.d. random subsets of a fixed set M of size m where each element of M belongs to each random subset with probability p, independently of all other elements in M. In 2000, Fill, Scheinerman, and Singer‐Cohen showed that the total variation distance between the random graph and the Erdös‐Rényi graph tends to 0 for any if , where is chosen so that the expected numbers of edges in the two graphs are the same. In this paper, it is proved that the total variation distance still tends to 0 for any whenever .

A large deviation principle for the Erdős–Rényi uniform random graph

Starting with the large deviation principle (LDP) for the Erdős–Renyi binomial random graph ${\mathcal G}(n,p)$ (edge indicators are i.i.d.), due to Chatterjee and Varadhan (2011), we derive the LDP for the uniform random graph ${\mathcal G}(n,m)$ (the uniform distribution over graphs with $n$ vertices and $m$ edges), at suitable $m=m_n$. Applying the latter LDP we find that tail decays for subgraph counts in ${\mathcal G}(n,m_n)$ are controlled by variational problems, which up to a constant shift, coincide with those studied by Kenyon et al. and Radin et al. in the context of constrained random graphs, e.g., the edge/triangle model.

Evolution of a Modified Binomial Random Graph by Agglomeration

In the classical Erd\"os-R\'enyi random graph G(n,p) there are n vertices and each of the possible edges is independently present with probability p. The random graph G(n,p) is homogeneous in the sense that all vertices have the same characteristics. On the other hand, numerous real-world networks are inhomogeneous in this respect. Such an inhomogeneity of vertices may influence the connection probability between pairs of vertices. The purpose of this paper is to propose a new inhomogeneous random graph model which is obtained in a constructive way from the Erd\"os-R\'enyi random graph G(n,p). Given a configuration of n vertices arranged in N subsets of vertices (we call each subset a super-vertex), we define a random graph with N super-vertices by letting two super-vertices be connected if and only if there is at least one edge between them in G(n,p). Our main result concerns the threshold for connectedness. We also analyze the phase transition for the emergence of the giant component and the degree distribution. Even though our model begins with G(n,p), it assumes the existence of some community structure encoded in the configuration. Furthermore, under certain conditions it exhibits a power law degree distribution. Both properties are important for real applications.

Clique coloring of dense random graphs

AbstractThe clique chromatic number of a graph is the minimum number of colors in a vertex coloring so that no maximal (with respect to containment) clique is monochromatic. We prove that the clique chromatic number of the binomial random graph is, with high probability, . This settles a problem of McDiarmid, Mitsche, and Prałat who proved that it is with high probability.

Contagious sets in random graphs

We consider the following activation process in undirected graphs: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it has at least $r$ active neighbors. A contagious set is a set whose activation results with the entire graph being active. Given a graph $G$, let $m(G,r)$ be the minimal size of a contagious set. We study this process on the binomial random graph $G:=G(n,p)$ with $p:=\frac{d}{n}$ and $1\ll d\ll (\frac{n\log\log n}{\log^{2}n})^{\frac{r-1}{r}}$. Assuming $r>1$ to be a constant that does not depend on $n$, we prove that \[m(G,r)=\Theta (\frac{n}{d^{\frac{r}{r-1}}\log d}),\] with high probability. We also show that the threshold probability for $m(G,r)=r$ to hold is $p^{*}=\Theta (\frac{1}{(n\log^{r-1}n)^{1/r}})$.

Embedding spanning bounded degree subgraphs in randomly perturbed graphs

We study the model of randomly perturbed dense graphs, which is the union of any graph Gα with minimum degree αn and the binomial random graph G(n,p). For p=ω(n−2/(Δ+1)), we show that Gα∪G(n,p) contains any single spanning graph with maximum degree Δ. As in previous results concerning this model, the bound for p we use is lower by a log-term in comparison to the bound known to be needed to find the same subgraph in G(n,p) alone.

On the number of r-transitive orientations of G(n,p)

We study the number of orientations Tr(n,p) of the binomial random graph G(n,p) in which every copy of Kr is transitive. We give the correct order of growth of log2⁡Tr(n,p) up to polylogarithmic factors. In particular, this significantly improves a result of Allen, Kohayakawa, Mota and Parente on T3(n,p).

The Game of Overprescribed Cops and Robbers Played on Graphs

We consider the effect on the length of the game of Cops and Robbers when more cops are added to the game play. In Overprescribed Cops and Robbers, as more cops are added, the capture time (the minimum length of the game assuming optimal play) monotonically decreases. We give the full range of capture times for any number of cops on trees, and classify the capture time for an asymptotic number of cops on grids, hypercubes, and binomial random graphs. The capture time of planar graphs with a number of cops at and far above the cop number is considered.

On an anti‐Ramsey threshold for sparse graphs with one triangle

AbstractFor graphs G and H, let denote the property that for every proper edge‐coloring of G (with an arbitrary number of colors) there is a rainbow copy of H in G, that is, a copy of H with no two edges of the same color. The authors (2014) proved that, for every graph H, the threshold function of this property for the binomial random graph is asymptotically at most , where denotes the so‐called maximum 2‐density of H. Nenadov et al. (2014) proved that if H is a cycle with at least seven vertices or a complete graph with at least 19 vertices, then . We show that there exists a fairly rich, infinite family of graphs F containing a triangle such that if for suitable constants and , where , then almost surely. In particular, for any such graph F.

