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Model for multi-messages spreading over complex networks considering the relationship between messages
Commun Nonlinear Sci Numer Simulat 48 (2017) 63–69
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Research paper
Model for multi-messages spreading over complex networks considering the relationship between messages Xingyuan Wang∗, Tianfang Zhao∗ Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian, 116024, China
a r t i c l e
i n f o
Article history: Received 28 May 2015 Revised 29 November 2016 Accepted 10 December 2016 Available online 14 December 2016 Keywords: Complex network Spreading Relationship Message
a b s t r a c t A novel messages spreading model is suggested in this paper. The model is a natural generalization of the SIS (susceptible-infective-susceptible) model, in which two relevant messages with same probability of acceptance may spread among nodes. One of the messages has a higher priority to be adopted than the other only in the sense that both messages act on the same node simultaneously. Node in the model is termed as supporter when it adopts either of messages. The transition probability allows that two kinds of supports may transform into each other with a certain rate, and it varies inversely with the associated levels which are discretely distributed in the symmetrical interval around original point. Results of numerical simulations show that individuals tend to believe the messages with a better consistency. If messages are conflicting with each other, the one with higher priority would be spread more and another would be ignored. Otherwise, the number of both supports remains at a uniformly higher level. Besides, in a network with lower connected degree, over a half of the individuals would keep neutral, and the message with lower priority becomes harder to diffuse than the prerogative one. This paper explores the propagation of multi-messages by considering their correlation degree, contributing to the understanding and predicting of the potential propagation trends. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Propagation is a common natural phenomenon that occurs on wherever there are connections or communication possibilities. It covers the spread of rumors, disease, internet virus, public opinion, information and so on. Understanding such a spreading process is essential to the diverse fields of medical science, biology, chemistry, physics, and sociology [1]. Meanwhile, complex networks have been widely utilized for describing the spreading dynamic in human world [2,3]. The incorporated population structures feature between the random networks and regular networks, which dramatically alter dynamical properties of the diffusion process therein [4]. Noticeably, the diffusion of information in the complex social network is widely studied, which related to email contents, rumors, advertising messages, ideas, innovations and so on. Zhao et al. studied the effect of message spreading in epidemic preventive control [5]. Wang et al. discussed a diffusive logistic prediction model based on information diffusion over online social networks [6–8]. Trpevski et al explored the rumors propagation in multiple networks [9]. Currently, the SIR and SIS models are most studied in the field of propagation in complex networks, which are transfered from epidemics spreading models. The SIR model considers that infected people get lifelong immunity after recovery ∗
Corresponding author. E-mail addresses: [email protected] (X. Wang), [email protected] (T. Zhao).
http://dx.doi.org/10.1016/j.cnsns.2016.12.019 1007-5704/© 2016 Elsevier B.V. All rights reserved.
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X. Wang, T. Zhao / Commun Nonlinear Sci Numer Simulat 48 (2017) 63–69
Fig. 1. The transition process of node status.
[10], while SIS model allows repeated infection [11]. DK model (named by Daley and Kendall), conceptually similar to the SIR model, is proposed by Daley and Kendall [12], in which population was divided into three types: ignorant, spreader and stifler, respectively corresponded to susceptible, infective and recovered people in SIR model. Further developing the DK model, Moreno assumes that ignorant may transforms into spreader with probability λ while meeting spreaders, and spreader may transforms into stifler with probability α sswhile contacting with other spreaders or stiflers [13]. The researches above are enlightened to our work, yet most of them focus little on the spreading of multiple messages except Trpevski et al. The latter explores the spreading of two types of rumors in synthetic networks and observes the fraction of nodes in the process of infection [9]. But the inadequacy of Trpevski’s work lies in the separate propagation and the ignoring of potential correlation between rumors. In this paper, we develop a novel model based on SIS model and DK model to fill the vacuum. The stifler in DK model is identified as ignorant and may be infected once again in the new model. The correlation parameter is assigned with a discrete value to mark the relationship between messages. The population structure is mapped to a network, where nodes represent individuals and links represent their informationexchange relationships. The scale-free networked population and small-world networked population are mainly studied. By analyzing the inner relationships, we can better understand the spreading dynamics of multi-messages and predict the potential propagation trend. This paper proceeds as follows. Section 2 defines the model formation and the required parameters. In Section 3, we develop numerical simulations on typical complex networks to investigate the behavior of model and analyze the results. The Section 4 concludes the paper then points out deficiencies and future directions. 