Biology Reference
In-Depth Information
[46 e 48] . In fact, in an SF network, where the scaling
exponent is 2
technological systems [5] . This ubiquitous topological
feature not only characterizes the architecture of a given
network, it also serves as an indicator for its formation
process. This idea is captured by the Barab´si e Albert
model, which attributes the emergence of an SF topology
to the presence of two fundamental formation processes:
network growth and preferential attachment [36] .By
growth we refer to the fact that networks are not static:
they evolve in time by constantly adding new nodes and
new links. By preferential attachment we mean that nodes
are more likely to link to already highly connected nodes.
For a more accurate definition, consider an evolving
network, where at each time step a single new node is
introduced, drawing m new links to any of the existing
nodes. According to the preferential attachment mecha-
nism the new node will choose to connect to an existing
node, x, with a probability proportional
< g <
3 the average path length satisfies
h
ln ln N, adding an additional logarithmic correction to
the average path length characteristic of Erd˝s e R´nyi
networks
l
i w
(Eq.
(1) ). For
g ¼
3tsf u dh t
lnN
ln lnN , and for g >
h
3 the result is like that of the
Erd˝s e R´nyi network. Cellular networks, for which the
scaling exponent is usually between 2 and 3, are thus ultra-
small worlds [49] .
The analysis above, regarding the importance of the
hubs as the structural backbone of SF networks, has some
surprising implications on the robustness of cellular
networks to random perturbations. Our intuition leads us to
view complex systems as highly intricate structures, which
depend strongly on the proper functionality of all of their
components. When a significant fraction of their nodes fail,
these systems are expected to become dysfunctional. In
contrast, biological networks prove to be astoundingly
resilient against component failure [50 e 54] . From the
topological perspective this can be attributed to their SF
topology and its hub-based backbone. Scale-free networks
have been shown to maintain their structural integrity even
under the deletion of as many as 80% of their nodes. The
remaining 20% will still form a connected component
[55 e 57] . This is although in an Erd ˝ s e R ´ nyi network the
removal of nodes beyond a certain fraction inevitably
results in the network disintegrating into small isolated
components [27] . The source of this topological resilience
of SF networks is rooted in their inherent non-uniformity.
The vast majority of nodes in SF networks have merely one
or two links, playing a marginal role in maintaining the
integrity of the network. Most random failures will occur
on these unimportant nodes and thus will not significantly
disrupt the network's functionality. The relative scarceness
of the hubs, and, on the other hand, their central role in
maintaining the network's structural integrity, ensures that
random failures will rarely break down the network.
The robustness of cellular networks, which relies
strongly on the hub nodes, is, however, a double-edged
sword. Despite allowing the networks to withstand a large
number of random failures, it makes them extremely
vulnerable to intentional interventions. The removal of just
a small number of key hubs will cause the SF network to
break down into isolated dysfunctional clusters [56 e 57] .
Supporting evidence for this comes from the small number
of lethal genes found in many organisms, and, on the other
hand, by the relatively large number of hubs found among
these genes [39,41,58 e 65] .
l
i w
to x's current
degree, namely
k x
P i k i ;
P
ð
x
Þ¼
(3)
where the sum in the denominator is over all nodes in the
current state of the network. These two processes, growth
and preferential attachment, give rise to the observed
power-law degree distributions. It can be shown that any
one of these processes alone is insufficient and does not
yield the desired SF topology. Network growth is required,
as otherwise the network reaches saturation and the degree
distribution becomes nearly Gaussian. The preferential
attachment mechanism is needed to support the formation
of hubs [36] . By this mechanism, if a node has many links it
is more likely to acquire new links, creating a state where
the rich get richer. The result is that the more connected
nodes gain new links at a higher rate, and eventually
emerge as hubs. Eliminating the preferential attachment
mechanism leads to an exponential distribution, much less
broad than a power-law.
Preferential Attachment in Biological
Networks
The realization of the Barab´si e Albert model in the
formation of cellular networks is rooted in the process of
gene duplication [66 e 71] . This process is clearly respon-
sible for network growth, as duplicated genes produce
duplicate proteins and thus introduce new nodes into the
network. The more delicate point is that gene duplication
also adheres to the rules of preferential attachment. To
understand this, consider an interaction network which
grows via node duplication. At each time step a random
node is chosen, say x, and an identical node,
The Origins of the Scale-Free Topology
The SF topology is a universal feature of many real
networks, both in the context of biology and in social and
x, is created.
This newly created duplicate node will have exactly the
same interactions as the original node. This means that each
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