10,356 research outputs found
Connectivity of Random Annulus Graphs and the Geometric Block Model
We provide new connectivity results for {\em vertex-random graphs} or {\em
random annulus graphs} which are significant generalizations of random
geometric graphs. Random geometric graphs (RGG) are one of the most basic
models of random graphs for spatial networks proposed by Gilbert in 1961,
shortly after the introduction of the Erd\H{o}s-R\'{en}yi random graphs. They
resemble social networks in many ways (e.g. by spontaneously creating cluster
of nodes with high modularity). The connectivity properties of RGG have been
studied since its introduction, and analyzing them has been significantly
harder than their Erd\H{o}s-R\'{en}yi counterparts due to correlated edge
formation.
Our next contribution is in using the connectivity of random annulus graphs
to provide necessary and sufficient conditions for efficient recovery of
communities for {\em the geometric block model} (GBM). The GBM is a
probabilistic model for community detection defined over an RGG in a similar
spirit as the popular {\em stochastic block model}, which is defined over an
Erd\H{o}s-R\'{en}yi random graph. The geometric block model inherits the
transitivity properties of RGGs and thus models communities better than a
stochastic block model. However, analyzing them requires fresh perspectives as
all prior tools fail due to correlation in edge formation. We provide a simple
and efficient algorithm that can recover communities in GBM exactly with high
probability in the regime of connectivity
Hardness and Algorithms for Rainbow Connectivity
An edge-colored graph G is rainbow connected if any two vertices are
connected by a path whose edges have distinct colors. The rainbow connectivity
of a connected graph G, denoted rc(G), is the smallest number of colors that
are needed in order to make G rainbow connected. In addition to being a natural
combinatorial problem, the rainbow connectivity problem is motivated by
applications in cellular networks. In this paper we give the first proof that
computing rc(G) is NP-Hard. In fact, we prove that it is already NP-Complete to
decide if rc(G) = 2, and also that it is NP-Complete to decide whether a given
edge-colored (with an unbounded number of colors) graph is rainbow connected.
On the positive side, we prove that for every > 0, a connected graph
with minimum degree at least has bounded rainbow connectivity,
where the bound depends only on , and the corresponding coloring can
be constructed in polynomial time. Additional non-trivial upper bounds, as well
as open problems and conjectures are also pre sented
Typical solution time for a vertex-covering algorithm on finite-connectivity random graphs
In this letter, we analytically describe the typical solution time needed by
a backtracking algorithm to solve the vertex-cover problem on
finite-connectivity random graphs. We find two different transitions: The first
one is algorithm-dependent and marks the dynamical transition from linear to
exponential solution times. The second one gives the maximum computational
complexity, and is found exactly at the threshold where the system undergoes an
algorithm-independent phase transition in its solvability. Analytical results
are corroborated by numerical simulations.Comment: 4 pages, 2 figures, to appear in Phys. Rev. Let
Sampling-based Algorithms for Optimal Motion Planning
During the last decade, sampling-based path planning algorithms, such as
Probabilistic RoadMaps (PRM) and Rapidly-exploring Random Trees (RRT), have
been shown to work well in practice and possess theoretical guarantees such as
probabilistic completeness. However, little effort has been devoted to the
formal analysis of the quality of the solution returned by such algorithms,
e.g., as a function of the number of samples. The purpose of this paper is to
fill this gap, by rigorously analyzing the asymptotic behavior of the cost of
the solution returned by stochastic sampling-based algorithms as the number of
samples increases. A number of negative results are provided, characterizing
existing algorithms, e.g., showing that, under mild technical conditions, the
cost of the solution returned by broadly used sampling-based algorithms
converges almost surely to a non-optimal value. The main contribution of the
paper is the introduction of new algorithms, namely, PRM* and RRT*, which are
provably asymptotically optimal, i.e., such that the cost of the returned
solution converges almost surely to the optimum. Moreover, it is shown that the
computational complexity of the new algorithms is within a constant factor of
that of their probabilistically complete (but not asymptotically optimal)
counterparts. The analysis in this paper hinges on novel connections between
stochastic sampling-based path planning algorithms and the theory of random
geometric graphs.Comment: 76 pages, 26 figures, to appear in International Journal of Robotics
Researc
Substructure Discovery Using Minimum Description Length and Background Knowledge
The ability to identify interesting and repetitive substructures is an
essential component to discovering knowledge in structural data. We describe a
new version of our SUBDUE substructure discovery system based on the minimum
description length principle. The SUBDUE system discovers substructures that
compress the original data and represent structural concepts in the data. By
replacing previously-discovered substructures in the data, multiple passes of
SUBDUE produce a hierarchical description of the structural regularities in the
data. SUBDUE uses a computationally-bounded inexact graph match that identifies
similar, but not identical, instances of a substructure and finds an
approximate measure of closeness of two substructures when under computational
constraints. In addition to the minimum description length principle, other
background knowledge can be used by SUBDUE to guide the search towards more
appropriate substructures. Experiments in a variety of domains demonstrate
SUBDUE's ability to find substructures capable of compressing the original data
and to discover structural concepts important to the domain. Description of
Online Appendix: This is a compressed tar file containing the SUBDUE discovery
system, written in C. The program accepts as input databases represented in
graph form, and will output discovered substructures with their corresponding
value.Comment: See http://www.jair.org/ for an online appendix and other files
accompanying this articl
Colouring random graphs and maximising local diversity
We study a variation of the graph colouring problem on random graphs of
finite average connectivity. Given the number of colours, we aim to maximise
the number of different colours at neighbouring vertices (i.e. one edge
distance) of any vertex. Two efficient algorithms, belief propagation and
Walksat are adapted to carry out this task. We present experimental results
based on two types of random graphs for different system sizes and identify the
critical value of the connectivity for the algorithms to find a perfect
solution. The problem and the suggested algorithms have practical relevance
since various applications, such as distributed storage, can be mapped onto
this problem.Comment: 10 pages, 10 figure
Threshold phenomena in random graphs
In the 1950s, random graphs appeared for the first time in a result of the prolific hungarian mathematician Pál Erd\H{o}s. Since then, interest in random graph theory has only grown up until now. In its first stages, the basis of its theory were set, while they were mainly used in probability and combinatorics theory. However, with the new century and the boom of technologies like the World Wide Web, random graphs are even more important since they are extremely useful to handle problems in fields like network and communication theory. Because of this fact, nowadays random graphs are widely studied by the mathematical community around the world and new promising results have been recently achieved, showing an exciting future for this field. In this bachelor thesis, we focus our study on the threshold phenomena for graph properties within random graphs
Improved Parallel Algorithms for Spanners and Hopsets
We use exponential start time clustering to design faster and more
work-efficient parallel graph algorithms involving distances. Previous
algorithms usually rely on graph decomposition routines with strict
restrictions on the diameters of the decomposed pieces. We weaken these bounds
in favor of stronger local probabilistic guarantees. This allows more direct
analyses of the overall process, giving: * Linear work parallel algorithms that
construct spanners with stretch and size in unweighted
graphs, and size in weighted graphs. * Hopsets that lead
to the first parallel algorithm for approximating shortest paths in undirected
graphs with work
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