On Connectivity and Robustness in Random Intersection Graphs

Random intersection graphs have received much attention recently and been used in a wide range of applications ranging from key predistribution in wireless sensor networks to modeling social networks. For these graphs, each node is equipped with a set of objects in a random manner, and two nodes have an undirected edge in between if they have at least one object in common. In this paper, we investigate connectivity and robustness in a general random intersection graph model. Specifically, we establish sharp asymptotic zero-one laws for k-connectivity and k-robustness, as well as the asymptotically exact probability of k-connectivity, for any positive integer k. The k-connectivity property quantifies how resilient is the connectivity of a graph against node or edge failures, while k-robustness measures the effectiveness of local-information-based consensus algorithms (that do not use global graph topology information) in the presence of adversarial nodes. In addition to presenting the results under the general random intersection graph model, we consider two special cases of the general model, a binomial random intersection graph and a uniform random intersection graph, which both have numerous applications as well. For these two specialized graphs, our results on asymptotically exact probabilities of k-connectivity and asymptotic zero-one laws for k-robustness are also novel in the literature.

Waiter–Client and Client–Waiter Hamiltonicity games on random graphs

We study two types of two player, perfect information games with no chance moves, played on the edge set of the binomial random graph G(n,p). In each round of the (1:q) Waiter–Client Hamiltonicity game, the first player, called Waiter, offers the second player, called Client, q+1 edges of G(n,p) which have not been offered previously. Client then chooses one of these edges, which he claims, and the remaining q edges go back to Waiter. Waiter wins this game if by the time every edge of G(n,p) has been claimed by some player, the graph consisting of Client’s edges is Hamiltonian; otherwise Client is the winner. Client–Waiter games are defined analogously, the main difference being that Client wins the game if his graph is Hamiltonian and Waiter wins otherwise. In this paper we determine a sharp threshold for both games. Namely, for every fixed positive integer q, we prove that the smallest edge probability p for which a.a.s. Waiter has a winning strategy for the (1:q) Waiter–Client Hamiltonicity game is (1+o(1))logn/n, and the smallest p for which a.a.s. Client has a winning strategy for the (1:q) Client–Waiter Hamiltonicity game is (q+1+o(1))logn/n.

Burning Graphs: A Probabilistic Perspective

In this paper, we study a graph parameter that was recently introduced, the burning number, focusing on a few probabilistic aspects of the problem. The original burning number is revisited and analyzed for binomial random graphs $${\mathcal {G}}(n,p)$$G(n,p), random geometric graphs, and the Cartesian product of paths. Moreover, new variants of the burning number are introduced in which a burning sequence of vertices is selected according to some probabilistic rules. We analyze these new graph parameters for paths.

On the Chromatic Number of Non-Sparse Random Intersection Graphs

An intersection graph of n vertices assumes that each vertex is equipped with a subset of a global label set. Two vertices share an edge when their label sets intersect. Random intersection graphs (RIGs) (as defined in Karonski et al. Comb. Probab. Comput. J. 8, 131---159 (1999); Singer-Cohen (1995)) consider label sets formed by the following experiment: each vertex, independently and uniformly, examines all the labels (m in total) one by one. Each examination is independent and the vertex succeeds to put the label in her set with probability p. Such graphs can capture interactions in networks due to sharing of resources among nodes. In this paper, we discuss various structural and algorithmic results concerning random intersection graphs and we focus on the computational problem of properly coloring random instances of the binomial random intersection graphs model. For the latter, we consider a range of parameters m, p for which RIGs differ substantially from Erdoźs-Renyi random graphs and for which greedy approaches fail.

The Threshold Probability for Long Cycles

For a given graph G of minimum degree at least k, let Gp denote the random spanning subgraph of G obtained by retaining each edge independently with probability p = p(k). We prove that if p ⩾ (logk + loglogk + ωk(1))/k, where ωk(1) is any function tending to infinity with k, then Gp asymptotically almost surely contains a cycle of length at least k + 1. When we take G to be the complete graph on k + 1 vertices, our theorem coincides with the classic result on the threshold probability for the existence of a Hamilton cycle in the binomial random graph.

Near-critical SIR epidemic on a random graph with given degrees

Emergence of new diseases and elimination of existing diseases is a key public health issue. In mathematical models of epidemics, such phenomena involve the process of infections and recoveries passing through a critical threshold where the basic reproductive ratio is 1. In this paper, we study near-critical behaviour in the context of a susceptible-infective-recovered epidemic on a random (multi)graph on n vertices with a given degree sequence. We concentrate on the regime just above the threshold for the emergence of a large epidemic, where the basic reproductive ratio is 1 + omega (n) n^{-1/3}, with omega (n) tending to infinity slowly as the population size, n, tends to infinity. We determine the probability that a large epidemic occurs, and the size of a large epidemic. Our results require basic regularity conditions on the degree sequences, and the assumption that the third moment of the degree of a random susceptible vertex stays uniformly bounded as n rightarrow infty . As a corollary, we determine the probability and size of a large near-critical epidemic on a standard binomial random graph in the ‘sparse’ regime, where the average degree is constant. As a further consequence of our method, we obtain an improved result on the size of the giant component in a random graph with given degrees just above the critical window, proving a conjecture by Janson and Luczak.

The Total Acquisition Number of Random Graphs

Let $G$ be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, provided that the weight on $v$ is at least as large as the weight on $u$. The total acquisition number of $G$, denoted by $a_t(G)$, is the minimum possible size of the set of vertices with positive weight at the end of the process.LeSaulnier, Prince, Wenger, West, and Worah asked for the minimum value of $p=p(n)$ such that $a_t(\mathcal{G}(n,p)) = 1$ with high probability, where $\mathcal{G}(n,p)$ is a binomial random graph. We show that $p = \frac{\log_2 n}{n} \approx 1.4427 \ \frac{\log n}{n}$ is a sharp threshold for this property. We also show that almost all trees $T$ satisfy $a_t(T) = \Theta(n)$, confirming a conjecture of West.