The impact of multiple information on coupled awareness-epidemic dynamics in multiplex networks

Accepted Manuscript The impact of multiple information on coupled awareness-epidemic dynamics in multiplex networks Yaohui Pan, Zhijun Yan

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S0378-4371(17)30813-0 http://dx.doi.org/10.1016/j.physa.2017.08.082 PHYSA 18528

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Physica A

Received date : 20 February 2017 Revised date : 16 June 2017 Please cite this article as: Y. Pan, Z. Yan, The impact of multiple information on coupled awareness-epidemic dynamics in multiplex networks, Physica A (2017), http://dx.doi.org/10.1016/j.physa.2017.08.082 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Highlights (for review)

HIGHLIGHTS  We study the spread of awareness and epidemic considering individual heterogeneity.  Precaution levels are heterogeneous and vary with contact and epidemic information.  Epidemic threshold is affected by contact information but not epidemic information.  Local and global epidemic information have the same impact on epidemic spreading.  Infected individuals’ altruistic behaviors can suppress an epidemic effectively.

*Manuscript Click here to view linked References

The impact of multiple information on coupled awareness-epidemic dynamics in multiplex networks Yaohui Pana, Zhijun Yana,b* a b

School of Management and Economics, Beijing Institute of Technology, Beijing, China Sustainable Development Research Institute for Economy and Society of Beijing, Beijing, China

ABSTRACT Growing interest has emerged in the study of the interplay between awareness and epidemics in multiplex networks. However, previous studies on this issue usually assume that all aware individuals take the same level of precautions, ignoring individual heterogeneity. In this paper, we investigate the coupled awareness-epidemic dynamics in multiplex networks considering individual heterogeneity. Here, the precaution levels are heterogeneous and depend on three types of information: contact information and local and global prevalence information. The results show that contact-based precautions can decrease the epidemic prevalence and augment the epidemic threshold, but prevalence-based precautions, regardless of local or global information, can only decrease the epidemic prevalence. Moreover, unlike previous studies in single-layer networks, we do not find a greater impact of local prevalence information on the epidemic prevalence compared to global prevalence information. In addition, we find that the altruistic behaviors of infected individuals can effectively suppress epidemic spreading, especially when the level of contact-based precaution is high. KEYWORDS: Contact information; Prevalence information; Awareness; Epidemics; Multiplex networks 1. INTRODUCTION The awareness of epidemics can trigger changes in humans’ behaviors to reduce their risk of being infected, e.g., wearing masks, reducing outdoor activities, taking antiviral drugs or seeking early treatment, which in turn affects the epidemic spreading process [1, 2]. In contrast to epidemic spreading by contact, individual awareness can spread through various channels, such as word of mouth (WOM), online social networks or mass media. Therefore, there has been growing interest in studying the interplay between awareness and epidemic spreading in multiplex networks [3-16]. Previous studies have typically modeled the interplay between awareness and epidemics as two competing spreading processes in multiplex networks. That is, just like epidemic spreading in contact networks, awareness also propagates from aware individuals to their neighbors in communication networks, which suppresses epidemic spreading by promoting more individuals to participate in preventive behaviors. Following this thought, Funk et al first incorporated awareness spreading in communication networks into the spread of epidemics in contact networks and found that awareness spreading can mitigate and even prevent epidemic spreading, especially when communication networks overlap with contact networks [3]. Granell et al proposed a UAU-SIS model to depict the dynamical interplay between awareness and epidemics in multiplex networks. The analysis using the Microscopic Markov Chain Approach (MMCA) reveals that there exists a meta-critical point of awareness transmission rate under which the epidemic threshold is not affected by awareness spreading [4]. They further investigate the effect of self-awareness and the mass media and find that the mass media make the meta-critical point disappear [5]. Furthermore, other studies extend the coupled model of awareness and epidemics by introducing additional sources and routes of awareness transmission [6], taking different types of behavioral changes into account (e.g., reducing susceptibility and infectivity, seeking early treatment or changing contact pattern) [6-8], adopting different models to describe the spread of awareness and epidemics (e.g., threshold model, SIR/SIRV model) [9-13], and adding complexity into awareness spreading (e.g., social reinforcement effect, *

