13,730 research outputs found
Unravelling small world networks
New classes of random graphs have recently been shown to exhibit the small world phenomenon - they are clustered like regular lattices and yet have small average pathlengths like traditional random graphs. Small world behaviour has been observed in a number of real life networks, and hence these random graphs represent a useful modelling tool. In particular, Grindrod [Phys. Rev. E 66 (2002) 066702-1] has proposed a class of range dependent random graphs for modelling proteome networks in bioinformatics. A property of these graphs is that, when suitably ordered, most edges in the graph are short-range, in the sense that they connect near-neighbours, and relatively few are long-range. Grindrod also looked at an inverse problem - given a graph that is known to be an instance of a range dependent random graph, but with vertices in arbitrary order, can we reorder the vertices so that the short-range/long-range connectivity structure is apparent? When the graph is viewed in terms of its adjacency matrix, this becomes a problem in sparse matrix theory: find a symmetric row/column reordering that places most nonzeros close to the diagonal. Algorithms of this general nature have been proposed for other purposes, most notably for reordering to reduce fill-in and for clustering large data sets. Here, we investigate their use in the small world reordering problem. Our numerical results suggest that a spectral reordering algorithm is extremely promising, and we give some theoretical justification for this observation via the maximum likelihood principle
Deterministic Small-World Networks
Many real life networks, such as the World Wide Web, transportation systems,
biological or social networks, achieve both a strong local clustering (nodes
have many mutual neighbors) and a small diameter (maximum distance between any
two nodes). These networks have been characterized as small-world networks and
modeled by the addition of randomness to regular structures. We show that
small-world networks can be constructed in a deterministic way. This exact
approach permits a direct calculation of relevant network parameters allowing
their immediate contrast with real-world networks and avoiding complex computer
simulations.Comment: 6 pages, 1 figur
Small-World Networks in Geophysics
Many geophysical processes can be modelled by using interconnected networks.
The small-world network model has recently attracted much attention in physics
and applied sciences. In this paper, we try to use and modify the small-world
theory to model geophysical processes such as diffusion and transport in
disordered porous rocks. We develop an analytical approach as well as numerical
simulations to try to characterize the pollutant transport and percolation
properties of small-world networks. The analytical expression of system
saturation time and fractal dimension of small-world networks are given and
thus compared with numerical simulations
Chaos in Small-World Networks
A nonlinear small-world network model has been presented to investigate the
effect of nonlinear interaction and time delay on the dynamic properties of
small-world networks. Both numerical simulations and analytical analysis for
networks with time delay and nonlinear interaction show chaotic features in the
system response when nonlinear interaction is strong enough or the length scale
is large enough. In addition, the small-world system may behave very
differently on different scales. Time-delay parameter also has a very strong
effect on properties such as the critical length and response time of
small-world networks
Stations, trains and small-world networks
The clustering coefficient, path length and average vertex degree of two
urban train line networks have been calculated. The results are compared with
theoretical predictions for appropriate random bipartite graphs. They have also
been compared with one another to investigate the effect of architecture on the
small-world properties.Comment: 6 pages, prepared in RevTe
Dynamic rewiring in small world networks
We investigate equilibrium properties of small world networks, in which both
connectivity and spin variables are dynamic, using replicated transfer matrices
within the replica symmetric approximation. Population dynamics techniques
allow us to examine order parameters of our system at total equilibrium,
probing both spin- and graph-statistics. Of these, interestingly, the degree
distribution is found to acquire a Poisson-like form (both within and outside
the ordered phase). Comparison with Glauber simulations confirms our results
satisfactorily.Comment: 21 pages, 5 figure
Nonextensive aspects of small-world networks
Nonextensive aspects of the degree distribution in Watts-Strogatz (WS)
small-world networks, , have been discussed in terms of a
generalized Gaussian (referred to as {\it -Gaussian}) which is derived by
the three approaches: the maximum-entropy method (MEM), stochastic differential
equation (SDE), and hidden-variable distribution (HVD). In MEM, the degree
distribution in complex networks has been obtained from -Gaussian
by maximizing the nonextensive information entropy with constraints on averages
of and in addition to the normalization condition. In SDE,
-Gaussian is derived from Langevin equations subject to additive and
multiplicative noises. In HVD, -Gaussian is made by a superposition of
Gaussians for random networks with fluctuating variances, in analogy to
superstatistics. Interestingly, {\it a single} may describe, with an
accuracy of \mid P_{SW}(k)-P_Q(k)\mid \siml 10^{-2} , main parts of degree
distributions of SW networks, within which about 96-99 percents of all
states are included. It has been demonstrated that the overall behavior of
including its tails may be well accounted for if the -dependence
is incorporated into the entropic index in MEM, which is realized in
microscopic Langevin equations with generalized multiplicative noises.Comment: 22 pages, 11 figures, accepted in Physca A with some augmentation
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