Biomedical Engineering Reference
In-Depth Information
bly die out. However, in scale-free networks the epidemic threshold is reduced
to zero (43). Thus, as some social and sexual networks are known to exhibit a
scale-free topology (33), even extremely weakly infectious viruses can spread
and prevail, making random immunization ineffective.
3.3. Hierarchical Networks
Many real networks are expected to be fundamentally modular, meaning
that the network can be seamlessly partitioned into a collection of modules.
Each module is expected to perform an identifiable task, separable from the
function of other modules (19,31,50,61). Therefore, we must reconcile the scale-
free property with the network's potential modularity. Numerical simulations
indicate that neither the random nor the scale-free network model are modular.
In order to account for the coexistence of modularity, local clustering, and
scale-free topology in real systems, we have to assume that clusters combine in
an iterative manner, generating a hierarchical network (7,46). Such networks
emerge from an iterative duplication and integration of clustered nodes, a proc-
ess that in principle can be repeated indefinitely. Our starting point is a small
cluster of four densely linked nodes. Next we generate three replicas of this hy-
pothetical module and connect the three external nodes of the replicated clusters
to the central node of the old cluster, obtaining a large 16-node module. Subse-
quently, we again generate three replicas of this 16-node module and connect
the 16 peripheral nodes to the central node of the old module, obtaining a new
module of 64 nodes (third column of Figure 3).
The hierarchical network model seamlessly integrates a scale-free topology
with an inherent modular structure by generating a network that has a power-law
degree distribution with degree exponent H = 1 + ln 4/ln 3 = 2.26. Yet the most
important signature of this hierarchical modularity is the fact that the clustering
coefficient,
C
(
k
), scales as
k
-1
(third column of Figure 4). Note that for a network
generated by the ER and BA models
C
(
k
) is independent of
k
.
Modularity does not, however, imply clear-cut subnetworks that are linked
in well-defined ways. In fact, the boundaries of modules are often considerably
blurred, triggered by highly connected nodes that interconnect modules.
4.
BIOLOGICAL NETWORKS
4.1. Metabolic Networks
The structure of metabolic networks was addressed by two independent
studies by Fell and Wagner and Jeong et al. Fell and Wagner assembled a list of
stoichiometric equations that represent the central routes of the energy metabo-
