460 research outputs found
Modularity of regular and treelike graphs
Clustering algorithms for large networks typically use modularity values to
test which partitions of the vertex set better represent structure in the data.
The modularity of a graph is the maximum modularity of a partition. We consider
the modularity of two kinds of graphs.
For -regular graphs with a given number of vertices, we investigate the
minimum possible modularity, the typical modularity, and the maximum possible
modularity. In particular, we see that for random cubic graphs the modularity
is usually in the interval , and for random -regular graphs
with large it usually is of order . These results help to
establish baselines for statistical tests on regular graphs.
The modularity of cycles and low degree trees is known to be close to 1: we
extend these results to `treelike' graphs, where the product of treewidth and
maximum degree is much less than the number of edges. This yields for example
the (deterministic) lower bound mentioned above on the modularity of
random cubic graphs.Comment: 25 page
Factors of IID on Trees
Classical ergodic theory for integer-group actions uses entropy as a complete
invariant for isomorphism of IID (independent, identically distributed)
processes (a.k.a. product measures). This theory holds for amenable groups as
well. Despite recent spectacular progress of Bowen, the situation for
non-amenable groups, including free groups, is still largely mysterious. We
present some illustrative results and open questions on free groups, which are
particularly interesting in combinatorics, statistical physics, and
probability. Our results include bounds on minimum and maximum bisection for
random cubic graphs that improve on all past bounds.Comment: 18 pages, 1 figur
On Vertex Bisection Width of Random -Regular Graphs
Vertex bisection is a graph partitioning problem in which the aim is to find
a partition into two equal parts that minimizes the number of vertices in one
partition set that have a neighbor in the other set. We are interested in
giving upper bounds on the vertex bisection width of random -regular graphs
for constant values of . Our approach is based on analyzing a greedy
algorithm by using the Differential Equations Method. In this way, we obtain
the first known upper bounds for the vertex bisection width in random regular
graphs. The results are compared with experimental ones and with lower bounds
obtained by Kolesnik and Wormald, (Lower Bounds for the Isoperimetric Numbers
of Random Regular Graphs, SIAM J. on Disc. Math. 28(1), 553-575, 2014).Comment: 31 pages, 2 figure
Maximum edge-cuts in cubic graphs with large girth and in random cubic graphs
We show that for every cubic graph G with sufficiently large girth there
exists a probability distribution on edge-cuts of G such that each edge is in a
randomly chosen cut with probability at least 0.88672. This implies that G
contains an edge-cut of size at least 1.33008n, where n is the number of
vertices of G, and has fractional cut covering number at most 1.127752. The
lower bound on the size of maximum edge-cut also applies to random cubic
graphs. Specifically, a random n-vertex cubic graph a.a.s. contains an edge cut
of size 1.33008n
On the minimum bisection of random regular graphs
In this paper we give new asymptotically almost sure lower and upper bounds
on the bisection width of random regular graphs. The main contribution is a
new lower bound on the bisection width of , based on a first moment
method together with a structural decomposition of the graph, thereby improving
a 27 year old result of Kostochka and Melnikov. We also give a complementary
upper bound of , combining known spectral ideas with original
combinatorial insights. Developping further this approach, with the help of
Monte Carlo simulations, we obtain a non-rigorous upper bound of .Comment: 48 pages, 20 figure
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