287 research outputs found

### Fractal and Transfractal Recursive Scale-Free Nets

We explore the concepts of self-similarity, dimensionality, and
(multi)scaling in a new family of recursive scale-free nets that yield
themselves to exact analysis through renormalization techniques. All nets in
this family are self-similar and some are fractals - possessing a finite
fractal dimension - while others are small world (their diameter grows
logarithmically with their size) and are infinite-dimensional. We show how a
useful measure of "transfinite" dimension may be defined and applied to the
small world nets. Concerning multiscaling, we show how first-passage time for
diffusion and resistance between hub (the most connected nodes) scale
differently than for other nodes. Despite the different scalings, the Einstein
relation between diffusion and conductivity holds separately for hubs and
nodes. The transfinite exponents of small world nets obey Einstein relations
analogous to those in fractal nets.Comment: Includes small revisions and references added as result of readers'
feedbac

### Explosive Percolation in the Human Protein Homology Network

We study the explosive character of the percolation transition in a
real-world network. We show that the emergence of a spanning cluster in the
Human Protein Homology Network (H-PHN) exhibits similar features to an
Achlioptas-type process and is markedly different from regular random
percolation. The underlying mechanism of this transition can be described by
slow-growing clusters that remain isolated until the later stages of the
process, when the addition of a small number of links leads to the rapid
interconnection of these modules into a giant cluster. Our results indicate
that the evolutionary-based process that shapes the topology of the H-PHN
through duplication-divergence events may occur in sudden steps, similarly to
what is seen in first-order phase transitions.Comment: 13 pages, 6 figure

### Vertex routing models

A class of models describing the flow of information within networks via
routing processes is proposed and investigated, concentrating on the effects of
memory traces on the global properties. The long-term flow of information is
governed by cyclic attractors, allowing to define a measure for the information
centrality of a vertex given by the number of attractors passing through this
vertex. We find the number of vertices having a non-zero information centrality
to be extensive/sub-extensive for models with/without a memory trace in the
thermodynamic limit. We evaluate the distribution of the number of cycles, of
the cycle length and of the maximal basins of attraction, finding a complete
scaling collapse in the thermodynamic limit for the latter. Possible
implications of our results on the information flow in social networks are
discussed.Comment: 12 pages, 6 figure

### Laws of Population Growth

An important issue in the study of cities is defining a metropolitan area, as
different definitions affect the statistical distribution of urban activity. A
commonly employed method of defining a metropolitan area is the Metropolitan
Statistical Areas (MSA), based on rules attempting to capture the notion of
city as a functional economic region, and is constructed using experience. The
MSA is time-consuming and is typically constructed only for a subset (few
hundreds) of the most highly populated cities. Here, we introduce a new method
to designate metropolitan areas, denoted the "City Clustering Algorithm" (CCA).
The CCA is based on spatial distributions of the population at a fine
geographic scale, defining a city beyond the scope of its administrative
boundaries. We use the CCA to examine Gibrat's law of proportional growth,
postulating that the mean and standard deviation of the growth rate of cities
are constant, independent of city size. We find that the mean growth rate of a
cluster utilizing the CCA exhibits deviations from Gibrat's law, and that the
standard deviation decreases as a power-law with respect to the city size. The
CCA allows for the study of the underlying process leading to these deviations,
shown to arise from the existence of long-range spatial correlations in the
population growth. These results have socio-political implications, such as
those pertaining to the location of new economic development in cities of
varied size.Comment: 30 pages, 8 figure

### Portraits of Complex Networks

We propose a method for characterizing large complex networks by introducing
a new matrix structure, unique for a given network, which encodes structural
information; provides useful visualization, even for very large networks; and
allows for rigorous statistical comparison between networks. Dynamic processes
such as percolation can be visualized using animations. Applications to graph
theory are discussed, as are generalizations to weighted networks, real-world
network similarity testing, and applicability to the graph isomorphism problem.Comment: 6 pages, 9 figure

