144,033 research outputs found
Growing Scale-Free Networks with Small World Behavior
In the context of growing networks, we introduce a simple dynamical model
that unifies the generic features of real networks: scale-free distribution of
degree and the small world effect. While the average shortest path length
increases logartihmically as in random networks, the clustering coefficient
assumes a large value independent of system size. We derive expressions for the
clustering coefficient in two limiting cases: random (C ~ (ln N)^2 / N) and
highly clustered (C = 5/6) scale-free networks.Comment: 4 pages, 4 figure
On the influence of topological characteristics on robustness of complex networks
In this paper, we explore the relationship between the topological
characteristics of a complex network and its robustness to sustained targeted
attacks. Using synthesised scale-free, small-world and random networks, we look
at a number of network measures, including assortativity, modularity, average
path length, clustering coefficient, rich club profiles and scale-free exponent
(where applicable) of a network, and how each of these influence the robustness
of a network under targeted attacks. We use an established robustness
coefficient to measure topological robustness, and consider sustained targeted
attacks by order of node degree. With respect to scale-free networks, we show
that assortativity, modularity and average path length have a positive
correlation with network robustness, whereas clustering coefficient has a
negative correlation. We did not find any correlation between scale-free
exponent and robustness, or rich-club profiles and robustness. The robustness
of small-world networks on the other hand, show substantial positive
correlations with assortativity, modularity, clustering coefficient and average
path length. In comparison, the robustness of Erdos-Renyi random networks did
not have any significant correlation with any of the network properties
considered. A significant observation is that high clustering decreases
topological robustness in scale-free networks, yet it increases topological
robustness in small-world networks. Our results highlight the importance of
topological characteristics in influencing network robustness, and illustrate
design strategies network designers can use to increase the robustness of
scale-free and small-world networks under sustained targeted attacks
Clustering of random scale-free networks
We derive the finite size dependence of the clustering coefficient of
scale-free random graphs generated by the configuration model with degree
distribution exponent . Degree heterogeneity increases the presence
of triangles in the network up to levels that compare to those found in many
real networks even for extremely large nets. We also find that for values of
, clustering is virtually size independent and, at the same
time, becomes a {\it de facto} non self-averaging topological property. This
implies that a single instance network is not representative of the ensemble
even for very large network sizes
Scale-Free Networks Generated By Random Walkers
We present a simple mechanism for generating undirected scale-free networks
using random walkers, where the network growth is determined by choosing parent
vertices by sequential random walks. We show that this mechanism produces
scale-free networks with degree exponent gamma=3 and clustering coefficients
depending on random walk length. The mechanism can be interpreted in terms of
preferential attachment without explicit knowledge of node degrees.Comment: 10 pages, 3 figure
Percolation in self-similar networks
We provide a simple proof that graphs in a general class of self-similar
networks have zero percolation threshold. The considered self-similar networks
include random scale-free graphs with given expected node degrees and zero
clustering, scale-free graphs with finite clustering and metric structure,
growing scale-free networks, and many real networks. The proof and the
derivation of the giant component size do not require the assumption that
networks are treelike. Our results rely only on the observation that
self-similar networks possess a hierarchy of nested subgraphs whose average
degree grows with their depth in the hierarchy. We conjecture that this
property is pivotal for percolation in networks.Comment: 4 pages, 3 figure
Generation of uncorrelated random scale-free networks
Uncorrelated random scale-free networks are useful null models to check the
accuracy an the analytical solutions of dynamical processes defined on complex
networks. We propose and analyze a model capable to generate random
uncorrelated scale-free networks with no multiple and self-connections. The
model is based on the classical configuration model, with an additional
restriction on the maximum possible degree of the vertices. We check
numerically that the proposed model indeed generates scale-free networks with
no two and three vertex correlations, as measured by the average degree of the
nearest neighbors and the clustering coefficient of the vertices of degree ,
respectively
Efficiency of Scale-Free Networks: Error and Attack Tolerance
The concept of network efficiency, recently proposed to characterize the
properties of small-world networks, is here used to study the effects of errors
and attacks on scale-free networks. Two different kinds of scale-free networks,
i.e. networks with power law P(k), are considered: 1) scale-free networks with
no local clustering produced by the Barabasi-Albert model and 2) scale-free
networks with high clustering properties as in the model by Klemm and Eguiluz,
and their properties are compared to the properties of random graphs
(exponential graphs). By using as mathematical measures the global and the
local efficiency we investigate the effects of errors and attacks both on the
global and the local properties of the network. We show that the global
efficiency is a better measure than the characteristic path length to describe
the response of complex networks to external factors. We find that, at variance
with random graphs, scale-free networks display, both on a global and on a
local scale, a high degree of error tolerance and an extreme vulnerability to
attacks. In fact, the global and the local efficiency are unaffected by the
failure of some randomly chosen nodes, though they are extremely sensititive to
the removal of the few nodes which play a crucial role in maintaining the
network's connectivity.Comment: 23 pages, 10 figure
Scale-free brain functional networks
Functional magnetic resonance imaging (fMRI) is used to extract {\em
functional networks} connecting correlated human brain sites. Analysis of the
resulting networks in different tasks shows that: (a) the distribution of
functional connections, and the probability of finding a link vs. distance are
both scale-free, (b) the characteristic path length is small and comparable
with those of equivalent random networks, and (c) the clustering coefficient is
orders of magnitude larger than those of equivalent random networks. All these
properties, typical of scale-free small world networks, reflect important
functional information about brain states.Comment: 4 pages, 5 figures, 2 table
Ising Model on Edge-Dual of Random Networks
We consider Ising model on edge-dual of uncorrelated random networks with
arbitrary degree distribution. These networks have a finite clustering in the
thermodynamic limit. High and low temperature expansions of Ising model on the
edge-dual of random networks are derived. A detailed comparison of the critical
behavior of Ising model on scale free random networks and their edge-dual is
presented.Comment: 23 pages, 4 figures, 1 tabl
Epidemic threshold in structured scale-free networks
We analyze the spreading of viruses in scale-free networks with high
clustering and degree correlations, as found in the Internet graph. For the
Suscetible-Infected-Susceptible model of epidemics the prevalence undergoes a
phase transition at a finite threshold of the transmission probability.
Comparing with the absence of a finite threshold in networks with purely random
wiring, our result suggests that high clustering and degree correlations
protect scale-free networks against the spreading of viruses. We introduce and
verify a quantitative description of the epidemic threshold based on the
connectivity of the neighborhoods of the hubs.Comment: 4 pages, 4 figure
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