286 research outputs found
Finding local community structure in networks
Although the inference of global community structure in networks has recently
become a topic of great interest in the physics community, all such algorithms
require that the graph be completely known. Here, we define both a measure of
local community structure and an algorithm that infers the hierarchy of
communities that enclose a given vertex by exploring the graph one vertex at a
time. This algorithm runs in time O(d*k^2) for general graphs when is the
mean degree and k is the number of vertices to be explored. For graphs where
exploring a new vertex is time-consuming, the running time is linear, O(k). We
show that on computer-generated graphs this technique compares favorably to
algorithms that require global knowledge. We also use this algorithm to extract
meaningful local clustering information in the large recommender network of an
online retailer and show the existence of mesoscopic structure.Comment: 7 pages, 6 figure
Finding community structure in very large networks
The discovery and analysis of community structure in networks is a topic of
considerable recent interest within the physics community, but most methods
proposed so far are unsuitable for very large networks because of their
computational cost. Here we present a hierarchical agglomeration algorithm for
detecting community structure which is faster than many competing algorithms:
its running time on a network with n vertices and m edges is O(m d log n) where
d is the depth of the dendrogram describing the community structure. Many
real-world networks are sparse and hierarchical, with m ~ n and d ~ log n, in
which case our algorithm runs in essentially linear time, O(n log^2 n). As an
example of the application of this algorithm we use it to analyze a network of
items for sale on the web-site of a large online retailer, items in the network
being linked if they are frequently purchased by the same buyer. The network
has more than 400,000 vertices and 2 million edges. We show that our algorithm
can extract meaningful communities from this network, revealing large-scale
patterns present in the purchasing habits of customers
Analysis of weighted networks
The connections in many networks are not merely binary entities, either
present or not, but have associated weights that record their strengths
relative to one another. Recent studies of networks have, by and large, steered
clear of such weighted networks, which are often perceived as being harder to
analyze than their unweighted counterparts. Here we point out that weighted
networks can in many cases be analyzed using a simple mapping from a weighted
network to an unweighted multigraph, allowing us to apply standard techniques
for unweighted graphs to weighted ones as well. We give a number of examples of
the method, including an algorithm for detecting community structure in
weighted networks and a new and simple proof of the max-flow/min-cut theorem.Comment: 9 pages, 3 figure
Efficient modularity optimization by multistep greedy algorithm and vertex mover refinement
Identifying strongly connected substructures in large networks provides
insight into their coarse-grained organization. Several approaches based on the
optimization of a quality function, e.g., the modularity, have been proposed.
We present here a multistep extension of the greedy algorithm (MSG) that allows
the merging of more than one pair of communities at each iteration step. The
essential idea is to prevent the premature condensation into few large
communities. Upon convergence of the MSG a simple refinement procedure called
"vertex mover" (VM) is used for reassigning vertices to neighboring communities
to improve the final modularity value. With an appropriate choice of the step
width, the combined MSG-VM algorithm is able to find solutions of higher
modularity than those reported previously. The multistep extension does not
alter the scaling of computational cost of the greedy algorithm.Comment: 7 pages, parts of text rewritten, illustrations and pseudocode
representation of algorithms adde
Stochastic blockmodels and community structure in networks
Stochastic blockmodels have been proposed as a tool for detecting community
structure in networks as well as for generating synthetic networks for use as
benchmarks. Most blockmodels, however, ignore variation in vertex degree,
making them unsuitable for applications to real-world networks, which typically
display broad degree distributions that can significantly distort the results.
