8,679 research outputs found
Approximate Closest Community Search in Networks
Recently, there has been significant interest in the study of the community
search problem in social and information networks: given one or more query
nodes, find densely connected communities containing the query nodes. However,
most existing studies do not address the "free rider" issue, that is, nodes far
away from query nodes and irrelevant to them are included in the detected
community. Some state-of-the-art models have attempted to address this issue,
but not only are their formulated problems NP-hard, they do not admit any
approximations without restrictive assumptions, which may not always hold in
practice.
In this paper, given an undirected graph G and a set of query nodes Q, we
study community search using the k-truss based community model. We formulate
our problem of finding a closest truss community (CTC), as finding a connected
k-truss subgraph with the largest k that contains Q, and has the minimum
diameter among such subgraphs. We prove this problem is NP-hard. Furthermore,
it is NP-hard to approximate the problem within a factor , for
any . However, we develop a greedy algorithmic framework,
which first finds a CTC containing Q, and then iteratively removes the furthest
nodes from Q, from the graph. The method achieves 2-approximation to the
optimal solution. To further improve the efficiency, we make use of a compact
truss index and develop efficient algorithms for k-truss identification and
maintenance as nodes get eliminated. In addition, using bulk deletion
optimization and local exploration strategies, we propose two more efficient
algorithms. One of them trades some approximation quality for efficiency while
the other is a very efficient heuristic. Extensive experiments on 6 real-world
networks show the effectiveness and efficiency of our community model and
search algorithms
On the Spectrum of Hecke Type Operators related to some Fractal Groups
We give the first example of a connected 4-regular graph whose Laplace
operator's spectrum is a Cantor set, as well as several other computations of
spectra following a common ``finite approximation'' method. These spectra are
simple transforms of the Julia sets associated to some quadratic maps. The
graphs involved are Schreier graphs of fractal groups of intermediate growth,
and are also ``substitutional graphs''. We also formulate our results in terms
of Hecke type operators related to some irreducible quasi-regular
representations of fractal groups and in terms of the Markovian operator
associated to noncommutative dynamical systems via which these fractal groups
were originally defined. In the computations we performed, the self-similarity
of the groups is reflected in the self-similarity of some operators; they are
approximated by finite counterparts whose spectrum is computed by an ad hoc
factorization process.Comment: 1 color figure, 2 color diagrams, many figure
Growth Tight Actions
We introduce and systematically study the concept of a growth tight action.
This generalizes growth tightness for word metrics as initiated by Grigorchuk
and de la Harpe. Given a finitely generated, non-elementary group acting on
a --space , we prove that if contains a strongly
contracting element and if is not too badly distorted in ,
then the action of on is a growth tight action. It follows
that if is a cocompact, relatively hyperbolic --space, then
the action of on is a growth tight action. This generalizes
all previously known results for growth tightness of cocompact actions: every
already known example of a group that admits a growth tight action and has some
infinite, infinite index normal subgroups is relatively hyperbolic, and,
conversely, relatively hyperbolic groups admit growth tight actions. This also
allows us to prove that many CAT(0) groups, including flip-graph-manifold
groups and many Right Angled Artin Groups, and snowflake groups admit
cocompact, growth tight actions. These provide first examples of non-relatively
hyperbolic groups admitting interesting growth tight actions. Our main result
applies as well to cusp uniform actions on hyperbolic spaces and to the action
of the mapping class group on Teichmueller space with the Teichmueller metric.
Towards the proof of our main result, we give equivalent characterizations of
strongly contracting elements and produce new examples of group actions with
strongly contracting elements.Comment: 29 pages, 4 figures v2 added references v3 40 pages, 6 figures,
expanded preliminary sections to make paper more self-contained, other minor
improvements v4 updated bibliography, to appear in Pacific Journal of
Mathematic
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