IEEE Circuits and Systems Magazine - Q4 2021 - 13

The first approaches to dynamic networks rely on time-scale separation.
Recently, several efforts have been made to overcome this limitation using
temporal-switching, activity-driven, and edge-Markovian networks.
that the epidemic process and the network dynamics
evolve at different paces, as illustrated in Fig. 7. If the
epidemic process evolves much faster than the network
of interactions, the system is in the so called quenched
regime. In this regime, static networks are accurate proxies
of slowly switching topologies, and the corresponding
results presented and discussed in the previous section
are thus used to study the evolution of the epidemic outbreak.
On the other extreme, we encounter the annealed
regime, in which the evolution of the network is assumed
to be much faster than the disease spreading process
[67]. In this regime, if the following limit exists
T
r
A =
Z
[
\
]
]
]]
lim
lim
T
"
"
T
3
3
1
1
T t=
T t= 0
/
#
! + 2r
r
E ij
a
At
T
1
()
At dt
()
if
if
t
t
!
!
Z
R
+
+
,
(18)
,
r
=
r
then we can define the average graph GV E (, ,)A ,
with (, ),ij
0 and the dynamic network can
be effectively represented and studied by means of its
average graph. Also in the annealed regime, results
on static networks are applied to the average graph to
study the behavior of the dynamical system.
In the physics community, epidemics on annealed
networks have been extensively studied, aiming at computing-or
approximating-the epidemic threshold of
the model on a network with known degree distribution,
without going through the explicit computation of the
spectral radius of the average adjacency matrix. Specifically,
by relying on a degree-based mean-field approach,
the epidemic threshold for the SIS model on unweighted
networks has been computed in [68] as a function of the
degree distribution of the network. Note that such an approximation
is exact in the limit of large-scale systems
n " 3. Specifically, let
GH 11 k2
2
k ,,
n
==
!!
kki
GH
iVV
n i
denote the mean and the second moment of the degree
distribution of the average network
GV E r
r
r ? ij
ij
rr ).
r = (, ,),A
i
rr rr
// (19)
Epidemics is Faster
Quenched Regime Co-Evolution Annealed Regime
respectively.
Then, the following expression for the epidemic
threshold can be obtained for uncorrelated annealed
networks (that is, if ak k
Theorem 4. Consider an SIS model on a dynamic network
in the uncorrelated annealed regime with imm= and
nn i
= , for all i V! . Let us define the following epidemic
threshold:
FOURTH QUARTER 2021
Figure 7. Modeling paradigms for dynamic networks. Under
the assumption of time-scale separation, quenched and
annealed regimes can be found. Between these regimes,
several paradigms have been proposed to capture the coevolution
of the epidemic process and the network structure
at comparable time-scales, including temporal-switching,
activity-driven, and edge-Markovian networks, detailed in
this survey.
IEEE CIRCUITS AND SYSTEMS MAGAZINE
13
ways spread on large scale-free networks. More details
on this approach and on further extensions of these
techniques to more general networks (including correlated
networks) and to more complex epidemic models
(including the SIR model) can be found in [24].
The quenched and annealed regimes discussed in the
above rely on the assumption that the epidemic process
and the network evolve at different paces and, thus, on
different time-scales. However, the arguments raised at
the beginning of this section to motivate the need for dynamic
networks provide evidence that such a time-scale
separation is often restrictive and unrealistic, since the
contagion process and the network evolution are often
intertwined and thus often evolve at comparable timescales.
In the last few years, several efforts have been
Network is Faster
v =
,
GH
GH
r
k
r
k
2
.
(20)
Then, in the limit n " 3 if /,1mn v the disease-free equilibrium
is asymptotically stable; otherwise, if /,2mn v the
disease-free equilibrium is unstable.
From Theorem 4, we establish that, below the epidemic
threshold in Eq. (20), the epidemic is in the fast
extinction regime while, above the epidemic threshold,
the epidemic becomes endemic. An important implication
of Eq. (20) applies to scale-free networks, whose
degrees follow a power-law distribution. In fact, in many
real-world applications, complex networked systems
are modeled by scale-free networks with the power-law
exponent between 2 and 3 [69]. In these scenarios, the
expression of v in Eq. (20) vanishes as
n " 3 implying
,
that for any nonzero contagion rate m 2 0 , epidemics al
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