SESSION GRAPH BASED AND TREE METHODS + RELATED ...

10 Int'l Conf. Foundations of Computer Science | FCS'12 |
‘strength’ of a minority of links is the primary reason for
a network’s assortativity, then we could predict that, if the
network evolves, its assortativity may change rapidly. Link
assortativity can be an indicator of the importance of links in
the network, particularly if the links are highly assortative.
Furthermore, profiles of link assortativity will provide us
with yet another tool to classify networks.
Our paper is structured as follows: in the next section,
we will introduce the concept of link assortativity for directed
networks. We will use the definition of assortativity
described in [7] to establish this concept, since this form of
definition is most conducive for link-based decomposition.
Then we will analyze the link-based assortativity profiles of
a number of synthesized and real world directed networks.
These include citation networks, Gene regulatory networks,
transcription networks, foodwebs and neural networks. We
will draw observations from this analysis, showing that a
network could be either assortative, disassortative or nonassortative,
due to a number of combinations between the
ratio of ‘positive’ and ‘negative’ links, and the average
strength of such links. We will also look at link-assortativity
distributions of a number of networks, and discuss what
insights can be gained from these about the evolution and
functionality of these networks. Finally we will present our
summary and conclusions.
2. Link assortativity of directed networks
Degree assortativity has been defined by Newman, as a
Pearson correlation between the ‘expected degree’ distribution
qk, and the ‘joint degree’ distribution ej,k [3]. The
expected degree distribution is the probability distribution of
traversing the links of the network, and finding nodes with
degree k at the end of the links. Similarly, the ‘joint degree’
distribution is the probability distribution of an link having
degree j on one end and degree k on the other end. In the
undirected case, the normalized Pearson coefficient of ej,k
and qk gives us the assortativity coefficient of the network,
r.
If a network has perfect assortativity (r = 1), then all
nodes connect only with nodes with the same degree. For
example, the joint distribution ej,k = qkδj,k where δj,k is
the Kronecker delta function, produces a perfectly assortative
network. If the network has no assortativity (r = 0), then any
node can randomly connect to any other node. A sufficiency
condition for a non-assortative network is ej,k = qjqk. If
a network is perfectly disassortative (r = −1), all nodes
will have to connect to nodes with different degrees. A star
network is an example of a perfectly disassortative network,
and complex networks with star ‘motifs’ in them tend to be
disassortative.
In the case of directed networks, the definition is a
bit more involved due to the existence of in-degrees and
out-degrees. Therefore, we must consider the probability
distribution of links going out of source nodes with j out-
degrees, denoted as q out
j , and the probability distribution of
links going into target nodes with k in-degrees, denoted q in
k .
In addition, we may consider the probability distribution of
links going into target nodes with k out-degrees, denoted
˘q out
k , and the probability distribution of links going out of
source nodes with j in-degrees, denoted ˘q in
j . In general,
qout k ̸= ˘q out
k and qin j ̸= ˘q in
j [6].
We can also consider distribution e out,out
j,k , abbreviated as
eout j,k , as the joint probability distribution of links going into
target nodes with k out-degrees, and out of source nodes of
j out-degrees (i.e., the joint distribution in terms of outdegrees).
Similarly, ein j,k = ein,in
j,k is the joint probability
distribution of links going into target nodes of k in-degrees,
and out of source nodes of j in-degrees (i.e., the joint
distribution of in-degrees).
we can therefore define the out-assortativity of directed
networks, as the tendency of nodes with similar out-degrees
to connect to each other. Using the above distributions, outassortativity
is formally defined, for out-degrees j and k, by
Piraveenan et al [6] as
rout =
1
σout q σout ⎡⎛
⎣⎝
˘q
∑
jke out
⎞
⎠
j,k − µ out
jk
q µ out
˘q
⎤
⎦ (1)
where µ out
q is the mean of qout k , µ˘q is the mean of ˘q out
k , σout q
is the standard deviation of qout k , and σout
˘q is the standard
deviation of ˘q out
k .
Similarly, Piraveenan et al. [6] defined in-assortativity as
the tendency in a network for nodes with similar in-degrees
to connect to each other, and this was formally specified as:
1
rin =
σin q σin ⎡⎛
⎣⎝
˘q
∑
jke in
⎞
⎠
j,k − µ in
q µ in
⎤
⎦
˘q (2)
jk
where µ in
q is the mean of qin k , µin
˘q is the mean of ˘q in
k , σin q is
the standard deviation of qin k , σin
˘q is the standard deviation
of ˘q in
k .
Meanwhile, Foster et al. [7] also defined assortativity of
directed networks in terms of the above distributions. While
they used a different set of notations, using our notation their
definition for out-assortativity can be written as
rout =
M −1
σ out
q σ out
˘q
[ ∑
i
(j out
i
− µ out
q )(k out
i − µ out
˘q )
where M is the number of links. Similarly, the definition of
Foster et al. for in-assortativity can be written as
[
−1 M ∑
rin = (j in
i − µ in
q )(k in
i − µ in
]
˘q ) (4)
σ in
q σ in
˘q
i
]
(3)