190,369 research outputs found
Network Information Flow in Small World Networks
Recent results from statistical physics show that large classes of complex
networks, both man-made and of natural origin, are characterized by high
clustering properties yet strikingly short path lengths between pairs of nodes.
This class of networks are said to have a small-world topology. In the context
of communication networks, navigable small-world topologies, i.e. those which
admit efficient distributed routing algorithms, are deemed particularly
effective, for example in resource discovery tasks and peer-to-peer
applications. Breaking with the traditional approach to small-world topologies
that privileges graph parameters pertaining to connectivity, and intrigued by
the fundamental limits of communication in networks that exploit this type of
topology, we investigate the capacity of these networks from the perspective of
network information flow. Our contribution includes upper and lower bounds for
the capacity of standard and navigable small-world models, and the somewhat
surprising result that, with high probability, random rewiring does not alter
the capacity of a small-world network.Comment: 23 pages, 8 fitures, submitted to the IEEE Transactions on
Information Theory, November 200
Collective versus hub activation of epidemic phases on networks
We consider a general criterion to discern the nature of the threshold in
epidemic models on scale-free (SF) networks. Comparing the epidemic lifespan of
the nodes with largest degrees with the infection time between them, we propose
a general dual scenario, in which the epidemic transition is either ruled by a
hub activation process, leading to a null threshold in the thermodynamic limit,
or given by a collective activation process, corresponding to a standard phase
transition with a finite threshold. We validate the proposed criterion applying
it to different epidemic models, with waning immunity or heterogeneous
infection rates in both synthetic and real SF networks. In particular, a waning
immunity, irrespective of its strength, leads to collective activation with
finite threshold in scale-free networks with large exponent, at odds with
canonical theoretical approaches.Comment: Revised version accepted for publication in PR
Subgraphs in preferential attachment models
We consider subgraph counts in general preferential attachment models with
power-law degree exponent . For all subgraphs , we find the scaling
of the expected number of subgraphs as a power of the number of vertices. We
prove our results on the expected number of subgraphs by defining an
optimization problem that finds the optimal subgraph structure in terms of the
indices of the vertices that together span it and by using the representation
of the preferential attachment model as a P\'olya urn model
On the Analysis of a Label Propagation Algorithm for Community Detection
This paper initiates formal analysis of a simple, distributed algorithm for
community detection on networks. We analyze an algorithm that we call
\textsc{Max-LPA}, both in terms of its convergence time and in terms of the
"quality" of the communities detected. \textsc{Max-LPA} is an instance of a
class of community detection algorithms called \textit{label propagation}
algorithms. As far as we know, most analysis of label propagation algorithms
thus far has been empirical in nature and in this paper we seek a theoretical
understanding of label propagation algorithms. In our main result, we define a
clustered version of \er random graphs with clusters where
the probability , of an edge connecting nodes within a cluster is
higher than , the probability of an edge connecting nodes in distinct
clusters. We show that even with fairly general restrictions on and
( for any , , where is the number of nodes), \textsc{Max-LPA} detects the
clusters in just two rounds. Based on this and on empirical
results, we conjecture that \textsc{Max-LPA} can correctly and quickly identify
communities on clustered \er graphs even when the clusters are much sparser,
i.e., with for some .Comment: 17 pages. Submitted to ICDCN 201
Variational principle for scale-free network motifs
For scale-free networks with degrees following a power law with an exponent
, the structures of motifs (small subgraphs) are not yet well
understood. We introduce a method designed to identify the dominant structure
of any given motif as the solution of an optimization problem. The unique
optimizer describes the degrees of the vertices that together span the most
likely motif, resulting in explicit asymptotic formulas for the motif count and
its fluctuations. We then classify all motifs into two categories: motifs with
small and large fluctuations
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