107,676 research outputs found
Distances in random graphs with infinite mean degrees
We study random graphs with an i.i.d. degree sequence of which the tail of
the distribution function is regularly varying with exponent . Thus, the degrees have infinite mean. Such random graphs can serve as
models for complex networks where degree power laws are observed.
The minimal number of edges between two arbitrary nodes, also called the
graph distance or the hopcount, in a graph with nodes is investigated when
. The paper is part of a sequel of three papers. The other two
papers study the case where , and
respectively.
The main result of this paper is that the graph distance converges for
to a limit random variable with probability mass exclusively on
the points 2 and 3. We also consider the case where we condition the degrees to
be at most for some For
, the hopcount converges to 3 in probability,
while for , the hopcount converges to the same limit as
for the unconditioned degrees. Our results give convincing asymptotics for the
hopcount when the mean degree is infinite, using extreme value theory.Comment: 20 pages, 2 figure
Sharpest possible clustering bounds using robust random graph analysis
Complex network theory crucially depends on the assumptions made about the
degree distribution, while fitting degree distributions to network data is
challenging, in particular for scale-free networks with power-law degrees. We
present a robust assessment of complex networks that does not depend on the
entire degree distribution, but only on its mean, range and dispersion: summary
statistics that are easy to obtain for most real-world networks. By solving
several semi-infinite linear programs, we obtain tight (the sharpest possible)
bounds for correlation and clustering measures, for all networks with degree
distributions that share the same summary statistics. We identify various
extremal random graphs that attain these tight bounds as the graphs with
specific three-point degree distributions. We leverage the tight bounds to
obtain robust laws that explain how degree-degree correlations and local
clustering evolve as function of node degrees and network size. These robust
laws indicate that power-law networks with diverging variance are among the
most extreme networks in terms of correlation and clustering, building further
theoretical foundation for widely reported scale-free network phenomena such as
correlation and clustering decay
Systematic Topology Analysis and Generation Using Degree Correlations
We present a new, systematic approach for analyzing network topologies. We
first introduce the dK-series of probability distributions specifying all
degree correlations within d-sized subgraphs of a given graph G. Increasing
values of d capture progressively more properties of G at the cost of more
complex representation of the probability distribution. Using this series, we
can quantitatively measure the distance between two graphs and construct random
graphs that accurately reproduce virtually all metrics proposed in the
literature. The nature of the dK-series implies that it will also capture any
future metrics that may be proposed. Using our approach, we construct graphs
for d=0,1,2,3 and demonstrate that these graphs reproduce, with increasing
accuracy, important properties of measured and modeled Internet topologies. We
find that the d=2 case is sufficient for most practical purposes, while d=3
essentially reconstructs the Internet AS- and router-level topologies exactly.
We hope that a systematic method to analyze and synthesize topologies offers a
significant improvement to the set of tools available to network topology and
protocol researchers.Comment: Final versio
Quick Detection of High-degree Entities in Large Directed Networks
In this paper, we address the problem of quick detection of high-degree
entities in large online social networks. Practical importance of this problem
is attested by a large number of companies that continuously collect and update
statistics about popular entities, usually using the degree of an entity as an
approximation of its popularity. We suggest a simple, efficient, and easy to
implement two-stage randomized algorithm that provides highly accurate
solutions for this problem. For instance, our algorithm needs only one thousand
API requests in order to find the top-100 most followed users in Twitter, a
network with approximately a billion of registered users, with more than 90%
precision. Our algorithm significantly outperforms existing methods and serves
many different purposes, such as finding the most popular users or the most
popular interest groups in social networks. An important contribution of this
work is the analysis of the proposed algorithm using Extreme Value Theory -- a
branch of probability that studies extreme events and properties of largest
order statistics in random samples. Using this theory, we derive an accurate
prediction for the algorithm's performance and show that the number of API
requests for finding the top-k most popular entities is sublinear in the number
of entities. Moreover, we formally show that the high variability among the
entities, expressed through heavy-tailed distributions, is the reason for the
algorithm's efficiency. We quantify this phenomenon in a rigorous mathematical
way
Local Difference Measures between Complex Networks for Dynamical System Model Evaluation
Acknowledgments We thank Reik V. Donner for inspiring suggestions that initialized the work presented herein. Jan H. Feldhoff is credited for providing us with the STARS simulation data and for his contributions to fruitful discussions. Comments by the anonymous reviewers are gratefully acknowledged as they led to substantial improvements of the manuscript.Peer reviewedPublisher PD
Statistical significance of communities in networks
Nodes in real-world networks are usually organized in local modules. These
groups, called communities, are intuitively defined as sub-graphs with a larger
density of internal connections than of external links. In this work, we
introduce a new measure aimed at quantifying the statistical significance of
single communities. Extreme and Order Statistics are used to predict the
statistics associated with individual clusters in random graphs. These
distributions allows us to define one community significance as the probability
that a generic clustering algorithm finds such a group in a random graph. The
method is successfully applied in the case of real-world networks for the
evaluation of the significance of their communities.Comment: 9 pages, 8 figures, 2 tables. The software to calculate the C-score
can be found at http://filrad.homelinux.org/cscor
Random Tensors and Quantum Gravity
We provide an informal introduction to tensor field theories and to their
associated renormalization group. We focus more on the general motivations
coming from quantum gravity than on the technical details. In particular we
discuss how asymptotic freedom of such tensor field theories gives a concrete
example of a natural "quantum relativity" postulate: physics in the deep
ultraviolet regime becomes asymptotically more and more independent of any
particular choice of Hilbert basis in the space of states of the universe.Comment: Section 6 is essentially reproduced from author's arXiv:1507.04190
for self-contained purpose of the revie
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