Here we demonstrate how the generalization of blockmodels to incorporate this
missing element leads to an improved objective function for community detection
in complex networks. We also propose a heuristic algorithm for community
detection using this objective function or its non-degree-corrected counterpart
and show that the degree-corrected version dramatically outperforms the
uncorrected one in both real-world and synthetic networks.Comment: 11 pages, 3 figure
Detecting community structure in networks using edge prediction methods
Community detection and edge prediction are both forms of link mining: they
are concerned with discovering the relations between vertices in networks. Some
of the vertex similarity measures used in edge prediction are closely related
to the concept of community structure. We use this insight to propose a novel
method for improving existing community detection algorithms by using a simple
vertex similarity measure. We show that this new strategy can be more effective
in detecting communities than the basic community detection algorithms.Comment: 5 pages, 2 figure
An efficient and principled method for detecting communities in networks
A fundamental problem in the analysis of network data is the detection of
network communities, groups of densely interconnected nodes, which may be
overlapping or disjoint. Here we describe a method for finding overlapping
communities based on a principled statistical approach using generative network
models. We show how the method can be implemented using a fast, closed-form
expectation-maximization algorithm that allows us to analyze networks of
millions of nodes in reasonable running times. We test the method both on
real-world networks and on synthetic benchmarks and find that it gives results
competitive with previous methods. We also show that the same approach can be
used to extract nonoverlapping community divisions via a relaxation method, and
demonstrate that the algorithm is competitively fast and accurate for the
nonoverlapping problem.Comment: 14 pages, 5 figures, 1 tabl
d_c=4 is the upper critical dimension for the Bak-Sneppen model
Numerical results are presented indicating d_c=4 as the upper critical
dimension for the Bak-Sneppen evolution model. This finding agrees with
previous theoretical arguments, but contradicts a recent Letter [Phys. Rev.
Lett. 80, 5746-5749 (1998)] that placed d_c as high as d=8. In particular, we
find that avalanches are compact for all dimensions d<=4, and are fractal for
d>4. Under those conditions, scaling arguments predict a d_c=4, where
hyperscaling relations hold for d<=4. Other properties of avalanches, studied
for 1<=d<=6, corroborate this result. To this end, an improved numerical
algorithm is presented that is based on the equivalent branching process.Comment: 4 pages, RevTex4, as to appear in Phys. Rev. Lett., related papers
available at http://userwww.service.emory.edu/~sboettc
Finding community structure in networks using the eigenvectors of matrices
We consider the problem of detecting communities or modules in networks,
groups of vertices with a higher-than-average density of edges connecting them.
Previous work indicates that a robust approach to this problem is the
maximization of the benefit function known as "modularity" over possible
divisions of a network. Here we show that this maximization process can be
written in terms of the eigenspectrum of a matrix we call the modularity
matrix, which plays a role in community detection similar to that played by the
graph Laplacian in graph partitioning calculations. This result leads us to a
number of possible algorithms for detecting community structure, as well as
several other results, including a spectral measure of bipartite structure in
networks and a new centrality measure that identifies those vertices that
occupy central positions within the communities to which they belong. The
algorithms and measures proposed are illustrated with applications to a variety
of real-world complex networks.Comment: 22 pages, 8 figures, minor corrections in this versio
New Kludge Scheme for the Construction of Approximate Waveforms for Extreme-Mass-Ratio Inspirals
We introduce a new kludge scheme to model the dynamics of generic extreme
mass-ratio inspirals (stellar compact objects spiraling into a spinning
supermassive black hole) and to produce the gravitational waveforms that
describe the gravitational-wave emission of these systems. This scheme combines
tools from different techniques in General Relativity: It uses a multipolar,
post-Minkowskian (MPM) expansion for the far-zone metric perturbation (which
provides the gravitational waveforms, here taken up to mass hexadecapole and
current octopole order) and for the local prescription of the self-force (since
we are lacking a general prescription for it); a post-Newtonian expansion for
the computation of the multipole moments in terms of the trajectories; and a BH
perturbation theory expansion when treating the trajectories as a sequence of
self-adjusting Kerr geodesics. The orbital evolution is thus equivalent to
solving the geodesic equations with time-dependent orbital elements, as
dictated by the MPM radiation-reaction prescription. To complete the scheme,
both the orbital evolution and wave generation require to map the
Boyer-Lindquist coordinates of the orbits to the harmonic coordinates in which
the different MPM quantities have been derived, a mapping that we provide
explicitly in this paper. This new kludge scheme is thus a combination of
approximations that can be used to model generic inspirals of systems with
extreme mass ratios to systems with more moderate mass ratios, and hence can
provide valuable information for future space-based gravitational-wave
observatories like LISA and even for advanced ground detectors. Finally, due to
the local character in time of our MPM self-force, this scheme can be used to
perform studies of the possible appearance of transient resonances in generic
inspirals.Comment: RevTeX 4.1. 35 pages, 10 Figures, 3 Tables. Version to match the
published article in Physical Review
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