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k
i
are the degrees of vertices at the ends of the i
th
edge. Positive values of r indicate
frequent connections between nodes of similar degree, while negative values indicate
relationships between nodes of dierent degree. When r = 1, the network is said to
have perfect assortative mixing, implying a high modularity of the network and the
presence of isolated clusters composed by nodes with the same degree i.e. layers.
On the converse, for r =1 the network is said to be completely disassortative;
there are no more layers, and connections maintain compact the network.
P
P
i
L
1
i
j
i
k
i
[L
1
1
2
(j
i
+ k
i
)]
2
P
i
P
i
r =
2
(j
i
+ k
i
)]
2
:
(2.1)
1
2
(j
i
+ k
i
)[L
1
1
L
1
The PIN in yeast turned out to be disassortative i.e. hubs tend to be connected
to more peripheral nodes. The same has been observed for a network of the synap-
tic connections in the neural network of the nematode C. elegans and for most of
the biological networks studied so far [90, 91]. Newman [90, 91] has observed that
several key properties of networks change when the model used to simulate network
growth includes dierent levels of assortative mixing and may thus be important
in biological networks evolution. One of the properties varying much is the size
of the giant component that is smaller if the network is assortative, reecting a
better percolation of assortative networks. Assortativity with respect to degree can
tell us several information on network topology; however, the most important fea-
ture of nodes in biological networks is that they represent real biological entities
participating in the physiology of the cell. Thus it is interesting to study the cor-
relations existing between properties of the nodes when a certain topology of the
network is specied, i.e. the PIN or the TRN. The categories can be the molecular
process in which proteins are involved. The work of Newman [91] improves the pre-
vious coecient developing a way to quantify the assortativity when discrete nodes
characteristics are accounted for.
With a similar purpose Park and Barabasi introduced in the biological sciences
the concept of dyadicity [92]. The simplest case is of un undirected graph where
each node can take only two values, say 1 or 0 (e.g. is this gene essential? Is
this gene involved in `cell-cycle'?). Let us call n
1
(n
0
) the number of nodes of
each group (N = n
1
+ n
0
). This allows for three kinds of dyads: (1-1), (0-1),
and (0-0) that are present in the network in m
11
, m
10
, m
00
fractions, respectively
(M = m
11
+ m
10
+ m
00
). The expected values of m
11
and m
10
[92] under the null
hypothesis that the property is distributed randomly on nodes can be obtained with
the hypergeometric distribution:
n
1
2
n
1
(n
1
1)
2
m
11
=
p =
p
(2.2)
and:
n
1
1
n
0
1
m
10
=
p = n
1
(Nn
1
)p
(2.3)