Corresponding author. E-mail: [email protected] 1

self-initiated awareness mechanism) [13-15]. However, the above studies assume that precaution levels are identical across all aware individuals, ignoring the effect of individual heterogeneity. In fact, individuals may take different levels of precautions to protect themselves from being infected based on the severity of epidemics or their own characteristics, such as epidemic prevalence and individual susceptibility. Some studies have explored the impact of the heterogeneity of individual precautions on epidemic spreading in single-layer networks. Bagnoli et al and Kitchovitch et al indicate that individual perceived risk and precaution level increase with the fraction of infected individuals among neighbors and found that local prevalence-based precautions can stop an epidemic [17, 18]. Zhang et al assume that the individual precaution level depends on the number of infected neighbors and found that local information-based precautions can effectively suppress the spread of an epidemic. Wu et al and Shang propose that the individual precaution level increases with the number of contacts and local and global infection density [19-22]. They found that both local infection information and contact information raise the epidemic threshold, but global infection information only decreases the epidemic prevalence. Nevertheless, few studies have investigated the impact of individual heterogeneity on the interplay of awareness and epidemics in multiplex networks. Liu et al indicate that the individual precaution level varies with the size of the community that one belongs to and studied the impact of community size and communities’ overlap between two-layer networks on epidemic spreading [23]. Guo et al assume that aware individuals with a larger degree or k-core take stronger precautions and studied the impact of preventive behaviors based on degree and k-core measures on epidemic spreading [24]. Although these studies incorporated individual heterogeneity, they only investigated the effect of network structure on one’s preventive behaviors, neglecting the impact of epidemic-related information. Therefore, we examine the impact of individual precautions based on multiple information on the interplay between awareness and epidemics in multiplex networks. In this paper, the coupled dynamics of awareness and epidemics are modeled as two competing spreading processes in two-layer networks. Aware individuals will take different levels of precautions to reduce the risk of being infected based on three types of information, including contact information and local and global epidemic prevalence information. The main concerns in the paper are as follows. First, how does awareness spreading affect the spread of epidemics with multiple information? Second, is there any difference between the impacts of contact information and prevalence information on the interplay between awareness and epidemics in multiplex networks? Third, compared with global prevalence information, does local prevalence information have a greater impact on the coupled dynamics of awareness and epidemics in multiplex networks? The rest of this paper is organized as follows. In Section 2, we describe the model of awareness and epidemic spreading in multiplex networks that incorporates multiple information. In Section 3, we conduct a theoretical analysis of the model using the MMCA. In Section 4, we testify to the accuracy of the MMCA in solving the proposed model and investigate the impact of contact information, prevalence information, awareness spreading and infected individuals’ altruistic behaviors on epidemic spreading dynamics. In Section 5, we present the conclusions. 2. MODEL Here, we model the coupled spreading dynamics of awareness and epidemics under the framework of the UAU-SIS model proposed by Granell et al [4, 5]. A sketch of the model is shown in Fig. 1. We apply the model in two-layer networks with the same nodes but with different connectivity on each layer. The epidemic spreads in the physical contact layer, where links represent the physical contacts with people you see in the family, office, school or other places. At the same time, the spreading of awareness occurs in the virtual communication layer, where links correspond to the relationships with people you regularly share information with. Note that the links in the communication network do not necessarily overlap with those in the physical contact network. 2

Physical contact layer I

S

I

I

Self-awareness

S

S S

Individual behaviors

S

I

S

U

A

A

A

U

A A U

U

U

Virtual communication layer

Fig. 1. Model description for the UAU-SIS dynamics in two-layer networks. The upper layer supports the spreading of epidemics. Individuals are susceptible (S) or infected (I) in this layer. The lower layer corresponds to the network where awareness spreading occurs. The individuals are same as those in the upper layer, but their states can be unaware (U) or aware (A).

