DARPA ULTRALOG Final Report - Industrial and Manufacturing ...

line of research that deals with networks of identical chaotic map is coupled map lattices
(Kaneko and Tsuda 1996) and cellular automata (Wolfram 1994). However these systems have
been used mainly as test-beds for exploring spatio-temporal chaos and pattern formation in the
simplest mathematical settings, rather than as models of real systems.
The second area in network dynamics is concerned about characterizing the network structure.
Network structure or topologies in general can vary from completely regular like chains, grids,
lattices and fully connected to completely random. Moreover the graphs can be directed or
undirected and cyclic or acyclic. In order to characterize topological properties of the graphs,
various statistical quantities have been defined. Most important of them include average path
length, clustering coefficient, degree distributions, size of giant component and various spectral
properties. A review of the main models and analytical tools, covering regular graphs, random
graphs, generalized random graphs, small-world and scale-free networks, as well as the interplay
between topology and the network's robustness against failures and attacks can be found in
(Albert and Barabasi 2002, Dorogovtsev and Mendes 2002).
The classic random graphs were introduced by Erdos and Renyi (Bollobas 1985) and have been
the most thoroughly studied models of networks. Such graphs have Poisson degree distribution
and statistically uncorrelated vertices. At large N (total number of nodes in the graph) and large
enough p (probability that two arbitrary vertices are connected), a giant connected component
appears in the network, a process known as percolation. The random graphs exhibit low average
path length, and low clustering coefficient. The regular networks on other hand show high
clustering coefficient and also a greater average path length compared to the random graphs of
similar size. The networks found in real world, however are neither completely regular nor
completely random. This has been recently discovered in the form of “small world” and “scale
free” characteristics, for many real networks like: social networks, internet, WWW, power grids,
collaboration networks, ecological and metabolic networks to name a few.
In order to describe the transition from a regular network to a random network, Watts and
Strogatz introduced the so-called small-world graphs as models of social networks (Watts and
Strogatz 1998) and (Newman 2000). This model exhibits a high degree of clustering as in the
regular network and a small average distance between vertices as in the classic random graphs. A
common feature of this model with random graph model is that the connectivity distribution of
the network peaks at an average value and decays exponentially. Such an exponential network is
homogeneous in nature: each node has roughly the same number of connections. Due to high
degree of clustering the models of dynamical systems with small-world coupling display
enhanced signal-propagation speed, rapid disease propagation, and synchronizability (Watts and
Strogatz 1998).
Another significant recent discovery in the field of complex networks is the observation that
the connectivity distributions of a number of large-scale and complex networks, including the
−γ
WWW, Internet, and metabolic network, have the power law form P ( k)
≈ k , where P(k)
is
the probability that a node in the network is connected to k other nodes, and γ is a positive real
number (Barabasi et al. 2000, Barabasi 2001). Since power-laws are free of characteristic scale,
such networks are called “scale-free network”. A scale-free network is inhomogeneous in nature:
most nodes have few connections but small number (but statistically significant) have many
connections. The average path length is smaller in the scale free network than in a random graph,
indicating that the heterogeneous scale-free topology is more efficient in bringing the nodes
closer than homogenous topology of the random graphs. The clustering coefficient of the scalefree
network is about 5 times higher than that of the random graph, and this factor slowly
increases with the number of nodes. It has been shown that it is practically impossible to achieve
synchronization in a nearest-neighbor coupled network (regular connectivity) if the network is
sufficiently large. However, it is quite easy to achieve synchronization in a scale-free dynamical
network no matter how large the network is (Weng and Chen, 2002). Moreover, the