2. Definition of the model This section defines the model about how the multiple messages spread and influence each other. To simplify the formulation, the relationships among more than two kinds of messages are translated to the correlations between messages. The model is defined in the text below. First consider a closed and mixed population of N individuals. The individual is represented by vertex (denoted as V) and contact is represented by edge (denoted as E). Then we get an undirected and un-weighted graph G = (V, E). The nodes communicate with each other through directly or indirectly connected links. At each time step, the node i(i ∈ [1, 2, ..., N]) may be in either of possible states: susceptible or infected. Nodes are vulnerable and easy to be infected in the former states and possess the will to disseminate information in the latter states; If the neighboring node is susceptible, it would be infected at a certain probability; The infectious nodes may forget or ignore the messages and then change to the susceptible ones again. Further, let A denotes the adjacency matrix of the graph G. λij is the element of A. If there is a directly connected edge between vertex i and vertex j, set λij = 1; otherwise set λij = 0. At time t, each node i (i ∈ [1, 2, ..., N]) maybe in one of three statuses: the ignorant (status 0), the spreader of message 1 (status 1), the spreader of message 2 (status 2). The statuses are described as a status vector, containing a single 1 in the position, and 0 everywhere else. Let
Stat usi (t ) = [s0i (t ), s1i (t ), s2i (t )]. The state transition process is shown in Fig. 1. The probability function of each status is set by
Pr ob(t ) = [ p0i (t ), p1i (t ), p2i (t )]. Evolution of the model is given by:
Pi0 (t + 1 ) = Si0 (t )(1 − ui )(1 − vi ) + Si1 (t )(1 − λ1 )(1 − vi f ) + Si2 (t )(1 − λ2 )(1 − ui f ) , Pi1 (t + 1 ) = Si0 (t )ui + Si1 (t )λ1 + Si2 (t )(1 − λ2 )ui f Pi2 (t + 1 ) = Si0 (t )(1 − ui )vi + Si1 (t )(1 − λ1 )vi f + Si2 (t )λ2
(1)
X. Wang, T. Zhao / Commun Nonlinear Sci Numer Simulat 48 (2017) 63–69
65
Fig. 2. The transition process of node status.
Stat usi (t + 1 ) = Mult iRealize[Pr obi (t + 1 )],
(2)
where Multi Real ize is a random realization of mapping from probability to status. For instance, suppose the state probability of node i equals (0.1, 0.3, 0.6) at time t, then the node i may be an ignorant, spreader 1 or spreader 2 with probability 0.1, 0.3 or 0.6 respectively. Only one of the statuses is allowed. For each node i, ui (t) or vi (t) respectively denotes the synthetic probability that vertex i accepts message 1 or message 2 source from any of its adjacent nodes, which are defined as
⎧ N ⎪ ⎪ 1 − βλi j s1j (t ) ⎨ui (t ) = 1 − j=1
N . ⎪ ⎪ 1 − γ λi j s2j (t ) ⎩vi (t ) = 1 −
(3)
j=1
At each time t, each node in status 1 or 2 tries to spread the corresponding message to its neighboring nodes and each try is successful with a rate β or γ , where 0 ≤ β ≤ 1, 0 ≤ γ ≤ 1. If j ∈ A, set λij = 1 and λij = 0 otherwise. f represents the transition probability between two messages:
f (d ) =
ed , 1 + ed
(4)
which is featured by three characteristics: 1) Its value is within the range of 0 to 1, which accords with the basic characteristic of probability; 2) Parameter d changes from −5 to 5; The negative values represent the negative correlation, while the positive represent the positive correlation; d = 0 means that two messages are completely irrelevant; 3) The function is constructed by the exponential function for its scalability and flexibility. As is shown in Fig. 2, the blue line represents the conversion rate, which presents a monotonically increasing trend; the dash line work as a basic line which can be obtained only when two messages are totally unconnected. For messages X and message Y the parameter estimation of distances between them is technically measured by euclidean distance:
dXY = g(X, Y )X − Y 2 = g(X, Y )
(x1 − y1 )2 + (x2 − y2 )2 + ... + (xn − yn )2 ,
(5)
where random variable xi and yi respectively represent the information characteristics of messages. If the two messages are positively related then set g(X, Y) = 1, otherwise set g(X, Y) = −1. The dynamic process of total number is given by Eqs. (1) and (2). In the following, we rewrite the model by construct probability transition equations.
Pi0 (t + 1 ) = Pi0 (t )(1 − ui )(1 − vi ) + Pi1 (t )(1 − λ1 )(1 − vi f ) + Pi2 (t )(1 − λ2 )(1 − ui f ) . Pi1 (t + 1 ) = Pi0 (t )ui + Pi1 (t )λ1 + Pi2 (t )(1 − λ2 )ui f Pi2 (t + 1 ) = Pi0 (t )(1 − ui )vi + Pi1 (t )(1 − λ1 )vi f + Pi2 (t )λ2
(6)
And ui (t) and vi (t) is calculated by:
⎧ N ⎪ ⎪ 1 − βλi j p1j (t ) ⎨ui (t ) = 1 − j=1 N . ⎪v (t ) = 1 − ⎪ 1 − γ λi j p2j (t ) ⎩ i
(7)
j=1
Let N represent the total number of nodes. At time t, the total number of individuals in status 0, 1 and 2 are given by
N0 (t ) =
N i=1
Si0 (t ),
N1 (t ) =
N i=1
Si1 (t ),
N2 (t ) =
N i=1
Si2 (t ).
