Percolation on dense graph sequences

In this paper we determine the percolation threshold for an arbitrary sequence of dense graphs $(G_n)$. Let $\lambda_n$ be the largest eigenvalue of the adjacency matrix of $G_n$, and let $G_n(p_n)$ be the random subgraph of $G_n$ obtained by keeping each edge independently with probability $p_n$. We show that the appearance of a giant component in $G_n(p_n)$ has a sharp threshold at $p_n=1/\lambda_n$. In fact, we prove much more: if $(G_n)$ converges to an irreducible limit, then the density of the largest component of $G_n(c/n)$ tends to the survival probability of a multi-type branching process defined in terms of this limit. Here the notions of convergence and limit are those of Borgs, Chayes, Lov\'asz, S\'os and Vesztergombi. In addition to using basic properties of convergence, we make heavy use of the methods of Bollob\'as, Janson and Riordan, who used multi-type branching processes to study the emergence of a giant component in a very broad family of sparse inhomogeneous random graphs.Comment: Published in at http://dx.doi.org/10.1214/09-AOP478 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Limits of dense graph sequences

We show that if a sequence of dense graphs has the property that for every fixed graph F, the density of copies of F in these graphs tends to a limit, then there is a natural ``limit object'', namely a symmetric measurable 2-variable function on [0,1]. This limit object determines all the limits of subgraph densities. We also show that the graph parameters obtained as limits of subgraph densities can be characterized by ``reflection positivity'', semidefiniteness of an associated matrix. Conversely, every such function arises as a limit object. Along the lines we introduce a rather general model of random graphs, which seems to be interesting on its own right.Comment: 27 pages; added extension of result (Sept 22, 2004

Γ\Gamma-limit of the cut functional on dense graph sequences

A sequence of graphs with diverging number of nodes is a dense graph sequence if the number of edges grows approximately as for complete graphs. To each such sequence a function, called graphon, can be associated, which contains information about the asymptotic behavior of the sequence. Here we show that the problem of subdividing a large graph in communities with a minimal amount of cuts can be approached in terms of graphons and the $\Gamma$-limit of the cut functional, and discuss the resulting variational principles on some examples. Since the limit cut functional is naturally defined on Young measures, in many instances the partition problem can be expressed in terms of the probability that a node belongs to one of the communities. Our approach can be used to obtain insights into the bisection problem for large graphs, which is known to be NP-complete.Comment: 25 pages, 5 figure

Chromatic roots and limits of dense graphs

In this short note we observe that recent results of Abert and Hubai and of Csikvari and Frenkel about Benjamini--Schramm continuity of the holomorphic moments of the roots of the chromatic polynomial extend to the theory of dense graph sequences. We offer a number of problems and conjectures motivated by this observation.Comment: 9 page

Glauber Dynamics for Ising Model on Convergent Dense Graph Sequences

We study the Glauber dynamics for Ising model on (sequences of) dense graphs. We view the dense graphs through the lens of graphons. For the ferromagnetic Ising model with inverse temperature beta on a convergent sequence of graphs G_n with limit graphon W we show fast mixing of the Glauber dynamics if beta * lambda_1(W) 1 (where lambda_1(W)is the largest eigenvalue of the graphon). We also show that in the case beta * lambda_1(W) = 1 there is insufficient information to determine the mixing time (it can be either fast or slow)