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(2) the average out-degree double from endogenous to exogenous conditions: each
TF has greater regulatory inuence by targeting more genes simultaneously;
(3) the ASPL (here equal to the number of intermediate regulators between a TF
and a target gene) halves from endogenous to exogenous conditions, implying
a faster propagation of the regulatory signal;
(4) the average clustering coecient (indicating the level of transcription factor
inter-regulation) nearly halves from endogenous to exogenous conditions.
Clustering Coecient: The clustering coecient has been introduced by Watts
and Strogatz in 1988 to determine whether or not a graph has small-world properties
[86]. Roughly speaking, the clustering coecient of a node depends on the number
of a node's neighbors that share a connection. More rigorously, C
I
= 2n
I
=k(k1),
where n
I
is the number of links connecting the k
I
neighbors of node I to each other.
C
I
gives the number of \triangles" going through node I, whereas k
I
(k
I
1)=2 is
the total number of triangles passing through node I if all of I's neighbors are
interconnected. The average clustering coecient (hCi) indicates the tendency of
the nodes of a network to form well-dened clusters. The average clustering coe-
cient can also be calculated by considering nodes with the same degree (C(k)), and
can be used as an indication of hierarchical behavior of a network [87, 88]. Network
topology may be said to correspond to a small world if the network's clustering coef-
cient is much greater than that of equivalent random controls CC
random
, while
their path lengths are comparable ASP LASP L
random
. The small-world index
small world
, introduced by [89], is dened as:
small world
=
C
random
ASP L
C
ASP L
random
.
Protein-protein interaction and gene regulatory networks often have small-world
topology; one explanation for an evolutionary advantage of this topology is that it
is very robust to perturbations. If this is the case, it would provide an advantage to
biological systems that are subject to damage e.g. by mutation or viral infection.
Moreover, in power law small-world networks the deletion of a random node rarely
causes a dramatic increase in ASPL (or a signicant decrease of C). This is because
(i) most paths pass through the hubs, and (ii) hubs are very rare, which means that
most of the times random deletions will involve loosely connected nodes, maintaining
unaltered the paths in the large majority of the network.
Assortativity Coecient: Assortativity is a measure of how much edges in a
network tend to connect similar nodes with respect to a certain feature [90, 91].
The so-called assortative mixing of complex networks has recently gained a lot
of attraction because it has a profound impact on the topological properties of a
network. In the case of perfect assortative mixing there are parts of the network
that are completely isolated, but within those parts there are dense interconnections.
The most basic form of assortativity concerns degree-degree correlation; in this case
the assortativity coecient (r2[1; 1]) is the correlation coecient between the
degrees of connected nodes, see Eq. 2.1 where L is the total number of edges, j
i
and