In the physical contact network, we use the susceptible-infected-susceptible (SIS) model to describe the epidemic spreading process. Individuals are either susceptible (S) or infected (I). The infection propagates from infected individuals to their neighbors with a certain infection rate and infected individuals recover from infection with rate . As proposed in Refs. [20, 25, 26], the infection rate between individual i and his/her neighbor j depends on both the susceptible individual’s susceptibility and the infected individual’s transmissibility, which can be defined as:  AT i j , i is susceptible and j is infectious;  qij  Ti Aj , i is infectious and j is susceptible;  otherwise. 0,

(1)

Here, denotes the admission rate that susceptible individual i would admit an infection through one contact with an infected individual and denotes the transmission rate that the individual would transmit an infection through one contact with a susceptible individual. It is assumed that when individual i is unaware and varies with i’s precaution level when i is aware. In the virtual communication network, awareness spreads following an unaware-aware-unaware (UAU) process similar to the SIS process. Individuals are either unaware (U) or aware (A) of the epidemic. Those who are infected in the physical layer will become aware in the virtual layer with probability . Then, aware individuals will transmit awareness to their unaware neighbors with rate and lose awareness with rate . After becoming aware, susceptible individuals will take precautions to reduce their susceptibility. Here, assuming that the individual precaution level increases with the number of contacts, the infection fraction among the neighbors (local prevalence) and the whole population (global prevalence) [20], we define the admission rate of aware individual as: Ai  ki  1  a‎ i  1  b   0  a, b  1,  0 , (2) where is the degree of individual i, is the infection density among individual i’s neighbors and is the infection density in the whole population. , a and b denote the impact strength of contact information, local epidemic information and global epidemic information on one’s precaution level, respectively. 3

3. THEORETICAL ANALYSIS USING THE MMCA In this section, we conduct a theoretical analysis of our model using the MMCA because of the high accuracy of the MMCA in solving the spreading dynamics in quenched networks [4, 5, 27-34]. According to the proposed model, at time t, each individual i has a certain probability of being in one of the four states: unaware and susceptible (US), aware and susceptible (AS), unaware and infected (UI), or aware and infected (AI), denoted by , , and . Obviously, . Let , be the adjacency matrices of the physical contact network and virtual communication network, respectively. Then, on the virtual communication layer, we define the probability of unaware individual i not being informed by any neighbors as . On the physical contact layer, we define the probabilities of individual i not being infected by any neighbors if i is unaware as and not being infected by any neighbors if i is aware as . Assuming the absence of dynamical correlations [4, 5, 10, 35], we have the following equations: ri (t )   1  v ji p Aj  t   , (3) j

qiU (t )   1  c ji p Ij  t   ,

(4)

qiA (t )   1  c ji p Ij  t  Ai  ,   j

(5)

j

where and . Therefore, for each individual i, the transition probability trees of all possible changes of the four states are presented in Fig. 2. (a)

(b)

US UI

US

US

UI US

AS

UI AS

AI

AS

AS AI

US UI

US

(d) UI

AS

UI

UI

UI UI

AI

AS AI

(c)

UI

US

UI AI

AS

AI

AI

AI

AI AI

AI

Fig. 2. Transition probability trees for the four states of UAU-SIS dynamics per time step for each individual i. The states include US, AS, UI and AI. The root of each tree represents the state of individual i at time t and the leaves represent the states of individual i at time t+1. Each time step is subdivided into three phases: awareness spreading (UAU process), epidemic spreading (SIS process) and the self-awareness generating process.

Combining Eqs. (3)-(5) with the schema presented in Fig. 2, we can develop the evolution equations of the four states for each individual i using the MMCA as

piUS  t  1  piAI  +piUS  t  ri t  qiU t   piUI t  ri t    piAS  qiU t  ,

(6)

piAS  t  1  piAI 1     +piUS  t  1  ri  t   qiA  t   piUI  t  1  ri  t     piAS 1    qiA  t  ,

(7)

4

piUI  t  1  piAI  1   1    +piUS  t  ri  t  1  qiU  t  1     piUI  t  ri  t 1   1     piAS  1  qiU  t  1    ,



(8)

  t . (9)

piAI  t  1  piAI  1      1   1     +piUS  t  ri  t  1  qiU  t     1  ri  t   1  qiA  t  







 piUI  t  ri  t 1      1  ri  t   1     piAS  1  qiU  t     1    1  qiA .

