Biology Reference
In-Depth Information
of x's nearest neighbors will receive a new edge. Therefore,
the distribution of new links in the network is biased
towards the more connected nodes. Indeed, a node with
many nearest neighbors is more likely to have one of its
neighbors chosen for duplication. In fact, for a given node
with degree k, the probability for a randomly chosen node
to be linked to it is directly proportional to k. Thus its
probability to gain a link in the growth process is also
proportional to k, consistently with Eq.
(3)
.
One of the predictions of the Barab
´
si
e
Albert model is
that nodes can become well connected by virtue of being
older. A node that was introduced early in the history of the
network will have more time to accumulate links, and, by
the 'rich get richer' mechanism, enhance its chances of
becoming a hub
[36]
. In metabolic networks, we find that
the hubs do, indeed, tend to be older. Some examples are
coenzyme A, NAD and GTP, remnants of the RNA world,
which are among the most connected substrates of the
metabolic network
[34]
. Similar findings rise from the
analysis of protein
e
protein interaction networks, where, on
average, the evolutionary ancient proteins are characterized
by higher degrees
[72
e
73]
. This offers direct empirical
evidence for the preferential attachment hypothesis.
many sub-graphs that would otherwise be isolated. The
quantifiable fingerprint of such a hierarchical design can be
found in C
, which describes the dependence of the
clustering coefficient on the degree
[35,45,79
e
80]
.Low-
degree nodes will tend to belong to a specific module and
thus feature a high clustering coefficient
e
indeed, almost
all their neighbors will themselves be part of the same
module. The hubs, on the other hand, will be connected to
many nodes from different modules, and accordingly will
tend to have a low clustering coefficient.
The analysis of cellular networks shows clear evidence
of hierarchical topology. The dependence of the clustering
coefficient on the degree features a power-law scaling,
C
ð
k
Þ
k
b
. This has been observed for metabolic networks
[35]
, protein
e
protein interaction networks
[32]
and regu-
latory networks, with
b
taking values typically between
1 and 2.
ð
k
Þ
w
PARTY VS. DATE HUBS
We have already acknowledged the crucial role that the
hubs play in the integration of the network. In the above
discussion we further emphasized their importance when
the network has a modular structure, as the mediators
between separate modules. In this context, an interesting
distinction between two types of hubs has been proposed
[81]
. The first type, named party hubs, corresponds to our
usual perception of hubs as nodes that interact with many
other nodes simultaneously. The second type, date hubs,
bind to their partners at different times or at different
cellular locations. While the party hubs tend to interact
within a module, it is the date hubs that typically connect
between separate modules. So that it is mainly the latter
that serve as the integrators of the network. In the analysis
of the yeast protein
e
protein interaction network these two
types of hubs were indeed identified
[81]
. When the date
hubs were systematically removed, the network split into
small disconnected modules. In contrast, the removal of
party hubs, despite diluting the modules themselves,
harmed the overall integrity of the network to a much lesser
extent.
HIERARCHY AND MODULARITY
The ability of complex systems to function properly and
carry out vital tasks requires the cooperation of many
independent components. In many artificial networks this is
commonly achieved by relying on a hierarchical design.
The network is layered, and nodes at one level orchestrate
the behavior of their subordinates belonging to a level
below. In that sense we tend to picture network hierarchy as
a tree-like topology. However, the idea of having distinct
hierarchical layers of nodes stands in sharp contrast to the
scale-free nature of the cellular networks. The presence of
hubs, which connect directly to a large fraction of the nodes
in the network, will inevitably break down the layered
topology. We therefore have to adopt a different notion of
hierarchy to account for the functional design of biological
networks.
The conceptual idea is that the functionality of these
elaborate networks can be broken into distinct, relatively
isolated tasks
[35,74
e
78]
. From a structural point of view,
this will be expressed in networks composed of highly
interconnected sub-graphs, or modules. The hierarchical
disposition of a given node can be characterized by the
number of such sub-graphs to which it belongs. This way,
a node which is placed low in the hierarchy will participate
in just one functional task, and hence belong to just one
module. Higher in the hierarchy we find nodes that bridge
between two or three different modules. Eventually, at the
highest level of the hierarchy will reside the hubs, which do
not belong to any specific sub-graph but rather connect
DEGREE CORRELATIONS
It is commonly observed in networks that similar nodes
tend to connect to one another. This feature, termed
assortative mixing, can be related to any characteristic of
a node, and in particular to the node's degree. For instance,
in social networks individuals with many friends tend to
link to others who too have a high degree. However, as
shown in
Figure 9.2
(b), in the featured protein
e
protein
interaction network the opposite is true: the network is
disassortative, which means that the hubs tend to avoid
each other, leading to a network where highly connected
