Identifying epidemic threshold by temporal
profile of outbreaks on networks
Cite as: Chaos 29, 103141 (2019); https://doi.org/10.1063/1.5120491
Submitted: 18 July 2019 • Accepted: 30 September 2019 • Published Online: 25 October 2019
Yizhan Xu,
Ming Tang, Ying Liu, et al.
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Chaos 29, 103141 (2019); https://doi.org/10.1063/1.5120491
© 2019 Author(s).
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Identifying epidemic threshold by temporal
profile of outbreaks on networks
Cite as: Chaos 29, 103141 (2019); doi: 10.1063/1.5120491
Submitted: 18 July 2019 · Accepted: 30 September 2019 ·
Published Online: 25 October 2019
Yizhan Xu,1 Ming Tang,2,3,a)
Ying Liu,4,5,b) Yong Zou,6
View Online
Export Citation
CrossMark
and Zonghua Liu6
AFFILIATIONS
1
School of Communication and Electronic Engineering, East China Normal University, Shanghai 200241, China
School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China
3
Shanghai Key Laboratory of Multidimensional Information Processing, East China Normal University, Shanghai 200241, China
4
School of Computer Science, Southwest Petroleum University, Chengdu 610500, China
5
Big Data Research Center, University of Electronic Science and Technology of China, Chengdu 611731, China
6
School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China
2
a)
b)
tangminghan007@gmail.com
shinningliu@163.com
ABSTRACT
Identifying epidemic threshold is of great significance in preventing and controlling disease spreading on real-world networks. Previous studies
have proposed different theoretical and numerical approaches to determine the epidemic threshold for the susceptible-infected-recovered (SIR)
model, but the numerical study of the critical points on networks by utilizing temporal characteristics of epidemic outbreaks is still lacking.
Here, we study the temporal profile of epidemic outbreaks, i.e., the average avalanche shapes of a fixed duration. At the critical point, the
rescaled average terminating and nonterminating avalanche shapes for different durations collapse onto two universal curves, respectively,
while the average number of subsequent events essentially remains constant. We propose two numerical measures to determine the epidemic
threshold by analyzing the convergence of the rescaled average nonterminating avalanche shapes for varying durations and the stability of the
average number of subsequent events, respectively. Extensive numerical simulations demonstrate that our methods can accurately identify the
numerical threshold for the SIR dynamics on synthetic and empirical networks. Compared with traditional numerical measures, our methods
are more efficient due to the constriction of observation duration and thus are more applicable to large-scale networks. This work helps one to
understand the temporal profile of disease propagation and would promote further studies on the phase transition of epidemic dynamics.
Published under license by AIP Publishing. https://doi.org/10.1063/1.5120491
Epidemic threshold plays an important role in disease spreading,
which recently has been attracting a lot of attention. Intense theoretical researches and numerical efforts have been devoted to
determining the epidemic threshold for the susceptible-infectedrecovered (SIR) dynamics, while the numerical study of the critical points on networks by exploiting temporal characteristics
of epidemic outbreaks is still lacking. Thus, we systematically
study the temporal profile of epidemic outbreaks, i.e., the average
avalanche shapes of a fixed duration. At criticality, it is found that
the rescaled average terminating and nonterminating avalanche
shapes for different durations collapse onto two respective universal curves, while the average number of subsequent events essentially remains constant. Two numerical measures are proposed to
identify the epidemic threshold by analyzing the convergence of
Chaos 29, 103141 (2019); doi: 10.1063/1.5120491
Published under license by AIP Publishing.
the rescaled average nonterminating avalanche shapes for varying
durations and the stability of the average number of subsequent
events. We experimentally validate the effectiveness of the numerical measures on synthetic networks and further apply them to
predict the epidemic threshold on real-world networks. Extensive simulation results demonstrate that our measures successfully
determine the numerical threshold for the SIR dynamics on complex networks and are in good agreement with the variability
measure. Traditional criticality detectors are based on the heterogeneity of epidemic outbreak sizes near criticality and thus
require large durations to get the substantial statistics. With the
constriction of observation duration, our measures can quantify
the system’s criticality at the early stage and thus are more efficient
to study criticality on large-scale networks. These findings help
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to understand the temporal profile of disease propagation and
would promote further studies on the phase transition of epidemic
dynamics.
