8,090 research outputs found
The topology of large Open Connectome networks for the human brain
The structural human connectome (i.e.\ the network of fiber connections in
the brain) can be analyzed at ever finer spatial resolution thanks to advances
in neuroimaging. Here we analyze several large data sets for the human brain
network made available by the Open Connectome Project. We apply statistical
model selection to characterize the degree distributions of graphs containing
up to nodes and edges. A three-parameter
generalized Weibull (also known as a stretched exponential) distribution is a
good fit to most of the observed degree distributions. For almost all networks,
simple power laws cannot fit the data, but in some cases there is statistical
support for power laws with an exponential cutoff. We also calculate the
topological (graph) dimension and the small-world coefficient of
these networks. While suggests a small-world topology, we found that
showing that long-distance connections provide only a small correction
to the topology of the embedding three-dimensional space.Comment: 14 pages, 6 figures, accepted version in Scientific Report
Catching the head, tail, and everything in between: a streaming algorithm for the degree distribution
The degree distribution is one of the most fundamental graph properties of
interest for real-world graphs. It has been widely observed in numerous domains
that graphs typically have a tailed or scale-free degree distribution. While
the average degree is usually quite small, the variance is quite high and there
are vertices with degrees at all scales. We focus on the problem of
approximating the degree distribution of a large streaming graph, with small
storage. We design an algorithm headtail, whose main novelty is a new estimator
of infrequent degrees using truncated geometric random variables. We give a
mathematical analysis of headtail and show that it has excellent behavior in
practice. We can process streams will millions of edges with storage less than
1% and get extremely accurate approximations for all scales in the degree
distribution.
We also introduce a new notion of Relative Hausdorff distance between tailed
histograms. Existing notions of distances between distributions are not
suitable, since they ignore infrequent degrees in the tail. The Relative
Hausdorff distance measures deviations at all scales, and is a more suitable
distance for comparing degree distributions. By tracking this new measure, we
are able to give strong empirical evidence of the convergence of headtail
The Zipf-Polylog distribution: Modeling human interactions through social networks
The Zipf distribution attracts considerable attention because it helps describe data from natural as well as man-made systems. Nevertheless, in most of the cases the Zipf is only appropriate to fit data in the upper tail. This is why it is important to dispose of Zipf extensions that allow to fit the data in its entire range. In this paper, we introduce the Zipf-Polylog family of distributions as a two-parameter generalization of the Zipf. The extended family contains the Zipf, the geometric, the logarithmic series and the shifted negative binomial with two successes, as particular distributions. We deduce important properties of the new family and demonstrate its suitability by analyzing the degree sequence of two real networks in all its range.Peer ReviewedPostprint (author's final draft
Analysis and Assembling of Network Structure in Mutualistic Systems
It has been observed that mutualistic bipartite networks have a nested
structure of interactions. In addition, the degree distributions associated
with the two guilds involved in such networks (e.g. plants & pollinators or
plants & seed dispersers) approximately follow a truncated power law. We show
that nestedness and truncated power law distributions are intimately linked,
and that any biological reasons for such truncation are superimposed to finite
size effects . We further explore the internal organization of bipartite
networks by developing a self-organizing network model (SNM) that reproduces
empirical observations of pollination systems of widely different sizes. Since
the only inputs to the SNM are numbers of plant and animal species, and their
interactions (i.e., no data on local abundance of the interacting species are
needed), we suggest that the well-known association between species frequency
of interaction and species degree is a consequence rather than a cause, of the
observed network structure.Comment: J. of. Theor. Biology, in pres
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