(8)
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X. Wang, T. Zhao / Commun Nonlinear Sci Numer Simulat 48 (2017) 63–69
(a)
(b)
(c)
Fig. 3. Evolution of Eqs. (1) and (6) with d = 0 (a), d = 5 (b) and d =−5 (c). Results are based on a 10 0 0-node BA network generated with m = 2. “Dashed line” represents the average number of nodes in each status through Eq. (8). “Solid line” denotes the number of nodes through Eq. (9).
Equivalently, the sum of probabilities are given by
M0 (t ) =
N
Pi0 (t ),
i=1
M1 (t ) =
N i=1
Pi1 (t ),
M2 (t ) =
N
Pi2 (t ).
(9)
i=1
3. Numerical simulation This section presents the result of model behavior on complex network topologies. The two messages spread in the networks and finally come to steady states. All experiments initially begin with the (N/3)th nodes in state 1 and (2N/3)th nodes in status 2. Any of the nodes may change its status as time changes. N represents the total number of nodes in the networks. Moreover, in almost all experiments two messages have the same parameters values, so as to observe their spread more scientifically. 3.1. Scale-free networks In this section, results on BA scale-free network are presented. BA model was first proposed by Barabasi and Albert [14]. It put forward the power law degree distribution and the growth of an open system. The network system first starts with a small group of core nodes and then extends under the scaling laws. The processes are as follows: (a) Network growth. Network begins with m0 nodes, adding a new node to the network at each time step and selecting m (m ≤ m0 ) nodes to connect to this new node. (b) Preferential attachment. Pi , representing the probability that a new node connects with an existing node i, and ki , representing the degree of a node, is given by: Pi = ki / ki . In this network, average degree k equals 2m; degree distribution Pk is approximate to k3 scale. First, let N = 10 0 0, m = 2, and we generate scale-free network by means of BA algorithm. Change the parameter estimation of distances d and remain other variables unchanged, then we get the spreading dynamic of both messages. As is shown in Figs. 3 and 1) when d = 0, the number of individuals in status 0, 1 and 2 would achieve a similar level, and messages 1 take an weak advantage for its higher priority; 2) when d = 5, two messages achieve the same level, and the number of nodes in status 0 come to a minimum value. It accords with Goebbels effect: repeat a lie often enough and it will believed; 3) when d = −5, the number of node in status 0 achieve a higher level, and the number of nodes in status 0 keep a similar level as in Fig. 3(a). In the purple curve of Fig. 3(c), a wave crest merges in the beginning and then disappears. It indicates that the message 2 is greatly refrained while conflicting with message 1. In short, individuals tend to believe the messages with a better consistency. If messages are conflicting with each other, the one with a minor priority would be spread more and another would be ignored soon. Fig. 4 depicts the steady-state behavior of the model when the correlation parameter d varies. The networks are generated with different values of m. The number of nodes in different status is given. Evidently, the spreaders of two messages achieve a similar level when the correlation parameter is positive. If negative, the number of nodes in status 2 is close to zero while the number of nodes in status 0 achieves a higher level. Furthermore, for the particular value of the network parameters like m = 6, the number of nodes in status 2 keep constant. It illustrates that messages 2 is hard to diffuse in the network with high connected degree. More specific results are shown in Fig. 5. There we show how the fraction of nodes is affected by the correlation parameter d and network parameter m. The number of nodes in status 0 keeps a higher level in the case that two messages are irrelevant or negatively relevant and the networks keep a lower average degree (first panel). If the messages are irrelevant and there is a low connected degree in the network, the effect of message 1 would be reduced (second panel). When the two messages are positively relevant, the effect of message 2 is greatly enhanced and the change is barely affected by the average degree (third panel).
X. Wang, T. Zhao / Commun Nonlinear Sci Numer Simulat 48 (2017) 63–69
(a)
67
(b)
Fig. 4. The steady-state behavior of the model for different values of d. Results obtained by averaging over 100 network realizations, run for 200 time units. (a) Network generated with m = 2. (b) Network generated with m = 6.
Fig. 5. The color map of steady-state behavior of the model for different values of d and m. Results obtained by averaging over 100 network realizations, run for 200 time units. Other parameters are: λ1 = 0.8, λ2 = 0.8, α = 0.3, β = 0.3. (a) Fraction of status 0. (b) Fraction of status 1. (c) Fraction of status 2.
Fig. 6. Evolution of Eqs. (6) and (9) with d = 0 (a), d = 5 (b) and d =−5 (c). Results are based on a 10 0 0-nodes BA network generated with K = 4, p = 0.01. “Solid line” denotes the number of nodes in status 0, 1 and 2.
3.2. Small-world networks In this part, we consider the model behavior on small-world networks. The networks are generated by WS algorithm [15], which firstly distributed in an irregular circular shape. Specially, first contains N nodes in the network and every node connects to K nodes nearby, with K/2 in the clockwise, K/2 in the anticlockwise and N > K > 1. Later rewire each edge at probability p. Chang the parameter p from 0 to 1 then the generating network is translated from regular network into random network. Self-loops or multiple edges between nodes are not allowed. As is shown in Fig. 6, the evolution processes of model in small-world networks obtain the similar results as in scale-free networks. There is a clear difference that the response approaches steady state more slowly in the small-world network.
