1,133 research outputs found
Upper bounds on the bisection width of 3- and 4-regular graphs
AbstractWe derive new upper bounds on the bisection width of graphs which have a regular vertex degree. We show that the bisection width of sufficiently large 3-regular graphs with |V| vertices is at most (16+ε)|V|, ε>0. For the bisection width of sufficiently large 4-regular graphs we show an upper bound of (25+ε)|V|, ε>0
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
The Peculiar Phase Structure of Random Graph Bisection
The mincut graph bisection problem involves partitioning the n vertices of a
graph into disjoint subsets, each containing exactly n/2 vertices, while
minimizing the number of "cut" edges with an endpoint in each subset. When
considered over sparse random graphs, the phase structure of the graph
bisection problem displays certain familiar properties, but also some
surprises. It is known that when the mean degree is below the critical value of
2 log 2, the cutsize is zero with high probability. We study how the minimum
cutsize increases with mean degree above this critical threshold, finding a new
analytical upper bound that improves considerably upon previous bounds.
Combined with recent results on expander graphs, our bound suggests the unusual
scenario that random graph bisection is replica symmetric up to and beyond the
critical threshold, with a replica symmetry breaking transition possibly taking
place above the threshold. An intriguing algorithmic consequence is that
although the problem is NP-hard, we can find near-optimal cutsizes (whose ratio
to the optimal value approaches 1 asymptotically) in polynomial time for
typical instances near the phase transition.Comment: substantially revised section 2, changed figures 3, 4 and 6, made
minor stylistic changes and added reference
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
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