I. INTRODUCTION
Models of epidemic spreading provide powerful tools for us
to understand the interplay between network topology and epidemic dynamics. Two classical epidemic models are the susceptibleinfected-susceptible (SIS) model and the susceptible-infectedrecovered (SIR) model.1–3 The SIR model has been used to study a
wide range of disease spreading allowing immunity or death4 and is
further extended to model knowledge and information diffusion.5,6
In the SIR model, an infected node spreads a disease to each of its
susceptible neighbors with infection rate β, while it spontaneously
recovers with recovery rate µ. A critical value λc of the effective
transmission rate λ = β/µ separates the absorbing phase from the
active one. In previous studies, it is believed that the effective epidemic threshold is inversely proportional to the average connectivity within a homogeneous mixing approximation.7 Considering
the fluctuations of network connectivity, the heterogeneous meanfield (HMF) theory8–10 is employed to analyze the behavior of the
SIR dynamics and the effective epidemic
P n threshold is predicted to
n
be λHMF
= hk2hki
c
i−hki , where hk i = k k P(k) is the nth moment of
degree distribution P(k).11 For networks with power-law degree distribution P(k) ∼ k−γ , the HMF theoretical prediction of the epidemic threshold is finite when the degree exponent γ > 3, while it
becomes vanishing in the thermodynamic limit for γ ≤ 3.4 Although
the HMF theory is in general precise for uncorrelated networks,
it neglects both topological and dynamical correlations among the
neighbors and thus may be inaccurate for correlated networks.12–15
The connection between the static properties of the SIR dynamics
and bond percolation was recognized long ago.16 By mapping the SIR
model to a bond percolation process,17 the effective epidemic thresh, which is a more exact estimate for
old is derived as λc = hk2 ihki
−2hki
the epidemic threshold. For the SIR dynamics with arbitrary recovery
rate, the edge-based compartmental theory is proposed to predict the
hki
.
epidemic threshold,18 which can be expressed by λEc = hk2 i−(2−µ)hki
In particular, it reproduces the HMF prediction when µ = 1, while
it approaches the bond percolation prediction when µ → 0.
Since the topological or dynamical correlations have been
largely disregarded in the above theories,15,19 some numerical measures such as susceptibility,20 lifetime measure,21 and variability22 are
proposed to validate the accuracy of different theoretical predictions for the epidemic spreading models. The susceptibility measure
was originally proposed to explore the nonequilibrium phase transition phenomena in statistical physics, aiming at determining the
critical point and associated critical exponents.23,24 In the context of
epidemics on complex networks,13 it is defined as
χ =N
ρ 2 − hρi2
,
hρi
(1)
where ρ and N denote the final epidemic outbreak size and network
size, respectively. The susceptibility measure predicts the epidemic
threshold by analyzing the peak of the epidemic susceptibility and
Chaos 29, 103141 (2019); doi: 10.1063/1.5120491
Published under license by AIP Publishing.
scitation.org/journal/cha
its effectiveness for the SIS model has been confirmed on various
types of networks in Ref. 13. However, the susceptibility measure
overestimates the SIR epidemic threshold due to the bimodal distribution of the epidemic outbreak sizes for λ > λc .25,26 Considering
the limitation of the susceptibility measure, the variability measure
was proposed to predict the epidemic threshold by analyzing the
epidemic variability,25 which is given by
p
hρ 2 i − hρi2
1=
.
(2)
hρi
Since the variability measure well captures the strong heterogeneity of
outbreak sizes near the critical point, it is hence a more suitable criticality detector for the SIR dynamics on networks.25,26 Nevertheless,
both the susceptibility and variability measures require large-scale
numerical simulations to obtain the final outbreak sizes distribution,
which, therefore, show high computational complexity and poor time
efficiency.