When the system reaches the stationary state, Eqs. (6)-(9) satisfy , , , . Using stationarity, we can further compute the epidemic threshold . If , the epidemic dies out in the population. Otherwise, the epidemic persists in the population. Near the epidemic threshold, the probability of nodes being in the infected state is close to zero, namely, . Accordingly, and are approximated as qiU  1    c ji j , (10) j

q  1    c ji Ai j . A i

(11)

j

Noting that , we get Therefore, inserting Eqs. (10) and (11) in Eqs. (4)-(9) and removing

,

. terms, we have

piU  piU ri  piA ,

(12)

p  p 1  ri   p 1    ,

(13)

 i  piUS  ri 1  qiU   1  ri  1  qiA   piAS  1  qiU   1    1  qiA    piU ri  piA 1  qiU    piU 1  ri   piA 1     1  qiA 

(14)

A i

U i

A i





=  c ji  j piU  Ai piA . j

Substituting

and Eq. (2) in Eq. (14), and removing

   i    c ji  j 1  piA  ki 1  a  c ji  j j j      c ji  j 1  p  k A i

 i

c j

p .

ji

terms, we get

  A  1  b  j N  pi  j   

(15)

A i

j

Therefore,    c ji   ji  j  0    j  is the element of the identity matrix. Defining matrix H with elements 

 1  1  k   p

where

 i

A i

(16)

hij  1  1  ki  piA  c ji

(17)

the solution of the epidemic threshold reduces to an eigenvalue problem for the matrix H. The epidemic threshold is the minimum value of satisfying Eq. (8) and is written as c 



(18)

 max  H 

where is the largest eigenvalue of H. According to Eq. (18), explicitly depends on the dynamics in the virtual communication network ( ), the contact-based precautions ( ) and the network topology in the physical contact layer. Moreover, is obviously independent of prevalence-based precautions regardless of local or global prevalence information. 4. RESULTS

5

In Section 3, we list the evolution equations of different states for each individual using the MMCA. Given the initial condition, we can obtain the probabilities of being in different states for each individual at any time by iteration. The fractions of infected individuals and aware individuals in the steady state are calculated as and , respectively. In this section, we first testify to the accuracy of the MMCA in solving the proposed model. Then, we present the numerical results obtained using the MMCA and investigate the impact of awareness spreading and individual behaviors on the coupled dynamics of awareness and epidemics, mainly on two quantities: and . First, we testify to the accuracy of the MMCA in solving the proposed mode by comparing the numerical results obtained by the MMCA with the results obtained by Monte Carlo simulations. In Fig. 3, we plot the comparison between the results obtained using the MMCA and Monte Carlo simulations. The multiplex networks we use are the two-layer Watts-Strogatz (WS) small world network [36] or the Barabási-Albert (BA) scale free network [37] with 1000 nodes. The two-layer WS network is constructed as follows: the contact network in the physical layer is a WS small world network with a rewiring probability of 0.2 and an average degree , and the communication network in the virtual layer is the same network as the contact network but with 400 extra (non-overlapping with previous) random links [4, 5]. For the two-layer BA network, the contact network is a BA scale-free network that begins with m0=3 connected nodes and adds m=3 edges to the existing nodes for each new added node to ensure that the average degree is 6. The communication network is the same network as the contact network but with 400 extra (non-overlapping with previous) random links. The networks are same throughout the whole paper for the sake of consistency. In addition, the initial condition is set to be that 20% of the nodes are infected in the physical layer, and all nodes are unaware in the virtual layer [4, 5]. As shown in Fig. 3, we find good agreement between the MMCA method and Monte Carlo simulations in calculating the epidemic and awareness prevalence, verifying the accuracy and suitability of the MMCA in solving the coupled dynamics proposed in our paper. In addition, we also observe the discrepancy between the MMCA and Monte Carlo simulations. This discrepancy can be attributed to the assumption of the absence of dynamical correlations in the MMCA [4, 5, 27]. Under this assumption, the states of the nodes among neighbors should be independent [31, 32, 38]. However, dynamical correlations may exist among the states of neighbors, especially in networks with a high clustering coefficient [31]. In this situation, the values of , and in the MMCA are smaller than that in Monte Carlo simulations [10]. Therefore, the MMCA always overestimates the results in Monte Carlo simulations, especially in two-layer WS networks.

Fig. 3. Comparison of and as a function of obtained using the MMCA and Monte Carlo (MC) simulations in two-layer (a) WS networks and (b) BA networks. The Monte Carlo simulations are averaged over 100 realizations. The initial fraction of infected nodes I0 is set to be 0.2. Other parameter values are: , , , , , , .

