1,244 research outputs found
Generalized Markov stability of network communities
We address the problem of community detection in networks by introducing a
general definition of Markov stability, based on the difference between the
probability fluxes of a Markov chain on the network at different time scales.
The specific implementation of the quality function and the resulting optimal
community structure thus become dependent both on the type of Markov process
and on the specific Markov times considered. For instance, if we use a natural
Markov chain dynamics and discount its stationary distribution -- that is, we
take as reference process the dynamics at infinite time -- we obtain the
standard formulation of the Markov stability. Notably, the possibility to use
finite-time transition probabilities to define the reference process naturally
allows detecting communities at different resolutions, without the need to
consider a continuous-time Markov chain in the small time limit. The main
advantage of our general formulation of Markov stability based on dynamical
flows is that we work with lumped Markov chains on network partitions, having
the same stationary distribution of the original process. In this way the form
of the quality function becomes invariant under partitioning, leading to a
self-consistent definition of community structures at different aggregation
scales
Personalized PageRank with Node-dependent Restart
Personalized PageRank is an algorithm to classify the improtance of web pages
on a user-dependent basis. We introduce two generalizations of Personalized
PageRank with node-dependent restart. The first generalization is based on the
proportion of visits to nodes before the restart, whereas the second
generalization is based on the probability of visited node just before the
restart. In the original case of constant restart probability, the two measures
coincide. We discuss interesting particular cases of restart probabilities and
restart distributions. We show that the both generalizations of Personalized
PageRank have an elegant expression connecting the so-called direct and reverse
Personalized PageRanks that yield a symmetry property of these Personalized
PageRanks
Clustering and Community Detection in Directed Networks: A Survey
Networks (or graphs) appear as dominant structures in diverse domains,
including sociology, biology, neuroscience and computer science. In most of the
aforementioned cases graphs are directed - in the sense that there is
directionality on the edges, making the semantics of the edges non symmetric.
An interesting feature that real networks present is the clustering or
community structure property, under which the graph topology is organized into
modules commonly called communities or clusters. The essence here is that nodes
of the same community are highly similar while on the contrary, nodes across
communities present low similarity. Revealing the underlying community
structure of directed complex networks has become a crucial and
interdisciplinary topic with a plethora of applications. Therefore, naturally
there is a recent wealth of research production in the area of mining directed
graphs - with clustering being the primary method and tool for community
detection and evaluation. The goal of this paper is to offer an in-depth review
of the methods presented so far for clustering directed networks along with the
relevant necessary methodological background and also related applications. The
survey commences by offering a concise review of the fundamental concepts and
methodological base on which graph clustering algorithms capitalize on. Then we
present the relevant work along two orthogonal classifications. The first one
is mostly concerned with the methodological principles of the clustering
algorithms, while the second one approaches the methods from the viewpoint
regarding the properties of a good cluster in a directed network. Further, we
present methods and metrics for evaluating graph clustering results,
demonstrate interesting application domains and provide promising future
research directions.Comment: 86 pages, 17 figures. Physics Reports Journal (To Appear
Dynamics-based centrality for general directed networks
Determining the relative importance of nodes in directed networks is
important in, for example, ranking websites, publications, and sports teams,
and for understanding signal flows in systems biology. A prevailing centrality
measure in this respect is the PageRank. In this work, we focus on another
class of centrality derived from the Laplacian of the network. We extend the
Laplacian-based centrality, which has mainly been applied to strongly connected
networks, to the case of general directed networks such that we can
quantitatively compare arbitrary nodes. Toward this end, we adopt the idea used
in the PageRank to introduce global connectivity between all the pairs of nodes
with a certain strength. Numerical simulations are carried out on some
networks. We also offer interpretations of the Laplacian-based centrality for
general directed networks in terms of various dynamical and structural
properties of networks. Importantly, the Laplacian-based centrality defined as
the stationary density of the continuous-time random walk with random jumps is
shown to be equivalent to the absorption probability of the random walk with
sinks at each node but without random jumps. Similarly, the proposed centrality
represents the importance of nodes in dynamics on the original network supplied
with sinks but not with random jumps.Comment: 7 figure
Ranking algorithms on directed configuration networks
This paper studies the distribution of a family of rankings, which includes
Google's PageRank, on a directed configuration model. In particular, it is
shown that the distribution of the rank of a randomly chosen node in the graph
converges in distribution to a finite random variable that can
be written as a linear combination of i.i.d. copies of the endogenous solution
to a stochastic fixed point equation of the form where is a
real-valued vector with , , and the are i.i.d. copies of ,
independent of . Moreover, we
provide precise asymptotics for the limit , which when the
in-degree distribution in the directed configuration model has a power law
imply a power law distribution for with the same exponent
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