Avalanching dynamics have been studied in many disciplines
such as the power-grid blackouts, the Barkhausen noise in magnetic
materials, and the propagation of memes on social networks.27–29
Strenuous research efforts have been devoted to studying the
avalanche sizes and durations distribution, which approximately have
a power-law scaling near the critical point.30 This feature has, therefore, been exploited to indicate whether systems are critical or not.
However, it is pointed out that the heavy-tailed distribution can also
be observed in the subcritical case;31 thus, recently much attention
has turned to the temporal characteristics of avalanches,30,32 which
can also indicate the criticality of the system.
In this paper, we systematically study the avalanche dynamics of
the SIR model on networks and propose two numerical identification
measures of the SIR epidemic threshold by analyzing the temporal
profile of epidemic outbreaks of fixed duration, also called the average avalanche shapes. At criticality, the rescaled average terminating
and nonterminating avalanche shapes for different durations collapse
onto two different curves, while the average number of subsequent
events roughly remains constant. We validate the effectiveness of the
two measures on random regular networks (RRNs), where each node
has exactly the same degree and then apply them to determine the SIR
epidemic threshold for networks with finite size. Extensive numerical experiments on synthetic and empirical networks demonstrate
that our measures can successfully determine the numerical threshold for the SIR model and agree well with the predictions given by the
variability measure. These findings shed light on the temporal profile
of epidemic dynamics and help us to effectively quantify whether a
system operates in the critical state.31,32
II. MODELS AND METHODS
In this section, we focus on the avalanche dynamics of the SIR
model on networks, especially the temporal profile of avalanches at
criticality, and propose two numerical measures to determine the
SIR epidemic threshold on networks. In the SIR model, each node
is in one of three states: susceptible, infected, and recovered (immunized or dead). At the beginning, one node is randomly chosen to be
an infected seed, while all other nodes are susceptible. At each time
step, an infected node transmits the disease to its susceptible neighbors with probability λ1t and spontaneously turn to a recovered
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state with probability µ1t. Remarkably, there exists an effective epidemic threshold λc , above which the final fraction of recovered nodes
is finite.4 The updating is typically a synchronous process, i.e., the
recovery rate µ is set to unity. However, the recovery rate can affect
both the epidemic threshold and the final outbreak size,18 while the
synchronous updating with 1t 1 is more accurate to capture the
characteristic of the continuous-time Markov process.33 As a result,
we set 1t = 0.1 to keep the validity of the discrete-time updating
process.
A. Avalanche dynamics in the SIR model
An avalanche can be triggered by the failure of some nodes,
which can then catastrophically impact the entire network. Previous studies have investigated the features of avalanches and possible
mechanisms of generating avalanches,28,30 while the studies of temporal characteristics of avalanches are still insufficient. The temporal
profile of an avalanche is typically defined as the number of newly
activated nodes in each time step, also called the avalanche shape. The
average terminating avalanche shape is determined by averaging all
avalanche shapes of duration T, where t = T is the precise time when
the avalanches terminate. At the critical point, the rescaled average
terminating avalanche shapes for different durations are found to collapse onto a single universal curve.28,31 This feature has recently been
used as a sensitive test for criticality in a range of dynamics, including the Barkhausen effect in ferromagnetic materials, neuron firings
in the brain and memes spreading on social networks.30–32 It is noteworthy that the system’s criticality is indicated by the convergence of
the scaling avalanche shapes through macroscopic observation, while
no quantitative index is provided to determine the critical point.
In the context of epidemics on networks, we give a detailed
description of the avalanche temporal characteristics in the SIR
model. In each realization, the time evolution of subsequent events
is defined as the number of nodes that are newly infected in each
time step, and the spreading process terminates at a fixed time T. The
average terminating avalanche shape ST (t) is determined by averaging the temporal profile of all outbreaks that have a duration T while
the nonterminating avalanche shape S0T (t) is determined by averaging the temporal profile of all outbreaks that have not terminated by
a given observation time T. Besides, the average number of subsequent events A(t) is given by averaging the shapes over all outbreaks,
regardless of the avalanche terminating time. In Fig. 1, we show the
temporal profile of epidemic outbreaks in the SIR model. Obviously,
each avalanche shape is independently random, while the average
avalanche shape exhibits a symmetric shape as a function of time t.
