87,237 research outputs found
Self-similar scaling of density in complex real-world networks
Despite their diverse origin, networks of large real-world systems reveal a
number of common properties including small-world phenomena, scale-free degree
distributions and modularity. Recently, network self-similarity as a natural
outcome of the evolution of real-world systems has also attracted much
attention within the physics literature. Here we investigate the scaling of
density in complex networks under two classical box-covering
renormalizations-network coarse-graining-and also different community-based
renormalizations. The analysis on over 50 real-world networks reveals a
power-law scaling of network density and size under adequate renormalization
technique, yet irrespective of network type and origin. The results thus
advance a recent discovery of a universal scaling of density among different
real-world networks [Laurienti et al., Physica A 390 (20) (2011) 3608-3613.]
and imply an existence of a scale-free density also within-among different
self-similar scales of-complex real-world networks. The latter further improves
the comprehension of self-similar structure in large real-world networks with
several possible applications
True scale-free networks hidden by finite size effects
We analyze about two hundred naturally occurring networks with distinct
dynamical origins to formally test whether the commonly assumed hypothesis of
an underlying scale-free structure is generally viable. This has recently been
questioned on the basis of statistical testing of the validity of power law
distributions of network degrees by contrasting real data. Specifically, we
analyze by finite-size scaling analysis the datasets of real networks to check
whether purported departures from the power law behavior are due to the
finiteness of the sample size. In this case, power laws would be recovered in
the case of progressively larger cutoffs induced by the size of the sample. We
find that a large number of the networks studied follow a finite size scaling
hypothesis without any self-tuning. This is the case of biological protein
interaction networks, technological computer and hyperlink networks, and
informational networks in general. Marked deviations appear in other cases,
especially infrastructure and transportation but also social networks. We
conclude that underlying scale invariance properties of many naturally
occurring networks are extant features often clouded by finite-size effects due
to the nature of the sample data
Griffiths phases in infinite-dimensional, non-hierarchical modular networks
Griffiths phases (GPs), generated by the heterogeneities on modular networks,
have recently been suggested to provide a mechanism, rid of fine parameter
tuning, to explain the critical behavior of complex systems. One conjectured
requirement for systems with modular structures was that the network of modules
must be hierarchically organized and possess finite dimension. We investigate
the dynamical behavior of an activity spreading model, evolving on
heterogeneous random networks with highly modular structure and organized
non-hierarchically. We observe that loosely coupled modules act as effective
rare-regions, slowing down the extinction of activation. As a consequence, we
find extended control parameter regions with continuously changing dynamical
exponents for single network realizations, preserved after finite size
analyses, as in a real GP. The avalanche size distributions of spreading events
exhibit robust power-law tails. Our findings relax the requirement of
hierarchical organization of the modular structure, which can help to
rationalize the criticality of modular systems in the framework of GPs.Comment: 14 pages, 8 figure
Griffiths phases and localization in hierarchical modular networks
We study variants of hierarchical modular network models suggested by Kaiser
and Hilgetag [Frontiers in Neuroinformatics, 4 (2010) 8] to model functional
brain connectivity, using extensive simulations and quenched mean-field theory
(QMF), focusing on structures with a connection probability that decays
exponentially with the level index. Such networks can be embedded in
two-dimensional Euclidean space. We explore the dynamic behavior of the contact
process (CP) and threshold models on networks of this kind, including
hierarchical trees. While in the small-world networks originally proposed to
model brain connectivity, the topological heterogeneities are not strong enough
to induce deviations from mean-field behavior, we show that a Griffiths phase
can emerge under reduced connection probabilities, approaching the percolation
threshold. In this case the topological dimension of the networks is finite,
and extended regions of bursty, power-law dynamics are observed. Localization
in the steady state is also shown via QMF. We investigate the effects of link
asymmetry and coupling disorder, and show that localization can occur even in
small-world networks with high connectivity in case of link disorder.Comment: 18 pages, 20 figures, accepted version in Scientific Report
Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version)
Although the ``scale-free'' literature is large and growing, it gives neither
a precise definition of scale-free graphs nor rigorous proofs of many of their
claimed properties. In fact, it is easily shown that the existing theory has
many inherent contradictions and verifiably false claims. In this paper, we
propose a new, mathematically precise, and structural definition of the extent
to which a graph is scale-free, and prove a series of results that recover many
of the claimed properties while suggesting the potential for a rich and
interesting theory. With this definition, scale-free (or its opposite,
scale-rich) is closely related to other structural graph properties such as
various notions of self-similarity (or respectively, self-dissimilarity).
Scale-free graphs are also shown to be the likely outcome of random
construction processes, consistent with the heuristic definitions implicit in
existing random graph approaches. Our approach clarifies much of the confusion
surrounding the sensational qualitative claims in the scale-free literature,
and offers rigorous and quantitative alternatives.Comment: 44 pages, 16 figures. The primary version is to appear in Internet
Mathematics (2005
Measuring the dimension of partially embedded networks
Scaling phenomena have been intensively studied during the past decade in the
context of complex networks. As part of these works, recently novel methods
have appeared to measure the dimension of abstract and spatially embedded
networks. In this paper we propose a new dimension measurement method for
networks, which does not require global knowledge on the embedding of the
nodes, instead it exploits link-wise information (link lengths, link delays or
other physical quantities). Our method can be regarded as a generalization of
the spectral dimension, that grasps the network's large-scale structure through
local observations made by a random walker while traversing the links. We apply
the presented method to synthetic and real-world networks, including road maps,
the Internet infrastructure and the Gowalla geosocial network. We analyze the
theoretically and empirically designated case when the length distribution of
the links has the form P(r) ~ 1/r. We show that while previous dimension
concepts are not applicable in this case, the new dimension measure still
exhibits scaling with two distinct scaling regimes. Our observations suggest
that the link length distribution is not sufficient in itself to entirely
control the dimensionality of complex networks, and we show that the proposed
measure provides information that complements other known measures
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