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X. Wang, T. Zhao / Commun Nonlinear Sci Numer Simulat 48 (2017) 63–69
Fig. 7. The color map of steady-state behavior of the model. Results obtained by averaging over 100 network realizations, run for 200 time units. Other parameters are: λ1 = 0.8, λ2 = 0.8, α = 0.3, β = 0.3. The blue, red and purple map respectively presents fractions of nodes in status 0, 1 and 2. (a), (b) and (c): different value of d versus network parameter K. (d), (e) and (f): different value of d versus rewiring parameter p.
Compare the behavior of the model on small-world network with different generating parameter K and rewiring probability p. From the first three panels of Fig. 7, it can be observed that the status of population rarely change as long as K > 4. The last three panels show that when the rewiring probability is low enough (less than 0.01), over half of the population would keep an ignorant or neutral status; when the rewiring probability is over than 0.01, the distribution of population is more affected by the correlation parameter between two messages.
4. Conclusions In this paper, we make an attempt to study how the relationship between messages affects their propagation dynamics. When individuals receive two messages simultaneously, one of them is set to be considered first. The model provides the interactions among individuals. General results that reflect the interplay between messages are also given. The key points of paper are as follows: Firstly, a novel model is proposed, which is a natural generalization of epidemic SIS model. The individual statuses in the model are attributed into three types: status 0 (ignorant or stifler), status 1 (spreader 1) and status 2 (spreader 2). The relationship between two messages is defined as parameter d, which values in a closed interval discretely. And then, we perform experiments on scale-free networks and small-world networks. The results in scale-free networks show that individuals tend to believe the messages with a better consistency. If messages are conflicting with each other, the one with higher priority would be spread more and another would be ignored. Message with a lower priority is hard to diffuse in the highly-connected network. The results in small-world networks show that it takes more time to achieve the steady state than in the scale-free networks. Besides, if connected degree of network is low enough, over a half of the individuals would become ignorant or stifler. In this paper, we present how messages affect each other while spreading over complex network. But the evolution of message 1 is mainly discussed, and message 2 plays an auxiliary role. In the reality, both of messages may play distinctive roles on different aspect and take the same priority. Moreover, the parameter estimation of distances between messages is set artificially, which can be quantified on the basis of specific contents. Therefore researches about how the model could be applied to the specific events are in order.
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Acknowledgment This research is supported by the National Natural Science Foundation of China (Nos: 61672124, 61370145 and 61173183), and Program for Liaoning Excellent Talents in University (No: LR2012003). References [1] Yeruva Sujatha, Devi T, et al. Selection of influential spreaders in complex networks using Pareto Shell decomposition. Physica A Stat Mech Appl 2016;452:133–44. [2] Sun W, Hu T, Chen Z, et al. Impulsive synchronization of a general nonlinear coupled complex network. Commun Nonlinear Sci Numer Simul 2011;16(11):4501–7. [3] Gan C, Yang X, Liu W, et al. Propagation of computer virus both across the Internet and external computers: a complex-network approach. Commun Nonlinear Sci Numer Simul 2014;19(8):2785–92. [4] Li Xun, Cao Lang. Diffusion processes of fragmentary information on scale-free networks. Physica A Stat Mech Appl 2016;450:624–34. [5] Wang X, Zhao T, Qin X. Model of epidemic control based on quarantine and message delivery. Physica A 2016;458:168–78. [6] Zhu LH, Zhao HY, Wang HY. Bifurcation and control of a delayed diffusive logistic model in online social networks. In: 33rd Chinese control conference (CCC); 2014. p. 2773–8. [7] Wang F, Wang HY, Xu K. Diffusive logistic model towards predicting information diffusion in online social networks. In: 32nd international conference on distributed computing systems workshops (ICDCSW); 2012. p. 133–9. [8] Lei CX, Lin ZG, Wang HY. The free boundary problem describing information diffusion in online social networks. J Diff Eqns 2013;254(3):1326–41. [9] Trpevski D, Tang Wallace KS, Kocarev L. Model for rumor spreading over networks. Phys Rev E 2010;81(5):056102. [10] Bai Z, Zhang S. Traveling waves of a diffusive SIR epidemic model with a class of nonlinear incidence rates and distributed delay. Commun Nonlinear Sci Numer Simul 2015;22(1-3):1370–81. [11] Tomovski I, Trpevski I, Kocarev L. Topology independent SIS process: an engineering viewpoint. Commun Nonlinear Sci Numer Simul 2014;19(19):627–37. [12] Daley DJ, Kendall DG. Epidemics and rumours. Nature 1965;204(4963) 1118-1118. [13] Moreno Y, Nekovee M, Pacheco A. Dynamics of rumor spreading in complex networks. Phys Rev E 2004;69(6):066130. [14] Barabasi AL, Albert R. Emergence of scaling in random networks. Science 1999;286(5439):509–12. [15] Watts DJ, Strogatz SH. Collective dynamics of small-world networks. Nature 1998;393(6684):440–2.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Research paper