For other avalanche dynamics, the avalanche shapes can be defined
as the voltage pulse in Barkhausen noise,27,28 the number of users that
receive a given message in meme spreading,29 the number of neuros
that become firing in neuro firing,30 etc. Figures 2(a) and 2(b) show
the average avalanche shapes of the SIR dynamics on RRNs, where
the epidemic threshold predicted by the edge-based compartmental theory is accurate. The average terminating and nonterminating
avalanche shapes with varying durations show similarity at criticality.
Here, we define the rescaled average terminating and nonterminating avalanche shapes as AT and A0T , respectively. Figure 2(c) shows
that both AT and A0T for different durations collapse onto two universal curves (namely, the convergence). In particular, the rescaled
Chaos 29, 103141 (2019); doi: 10.1063/1.5120491
Published under license by AIP Publishing.
FIG. 1. Temporal profile of epidemic outbreaks in the SIR model. In each panel,
the black lines represent five individual avalanche shapes on RRNs. (a) The average terminating avalanche shape ST (t) (blue dashed curve) is determined by
averaging the temporal profile of all outbreaks that terminate exactly at time T. (b)
The average nonterminating avalanche shape ST0 (t) (blue dashed curve) is determined by averaging the temporal profile of all outbreaks that have not terminated
by time T. The RRNs have a network size N = 5 × 104 and a degree k = 10,
and the average avalanche shapes are determined by 107 individual avalanches
of duration T = 5.
average terminating avalanche shapes separate from each other at
the early stage and gradually collapse onto a single curve, while the
rescaled average nonterminating avalanche shapes always well collapse together. On one hand, the avalanches that have not terminated
by time T is typically much larger than those that terminate exactly at
T, and a larger number of avalanche shapes can better determine the
average avalanche shapes. On the other hand, the randomness at the
initial steps of avalanches leads to a large fluctuation of the average
terminating avalanche shapes of different durations.
Extensive simulations with more and greater observation durations also imply that the rescaled average nonterminating avalanche
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FIG. 2. The average avalanche shapes of the SIR dynamics on RRNs. (a) and (b) show the average terminating and nonterminating avalanche shapes for T = 3, 4, 5
at criticality, respectively. (c) Both the rescaled average terminating and nonterminating avalanche shapes of different durations show a universal scaling curve, where the
horizontal and vertical axes are rescaled to unit height and duration. (d) The average number of subsequent events for subcritical, critical, and supercritical case. The RRNs
have a network size N = 5×104 and a degree k = 10, and the average avalanche shapes are determined by 107 individual avalanches of given durations.
shapes have a better convergence than the rescaled average terminating avalanche shapes. Thus, we suggest that the convergence of the
rescaled average nonterminating avalanche shapes may be more suitable for detecting system criticality. As shown in Fig. 2(d), the average
number of subsequent events nearly remains constant (namely, the
stability) irrespective of time at criticality, while it rapidly increases
or decreases when the effective transmission rate is away from the
epidemic threshold, meaning that the disease propagation will break
or die out soon.
B. Numerical identification of epidemic threshold
by the temporal profile of outbreaks
Among the existing numerical methods, the susceptibility and
variability measures are two common criticality detectors,26,34 which
are based on the heterogeneity of epidemic outbreak sizes near criticality. However, these methods are not so efficient because a large
amount of realizations are required to get the substantial statistics
Chaos 29, 103141 (2019); doi: 10.1063/1.5120491
Published under license by AIP Publishing.
