145 research outputs found
Graph properties, graph limits and entropy
We study the relation between the growth rate of a graph property and the
entropy of the graph limits that arise from graphs with that property. In
particular, for hereditary classes we obtain a new description of the colouring
number, which by well-known results describes the rate of growth.
We study also random graphs and their entropies. We show, for example, that
if a hereditary property has a unique limiting graphon with maximal entropy,
then a random graph with this property, selected uniformly at random from all
such graphs with a given order, converges to this maximizing graphon as the
order tends to infinity.Comment: 24 page
On the Typical Structure of Graphs in a Monotone Property
Given a graph property , it is interesting to determine the
typical structure of graphs that satisfy . In this paper, we
consider monotone properties, that is, properties that are closed under taking
subgraphs. Using results from the theory of graph limits, we show that if
is a monotone property and is the largest integer for which
every -colorable graph satisfies , then almost every graph with
is close to being a balanced -partite graph.Comment: 5 page
On String Graph Limits and the Structure of a Typical String Graph
We study limits of convergent sequences of string graphs, that is, graphs
with an intersection representation consisting of curves in the plane. We use
these results to study the limiting behavior of a sequence of random string
graphs. We also prove similar results for several related graph classes.Comment: 18 page
Phase transitions in a complex network
We study a mean field model of a complex network, focusing on edge and
triangle densities. Our first result is the derivation of a variational
characterization of the entropy density, compatible with the infinite node
limit. We then determine the optimizing graphs for small triangle density and a
range of edge density, though we can only prove they are local, not global,
maxima of the entropy density. With this assumption we then prove that the
resulting entropy density must lose its analyticity in various regimes. In
particular this implies the existence of a phase transition between distinct
heterogeneous multipartite phases at low triangle density, and a phase
transition between these phases and the disordered phase at high triangle
density.Comment: Title of previous version was `A mean field analysis of the
fluid/solid phase transition
Ground States for Exponential Random Graphs
We propose a perturbative method to estimate the normalization constant in
exponential random graph models as the weighting parameters approach infinity.
As an application, we give evidence of discontinuity in natural parametrization
along the critical directions of the edge-triangle model.Comment: 12 pages, 3 figures, 1 tabl
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