Biomedical Engineering Reference
In-Depth Information
Figure 4
. (
a
) For the random graph, the degree distribution,
P
(
k
), which gives the probability
that a randomly selected node has exactly
k
edges, follows a Poisson distribution that is strongly
peaked at the average degree <
k
> and decays exponentially for large
k
. (
b,c
) The
P
(
k
) values of
a scale-free and a hierarchical network do not have a peak and decay as a power-law,
P
(
k
) ~
k
H
.
(
d,e
) For both a random and scale-free network, the
C
(
k
) function, which denotes the mean
clustering coefficient for nodes with exactly
k
links, is independent of
k
. (
f
) In contrast,
C
(
k
) of a
hierarchical network depends on
k
, decaying as
C
(
k
) ~
k
-1
. Insets correspond to the number of
underlying networks.
where
k
i
is the degree of node
i
. The network generated by this growth process
will be scale-free with degree exponent H = 3. In a scale-free network the prob-
ability that a node is highly connected (
k
>> <
k
>) is statistically more significant
than in a random graph. Thus, the network's properties are often determined by a
relatively small number of highly connected nodes or hubs. An important conse-
quence of the hubs is that scale-free networks exhibit high tolerance to random
perturbations but are sensitive to targeted attack upon the highly connected
nodes (3). Accordingly, failure of randomly selected nodes cannot destroy the
network's integrity. However, systematic removal of the hubs will rapidly frag-
ment the network. This feature is of particular importance for biological sys-
tems, since it reflects a biochemical network's resilience against random
mutations. Therefore, highly connected nodes in biochemical networks might be
potential candidates for drug targets.
The presence of hubs in a scale-free network has a fundamental impact on
virus spreading as well. Classical epidemiological models predict that infectious
diseases with transmission probability under an epidemic threshold will inevita-
