11 research outputs found
Dynamic networks and directed percolation
We introduce a model for dynamic networks, where the links or the strengths
of the links change over time. We solve the model by mapping dynamic networks
to the problem of directed percolation, where the direction corresponds to the
evolution of the network in time. We show that the dynamic network undergoes a
percolation phase transition at a critical concentration , which decreases
with the rate at which the network links are changed. The behavior near
criticality is universal and independent of . We find fundamental network
laws are changed. (i) For Erd\H{o}s-R\'{e}nyi networks we find that the size of
the giant component at criticality scales with the network size for all
values of , rather than as . (ii) In the presence of a broad
distribution of disorder, the optimal path length between two nodes in a
dynamic network scales as , compared to in a static network.Comment: 10 pages 5 figures; corrected metadata onl
Structural crossover of polymers in disordered media
We present a unified scaling theory for the structural behavior of polymers
embedded in a disordered energy substrate. An optimal polymer configuration is
defined as the polymer configuration that minimizes the sum of interacting
energies between the monomers and the substrate. The fractal dimension of the
optimal polymer in the limit of strong disorder (SD) was found earlier to be
larger than the fractal dimension in weak disorder (WD). We introduce a scaling
theory for the crossover between the WD and SD limits. For polymers of various
sizes in the same disordered substrate we show that polymers with a small
number of monomers, N << N*, will behave as in SD, while large polymers with
length N >> N* will behave as in WD. This implies that small polymers will be
relatively more compact compared to large polymers even in the same substrate.
The crossover length N* is a function of \nu and a, where \nu is the
percolation correlation length exponent and a is the parameter which controls
the broadness of the disorder. Furthermore, our results show that the crossover
between the strong and weak disorder limits can be seen even within the same
polymer configuration. If one focuses on a segment of size n << N* within a
long polymer (N >> N*) that segment will have a higher fractal dimension
compared to a segment of size n >> N*
Limited path percolation in complex networks
We study the stability of network communication after removal of
links under the assumption that communication is effective only if the shortest
path between nodes and after removal is shorter than where is the shortest path before removal. For a large
class of networks, we find a new percolation transition at
, where and
is the node degree. Below , only a fraction of
the network nodes can communicate, where , while above , order nodes can
communicate within the limited path length . Our analytical results
are supported by simulations on Erd\H{o}s-R\'{e}nyi and scale-free network
models. We expect our results to influence the design of networks, routing
algorithms, and immunization strategies, where short paths are most relevant.Comment: 11 pages, 3 figures, 1 tabl