相关论文: The phase transition in inhomogeneous random graph…

We study a point process describing the asymptotic behavior of sizes of the largest components of the random graph G(n,p) in the critical window p=n^{-1}+lambda n^{-4/3}. In particular, we show that this point process has a surprising…

The independence number of a sparse random graph G(n,m) of average degree d=2m/n is well-known to be \alpha(G(n,m))~2n ln(d)/d with high probability. Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size (1+o(1)) n…

We derive the full phase diagram for a large family of two-parameter exponential random graph models, each containing a first order transition curve ending in a critical point.

We study the asymptotics of large, moderate and normal deviations for the connected components of the sparse random graph by the method of stochastic processes. We obtain the logarithmic asymptotics of large deviations of the joint…

The energy level statistics of uniform random graphs are studied, by treating the graphs as random tight-binding lattices. The inherent random geometry of the graphs and their dynamical spatial dimensionality, leads to various quantum…

We consider bond percolation on $n$ vertices on a circle where edges are permitted between vertices whose spacing is at most some number L=L(n). We show that the resulting random graph gets a giant component when $L\gg(\log n)^2$ (when the…

Our general subject is the emergence of phases, and phase transitions, in large networks subjected to a few variable constraints. Our main result is the analysis, in the model using edge and triangle subdensities for constraints, of a sharp…

To provide a phenomenological theory for the various interesting transitions in restructuring networks we employ a statistical mechanical approach with detailed balance satisfied for the transitions between topological states. This enables…

We apply here methods of inhomogeneous random graphs to a class of random distance graphs. This provides an example outside of the rank 1 models which is still solvable as long as the largest connected component is concerned. In particular,…

A growing random graph is constructed by successively sampling without replacement an element from the pool of virtual vertices and edges. At start of the process the pool contains $N$ virtual vertices and no edges. Each time a vertex is…

Random graphs defined by an occurrence probability that is invariant under node aggregation have been identified recently in the context of network renormalization. The invariance property requires that edges are drawn with a specific…

Random non-commutative geometries are a novel approach to taking a non-perturbative path integral over geometries. They were introduced in arxiv.org/abs/1510.01377, where a first examination was performed. During this examination we found…

We give sufficient conditions under which a random graph with a specified degree sequence is symmetric or asymmetric. In the case of bounded degree sequences, our characterisation captures the phase transition of the symmetry of the random…

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…

We consider an infinite spatial inhomogeneous random graph model with an integrable connection kernel that interpolates nicely between existing spatial random graph models. Key examples are versions of the weight-dependent random connection…

Testing for independence between graphs is a problem that arises naturally in social network analysis and neuroscience. In this paper, we address independence testing for inhomogeneous Erd\H{o}s-R\'{e}nyi random graphs on the same vertex…

Recently, we adapted random walk arguments based on work of Nachmias and Peres, Martin-L\"of, Karp and Aldous to give a simple proof of the asymptotic normality of the size of the giant component in the random graph $G(n,p)$ above the phase…

Inhomogeneous random graphs are fundamental models for real-world networks, where prescribed degrees are imposed as soft constraints. A common assumption in such models is that the degree distribution follows a power-law, capturing the…

The interchange process on a finite graph is obtained by placing a particle on each vertex of the graph, then at rate 1, selecting an edge uniformly at random and swapping the two particles at either end of this edge. In this paper we…

We study the $k$-core of a random (multi)graph on $n$ vertices with a given degree sequence. In our previous paper [Random Structures Algorithms 30 (2007) 50--62] we used properties of empirical distributions of independent random variables…