975 research outputs found
Determination of multifractal dimensions of complex networks by means of the sandbox algorithm
Complex networks have attracted much attention in diverse areas of science
and technology. Multifractal analysis (MFA) is a useful way to systematically
describe the spatial heterogeneity of both theoretical and experimental fractal
patterns. In this paper, we employ the sandbox (SB) algorithm proposed by
T\'{e}l et al. (Physica A, 159 (1989) 155-166), for MFA of complex networks.
First we compare the SB algorithm with two existing algorithms of MFA for
complex networks: the compact-box-burning (CBB) algorithm proposed by Furuya
and Yakubo (Phys. Rev. E, 84 (2011) 036118), and the improved box-counting (BC)
algorithm proposed by Li et al. (J. Stat. Mech.: Theor. Exp., 2014 (2014)
P02020) by calculating the mass exponents tau(q) of some deterministic model
networks. We make a detailed comparison between the numerical and theoretical
results of these model networks. The comparison results show that the SB
algorithm is the most effective and feasible algorithm to calculate the mass
exponents tau(q) and to explore the multifractal behavior of complex networks.
Then we apply the SB algorithm to study the multifractal property of some
classic model networks, such as scale-free networks, small-world networks, and
random networks. Our results show that multifractality exists in scale-free
networks, that of small-world networks is not obvious, and it almost does not
exist in random networks.Comment: 17 pages, 2 table, 10 figure
Fractal and multifractal properties of a family of fractal networks
In this work, we study the fractal and multifractal properties of a family of
fractal networks introduced by Gallos {\it et al.} ({\it Proc. Natl. Acad. Sci.
U.S.A.}, 2007, {\bf 104}: 7746). In this fractal network model, there is a
parameter which is between and , and allows for tuning the level of
fractality in the network. Here we examine the multifractal behavior of these
networks, dependence relationship of fractal dimension and the multifractal
parameters on the parameter . First, we find that the empirical fractal
dimensions of these networks obtained by our program coincide with the
theoretical formula given by Song {\it et al.} ( {\it Nat. Phys}, 2006, {\bf
2}: 275). Then from the shape of the and curves, we find the
existence of multifractality in these networks. Last, we find that there exists
a linear relationship between the average information dimension and
the parameter .Comment: 12 pages, 7 figures, accepted by J. Stat. Mec
Multifractal analysis of weighted networks by a modified sandbox algorithm
Complex networks have attracted growing attention in many fields. As a
generalization of fractal analysis, multifractal analysis (MFA) is a useful way
to systematically describe the spatial heterogeneity of both theoretical and
experimental fractal patterns. Some algorithms for MFA of unweighted complex
networks have been proposed in the past a few years, including the sandbox (SB)
algorithm recently employed by our group. In this paper, a modified SB
algorithm (we call it SBw algorithm) is proposed for MFA of weighted
networks.First, we use the SBw algorithm to study the multifractal property of
two families of weighted fractal networks (WFNs): "Sierpinski" WFNs and "Cantor
dust" WFNs. We also discuss how the fractal dimension and generalized fractal
dimensions change with the edge-weights of the WFN. From the comparison between
the theoretical and numerical fractal dimensions of these networks, we can find
that the proposed SBw algorithm is efficient and feasible for MFA of weighted
networks. Then, we apply the SBw algorithm to study multifractal properties of
some real weighted networks ---collaboration networks. It is found that the
multifractality exists in these weighted networks, and is affected by their
edge-weights.Comment: 15 pages, 6 figures. Accepted for publication by Scientific Report
Fractal and multifractal analysis of complex networks: Estonian network of payments
Complex networks have gained much attention from different areas of knowledge
in recent years. Particularly, the structures and dynamics of such systems have
attracted considerable interest. Complex networks may have characteristics of
multifractality. In this study, we analyze fractal and multifractal properties
of a novel network: the large scale economic network of payments of Estonia,
where companies are represented by nodes and the payments done between
companies are represented by links. We present a fractal scaling analysis and
examine the multifractal behavior of this network by using a sandbox algorithm.
Our results indicate the existence of multifractality in this network and
consequently, the existence of multifractality in the Estonian economy. To the
best of our knowledge, this is the first study that analyzes multifractality of
a complex network of payments.Comment: 13 page
Dynamical variety of shapes in financial multifractality
The concept of multifractality offers a powerful formal tool to filter out
multitude of the most relevant characteristics of complex time series. The
related studies thus far presented in the scientific literature typically limit
themselves to evaluation of whether or not a time series is multifractal and
width of the resulting singularity spectrum is considered a measure of the
degree of complexity involved. However, the character of the complexity of time
series generated by the natural processes usually appears much more intricate
than such a bare statement can reflect. As an example, based on the long-term
records of S&P500 and NASDAQ - the two world leading stock market indices - the
present study shows that they indeed develop the multifractal features, but
these features evolve through a variety of shapes, most often strongly
asymmetric, whose changes typically are correlated with the historically most
significant events experienced by the world economy. Relating at the same time
the index multifractal singularity spectra to those of the component stocks
that form this index reflects the varying degree of correlations involved among
the stocks.Comment: 26 pages, 10 figure
Multifractal Characterization of Protein Contact Networks
The multifractal detrended fluctuation analysis of time series is able to
reveal the presence of long-range correlations and, at the same time, to
characterize the self-similarity of the series. The rich information derivable
from the characteristic exponents and the multifractal spectrum can be further
analyzed to discover important insights about the underlying dynamical process.
In this paper, we employ multifractal analysis techniques in the study of
protein contact networks. To this end, initially a network is mapped to three
different time series, each of which is generated by a stationary unbiased
random walk. To capture the peculiarities of the networks at different levels,
we accordingly consider three observables at each vertex: the degree, the
clustering coefficient, and the closeness centrality. To compare the results
with suitable references, we consider also instances of three well-known
network models and two typical time series with pure monofractal and
multifractal properties. The first result of notable interest is that time
series associated to proteins contact networks exhibit long-range correlations
(strong persistence), which are consistent with signals in-between the typical
monofractal and multifractal behavior. Successively, a suitable embedding of
the multifractal spectra allows to focus on ensemble properties, which in turn
gives us the possibility to make further observations regarding the considered
networks. In particular, we highlight the different role that small and large
fluctuations of the considered observables play in the characterization of the
network topology
Fractal Systems of Central Places Based on Intermittency of Space-filling
The central place models are fundamentally important in theoretical geography
and city planning theory. The texture and structure of central place networks
have been demonstrated to be self-similar in both theoretical and empirical
studies. However, the underlying rationale of central place fractals in the
real world has not yet been revealed so far. This paper is devoted to
illustrating the mechanisms by which the fractal patterns can be generated from
central place systems. The structural dimension of the traditional central
place models is d=2 indicating no intermittency in the spatial distribution of
human settlements. This dimension value is inconsistent with empirical
observations. Substituting the complete space filling with the incomplete space
filling, we can obtain central place models with fractional dimension D<d=2
indicative of spatial intermittency. Thus the conventional central place models
are converted into fractal central place models. If we further integrate the
chance factors into the improved central place fractals, the theory will be
able to well explain the real patterns of urban places. As empirical analyses,
the US cities and towns are employed to verify the fractal-based models of
central places.Comment: 30 pages, 8 figures, 5 table
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