Biology Reference
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nodes are surrounded by low-degree nodes [82] . This dis-
assortativity is observed in most biological networks,
including the metabolic and regulatory networks, and is, in
fact, a property shared by technological networks, such as
the power grid or the internet [83 e 84] .
To classify a network as assortative or disassortative we
first need to define our expectations of a neutral network.
What we are aiming at is to characterize the expected
correlations between the degrees of nearest neighbors in the
absence of any assortative bias. To do this we consider the
random selection of an edge in the network and calculate
the probability that at one of its ends resides a node with
degree of k and at the other a node with a degree of k 0 . Let
us first calculate the probability for the first node: i.e., we
are seeking the probability of finding a node with k links at
the end of a randomly selected edge. This is essentially
different from the direct selection of a random node, since it
gives an advantage to nodes with a higher degree. The
reason is simply because such nodes have a larger number
of edges to which they are attached. For instance, to reach
a node with a single edge through this procedure, one must
pinpoint the one edge leading to it. On the other hand, there
are k potential edges through which a k degree node can be
reached, making this outcome k times more likely. Thus the
desired probability is proportional to the abundance of k
degree nodes, as well as to the degree itself, i.e., it is
the hubs, so that for them K nn is much greater. Indeed, the
analysis shows that for this network K nn ð
k a ,where
k
Þ w
a z
24 [85] .
One can obtain an even more compact parameterization
for a network's assortativity, by referring to the Pearson
correlation coefficient measured between the degrees of
pairs of connected nodes. This can be explicitly done by
extracting the correlation coefficient for k and k 0
0
:
from the
distribution given by Q kk 0 . The result is [83]
r ¼ h
kk 0 ih
k 0 i
k
ih
;
(4)
s
2
2 is the variance obtained from the distribution q k .
The parameter r takes values between 1, for a perfectly
assortative network, and
where s
1, when the network is perfectly
disassortative. For the yeast protein e protein interaction
network shown in Figure 9.2 (a), it measures r
¼
:
0
156,
confirming that the network is, indeed, disassortative.
The mechanism responsible for the disassortative nature
of biological networks remains unclear. It cannot be
accounted for by the Barab´si e Albert mechanism, which
does not yield any degree correlations [83] . From a func-
tional point of view, it highlights the modular structure of
biological networks, possibly strengthening even further
the central role of the hubs. It was also shown that dis-
assortativity harms the resilience of the network and makes
it more vulnerable to the intentional removal of hubs, since
in such networks the majority of low-degree nodes are
connected solely to the hubs. On the other hand, dis-
assortativity has a positive contribution to the integrity of
the network when it is not under attack, as typically a dis-
assortative network will feature a larger giant connected
component than an assortative or a neutral one [83] . This
once again emphasizes the resilience of cellular networks
against random failure, compared to their vulnerability
against selected node removal.
kP ð k Þ
h
q k ¼
, where the denominator is used as a normali-
k
i
zation constant. In a neutral network, the degree distribu-
tion of the nodes that lie at the other end of the selected
edge is independent of q k . Thus, in the absence of degree
correlations, the probability that a randomly selected edge
links between two nodes with a degree of k and k 0
is simply
Q Neu
kk 0
q k q k 0 . To evaluate the assortativity of the network
we compare the observed probability Q kk 0
¼
to Q Neu
kk 0 . For an
assortative network, the observed probability will show
a positive bias along the diagonal, where the value of k is
close to that of k 0 . Disassortativity will be expressed as
a negative bias along the diagonal, and a tendency to have
more links where k s k 0 .
Another, more compact description of the degree
correlations can be viewed by observing the average
degree of a node's nearest neighbors. We denote this
average by K nn . We then average over all nodes with
agivendegree,k,toobtainK nn ð
HUMAN DISEASE NETWORK
The applications of graph theory to systems biology can go
beyond the mapping of the concrete network systems found
within the cell. Graphs could also be used as a means of
organizing biological information in a way that could
potentially spark new insights. An innovative example is
provided by the network approach to the study of human
diseases [86] . In this approach two networks are con-
structed. The first is the human disease network. In this
network the nodes represent genetic disorders and the edges
link disorders which are associated with mutations in the
same gene. The second network is the disease gene
network. Here the nodes represent genes, and the edges link
genes which are associated with the same disorder. Both
networks are found to be highly clustered, showing that
, namely the average
degree of the neighbors surrounding a typical node with k
links. In a neutral network, K nn should not depend on k,
but if degree correlations are present they will be
expressed as a monotonic increase or decrease in K nn ð
k
Þ
.
In the protein e protein interaction network displayed in
Figure 9.2 , this dependency is clearly visible: the average
degree of the hub's nearest neighbors is between 1 and 2,
and yet the low-degree nodes are almost all connected to
k
Þ
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