280 research outputs found
Renormalization group analysis of the small-world network model
We study the small-world network model, which mimics the transition between
regular-lattice and random-lattice behavior in social networks of increasing
size. We contend that the model displays a normal continuous phase transition
with a divergent correlation length as the degree of randomness tends to zero.
We propose a real-space renormalization group transformation for the model and
demonstrate that the transformation is exact in the limit of large system size.
We use this result to calculate the exact value of the single critical exponent
for the system, and to derive the scaling form for the average number of
"degrees of separation" between two nodes on the network as a function of the
three independent variables. We confirm our results by extensive numerical
simulation.Comment: 4 pages including 3 postscript figure
Why social networks are different from other types of networks
We argue that social networks differ from most other types of networks,
including technological and biological networks, in two important ways. First,
they have non-trivial clustering or network transitivity, and second, they show
positive correlations, also called assortative mixing, between the degrees of
adjacent vertices. Social networks are often divided into groups or
communities, and it has recently been suggested that this division could
account for the observed clustering. We demonstrate that group structure in
networks can also account for degree correlations. We show using a simple model
that we should expect assortative mixing in such networks whenever there is
variation in the sizes of the groups and that the predicted level of
assortative mixing compares well with that observed in real-world networks.Comment: 9 pages, 2 figure
Planning for Persistence in Marine Reserves: A Question of Catastrophic importance
Large-scale catastrophic events, although rare, lie generally beyond the control of local management and can prevent marine reserves from achieving biodiversity outcomes. We formulate a new conservation planning problem that aims to minimize the probability of missing conservation targets as a result of catastrophic events. To illustrate this approach we formulate and solve the problem of minimizing the impact of large-scale coral bleaching events on a reserve system for the Great Barrier Reef, Australia. We show that by considering the threat of catastrophic events as part of the reserve design problem it is possible to substantially improve the likely persistence of conservation features within reserve networks for a negligible increase in cost. In the case of the Great Barrier Reef, a 2% increase in overall reserve cost was enough to improve the long-run performance of our reserve network by >60%. Our results also demonstrate that simply aiming to protect the reefs at lowest risk of catastrophic bleaching does not necessarily lead to the best conservation outcomes, and enormous gains in overall persistence can be made by removing the requirement to represent all bioregions in the reserve network. We provide an explicit and well-defined method that allows the probability of catastrophic disturbances to be included in the site selection problem without creating additional conservation targets or imposing arbitrary presence/absence thresholds on existing data. This research has implications for reserve design in a changing climate
Percolation and epidemics in a two-dimensional small world
Percolation on two-dimensional small-world networks has been proposed as a
model for the spread of plant diseases. In this paper we give an analytic
solution of this model using a combination of generating function methods and
high-order series expansion. Our solution gives accurate predictions for
quantities such as the position of the percolation threshold and the typical
size of disease outbreaks as a function of the density of "shortcuts" in the
small-world network. Our results agree with scaling hypotheses and numerical
simulations for the same model.Comment: 7 pages, 3 figures, 2 table
Edge overload breakdown in evolving networks
We investigate growing networks based on Barabasi and Albert's algorithm for
generating scale-free networks, but with edges sensitive to overload breakdown.
the load is defined through edge betweenness centrality. We focus on the
situation where the average number of connections per vertex is, as the number
of vertices, linearly increasing in time. After an initial stage of growth, the
network undergoes avalanching breakdowns to a fragmented state from which it
never recovers. This breakdown is much less violent if the growth is by random
rather than preferential attachment (as defines the Barabasi and Albert model).
We briefly discuss the case where the average number of connections per vertex
is constant. In this case no breakdown avalanches occur. Implications to the
growth of real-world communication networks are discussed.Comment: To appear in Phys. Rev.
Properties of highly clustered networks
We propose and solve exactly a model of a network that has both a tunable
degree distribution and a tunable clustering coefficient. Among other things,
our results indicate that increased clustering leads to a decrease in the size
of the giant component of the network. We also study SIR-type epidemic
processes within the model and find that clustering decreases the size of
epidemics, but also decreases the epidemic threshold, making it easier for
diseases to spread. In addition, clustering causes epidemics to saturate
sooner, meaning that they infect a near-maximal fraction of the network for
quite low transmission rates.Comment: 7 pages, 2 figures, 1 tabl
Social games in a social network
We study an evolutionary version of the Prisoner's Dilemma game, played by
agents placed in a small-world network. Agents are able to change their
strategy, imitating that of the most successful neighbor. We observe that
different topologies, ranging from regular lattices to random graphs, produce a
variety of emergent behaviors. This is a contribution towards the study of
social phenomena and transitions governed by the topology of the community
Epidemics and percolation in small-world networks
We study some simple models of disease transmission on small-world networks,
in which either the probability of infection by a disease or the probability of
its transmission is varied, or both. The resulting models display epidemic
behavior when the infection or transmission probability rises above the
threshold for site or bond percolation on the network, and we give exact
solutions for the position of this threshold in a variety of cases. We confirm
our analytic results by numerical simulation.Comment: 6 pages, including 3 postscript figure
XY model in small-world networks
The phase transition in the XY model on one-dimensional small-world networks
is investigated by means of Monte-Carlo simulations. It is found that
long-range order is present at finite temperatures, even for very small values
of the rewiring probability, suggesting a finite-temperature transition for any
nonzero rewiring probability. Nature of the phase transition is discussed in
comparison with the globally-coupled XY model.Comment: 5 pages, accepted in PR
Scaling Properties of Random Walks on Small-World Networks
Using both numerical simulations and scaling arguments, we study the behavior
of a random walker on a one-dimensional small-world network. For the properties
we study, we find that the random walk obeys a characteristic scaling form.
These properties include the average number of distinct sites visited by the
random walker, the mean-square displacement of the walker, and the distribution
of first-return times. The scaling form has three characteristic time regimes.
At short times, the walker does not see the small-world shortcuts and
effectively probes an ordinary Euclidean network in -dimensions. At
intermediate times, the properties of the walker shows scaling behavior
characteristic of an infinite small-world network. Finally, at long times, the
finite size of the network becomes important, and many of the properties of the
walker saturate. We propose general analytical forms for the scaling properties
in all three regimes, and show that these analytical forms are consistent with
our numerical simulations.Comment: 7 pages, 8 figures, two-column format. Submitted to PR
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