34,307 research outputs found
Random graphs from a weighted minor-closed class
There has been much recent interest in random graphs sampled uniformly from
the n-vertex graphs in a suitable minor-closed class, such as the class of all
planar graphs. Here we use combinatorial and probabilistic methods to
investigate a more general model. We consider random graphs from a
`well-behaved' class of graphs: examples of such classes include all
minor-closed classes of graphs with 2-connected excluded minors (such as
forests, series-parallel graphs and planar graphs), the class of graphs
embeddable on any given surface, and the class of graphs with at most k
vertex-disjoint cycles. Also, we give weights to edges and components to
specify probabilities, so that our random graphs correspond to the random
cluster model, appropriately conditioned.
We find that earlier results extend naturally in both directions, to general
well-behaved classes of graphs, and to the weighted framework, for example
results concerning the probability of a random graph being connected; and we
also give results on the 2-core which are new even for the uniform (unweighted)
case.Comment: 46 page
A Quasi-Polynomial Time Partition Oracle for Graphs with an Excluded Minor
Motivated by the problem of testing planarity and related properties, we
study the problem of designing efficient {\em partition oracles}. A {\em
partition oracle} is a procedure that, given access to the incidence lists
representation of a bounded-degree graph and a parameter \eps,
when queried on a vertex , returns the part (subset of vertices) which
belongs to in a partition of all graph vertices. The partition should be
such that all parts are small, each part is connected, and if the graph has
certain properties, the total number of edges between parts is at most \eps
|V|. In this work we give a partition oracle for graphs with excluded minors
whose query complexity is quasi-polynomial in 1/\eps, thus improving on the
result of Hassidim et al. ({\em Proceedings of FOCS 2009}) who gave a partition
oracle with query complexity exponential in 1/\eps. This improvement implies
corresponding improvements in the complexity of testing planarity and other
properties that are characterized by excluded minors as well as sublinear-time
approximation algorithms that work under the promise that the graph has an
excluded minor.Comment: 13 pages, 1 figur
Connectivity for bridge-addable monotone graph classes
A class A of labelled graphs is bridge-addable if for all graphs G in A and
all vertices u and v in distinct connected components of G, the graph obtained
by adding an edge between u and u is also in A; the class A is monotone if for
all G in A and all subgraphs H of G, H is also in A. We show that for any
bridge-addable, monotone class A whose elements have vertex set 1,...,n, the
probability that a uniformly random element of A is connected is at least
(1-o_n(1)) e^{-1/2}, where o_n(1) tends to zero as n tends to infinity. This
establishes the special case of a conjecture of McDiarmid, Steger and Welsh
when the condition of monotonicity is added. This result has also been obtained
independently by Kang and Panagiotiou (2011).Comment: 11 page
Approximating the Spectrum of a Graph
The spectrum of a network or graph with adjacency matrix ,
consists of the eigenvalues of the normalized Laplacian . This set of eigenvalues encapsulates many aspects of the structure
of the graph, including the extent to which the graph posses community
structures at multiple scales. We study the problem of approximating the
spectrum , of in the regime where the graph is too
large to explicitly calculate the spectrum. We present a sublinear time
algorithm that, given the ability to query a random node in the graph and
select a random neighbor of a given node, computes a succinct representation of
an approximation , such that . Our algorithm has query complexity and running time ,
independent of the size of the graph, . We demonstrate the practical
viability of our algorithm on 15 different real-world graphs from the Stanford
Large Network Dataset Collection, including social networks, academic
collaboration graphs, and road networks. For the smallest of these graphs, we
are able to validate the accuracy of our algorithm by explicitly calculating
the true spectrum; for the larger graphs, such a calculation is computationally
prohibitive.
In addition we study the implications of our algorithm to property testing in
the bounded degree graph model
On the probability of planarity of a random graph near the critical point
Consider the uniform random graph with vertices and edges.
Erd\H{o}s and R\'enyi (1960) conjectured that the limit
\lim_{n \to \infty} \Pr\{G(n,\textstyle{n\over 2}) is planar}} exists
and is a constant strictly between 0 and 1. \L uczak, Pittel and Wierman (1994)
proved this conjecture and Janson, \L uczak, Knuth and Pittel (1993) gave lower
and upper bounds for this probability.
In this paper we determine the exact probability of a random graph being
planar near the critical point . For each , we find an exact
analytic expression for
In particular, we obtain .
We extend these results to classes of graphs closed under taking minors. As
an example, we show that the probability of being
series-parallel converges to 0.98003.
For the sake of completeness and exposition we reprove in a concise way
several basic properties we need of a random graph near the critical point.Comment: 10 pages, 1 figur
Network Models in Class C on Arbitrary Graphs
We consider network models of quantum localisation in which a particle with a
two-component wave function propagates through the nodes and along the edges of
an arbitrary directed graph, subject to a random SU(2) rotation on each edge it
traverses. The propagation through each node is specified by an arbitrary but
fixed S-matrix. Such networks model localisation problems in class C of the
classification of Altland and Zirnbauer, and, on suitable graphs, they model
the spin quantum Hall transition. We extend the analyses of Gruzberg, Ludwig
and Read and of Beamond, Cardy and Chalker to show that, on an arbitrary graph,
the mean density of states and the mean conductance may be calculated in terms
of observables of a classical history-dependent random walk on the same graph.
The transition weights for this process are explicitly related to the elements
of the S-matrices. They are correctly normalised but, on graphs with nodes of
degree greater than 4, not necessarily non-negative (and therefore
interpretable as probabilities) unless a sufficient number of them happen to
vanish. Our methods use a supersymmetric path integral formulation of the
problem which is completely finite and rigorous.Comment: 17 pages, 3 figure
Connectivity for random graphs from a weighted bridge-addable class
There has been much recent interest in random graphs sampled uniformly from
the n-vertex graphs in a suitable structured class, such as the class of all
planar graphs. Here we consider a general 'bridge-addable' class of graphs - if
a graph is in the class and u and v are vertices in different components then
the graph obtained by adding an edge (bridge) between u and v must also be in
the class.
Various bounds are known concerning the probability of a random graph from
such a class being connected or having many components, sometimes under the
additional assumption that bridges can be deleted as well as added. Here we
improve or amplify or generalise these bounds. For example, we see that the
expected number of vertices left when we remove a largest component is less
than 2. The generalisation is to consider 'weighted' random graphs, sampled
from a suitable more general distribution, where the focus is on the bridges.Comment: 23 page
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