Model for multi-messages spreading over complex networks considering the relationship between messages Xingyuan Wang∗, Tianfang Zhao∗ Faculty of Electronic Information and Electrical Engineering, Dalian University of Technology, Dalian, 116024, China
a r t i c l e
i n f o
Article history: Received 28 May 2015 Revised 29 November 2016 Accepted 10 December 2016 Available online 14 December 2016 Keywords: Complex network Spreading Relationship Message
a b s t r a c t A novel messages spreading model is suggested in this paper. The model is a natural generalization of the SIS (susceptible-infective-susceptible) model, in which two relevant messages with same probability of acceptance may spread among nodes. One of the messages has a higher priority to be adopted than the other only in the sense that both messages act on the same node simultaneously. Node in the model is termed as supporter when it adopts either of messages. The transition probability allows that two kinds of supports may transform into each other with a certain rate, and it varies inversely with the associated levels which are discretely distributed in the symmetrical interval around original point. Results of numerical simulations show that individuals tend to believe the messages with a better consistency. If messages are conflicting with each other, the one with higher priority would be spread more and another would be ignored. Otherwise, the number of both supports remains at a uniformly higher level. Besides, in a network with lower connected degree, over a half of the individuals would keep neutral, and the message with lower priority becomes harder to diffuse than the prerogative one. This paper explores the propagation of multi-messages by considering their correlation degree, contributing to the understanding and predicting of the potential propagation trends. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Propagation is a common natural phenomenon that occurs on wherever there are connections or communication possibilities. It covers the spread of rumors, disease, internet virus, public opinion, information and so on. Understanding such a spreading process is essential to the diverse fields of medical science, biology, chemistry, physics, and sociology [1]. Meanwhile, complex networks have been widely utilized for describing the spreading dynamic in human world [2,3]. The incorporated population structures feature between the random networks and regular networks, which dramatically alter dynamical properties of the diffusion process therein [4]. Noticeably, the diffusion of information in the complex social network is widely studied, which related to email contents, rumors, advertising messages, ideas, innovations and so on. Zhao et al. studied the effect of message spreading in epidemic preventive control [5]. Wang et al. discussed a diffusive logistic prediction model based on information diffusion over online social networks [6–8]. Trpevski et al explored the rumors propagation in multiple networks [9]. Currently, the SIR and SIS models are most studied in the field of propagation in complex networks, which are transfered from epidemics spreading models. The SIR model considers that infected people get lifelong immunity after recovery ∗
Corresponding author. E-mail addresses: [email protected] (X. Wang), [email protected] (T. Zhao).
http://dx.doi.org/10.1016/j.cnsns.2016.12.019 1007-5704/© 2016 Elsevier B.V. All rights reserved.
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X. Wang, T. Zhao / Commun Nonlinear Sci Numer Simulat 48 (2017) 63–69
Fig. 1. The transition process of node status.
[10], while SIS model allows repeated infection [11]. DK model (named by Daley and Kendall), conceptually similar to the SIR model, is proposed by Daley and Kendall [12], in which population was divided into three types: ignorant, spreader and stifler, respectively corresponded to susceptible, infective and recovered people in SIR model. Further developing the DK model, Moreno assumes that ignorant may transforms into spreader with probability λ while meeting spreaders, and spreader may transforms into stifler with probability α sswhile contacting with other spreaders or stiflers [13]. The researches above are enlightened to our work, yet most of them focus little on the spreading of multiple messages except Trpevski et al. The latter explores the spreading of two types of rumors in synthetic networks and observes the fraction of nodes in the process of infection [9]. But the inadequacy of Trpevski’s work lies in the separate propagation and the ignoring of potential correlation between rumors. In this paper, we develop a novel model based on SIS model and DK model to fill the vacuum. The stifler in DK model is identified as ignorant and may be infected once again in the new model. The correlation parameter is assigned with a discrete value to mark the relationship between messages. The population structure is mapped to a network, where nodes represent individuals and links represent their informationexchange relationships. The scale-free networked population and small-world networked population are mainly studied. By analyzing the inner relationships, we can better understand the spreading dynamics of multi-messages and predict the potential propagation trend. This paper proceeds as follows. Section 2 defines the model formation and the required parameters. In Section 3, we develop numerical simulations on typical complex networks to investigate the behavior of model and analyze the results. The Section 4 concludes the paper then points out deficiencies and future directions. 