such as the distribution of final outbreak sizes or lifetimes. Thus, we
turn to identify the epidemic threshold by analyzing the convergence
of the rescaled average nonterminating avalanche shapes and the stability of the average number of subsequent events. Inspired by the
coefficient of variation,22,35 we quantify the magnitude of the convergence of the rescaled average nonterminating avalanche shapes by the
convergence measure, defined as
χA0 =
T
σA0
X
t=0
T
0
hAT i
,
(3)
0
where σA0 and hAT i are the variance and average of the average
T
nonterminating avalanche shapes of different durations at time t,
respectively. Next, the stability of the average number of subsequent
events is given by
σA(t)
χA =
,
(4)
hA(t)i
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FIG. 3. Numerical identification of the SIR epidemic threshold on RRNs. (a) and (b) show the convergence and stability measures as a function of λ, respectively. In each
panel, the position of the dashed green line represents the threshold predicted by edge-based compartmental theory. (c) The threshold λc as a function of degree k, where N
is set to 5 × 104 . (d) The threshold λc as a function of degree N, where k is set to 10. The average avalanche shapes are determined by 107 individual avalanches of given
durations T, which vary from 3 to 5 with an interval of 0.1. The results are averaged over 10 independent realizations.
where σA(t) and hA(t)i is the variance and average of the average
number of subsequent events in the range 0 to T. The effectiveness of our measures is experimentally validated on RRNs, where
the threshold predicted by the edge-based compartmental theory
hki
λEc = hk2 i−(2−µ)hki
(µ = 0.1) is accurate. Figures 3(a) and 3(b) depict
χA0 and χA as a function of λ for RRNs with a given network size
and degree. With the increase of λ in a wide range, both the convergence and stability measure show a trend of decreasing first and then
increasing. Here, we take the λA0 and λA corresponding to the minimum values of the convergence and stability measures respectively
as the numerical estimate of the SIR epidemic threshold. It can been
seen that our measures can successfully identify the critical point for
the SIR model on RRNs and agree well with the edge-based compartmental theory prediction. Figure 3(c) depicts the epidemic thresholds
on RRNs with different degrees. The numerical thresholds decrease
with the increase of k due to the increasing connectivity on networks. We further investigate the relationship between the epidemic
Chaos 29, 103141 (2019); doi: 10.1063/1.5120491
Published under license by AIP Publishing.
thresholds and network size N in Fig. 3(d), where λA0 and λA nearly
remain constant with increasing network size N. It is worth mentioning that the susceptibility measure is quite far from the theoretical
thresholds due to the bimodal distribution of epidemic outbreak sizes
near criticality, indicating that it is unsuitable as a criticality detector
for the SIR model. From the above analysis, we conclude that both
the convergence and stability measures can effectively determine the
SIR epidemic threshold on RRNs and are in good agreement with the
standard methods, while the results predicted by the susceptibility
measure are not always accurate.
III. APPLICATION OF NUMERICAL IDENTIFICATION
METHOD
As a practical application of the convergence and stability measures, we apply them to predict the SIR epidemic threshold on
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scale-free networks (SFNs) and real-world networks, comparing
their performance with the variability measure.
A. Scale-free networks
Based on the uncorrelated configuration model (UCM),36 we
consider the SIR dynamic on scale-free networks with degree distribution P(k) ∼ k−γ . The configuration model constrains the possible
maximum degree of the nodes by kmax ∼ N 1/2 and guarantees the
absence of the degree correlations on SFNs.37 Here, the possible min√
imum and maximum degree are set to kmin = 3 and kmax = N,
respectively. As shown in Fig. 4(a), the numerical threshold increases
monotonically with the degree exponent γ . This is because the
FIG. 4. Numerical identification of the SIR epidemic threshold on SFNs. (a) The
threshold λc as a function of the degree √
exponent, where the parameters are set
to N = 5 × 104 , kmin = 3, and kmax ∼ N. (b) The threshold λc as a function
of networks size for γ = 2.5 and γ = 3.5. The average avalanche shapes are
given by 107 individual avalanches of given durations T, which vary from 3 to 5
with an interval of 0.1. The results are averaged over 10 independent realizations.
Chaos 29, 103141 (2019); doi: 10.1063/1.5120491
Published under license by AIP Publishing.
