Physics (Internal)
TRANSPORT AND PERCOLATION IN COMPLEX NETWORKS
This event is part of the PhD Final Oral Exams.
Dissertation Committee: H. Eugene Stanley, William Skocpol, Karl Ludwig, William Klein, Martin Schmaltz
Abstract:
To design complex networks with optimal transport properties such as
flow efficiency, we consider three approaches to understanding transport
and percolation in complex networks. We analyze the effects of randomizing
the strengths of connections, randomly adding long-range connections
to regular lattices and percolation of spatially constrained networks.
Various real-world networks often have links that are differentiated
in terms of their strength, intensity, or capacity. We study the distribution
$P(\sigma)$ of the equivalent conductance for Erd\H{o}s-R\'enyi (ER)
and scale-free (SF) weighted resistor networks with $N$ nodes, for which
links are assigned with conductance $\sigmai \equiv e^{-axi}$,
where $xi$ is a random variable with $0<xi<1$. We find, both
analytically and numerically, that $P(\sigma)$ for ER networks
exhibits two regimes: (i) For $\sigma < e^{-apc}$,
$P(\sigma)$ is independent of $N$ and scales as
$P(\sigma) \sim \sigma^{-\delta}$, where $\delta=1-\frac{1}{apc}$.
Here $pc=1/\av{k}$ is the critical percolation threshold of the network and $\av{k}$ is the average degree of the network.
(ii) For $ \sigma > e^{-apc}$,
$P(\sigma)$ has strong $N$ dependence and scales as
$P(\sigma) \sim f(\sigma,apc/N^{1/3})$.
Transport properties are greatly affected by the topology of networks.
We investigate the transport problem in lattices with long-range
connections and subject to a cost
constraint, seeking design principles for optimal transport networks.
Our network is built from a regular $d$-dimensional lattice to be improved by adding
long-range connections with probability $P{ij} \sim r{ij}^{-\alpha}$,
where $r{ij}$ is the lattice distance between site $i$ and $j$.
We introduce a cost constraint on the total length of the additional
links and find optimal transport in the system for $ \alpha=d+1 $,
established here for $d=1, 2$ and $3$ for regular lattices
and $d_f$ for fractals. Remarkably,
this cost constraint approach remains optimal, regardless of
the strategy used for transport, whether
based on local or global knowledge of the network structure.
To further understand the role that long-range
connections play in optimizing the transport of complex systems,
we study the percolation of spatially constrained networks.
We now consider originally empty lattices embedded in $d$ dimensions by
adding long-range connections with the same power law probability
$p(r) \sim r^{-\alpha}$.
We find that, for $\alpha \le d$, the percolation transition
belongs to the universality class of percolation in
ER networks, while for $\alpha >2d$ it belongs to the
universality class of percolation in regular lattices
(for one dimension, there is no percolation transition as one-dimensional regular lattice).
However for $d <\alpha < 2d$, the percolation properties
show new intermediate behavior different from ER networks,
with critical exponents that depend on $\alpha$.