2. Definition of the model This section defines the model about how the multiple messages spread and influence each other. To simplify the formulation, the relationships among more than two kinds of messages are translated to the correlations between messages. The model is defined in the text below. First consider a closed and mixed population of N individuals. The individual is represented by vertex (denoted as V) and contact is represented by edge (denoted as E). Then we get an undirected and un-weighted graph G = (V, E). The nodes communicate with each other through directly or indirectly connected links. At each time step, the node i(i ∈ [1, 2, ..., N]) may be in either of possible states: susceptible or infected. Nodes are vulnerable and easy to be infected in the former states and possess the will to disseminate information in the latter states; If the neighboring node is susceptible, it would be infected at a certain probability; The infectious nodes may forget or ignore the messages and then change to the susceptible ones again. Further, let A denotes the adjacency matrix of the graph G. λij is the element of A. If there is a directly connected edge between vertex i and vertex j, set λij = 1; otherwise set λij = 0. At time t, each node i (i ∈ [1, 2, ..., N]) maybe in one of three statuses: the ignorant (status 0), the spreader of message 1 (status 1), the spreader of message 2 (status 2). The statuses are described as a status vector, containing a single 1 in the position, and 0 everywhere else. Let
Stat usi (t ) = [s0i (t ), s1i (t ), s2i (t )]. The state transition process is shown in Fig. 1. The probability function of each status is set by
Pr ob(t ) = [ p0i (t ), p1i (t ), p2i (t )]. Evolution of the model is given by:
Pi0 (t + 1 ) = Si0 (t )(1 − ui )(1 − vi ) + Si1 (t )(1 − λ1 )(1 − vi f ) + Si2 (t )(1 − λ2 )(1 − ui f ) , Pi1 (t + 1 ) = Si0 (t )ui + Si1 (t )λ1 + Si2 (t )(1 − λ2 )ui f Pi2 (t + 1 ) = Si0 (t )(1 − ui )vi + Si1 (t )(1 − λ1 )vi f + Si2 (t )λ2
(1)
X. Wang, T. Zhao / Commun Nonlinear Sci Numer Simulat 48 (2017) 63–69
65
Fig. 2. The transition process of node status.
Stat usi (t + 1 ) = Mult iRealize[Pr obi (t + 1 )],
(2)
where Multi Real ize is a random realization of mapping from probability to status. For instance, suppose the state probability of node i equals (0.1, 0.3, 0.6) at time t, then the node i may be an ignorant, spreader 1 or spreader 2 with probability 0.1, 0.3 or 0.6 respectively. Only one of the statuses is allowed. For each node i, ui (t) or vi (t) respectively denotes the synthetic probability that vertex i accepts message 1 or message 2 source from any of its adjacent nodes, which are defined as
⎧ N ⎪ ⎪ 1 − βλi j s1j (t ) ⎨ui (t ) = 1 − j=1
N . ⎪ ⎪ 1 − γ λi j s2j (t ) ⎩vi (t ) = 1 −
(3)
j=1
At each time t, each node in status 1 or 2 tries to spread the corresponding message to its neighboring nodes and each try is successful with a rate β or γ , where 0 ≤ β ≤ 1, 0 ≤ γ ≤ 1. If j ∈ A, set λij = 1 and λij = 0 otherwise. f represents the transition probability between two messages:
f (d ) =
ed , 1 + ed
(4)
which is featured by three characteristics: 1) Its value is within the range of 0 to 1, which accords with the basic characteristic of probability; 2) Parameter d changes from −5 to 5; The negative values represent the negative correlation, while the positive represent the positive correlation; d = 0 means that two messages are completely irrelevant; 3) The function is constructed by the exponential function for its scalability and flexibility. As is shown in Fig. 2, the blue line represents the conversion rate, which presents a monotonically increasing trend; the dash line work as a basic line which can be obtained only when two messages are totally unconnected. For messages X and message Y the parameter estimation of distances between them is technically measured by euclidean distance:
dXY = g(X, Y )X − Y 2 = g(X, Y )
(x1 − y1 )2 + (x2 − y2 )2 + ... + (xn − yn )2 ,
(5)
where random variable xi and yi respectively represent the information characteristics of messages. If the two messages are positively related then set g(X, Y) = 1, otherwise set g(X, Y) = −1. The dynamic process of total number is given by Eqs. (1) and (2). In the following, we rewrite the model by construct probability transition equations.
Pi0 (t + 1 ) = Pi0 (t )(1 − ui )(1 − vi ) + Pi1 (t )(1 − λ1 )(1 − vi f ) + Pi2 (t )(1 − λ2 )(1 − ui f ) . Pi1 (t + 1 ) = Pi0 (t )ui + Pi1 (t )λ1 + Pi2 (t )(1 − λ2 )ui f Pi2 (t + 1 ) = Pi0 (t )(1 − ui )vi + Pi1 (t )(1 − λ1 )vi f + Pi2 (t )λ2
(6)
And ui (t) and vi (t) is calculated by:
⎧ N ⎪ ⎪ 1 − βλi j p1j (t ) ⎨ui (t ) = 1 − j=1 N . ⎪v (t ) = 1 − ⎪ 1 − γ λi j p2j (t ) ⎩ i
(7)
j=1
Let N represent the total number of nodes. At time t, the total number of individuals in status 0, 1 and 2 are given by
N0 (t ) =
N i=1
Si0 (t ),
N1 (t ) =
N i=1
Si1 (t ),
N2 (t ) =
N i=1
Si2 (t ).