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increasing network heterogeneity tends to accelerate the early outbreak stage. We further plot the epidemic threshold as a function
of networks size for fixed degree exponent γ = 2.5 and γ = 3.5 in
Fig. 4(b). The epidemic threshold decreases slowly with the increasing N when γ = 3.5, while it decreases rapidly for γ = 2.5 due to the
presence of more hubs. The numerical results confirm that our measures can effectively determine the SIR epidemic threshold on SFNs
and in general agree with the prediction based on the edge-based
compartmental theory and the variability measure. Furthermore, our
measures are fairly efficient because of the constriction of observation
duration T.
FIG. 5. Numerical identification of the SIR epidemic threshold on real-world networks. (a) The convergence and stability measures as a function of λ for Brightkite
network, which consists of 56 739 nodes and 212 945 edges. The positions of the
green and blue dashed line represent the threshold predicted by the variability
and the edge-based compartmental theory, respectively. (b) The epidemic thresholds given by different methods compared to the variability measure. The average
avalanche shapes are determined by 107 individual avalanches of given durations
T, which vary from 3 to 5 with an interval of 0.1. The results are averaged over 10
independent realizations.
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TABLE I. Structural features and epidemic thresholds of 25 real-world networks. The information of networks includes the network size N, the average degree hki, the average
clustering coefficient c, the degree correlation coefficient r, and the heterogeneity coefficient κ = hk 2 i/hki2 . The epidemic thresholds are given by the edge-based compartmental
theory (λE ), the variability measure (λ1 ), the convergence measure (λA0 ), and the stability measure (λA ), respectively.
N
hki
c
r
κ
λE
λ1
λA0
λA
17 903
34 401
27 400
403 364
56 739
26 475
365 154
23 166
317 080
12 495
75 877
224 832
63 392
105 722
196 591
62 561
15 763
23 613
30 817
325 729
10 680
255 265
22 322
145 145
1 134 885
22.00
24.46
25.69
12.11
7.51
4.03
9.43
7.70
6.62
7.93
10.69
3.02
25.77
43.83
9.67
4.73
18.85
3.32
13.84
6.69
4.55
15.21
2.85
9.04
5.26
0.63
0.29
0.31
0.42
0.17
0.21
0.18
0.27
0.63
0.12
0.14
0.08
0.22
0.09
0.24
0.01
0.52
0.17
0.13
0.23
0.27
0.62
0.07
0.60
0.08
0.20
−0.01
−0.03
−0.02
0.01
−0.19
−0.06
−0.06
0.27
−0.05
−0.04
−0.19
0.18
0.25
−0.03
−0.09
−0.12
−0.39
−0.17
−0.05
0.24
−0.12
−0.49
−0.06
−0.04
2.99
2.60
4.41
2.52
8.53
69.50
5.14
3.08
3.28
5.52
17.19
187.73
3.42
7.97
31.71
2.45
47.83
113.65
61.62
41.93
4.15
133.47
13.78
6.15
93.84
0.016
0.016
0.010
0.035
0.016
0.004
0.022
0.046
0.050
0.024
0.006
0.002
0.012
0.003
0.003
0.103
0.001
0.003
0.001
0.004
0.059
0.005
0.027
0.019
0.002
0.013
0.015
0.011
0.037
0.014
0.021
0.025
0.050
0.038
0.034
0.007
0.013
0.009
0.002
0.008
0.101
0.008
0.026
0.008
0.01
0.058
0.004
0.099
0.021
0.007
0.012
0.015
0.011
0.035
0.014
0.019
0.025
0.048
0.039
0.040
0.006
0.012
0.008
0.004
0.008
0.099
0.008
0.027
0.010
0.008
0.057
0.003
0.089
0.019
0.006
0.013
0.016
0.010
0.037
0.014
0.016
0.026
0.048
0.037
0.029
0.006
0.011
0.009
0.002
0.007
0.103
0.007
0.025
0.007
0.008
0.059
0.003
0.087
0.021
0.006
Networks
arXiv astro-ph
arXiv hep-ph
arXiv hep-th
Amazon (TWEB)
Brightkite
CAIDA
CiteSeer
Cora citation
DBLP coauthorship
DBLP
Epinions
EU institution
Facebook friendships
Flickr
Gowalla
Gnutella
Google.com internal
Google+
Linux
Notre Dame
Pretty Good Privacy
Stanford
Twitter lists
WordNet
Youtube links
B. Real-world networks
All the networks used in the above numerical simulations are
synthetic networks. However, synthetic networks are markedly different from empirical networks because of the absence of degree
correlations, community structure, and clustering.11,38 To validate the
performance of our methods, we use some representative samples of
real-world networks that are available for download on websites.39,40
Examples include social networks, citation networks, collaboration
networks, etc. Based on the largest connected component, Fig. 5(a)
depicts the convergence and stability measures as a function of λ for
Brightkite network, where one node represents a user and an edge
indicates a friendship between the users on Brightkite. The numerical results show that both the convergence and stability decrease at
first and then increase with increasing λ and the numerical threshold
is in excellent agreement with the variability measure prediction. It is
worth remarking that the edge-based compartmental theory prediction becomes incorrect due to the presence of degree correlations and
the dynamical correlations. Actually, the edge-based compartmental
theory is based on the cavity theory, which cannot perfectly capture
the dynamical correlations among the states of neighbors, and thus
its prediction accuracy becomes poor on many empirical networks.
