42,183 research outputs found
More on quasi-random graphs, subgraph counts and graph limits
We study some properties of graphs (or, rather, graph sequences) defined by
demanding that the number of subgraphs of a given type, with vertices in
subsets of given sizes, approximatively equals the number expected in a random
graph. It has been shown by several authors that several such conditions are
quasi-random, but that there are exceptions. In order to understand this
better, we investigate some new properties of this type. We show that these
properties too are quasi-random, at least in some cases; however, there are
also cases that are left as open problems, and we discuss why the proofs fail
in these cases.
The proofs are based on the theory of graph limits; and on the method and
results developed by Janson (2011), this translates the combinatorial problem
to an analytic problem, which then is translated to an algebraic problem.Comment: 35 page
Processes on Unimodular Random Networks
We investigate unimodular random networks. Our motivations include their
characterization via reversibility of an associated random walk and their
similarities to unimodular quasi-transitive graphs. We extend various theorems
concerning random walks, percolation, spanning forests, and amenability from
the known context of unimodular quasi-transitive graphs to the more general
context of unimodular random networks. We give properties of a trace associated
to unimodular random networks with applications to stochastic comparison of
continuous-time random walk.Comment: 66 pages; 3rd version corrects formula (4.4) -- the published version
is incorrect --, as well as a minor error in the proof of Proposition 4.10;
4th version corrects proof of Proposition 7.1; 5th version corrects proof of
Theorem 5.1; 6th version makes a few more minor correction
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