(8)
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(a)
(b)
(c)
Fig. 3. Evolution of Eqs. (1) and (6) with d = 0 (a), d = 5 (b) and d =−5 (c). Results are based on a 10 0 0-node BA network generated with m = 2. “Dashed line” represents the average number of nodes in each status through Eq. (8). “Solid line” denotes the number of nodes through Eq. (9).
Equivalently, the sum of probabilities are given by
M0 (t ) =
N
Pi0 (t ),
i=1
M1 (t ) =
N i=1
Pi1 (t ),
M2 (t ) =
N
Pi2 (t ).
(9)
i=1
3. Numerical simulation This section presents the result of model behavior on complex network topologies. The two messages spread in the networks and finally come to steady states. All experiments initially begin with the (N/3)th nodes in state 1 and (2N/3)th nodes in status 2. Any of the nodes may change its status as time changes. N represents the total number of nodes in the networks. Moreover, in almost all experiments two messages have the same parameters values, so as to observe their spread more scientifically. 3.1. Scale-free networks In this section, results on BA scale-free network are presented. BA model was first proposed by Barabasi and Albert [14]. It put forward the power law degree distribution and the growth of an open system. The network system first starts with a small group of core nodes and then extends under the scaling laws. The processes are as follows: (a) Network growth. Network begins with m0 nodes, adding a new node to the network at each time step and selecting m (m ≤ m0 ) nodes to connect to this new node. (b) Preferential attachment. Pi , representing the probability that a new node connects with an existing node i, and ki , representing the degree of a node, is given by: Pi = ki / ki . In this network, average degree k equals 2m; degree distribution Pk is approximate to k3 scale. First, let N = 10 0 0, m = 2, and we generate scale-free network by means of BA algorithm. Change the parameter estimation of distances d and remain other variables unchanged, then we get the spreading dynamic of both messages. As is shown in Figs. 3 and 1) when d = 0, the number of individuals in status 0, 1 and 2 would achieve a similar level, and messages 1 take an weak advantage for its higher priority; 2) when d = 5, two messages achieve the same level, and the number of nodes in status 0 come to a minimum value. It accords with Goebbels effect: repeat a lie often enough and it will believed; 3) when d = −5, the number of node in status 0 achieve a higher level, and the number of nodes in status 0 keep a similar level as in Fig. 3(a). In the purple curve of Fig. 3(c), a wave crest merges in the beginning and then disappears. It indicates that the message 2 is greatly refrained while conflicting with message 1. In short, individuals tend to believe the messages with a better consistency. If messages are conflicting with each other, the one with a minor priority would be spread more and another would be ignored soon. Fig. 4 depicts the steady-state behavior of the model when the correlation parameter d varies. The networks are generated with different values of m. The number of nodes in different status is given. Evidently, the spreaders of two messages achieve a similar level when the correlation parameter is positive. If negative, the number of nodes in status 2 is close to zero while the number of nodes in status 0 achieves a higher level. Furthermore, for the particular value of the network parameters like m = 6, the number of nodes in status 2 keep constant. It illustrates that messages 2 is hard to diffuse in the network with high connected degree. More specific results are shown in Fig. 5. There we show how the fraction of nodes is affected by the correlation parameter d and network parameter m. The number of nodes in status 0 keeps a higher level in the case that two messages are irrelevant or negatively relevant and the networks keep a lower average degree (first panel). If the messages are irrelevant and there is a low connected degree in the network, the effect of message 1 would be reduced (second panel). When the two messages are positively relevant, the effect of message 2 is greatly enhanced and the change is barely affected by the average degree (third panel).
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(a)
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(b)
Fig. 4. The steady-state behavior of the model for different values of d. Results obtained by averaging over 100 network realizations, run for 200 time units. (a) Network generated with m = 2. (b) Network generated with m = 6.
Fig. 5. The color map of steady-state behavior of the model for different values of d and m. Results obtained by averaging over 100 network realizations, run for 200 time units. Other parameters are: λ1 = 0.8, λ2 = 0.8, α = 0.3, β = 0.3. (a) Fraction of status 0. (b) Fraction of status 1. (c) Fraction of status 2.
Fig. 6. Evolution of Eqs. (6) and (9) with d = 0 (a), d = 5 (b) and d =−5 (c). Results are based on a 10 0 0-nodes BA network generated with K = 4, p = 0.01. “Solid line” denotes the number of nodes in status 0, 1 and 2.
3.2. Small-world networks In this part, we consider the model behavior on small-world networks. The networks are generated by WS algorithm [15], which firstly distributed in an irregular circular shape. Specially, first contains N nodes in the network and every node connects to K nodes nearby, with K/2 in the clockwise, K/2 in the anticlockwise and N > K > 1. Later rewire each edge at probability p. Chang the parameter p from 0 to 1 then the generating network is translated from regular network into random network. Self-loops or multiple edges between nodes are not allowed. As is shown in Fig. 6, the evolution processes of model in small-world networks obtain the similar results as in scale-free networks. There is a clear difference that the response approaches steady state more slowly in the small-world network.