Second, we investigate the impact of the awareness transmission rate on the UAU-SIS dynamics. Fig. 4(a) is the full-phase diagram ( ) of UAU-SIS dynamics in the two-layer WS network. As shown in Fig. 4(a), increasing the transmission rate of the awareness can decrease the final epidemic prevalence and augment the epidemic threshold. We further explore the impact of 6

on the epidemic threshold for different values of , and , see Fig. 4(b). Consistent with the results obtained by Granell et al[4, 5], a meta-critical point exists, under which does not vary with . This is because similar to epidemic spreading, a critical point also exists for the onset of awareness, which depends on awareness decay rate and the largest eigenvalue of the adjacent matrix of the virtual communication network . Therefore, when , awareness cannot spread in the population and Eq. (18) reduces to . In this situation, only depends on recovery rate and the network structure in the physical contact layer and does not vary with . When , awareness persists in the population and varies with , , and the network topology of physical contact network. Therefore, as shown in Fig. 4(b), the meta-critical point varies with , but not with or . Moreover, remains unchanged with and , but varies with when . When , it increases with , and .

Fig. 4. (a) The full-phase diagram of UAU-SIS dynamics in two-layer WS networks. becomes larger when the color varies from blue to red. Other parameter values are: I0=0.2, , , , , , . (b) Epidemic threshold as a function of for different values of parameters , and in two-layer WS networks. Other parameter values are: I0=0.2, , , .

Third, we explore individual precautions induced by three types of information, including contact information ( ), local epidemic prevalence (a) and global epidemic prevalence (b), on the UAU-SIS dynamics. By analyzing the stationary epidemic prevalence as a function of under different values of in two-layer WS networks, see Fig. 5(a), we find that increasing contact-based precautions can decrease the epidemic prevalence and augment the epidemic threshold effectively. This is consistent with the results obtained from single-layer networks [19, 20]. In Fig. 5(b), We further explore the impact of on the epidemic threshold under different values of . The result shows that the epidemic threshold increases linearly with and the slope increases with .

Fig. 5. (a) function of ,

as a function of for different values of with a fixed value of and (b) for different values of in two-layer WS networks. Other parameter values are: I0=0.2, , , . 7

as a ,

Similarly, we analyze the impact of local and global epidemic information on the stationary epidemic prevalence (Fig. 6). The results show that both local and global epidemic information have no influence on the epidemic threshold and only decrease the epidemic prevalence slightly, which is different from the results obtained from single-layer networks [20]. In sum, the epidemic threshold is affected by precautions induced by contact-based information but not by prevalence-based information, which is a consequence of the absence of a and b in Eqs. (17) and (18) to determine the epidemic threshold. This may be because both the local and the global epidemic prevalence approach 0 near . The precaution level based on prevalence information is too low to stop susceptible individuals with awareness from getting infected, making prevalence-based precautions negligible in affecting the coupled awareness-epidemic spreading process. In contrast, contact-based precautions are independent of epidemic severity and remain strong even when is very small, especially for high-degree individuals, resulting in an increased epidemic threshold.

Fig. 6. as a function of for different values of (a) a with a fixed value of b=0.5 and (b) b with a fixed value of a=0.5 in two-layer WS networks. Other parameter values are: I0=0.2, , , , .

Moreover, Wu et al[20] indicate that local prevalence information has a greater impact on the epidemic prevalence than global prevalence information in single-layer networks. Referring the quantity and its approximate calculation proposed by Wu et al[20], we analyze the discrepancy between the impact of local and global prevalence-based precautions on the final epidemic prevalence in multiplex networks, where F  a, b  

 I  I 1 I      a  , b    I  a, b  a b

 .

(19)

Because epidemic prevalence decreases with increasing a and b, indicates that local prevalence-based precautions have a greater impact on the epidemic prevalence than global prevalence-based precautions; otherwise, global prevalence-based precautions have a greater impact. In Fig. 7, we plot as a function of a and b in two-layer WS networks. The result shows that the impact of local prevalence-based precautions is not always stronger than global prevalence-based precautions. Local prevalence information has a greater impact than global prevalence information when a>b, but global prevalence information has a greater impact when b>a. Local and global prevalence information have a nearly equivalent impact on the epidemic prevalence when a=b.

8

Fig. 7. as a function of a and b for a fixed value of and in two-layer WS networks. becomes larger when the color varies from blue to red. The lines are contour lines of for different values of a and b. Other parameter values are: I0=0.2, , , , , .

