50 research outputs found
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
Universal transient behavior in large dynamical systems on networks
We analyze how the transient dynamics of large dynamical systems in the
vicinity of a stationary point, modeled by a set of randomly coupled linear
differential equations, depends on the network topology. We characterize the
transient response of a system through the evolution in time of the squared
norm of the state vector, which is averaged over different realizations of the
initial perturbation. We develop a mathematical formalism that computes this
quantity for graphs that are locally tree-like. We show that for unidirectional
networks the theory simplifies and general analytical results can be derived.
For example, we derive analytical expressions for the average squared norm for
random directed graphs with a prescribed degree distribution. These analytical
results reveal that unidirectional systems exhibit a high degree of
universality in the sense that the average squared norm only depends on a
single parameter encoding the average interaction strength between the
individual constituents. In addition, we derive analytical expressions for the
average squared norm for unidirectional systems with fixed diagonal disorder
and with bimodal diagonal disorder. We illustrate these results with numerical
experiments on large random graphs and on real-world networks.Comment: 19 pages, 7 figures. Substantially enlarged version. Submitted to
Physical Review Researc
Problems with classification, hypothesis testing, and estimator convergence in the analysis of degree distributions in networks
In their recent work "Scale-free networks are rare", Broido and Clauset
address the problem of the analysis of degree distributions in networks to
classify them as scale-free at different strengths of "scale-freeness." Over
the last two decades, a multitude of papers in network science have reported
that the degree distributions in many real-world networks follow power laws.
Such networks were then referred to as scale-free. However, due to a lack of a
precise definition, the term has evolved to mean a range of different things,
leading to confusion and contradictory claims regarding scale-freeness of a
given network. Recognizing this problem, the authors of "Scale-free networks
are rare" try to fix it. They attempt to develop a versatile and statistically
principled approach to remove this scale-free ambiguity accumulated in network
science literature. Although their paper presents a fair attempt to address
this fundamental problem, we must bring attention to some important issues in
it
The Brevity Law as a Scaling Law, and a Possible Origin of Zipf's Law for Word Frequencies
An important body of quantitative linguistics is constituted by a series of statistical laws about language usage. Despite the importance of these linguistic laws, some of them are poorly formulated, and, more importantly, there is no unified framework that encompasses all them. This paper presents a new perspective to establish a connection between different statistical linguistic laws. Characterizing each word type by two random variables-length (in number of characters) and absolute frequency-we show that the corresponding bivariate joint probability distribution shows a rich and precise phenomenology, with the type-length and the type-frequency distributions as its two marginals, and the conditional distribution of frequency at fixed length providing a clear formulation for the brevity-frequency phenomenon. The type-length distribution turns out to be well fitted by a gamma distribution (much better than with the previously proposed lognormal), and the conditional frequency distributions at fixed length display power-law-decay behavior with a fixed exponent and a characteristic-frequency crossover that scales as an inverse power of length, which implies the fulfillment of a scaling law analogous to those found in the thermodynamics of critical phenomena. As a by-product, we find a possible model-free explanation for the origin of Zipf's law, which should arise as a mixture of conditional frequency distributions governed by the crossover length-dependent frequency
Biyolojik ve biyolojik olmayan ağlar üzerine
With a general classification, there are two types of networks in the world: Biological and non-biological networks. We are unable to change the structure of biological networks. However, for networks such as social networks, technological networks and transportation networks, the architectures of non-biological networks are designed and can be changed by people. Networks can be classified as random networks, small-world networks and scale-free networks. However, we have problems with small-world networks and scale free networks. As some authors ask, “how small is a small-world network and how does it compare to other models?” Even the issue of scale-free networks are whether abundant or rare is still debated. Our main goal in this study is to investigate whether biological and non-biological networks have basic defining features. Especially if we can determine the properties of biological networks in a detailed way, then we may have the chance to design more robust and efficient non-biological networks. However, this research results shows that discussions on the properties of biological networks are not yet complete