7,909 research outputs found
Statistical Mechanics of Semi-Supervised Clustering in Sparse Graphs
We theoretically study semi-supervised clustering in sparse graphs in the
presence of pairwise constraints on the cluster assignments of nodes. We focus
on bi-cluster graphs, and study the impact of semi-supervision for varying
constraint density and overlap between the clusters. Recent results for
unsupervised clustering in sparse graphs indicate that there is a critical
ratio of within-cluster and between-cluster connectivities below which clusters
cannot be recovered with better than random accuracy. The goal of this paper is
to examine the impact of pairwise constraints on the clustering accuracy. Our
results suggests that the addition of constraints does not provide automatic
improvement over the unsupervised case. When the density of the constraints is
sufficiently small, their only impact is to shift the detection threshold while
preserving the criticality. Conversely, if the density of (hard) constraints is
above the percolation threshold, the criticality is suppressed and the
detection threshold disappears.Comment: 8 pages, 4 figure
Nucleation scaling in jigsaw percolation
Jigsaw percolation is a nonlocal process that iteratively merges connected
clusters in a deterministic "puzzle graph" by using connectivity properties of
a random "people graph" on the same set of vertices. We presume the
Erdos--Renyi people graph with edge probability p and investigate the
probability that the puzzle is solved, that is, that the process eventually
produces a single cluster. In some generality, for puzzle graphs with N
vertices of degrees about D (in the appropriate sense), this probability is
close to 1 or small depending on whether pD(log N) is large or small. The one
dimensional ring and two dimensional torus puzzles are studied in more detail
and in many cases the exact scaling of the critical probability is obtained.
The paper settles several conjectures posed by Brummitt, Chatterjee, Dey, and
Sivakoff who introduced this model.Comment: 39 pages, 3 figures. Moved main results to the introduction and
improved exposition of section
Indistinguishability of Percolation Clusters
We show that when percolation produces infinitely many infinite clusters on a
Cayley graph, one cannot distinguish the clusters from each other by any
invariantly defined property. This implies that uniqueness of the infinite
cluster is equivalent to non-decay of connectivity (a.k.a. long-range order).
We then derive applications concerning uniqueness in Kazhdan groups and in
wreath products, and inequalities for .Comment: To appear in Ann. Proba
Spectral asymptotics of the Laplacian on supercritical bond-percolation graphs
We investigate Laplacians on supercritical bond-percolation graphs with
different boundary conditions at cluster borders. The integrated density of
states of the Dirichlet Laplacian is found to exhibit a Lifshits tail at the
lower spectral edge, while that of the Neumann Laplacian shows a van Hove
asymptotics, which results from the percolating cluster. At the upper spectral
edge, the behaviour is reversed.Comment: 16 pages, typos corrected, to appear in J. Funct. Ana
Transience and recurrence of random walks on percolation clusters in an ultrametric space
We study existence of percolation in the hierarchical group of order ,
which is an ultrametric space, and transience and recurrence of random walks on
the percolation clusters. The connection probability on the hierarchical group
for two points separated by distance is of the form , with , non-negative constants , and . Percolation was proved in Dawson and Gorostiza
(2013) for , with
. In this paper we improve the result for the critical case by
showing percolation for . We use a renormalization method of the type
in the previous paper in a new way which is more intrinsic to the model. The
proof involves ultrametric random graphs (described in the Introduction). The
results for simple (nearest neighbour) random walks on the percolation clusters
are: in the case the walk is transient, and in the critical case
, there exists a critical
such that the walk is recurrent for and transient for
. The proofs involve graph diameters, path lengths, and
electric circuit theory. Some comparisons are made with behaviours of random
walks on long-range percolation clusters in the one-dimensional Euclidean
lattice.Comment: 27 page
Uniqueness and non-uniqueness in percolation theory
This paper is an up-to-date introduction to the problem of uniqueness versus
non-uniqueness of infinite clusters for percolation on and,
more generally, on transitive graphs. For iid percolation on ,
uniqueness of the infinite cluster is a classical result, while on certain
other transitive graphs uniqueness may fail. Key properties of the graphs in
this context turn out to be amenability and nonamenability. The same problem is
considered for certain dependent percolation models -- most prominently the
Fortuin--Kasteleyn random-cluster model -- and in situations where the standard
connectivity notion is replaced by entanglement or rigidity. So-called
simultaneous uniqueness in couplings of percolation processes is also
considered. Some of the main results are proved in detail, while for others the
proofs are merely sketched, and for yet others they are omitted. Several open
problems are discussed.Comment: Published at http://dx.doi.org/10.1214/154957806000000096 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Statistical mechanics of complex networks
Complex networks describe a wide range of systems in nature and society, much
quoted examples including the cell, a network of chemicals linked by chemical
reactions, or the Internet, a network of routers and computers connected by
physical links. While traditionally these systems were modeled as random
graphs, it is increasingly recognized that the topology and evolution of real
networks is governed by robust organizing principles. Here we review the recent
advances in the field of complex networks, focusing on the statistical
mechanics of network topology and dynamics. After reviewing the empirical data
that motivated the recent interest in networks, we discuss the main models and
analytical tools, covering random graphs, small-world and scale-free networks,
as well as the interplay between topology and the network's robustness against
failures and attacks.Comment: 54 pages, submitted to Reviews of Modern Physic
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