We further check the robustness of our measures via 25 real-world
networks and compare them with the existing theoretical and numerical methods. The topology information of 25 real-world networks
Chaos 29, 103141 (2019); doi: 10.1063/1.5120491
Published under license by AIP Publishing.
and their numerical thresholds are provided in Table I. Figure 5(b)
shows the prediction accuracy of different methods with respect to
the variability measure on real-world networks. It can be seen that
our measures can perform well even in complicated networks while
the prediction accuracy of the edge-based compartmental theory
prediction becomes poor on some networks due to dynamical correlations. In addition, our approaches are transient measures that
determine the system state without the final outbreak size and thus
can be applied to study the phase transition of epidemic dynamics on
large-scale networks. These findings help us to understand the temporal profile of epidemic outbreaks on real-world networks and may
find some applications in disease prevention and control.
IV. CONCLUSION AND DISCUSSION
In this paper, we studied the temporal profile of epidemic outbreaks for the SIR model on networks. At the critical point, the
rescaled average terminating and nonterminating avalanche shapes
of different durations collapse onto two universal curves, respectively,
and the convergence of the latter is even more significant. Considering the limited observation time and number of realizations, we suggest that the nonterminating avalanche shapes is a suitable temporal
feature to test the criticality of the system. Besides, the average number of subsequent events roughly remains constant near the epidemic
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threshold, while it rapidly increases or decreases when the effective
transmission rate is away from the critical point. By analyzing the
convergence of the nonterminating avalanche shapes and the stability of the average number of subsequent events, we proposed two
numerical measures to identify the SIR epidemic threshold, whose
minimum value provides a numerical estimate for the phase transition point. The effectiveness of the two numerical measures was validated on RRNs and their prediction accuracy was further compared
on synthetic and empirical networks. Extensive numerical simulations indicate that the proposed measures are good criticality detectors for the SIR dynamics on networks and are generally consistent
with the predictions given by the variability measure. Remarkably,
our approaches can be applied to study the phase transition phenomenon on large-scale networks because of its low computational
complexity and high prediction accuracy.
As an application of the temporal characteristics of outbreaks,
we determined the SIR epidemic threshold by utilizing the temporal profile of epidemic outbreaks. Further efforts should be devoted
to checking their validity on more complex networks such as temporal networks41 and multilayer networks.42–44 It is also hoped that
the temporal characteristics may find potential applications in other
irreversible avalanche dynamics such as Kuramoto model,45 Ising
model,46 and information cascade model.47 Moreover, the temporal
profile of reversible dynamics and real-world disease spreading are
still open questions.
ACKNOWLEDGMENTS
This work was supported by the National Natural Science
Foundation of China (NNSFC) (Grant Nos. 11575041, 11975099,
61802321, 11872182, 11675056, and 11835003), the Natural Science
Foundation of Shanghai (Grant No. 18ZR1412200), and the Science
and Technology Commission of Shanghai Municipality (Grant No.
18dz2271000).
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