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where the connectivity pattern dominates the system's
behavior as the above degree block variables define a mean-
field approximation within each degree class, relaxing
however the overall homogeneity assumption on the degree
distribution
[38]
. This framework, first introduced for the
description of epidemic processes, is the basis of the
heterogeneous mean-field (HMF) approach which allows
the analytical study of dynamical processes in complex
networks by writing mean-field dynamical equations for
each degree class variable. The HMF approach generalizes
to the case of networks with arbitrary degree distribution
the equations describing the dynamical process by
considering degree block variables grouping nodes within
the same degree class k. If we consider the SIS model, the
variables describing the system are i
k
,s
k
and r
k
that
represent the fraction of nodes with degree k in the
susceptible, infected and recovered class. The evolution
equation for the infected individual reads as
di
k
ð
The classic example of the effect of degree heterogeneity
on dynamical processes in complex networks is offered by
epidemic spreading. The previously discussed result of the
presence of an epidemic threshold in the SIR and SIS models
is obtained under the assumption that each individual in the
system has, at a first approximation, the same number of
connections k
.However,socialheterogeneityand
the existence of 'super-spreaders' have long been known in
the epidemics literature
[39]
.Generally,itispossibletoshow
that the reproductive rate R
0
is re-normalized by fluctuations
in
y
<
k
>
the
transmissibility
or
contact
pattern
as
R
0
/
is a positive and increasing
function of the standard deviation
n
of the individual trans-
missibility or connectivity pattern
[40]
.Inparticular,by
generalizing the dynamical equations of the SIS model, the
HMF approach yields that the disease will affect a finite
fraction of the population only if
b
R
0
ð
1
þ
f
ðyÞÞ
,wheref
ðyÞ
2
k
2
m
<
k
>
=<
>
[36,38]
. This readily points out that the topology of the
network enters the very definition of the epidemic threshold
through the ratio between the first and secondmoments of the
degree distribution. Furthermore, this implies that in heavy-
tailed networks such that
Þ
dt
¼m
t
i
k
ð
t
Þþl½
1
i
k
ð
t
Þ
k
Q
k
ð
t
Þ
Here the first term simply expresses the fact that any
node in the infected state may recover with rate
m
. The
second term, which generates new infected individuals, is
proportional to the probability of transmission
l
, the degree
k, the probability1-i
k
that a vertex with degree k is not
infected, and the density
Q
k
of infected neighbors of
vertices of degree k, i.e., the probability of contacting an
infected individual. As we are still assuming a mean-field
description of the system, the latter term is the average
probability that any given neighbor of a vertex of degree k
is
k
2
, in the limit of
a network of infinite size, we have a null epidemic threshold.
While this is not the case in any finite size real-world network
[41,42]
, larger heterogeneity levels lead to smaller epidemic
thresholds (see
Figure 27.3
). This is a very relevant result
indicating that heterogeneous networks behave very differ-
ently from homogeneous networks with respect to physical
and dynamical processes. Indeed, the heterogeneous
connectivity pattern of networks also affects the dynamical
progression of the epidemic process, resulting in striking
hierarchical dynamics inwhich the infection propagates from
higher to lower degree classes. The infection first takes
control of the large degree vertices in the network, then
rapidly invades the network via a cascade through progres-
sively smaller degree classes (
Figure 27.4
). It also turns out
that the time behavior of epidemic outbreaks and the growth
of the number of infected individuals are governed by a time
scale
<
>
/
N
infected. This
quantity
can
be
expressed
as
k
0
j
Q
k
ð
, which considers the average over all
possible degrees k
0
of the probability P
t
Þ¼
P
ð
k
Þ
i
k
0
ð
t
Þ
k
0
j
ð
Þ
that any edge
of a node of degree k is pointing to a node of degree k
0
times
the probability i
k
0
that the node is infected. This expression
can be further simplified by considering a random network
in which the conditional probability does not depend on
the originating node.
k
In this
case we have
that
P
ð
k
0
j
k
Þ¼
k
0
P
ð
k
0
Þ=<
k
>
simply descending from the fact
that any edge has a probability to point to a node with
degree k
0
which is proportional to the degree itself
[38]
.
The HMF technique is often the first line of attack in
understanding the effects of complex connectivity patterns
on dynamical processes and has been widely used in a wide
range of phenomena, although with different names and
specific assumptions depending on the problem at hand.
Although it contains several approximations, the HMF
approach readily shows that the heterogeneity found in the
connectivity pattern of many networks may drastically
affect the unfolding of the dynamical process, providing
novel and interesting features that depart from the common
picture we are used to in regular lattices and homogenous
population.
proportional to the ratio between the first and second
moments of the network's degree distribution, thus pointing
to a velocity of progression that is increasing with the
heterogeneity of the network
[43]
.
The change of framework induced by the network
heterogeneity in the case of epidemic processes has trig-
gered a large number of studies aimed at providing more
rigorous analytical basis to the results obtained with the
HMF and other approximate methods, and exploring
different spreading models
[44
s
48]
. In particular, the HMF
approach is exact for networks whose connections are fixed
only on average, annealed networks. For networks that
instead have fix connections, quenched networks, the HMF
approach does not always give correct results and inter-
pretations. Indeed, in these networks there are striking
e
