Convergent sequences of dense graphs II. Multiway cuts and statistical physics

Abstract

We consider sequences of graphs (Gn) and define various notions of convergence related to these sequences including "left-convergence," defined in terms of the densities of homomorphisms from small graphs into Gn, and "right-convergence," defined in terms of the densities of homomorphisms from Gn into small graphs.We show that right-convergence is equivalent to left-convergence, both for simple graphs Gn, and for graphs Gn with nontrivial nodeweights and edgeweights. Other equivalent conditions for convergence are given in terms of fundamental notions from combinatorics, such as maximum cuts and Szemerédi partitions, and fundamental notions from statistical physics, like energies and free energies. We thereby relate local and global properties of graph sequences. Quantitative forms of these results express the relationships among different measures of similarity of large graphs.

Web Science: Understanding the Emergence of Macro-Level Features on the World Wide Web

“…Borgs et al [60,61] have developed a graphical theory of limits, defined via graph homomorphisms, which allows the estimation of properties of large graphs, such as finding the approximate value of a parameter with associated probability, or determining whether the graph has a certain property. The limit property defined in this way has been shown to be equivalent to other well-known definitions of limits [57,62,63]. The result is that sampling and testing of a large graph (such as the Web, a graph so large that it cannot be completely described) can be performed with some confidence that key parameters have been preserved, and that bias can not only be defined, but also be eliminated with a determinable probability.…”

“…Secondly, there is a phase where the fit-get-richer, in which vertices of higher fitness grow faster than those of smaller fitness; the behaviour here is a power law within each fitness value, but the tail exponent decreases as the fitness increases. Finally, the graph moves towards an innovation-pays-off phase, where the competition for links results in a constant fraction of the links continuously shifting to ever larger fitness values [57,62].…”

Matrix and discrepancy view of generalized random and quasirandom graphs

“…Actually, the authors in [17] proved that for any k, the k largest absolute value normalized adjacency eigenvalues of a convergent graph sequence converge (to the corresponding eigenvalues of the limiting graphon). In [9] we proved the same for the normalized modularity spectra of convergent graph sequences.…”

“…In the k > 1 case, the deterministic counterparts of the generalized random graphs were first defined in [28] as graph sequences converging to a vertex-and edge-weighted graph (vertex-weights correspond to the relative sizes of the partition-members, whereas edge-weights to the probability matrix) in the sense of the homomorphism densities. Due to convergence facts on spectra [17], the generalized quasirandom graphs are spectrally equivalent to the generalized random graphs.…”

Weak regularity and finitely forcible graph limits

“…This line of research was sparked by limits of dense graphs [7][8][9]32], which we focus on here, followed by limits of other structures, e.g. permutations [22,23,28], sparse graphs [5,14] and partial orders [25].…”