Abstract. A straightforward linear time canonical labeling algorithm is shown to apply to almost all graphs (i .e. all but o(2 (2 >) of the 2 t 1 graphs on n vertices) . Hence, for almost all graphs X, any graph Y can be easily tested for isomorphism to X by an extremely naive linear time algorithm . This result is based on the following : In almost all graphs on n vertices, the largest n 0 .15...

Synonyms p* models, p-star models, p1 models, exponential family of random graphs, maximum entropy random networks, logit models, Markov graphs Glossary • Graph and network: the terms are used interchangeably in this essay. • Real-world network: (real network, observed network) means network data the researcher has collected and is interested in modelling. • Ensemble of graphs: means the set of...

Let G be any connected graph, and let λ1 and ρ denote the spectral radius of G and the universal cover of G, respectively. In [LP09], Linial and Puder have shown that almost every n-lift of G has all of its new eigenvalues bounded by O(λ 1/3 1 ρ 2/3). Friedman had conjectured that this bound can be improved to ρ+ on(1) (e.g., see [Fri03, HLW06]). In [LP09], Linial and Puder have formulated two ...

The random graph, or Rado’s graph (the unique countable homogeneous graph) has made an appearance in many parts of mathematics since its first occurrences in the early 1960s. In this paper I will discuss some old and new results on this remarkable structure.

We propose a principled framework for designing random graph models from minimal assumptions. Our central principle is to study the uniform measure over symmetric subsets of graphs. The symmetries we require are far less limiting than those in existing graph models and flexible enough to encompass both geometric features and properties of graph limits. Our main contribution is to derive natural...

Let Pn be the order determined by taking a random graph G on {l,2,...,n}, directing the edges from the lesser vertex to the greater (as integers), and then taking the transitive closure of this relation. We call such an ordered set a random graph order. We show that there exist constants c, and a, such that the expected height and set up number of Pn are sharply concentrated around en and cm re...

Vertices of an affiliation network are linked to features and two vertices are declared adjacent whenever they share a common feature. We introduce a random intersection graph process aimed at modeling sparse evolving affiliation networks. We establish the asymptotic degree distribution and provide explicit asymptotic formulas for assortativity and clustering coefficients showing how these edge...

Canonical labeling of a graph consists of assigning a unique label to each vertex such that the labels are invariant under isomorphism. Such a labeling can be used to solve the graph isomorphism problem. We give a simple, linear time, high probability algorithm for the canonical labeling of a G(n, p) random graph for p ∈ [ ω ( ln n/n ln lnn ) , 1− ω ( ln n/n ln lnn )] . Our result covers a gap ...