Related papers: Properties of stochastic Kronecker graphs

It is commonly believed that real networks are scale-free and fraction of nodes $P(k)$ with degree $k$ satisfies the power law $P(k) \propto k^{-\gamma} \text{ for } k > k_{min} > 0$. Preferential attachment is the mechanism that has been…

Rank 1 inhomogeneous random graphs are a natural generalization of Erd\H{o}s R\'enyi random graphs. In this generalization each node is given a weight. Then the probability that an edge is present depends on the product of the weights of…

An intuitive property of a random graph is that its subgraphs should also appear randomly distributed. We consider graphs whose subgraph densities exactly match their expected values. We call graphs with this property for all subgraphs with…

We prove a metric space scaling limit for a critical random graph with independent and identically distributed degrees having power-law tail behaviour with exponent $\alpha+1$, where $\alpha \in (1,2)$. The limiting components are…

Graph models have long been used in lieu of real data which can be expensive and hard to come by. A common class of models constructs a matrix of probabilities, and samples an adjacency matrix by flipping a weighted coin for each entry.…

We offer an alternative proof, using the Stein-Chen method, of Bollob\'{a}s' theorem concerning the distribution of the extreme degrees of a random graph. Our proof also provides a rate of convergence of the extreme degree to its asymptotic…

Random hyperbolic graphs were recently introduced by Krioukov et. al. [KPKVB10] as a model for large networks. Gugelmann, Panagiotou, and Peter [GPP12] then initiated the rigorous study of random hyperbolic graphs using the following model:…

Let $G$ be a $d$-regular graph on $n$ vertices. Frieze, Gould, Karo\'nski and Pfender began the study of the following random spanning subgraph model $H=H(G)$. Assign independently to each vertex $v$ of $G$ a uniform random number $x(v) \in…

We describe a new random greedy algorithm for generating regular graphs of high girth: Let $k\geq 3$ and $c \in (0,1)$ be fixed. Let $n \in \mathbb{N}$ be even and set $g = c \log_{k-1} (n)$. Begin with a Hamilton cycle $G$ on $n$ vertices.…

A uniformly random graph on $n$ vertices with a fixed degree sequence, obeying a $\gamma$ subpower law, is studied. It is shown that, for $\gamma>3$, in a subcritical phase with high probability the largest component size does not exceed…

The following random graph model was introduced for the evolution of protein-protein interaction networks: Let $\mathcal G = (G_n)_{n=n_0, n_0+1,...}$ be a sequence of random graphs, where $G_n = (V_n, E_n)$ is a graph with $|V_n|=n$…

We introduce the asymptotic spectrum of graphs and apply the theory of asymptotic spectra of Strassen (J. Reine Angew. Math. 1988) to obtain a new dual characterisation of the Shannon capacity of graphs. Elements in the asymptotic spectrum…

The famous P\'{o}sa-Seymour conjecture, confirmed in 1998 by Koml\'{o}s, S\'{a}rk\"{o}zy, and Szemer\'{e}di, states that for any $k \geq 2$, every graph on $n$ vertices with minimum degree $kn/(k + 1)$ contains the $k$-th power of a…

For a random intersection graph with a power law degree sequence having a finite mean and an infinite variance we show that the global clustering coefficient admits a tunable asymptotic distribution.

Recently several authors have proposed stochastic models of the growth of the Web graph that give rise to power-law distributions. These models are based on the notion of preferential attachment leading to the ``rich get richer''…

In the branch of mathematics known as graph theory, graphs are considered as a set of points, called vertices, with connections between these points, called edges. The purpose of this paper is to study mappings between two graphs that have…

We study the joint degree counts in proportional attachment random graphs and find a simple representation for the limit distribution in infinite sequence space. We show weak convergence with respect to the p-norm topology for appropriate p…

There has been much recent interest in random graphs sampled uniformly from the n-vertex graphs in a suitable minor-closed class, such as the class of all planar graphs. Here we use combinatorial and probabilistic methods to investigate a…

Scale-free percolation is a stochastic model for complex networks. In this spatial random graph model, vertices $x,y\in\mathbb{Z}^d$ are linked by an edge with probability depending on i.i.d.\ vertex weights and the Euclidean distance…

A classical result by Otter shows that the complete graph has an exponential number of non-isomorphic spanning trees. This was recently extended by Lee to every almost regular graph of sufficiently large degree. In this paper, we consider…