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Fig. 7. The color map of steady-state behavior of the model. Results obtained by averaging over 100 network realizations, run for 200 time units. Other parameters are: λ1 = 0.8, λ2 = 0.8, α = 0.3, β = 0.3. The blue, red and purple map respectively presents fractions of nodes in status 0, 1 and 2. (a), (b) and (c): different value of d versus network parameter K. (d), (e) and (f): different value of d versus rewiring parameter p.
Compare the behavior of the model on small-world network with different generating parameter K and rewiring probability p. From the first three panels of Fig. 7, it can be observed that the status of population rarely change as long as K > 4. The last three panels show that when the rewiring probability is low enough (less than 0.01), over half of the population would keep an ignorant or neutral status; when the rewiring probability is over than 0.01, the distribution of population is more affected by the correlation parameter between two messages.
4. Conclusions In this paper, we make an attempt to study how the relationship between messages affects their propagation dynamics. When individuals receive two messages simultaneously, one of them is set to be considered first. The model provides the interactions among individuals. General results that reflect the interplay between messages are also given. The key points of paper are as follows: Firstly, a novel model is proposed, which is a natural generalization of epidemic SIS model. The individual statuses in the model are attributed into three types: status 0 (ignorant or stifler), status 1 (spreader 1) and status 2 (spreader 2). The relationship between two messages is defined as parameter d, which values in a closed interval discretely. And then, we perform experiments on scale-free networks and small-world networks. The results in scale-free networks show that individuals tend to believe the messages with a better consistency. If messages are conflicting with each other, the one with higher priority would be spread more and another would be ignored. Message with a lower priority is hard to diffuse in the highly-connected network. The results in small-world networks show that it takes more time to achieve the steady state than in the scale-free networks. Besides, if connected degree of network is low enough, over a half of the individuals would become ignorant or stifler. In this paper, we present how messages affect each other while spreading over complex network. But the evolution of message 1 is mainly discussed, and message 2 plays an auxiliary role. In the reality, both of messages may play distinctive roles on different aspect and take the same priority. Moreover, the parameter estimation of distances between messages is set artificially, which can be quantified on the basis of specific contents. Therefore researches about how the model could be applied to the specific events are in order.
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Acknowledgment This research is supported by the National Natural Science Foundation of China (Nos: 61672124, 61370145 and 61173183), and Program for Liaoning Excellent Talents in University (No: LR2012003). References [1] Yeruva Sujatha, Devi T, et al. Selection of influential spreaders in complex networks using Pareto Shell decomposition. Physica A Stat Mech Appl 2016;452:133–44. [2] Sun W, Hu T, Chen Z, et al. Impulsive synchronization of a general nonlinear coupled complex network. Commun Nonlinear Sci Numer Simul 2011;16(11):4501–7. [3] Gan C, Yang X, Liu W, et al. Propagation of computer virus both across the Internet and external computers: a complex-network approach. Commun Nonlinear Sci Numer Simul 2014;19(8):2785–92. [4] Li Xun, Cao Lang. Diffusion processes of fragmentary information on scale-free networks. Physica A Stat Mech Appl 2016;450:624–34. [5] Wang X, Zhao T, Qin X. Model of epidemic control based on quarantine and message delivery. Physica A 2016;458:168–78. [6] Zhu LH, Zhao HY, Wang HY. Bifurcation and control of a delayed diffusive logistic model in online social networks. In: 33rd Chinese control conference (CCC); 2014. p. 2773–8. [7] Wang F, Wang HY, Xu K. Diffusive logistic model towards predicting information diffusion in online social networks. In: 32nd international conference on distributed computing systems workshops (ICDCSW); 2012. p. 133–9. [8] Lei CX, Lin ZG, Wang HY. The free boundary problem describing information diffusion in online social networks. J Diff Eqns 2013;254(3):1326–41. [9] Trpevski D, Tang Wallace KS, Kocarev L. Model for rumor spreading over networks. Phys Rev E 2010;81(5):056102. [10] Bai Z, Zhang S. Traveling waves of a diffusive SIR epidemic model with a class of nonlinear incidence rates and distributed delay. Commun Nonlinear Sci Numer Simul 2015;22(1-3):1370–81. [11] Tomovski I, Trpevski I, Kocarev L. Topology independent SIS process: an engineering viewpoint. Commun Nonlinear Sci Numer Simul 2014;19(19):627–37. [12] Daley DJ, Kendall DG. Epidemics and rumours. Nature 1965;204(4963) 1118-1118. [13] Moreno Y, Nekovee M, Pacheco A. Dynamics of rumor spreading in complex networks. Phys Rev E 2004;69(6):066130. [14] Barabasi AL, Albert R. Emergence of scaling in random networks. Science 1999;286(5439):509–12. [15] Watts DJ, Strogatz SH. Collective dynamics of small-world networks. Nature 1998;393(6684):440–2.