103 research outputs found
Percolation Critical Exponents in Scale-Free Networks
We study the behavior of scale-free networks, having connectivity
distribution P(k) k^-a, close to the percolation threshold. We show that for
networks with 3<a<4, known to undergo a transition at a finite threshold of
dilution, the critical exponents are different than the expected mean-field
values of regular percolation in infinite dimensions. Networks with 2<a<3
possess only a percolative phase. Nevertheless, we show that in this case
percolation critical exponents are well defined, near the limit of extreme
dilution (where all sites are removed), and that also then the exponents bear a
strong a-dependence. The regular mean-field values are recovered only for a>4.Comment: Latex, 4 page
A General Formalism for Inhomogeneous Random Graphs
We present and investigate an extension of the classical random graph to a
general class of inhomogeneous random graph models, where vertices come in
different types, and the probability of realizing an edge depends on the types
of its terminal vertices. This approach provides a general framework for the
analysis of a large class of models. The generic phase structure is derived
using generating function techniques, and relations to other classes of models
are pointed out.Comment: 7 pages, no figures. To appear in Phys. Rev.
Properties of Random Graphs with Hidden Color
We investigate in some detail a recently suggested general class of ensembles
of sparse undirected random graphs based on a hidden stub-coloring, with or
without the restriction to nondegenerate graphs. The calculability of local and
global structural properties of graphs from the resulting ensembles is
demonstrated. Cluster size statistics are derived with generating function
techniques, yielding a well-defined percolation threshold. Explicit rules are
derived for the enumeration of small subgraphs. Duality and redundancy is
discussed, and subclasses corresponding to commonly studied models are
identified.Comment: 14 pages, LaTeX, no figure
Universality in percolation of arbitrary Uncorrelated Nested Subgraphs
The study of percolation in so-called {\em nested subgraphs} implies a
generalization of the concept of percolation since the results are not linked
to specific graph process. Here the behavior of such graphs at criticallity is
studied for the case where the nesting operation is performed in an
uncorrelated way. Specifically, I provide an analyitic derivation for the
percolation inequality showing that the cluster size distribution under a
generalized process of uncorrelated nesting at criticality follows a power law
with universal exponent . The relevance of the result comes from
the wide variety of processes responsible for the emergence of the giant
component that fall within the category of nesting operations, whose outcome is
a family of nested subgraphs.Comment: 5 pages, no figures. Mistakes found in early manuscript have been
remove
Infinite-Order Percolation and Giant Fluctuations in a Protein Interaction Network
We investigate a model protein interaction network whose links represent
interactions between individual proteins. This network evolves by the
functional duplication of proteins, supplemented by random link addition to
account for mutations. When link addition is dominant, an infinite-order
percolation transition arises as a function of the addition rate. In the
opposite limit of high duplication rate, the network exhibits giant structural
fluctuations in different realizations. For biologically-relevant growth rates,
the node degree distribution has an algebraic tail with a peculiar rate
dependence for the associated exponent.Comment: 4 pages, 2 figures, 2 column revtex format, to be submitted to PRL 1;
reference added and minor rewording of the first paragraph; Title change and
major reorganization (but no result changes) in response to referee comments;
to be published in PR
Percolation in Directed Scale-Free Networks
Many complex networks in nature have directed links, a property that affects
the network's navigability and large-scale topology. Here we study the
percolation properties of such directed scale-free networks with correlated in-
and out-degree distributions. We derive a phase diagram that indicates the
existence of three regimes, determined by the values of the degree exponents.
In the first regime we regain the known directed percolation mean field
exponents. In contrast, the second and third regimes are characterized by
anomalous exponents, which we calculate analytically. In the third regime the
network is resilient to random dilution, i.e., the percolation threshold is
p_c->1.Comment: Latex, 5 pages, 2 fig
Range-based attack on links in scale-free networks: are long-range links responsible for the small-world phenomenon?
The small-world phenomenon in complex networks has been identified as being
due to the presence of long-range links, i.e., links connecting nodes that
would otherwise be separated by a long node-to-node distance. We find,
surprisingly, that many scale-free networks are more sensitive to attacks on
short-range than on long-range links. This result, besides its importance
concerning network efficiency and/or security, has the striking implication
that the small-world property of scale-free networks is mainly due to
short-range links.Comment: 4 pages, 4 figures, Revtex, published versio
Epidemic Incidence in Correlated Complex Networks
We introduce a numerical method to solve epidemic models on the underlying
topology of complex networks. The approach exploits the mean-field like rate
equations describing the system and allows to work with very large system
sizes, where Monte Carlo simulations are useless due to memory needs. We then
study the SIR epidemiological model on assortative networks, providing
numerical evidence of the absence of epidemic thresholds. Besides, the time
profiles of the populations are analyzed. Finally, we stress that the present
method would allow to solve arbitrary epidemic-like models provided that they
can be described by mean-field rate equations.Comment: 5 pages, 4 figures. Final version published in PR
Are randomly grown graphs really random?
We analyze a minimal model of a growing network. At each time step, a new
vertex is added; then, with probability delta, two vertices are chosen
uniformly at random and joined by an undirected edge. This process is repeated
for t time steps. In the limit of large t, the resulting graph displays
surprisingly rich characteristics. In particular, a giant component emerges in
an infinite-order phase transition at delta = 1/8. At the transition, the
average component size jumps discontinuously but remains finite. In contrast, a
static random graph with the same degree distribution exhibits a second-order
phase transition at delta = 1/4, and the average component size diverges there.
These dramatic differences between grown and static random graphs stem from a
positive correlation between the degrees of connected vertices in the grown
graph--older vertices tend to have higher degree, and to link with other
high-degree vertices, merely by virtue of their age. We conclude that grown
graphs, however randomly they are constructed, are fundamentally different from
their static random graph counterparts.Comment: 8 pages, 5 figure
Stability of shortest paths in complex networks with random edge weights
We study shortest paths and spanning trees of complex networks with random
edge weights. Edges which do not belong to the spanning tree are inactive in a
transport process within the network. The introduction of quenched disorder
modifies the spanning tree such that some edges are activated and the network
diameter is increased. With analytic random-walk mappings and numerical
analysis, we find that the spanning tree is unstable to the introduction of
disorder and displays a phase-transition-like behavior at zero disorder
strength . In the infinite network-size limit (), we
obtain a continuous transition with the density of activated edges
growing like and with the diameter-expansion coefficient
growing like in the regular network, and
first-order transitions with discontinuous jumps in and at
for the small-world (SW) network and the Barab\'asi-Albert
scale-free (SF) network. The asymptotic scaling behavior sets in when , where the crossover size scales as for the
regular network, for the SW network, and
for the SF network. In a
transient regime with , there is an infinite-order transition with
for the SW network
and for the SF network. It
shows that the transport pattern is practically most stable in the SF network.Comment: 9 pages, 7 figur
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