In addition, most previous researches have investigated the impact of susceptible individuals’ behavioral responses on epidemic spreading. Few have focused on infected individuals’ behaviors. However, in real situations, infected individuals also participate in some altruistic behaviors to reduce their infectivity, e.g., reducing outdoor activities, wearing masks or receiving early treatment [7]. Therefore, we also explore the impact of the altruistic behaviors of infected individuals on the UAU-SIS dynamics. In this case, both susceptible and infected individuals with awareness change their behaviors. We assume that the admission rate of susceptible individuals will reduce to and that the transmission rate of infected individuals will reduce to when they become aware. A smaller results in a smaller , reflecting a higher level of infected individuals’ altruistic behaviors. Fig. 8(a) presents the full-phase diagram ( ) of UAU-SIS dynamics for two-layer WS networks. The result shows that infected individuals’ altruistic behaviors can effectively suppress epidemic spreading by reducing the epidemic prevalence and increasing the epidemic threshold significantly. By further analyzing the impact of on the epidemic threshold (Fig. 8(b)), we find an inverse change of with and the change in amplitude increases with . Therefore, encouraging infected individuals to take certain measures is an effective way to control epidemics.

Fig. 8. (a) The full-phase diagram of UAU-SIS dynamics in two-layer WS networks with a fixed value of . becomes larger when the color varies from blue to red. (b) as a function of for different values of in two-layer WS networks. Other parameter values are: I0=0.2, , , , , , .

9

We have also explored the impact of self-awareness on the coupled awareness-epidemic dynamics. In addition, we perform the same analysis in two-layer BA networks for completion and obtained consistent results (see the Supplementary Material). 5. CONCLUSIONS We have investigated the impact of multiple information on the interplay between awareness and epidemics in multiplex networks. Assuming that the individual precaution level increases with contact number, local epidemic prevalence and global epidemic prevalence, we explore the impact of contact information, local and global prevalence information on the epidemic prevalence and threshold. The results show that contact-based precautions can augment the epidemic threshold and suppress the epidemic spreading effectively, but prevalence-based precautions, regardless of local or global epidemic information, can only decrease the epidemic prevalence. Furthermore, we find that local epidemic information has almost the same influence as global epidemic information on the epidemic prevalence and cannot alter the epidemic threshold. In addition, we also determine the existence of a meta-critical point of awareness transmission rate under which the epidemic threshold is not affected by awareness spreading. Finally, we find that the altruistic behaviors of infected individuals can decrease the epidemic prevalence and augment the epidemic threshold greatly. Our paper contributes to the literature in three aspects. First, it proposes a coupled dynamical spreading model of awareness and epidemics that incorporates the heterogeneity of individual precautions. In contrast to previous studies, we assume that individual precaution levels are not identical and vary with the severity of epidemics and their own personal characteristics. Second, it investigates the impact of epidemic prevalence information (including local and global epidemic prevalence) on the interplay between awareness and epidemics in multiplex networks and explores the difference between the effects of different types of information. Third, the findings provide some new insights into the control of epidemic spreading. Unlike the results in single-layer networks [17-20, 39, 40], we do not find a greater impact of local prevalence information on the epidemic spreading compared to global prevalence information in multiplex networks. The more effective way to suppress epidemic spreading in multiplex networks is to encourage high-degree individuals to take stronger precautions or appeal to infected individuals to participate in altruistic behaviors. There are also some limitations in our paper. First, we only consider individual heterogeneity in precaution levels. However, heterogeneity exists not only in precaution levels from person to person but also in individually perceived crisis awareness. Individuals with a high degree or susceptibility may be more likely to become aware and participate in preventive behaviors. Future research should consider the heterogeneity in individual awareness perception. Second, we only investigate the impact of multiple information on the interplay of awareness and epidemics in two-layer WS and BA networks. Future work could explore the above issue in networks with more diverse and complicated topologies. Third, we neglect the impact of the lag of information and human behavioral responses. In fact, there may be time delay in the spread of information on epidemics or human reactions, which can also be taken as an impact factor in future research. SUPPLEMENTARY MATERIAL See the supplementary material for the analysis of self-awareness the two-layer BA networks.

and the results obtained in

ACKNOWLEDGE This work is supported by the National Natural Science Foundation of China (No: 71572013, 71272057) and the Special Items Fund of the Beijing Municipal Commission of Education. REFERENCES 1. 2.

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