### Average distance in a hierarchical scale-free network: an exact solution

Various real systems simultaneously exhibit scale-free and hierarchical
structure. In this paper, we study analytically average distance in a
deterministic scale-free network with hierarchical organization. Using a
recursive method based on the network construction, we determine explicitly the
average distance, obtaining an exact expression for it, which is confirmed by
extensive numerical calculations. The obtained rigorous solution shows that the
average distance grows logarithmically with the network order (number of nodes
in the network). We exhibit the similarity and dissimilarity in average
distance between the network under consideration and some previously studied
networks, including random networks and other deterministic networks. On the
basis of the comparison, we argue that the logarithmic scaling of average
distance with network order could be a generic feature of deterministic
scale-free networks.Comment: Definitive version published in Journal of Statistical Mechanic

### A Hebbian approach to complex network generation

Through a redefinition of patterns in an Hopfield-like model, we introduce
and develop an approach to model discrete systems made up of many, interacting
components with inner degrees of freedom. Our approach clarifies the intrinsic
connection between the kind of interactions among components and the emergent
topology describing the system itself; also, it allows to effectively address
the statistical mechanics on the resulting networks. Indeed, a wide class of
analytically treatable, weighted random graphs with a tunable level of
correlation can be recovered and controlled. We especially focus on the case of
imitative couplings among components endowed with similar patterns (i.e.
attributes), which, as we show, naturally and without any a-priori assumption,
gives rise to small-world effects. We also solve the thermodynamics (at a
replica symmetric level) by extending the double stochastic stability
technique: free energy, self consistency relations and fluctuation analysis for
a picture of criticality are obtained

### Living with diabetes: An exploratory study of illness representation and medication adherence in Ghana

Background: Compared to other chronic conditions, non-adherence to medication in diabetes patients is very high. This study explores the relationship between illness representation and medication adherence in diabetes patients in Ghana. Method: A total of 196 type 2 diabetes patients purposively and conveniently sampled from a tertiary hospital in Ghana responded to the Revised Illness Perception Questionnaire (IPQ-R) and the Medication Adherence Report Scale (MARS-5). The Pearson Moment Product correlation and the hierarchical multiple regression statistical tools were used to analyse the data. Results: Illness consequence and emotional representation were negatively related to medication adherence, while personal control positively accounted for significant variance in medication adherence. However, none of the selected key demographic variables (i.e. age, illness duration, gender, religion and education) independently accounted for any significant variance in medication adherence. Conclusion: Diabetes has a telling consequence on patientsâ€™ life; the patient can do something to control diabetes; and the negative emotional representations concerning the disease have a significant influence on the degree of medication adherence by the patients. This observation has implications for the management and treatment plan of diabetes

### Anomalous behavior of trapping on a fractal scale-free network

It is known that the heterogeneity of scale-free networks helps enhancing the
efficiency of trapping processes performed on them. In this paper, we show that
transport efficiency is much lower in a fractal scale-free network than in
non-fractal networks. To this end, we examine a simple random walk with a fixed
trap at a given position on a fractal scale-free network. We calculate
analytically the mean first-passage time (MFPT) as a measure of the efficiency
for the trapping process, and obtain a closed-form expression for MFPT, which
agrees with direct numerical calculations. We find that, in the limit of a
large network order $V$, the MFPT $$ behaves superlinearly as $\sim
V^{{3/2}}$ with an exponent 3/2 much larger than 1, which is in sharp contrast
to the scaling $\sim V^{\theta}$ with $\theta \leq 1$, previously obtained
for non-fractal scale-free networks. Our results indicate that the degree
distribution of scale-free networks is not sufficient to characterize trapping
processes taking place on them. Since various real-world networks are
simultaneously scale-free and fractal, our results may shed light on the
understanding of trapping processes running on real-life systems.Comment: 6 pages, 5 figures; Definitive version accepted for publication in
EPL (Europhysics Letters

### Exact eigenvalue spectrum of a class of fractal scale-free networks

The eigenvalue spectrum of the transition matrix of a network encodes
important information about its structural and dynamical properties. We study
the transition matrix of a family of fractal scale-free networks and
analytically determine all the eigenvalues and their degeneracies. We then use
these eigenvalues to evaluate the closed-form solution to the eigentime for
random walks on the networks under consideration. Through the connection
between the spectrum of transition matrix and the number of spanning trees, we
corroborate the obtained eigenvalues and their multiplicities.Comment: Definitive version accepted for publication in EPL (Europhysics
Letters

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