61,780 research outputs found
Local Fiedler vector centrality for detection of deep and overlapping communities in networks
Abstract—In this paper, a new centrality called local Fiedler vector centrality (LFVC) is proposed to analyze the connectivity structure of a graph. It is associated with the sensitivity of algebraic connectivity to node or edge removals and features distributed computations via the associated graph Laplacian matrix. We prove that LFVC can be related to a monotonic submodular set function that guarantees that greedy node or edge removals come within a factor 11=e of the optimal non-greedy batch removal strategy. Due to the close relationship between graph topology and community structure, we use LFVC to detect deep and overlapping communities on real-world social network datasets. The results offer new insights on community detection by discovering new significant communities and key members in the network. Notably, LFVC is also shown to significantly out-perform other well-known centralities for community detection. I
Correlation Decay in Random Decision Networks
We consider a decision network on an undirected graph in which each node
corresponds to a decision variable, and each node and edge of the graph is
associated with a reward function whose value depends only on the variables of
the corresponding nodes. The goal is to construct a decision vector which
maximizes the total reward. This decision problem encompasses a variety of
models, including maximum-likelihood inference in graphical models (Markov
Random Fields), combinatorial optimization on graphs, economic team theory and
statistical physics. The network is endowed with a probabilistic structure in
which costs are sampled from a distribution. Our aim is to identify sufficient
conditions to guarantee average-case polynomiality of the underlying
optimization problem. We construct a new decentralized algorithm called Cavity
Expansion and establish its theoretical performance for a variety of models.
Specifically, for certain classes of models we prove that our algorithm is able
to find near optimal solutions with high probability in a decentralized way.
The success of the algorithm is based on the network exhibiting a correlation
decay (long-range independence) property. Our results have the following
surprising implications in the area of average case complexity of algorithms.
Finding the largest independent (stable) set of a graph is a well known NP-hard
optimization problem for which no polynomial time approximation scheme is
possible even for graphs with largest connectivity equal to three, unless P=NP.
We show that the closely related maximum weighted independent set problem for
the same class of graphs admits a PTAS when the weights are i.i.d. with the
exponential distribution. Namely, randomization of the reward function turns an
NP-hard problem into a tractable one
Fast Distributed PageRank Computation
Over the last decade, PageRank has gained importance in a wide range of
applications and domains, ever since it first proved to be effective in
determining node importance in large graphs (and was a pioneering idea behind
Google's search engine). In distributed computing alone, PageRank vector, or
more generally random walk based quantities have been used for several
different applications ranging from determining important nodes, load
balancing, search, and identifying connectivity structures. Surprisingly,
however, there has been little work towards designing provably efficient
fully-distributed algorithms for computing PageRank. The difficulty is that
traditional matrix-vector multiplication style iterative methods may not always
adapt well to the distributed setting owing to communication bandwidth
restrictions and convergence rates.
In this paper, we present fast random walk-based distributed algorithms for
computing PageRanks in general graphs and prove strong bounds on the round
complexity. We first present a distributed algorithm that takes O\big(\log
n/\eps \big) rounds with high probability on any graph (directed or
undirected), where is the network size and \eps is the reset probability
used in the PageRank computation (typically \eps is a fixed constant). We
then present a faster algorithm that takes O\big(\sqrt{\log n}/\eps \big)
rounds in undirected graphs. Both of the above algorithms are scalable, as each
node sends only small (\polylog n) number of bits over each edge per round.
To the best of our knowledge, these are the first fully distributed algorithms
for computing PageRank vector with provably efficient running time.Comment: 14 page
Distance matrices on the H-join of graphs: A general result and applications
Given a graph with vertices and a set of pairwise vertex disjoint graphs the vertex of is assigned to Let be the graph obtained from the graphs and the edges connecting each vertex of with all the vertices of for all edge of The graph is called the of . Let be a matrix on a graph . A general result on the eigenvalues of , when the all ones vector is an eigenvector of for , is given. This result is applied to obtain the distance eigenvalues, the distance Laplacian eigenvalues and as well as the distance signless Laplacian eigenvalues of when are regular graphs. Finally, we introduce the notions of the distance incidence energy and distance Laplacian-energy like of a graph and we derive sharp lower bounds on these two distance energies among all the connected graphs of prescribed order in terms of the vertex connectivity. The graphs for which those bounds are attained are characterized.publishe
Algebraic Connectivity and Degree Sequences of Trees
We investigate the structure of trees that have minimal algebraic
connectivity among all trees with a given degree sequence. We show that such
trees are caterpillars and that the vertex degrees are non-decreasing on every
path on non-pendant vertices starting at the characteristic set of the Fiedler
vector.Comment: 8 page
On the multiple unicast capacity of 3-source, 3-terminal directed acyclic networks
We consider the multiple unicast problem with three source-terminal pairs
over directed acyclic networks with unit-capacity edges. The three
pairs wish to communicate at unit-rate via network coding. The connectivity
between the pairs is quantified by means of a connectivity level
vector, such that there exist edge-disjoint paths between
and . In this work we attempt to classify networks based on the
connectivity level. It can be observed that unit-rate transmission can be
supported by routing if , for all . In this work,
we consider, connectivity level vectors such that . We present either a constructive linear network coding scheme or an
instance of a network that cannot support the desired unit-rate requirement,
for all such connectivity level vectors except the vector (and its
permutations). The benefits of our schemes extend to networks with higher and
potentially different edge capacities. Specifically, our experimental results
indicate that for networks where the different source-terminal paths have a
significant overlap, our constructive unit-rate schemes can be packed along
with routing to provide higher throughput as compared to a pure routing
approach.Comment: To appear in the IEEE/ACM Transactions on Networkin
- …