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Rumor diffusion model with spatio-temporal diffusion and uncertainty of behavior decision in complex social networks
Accepted Manuscript Rumor diffusion model with spatio-temporal diffusion and uncertainty of behavior decision in complex social networks Liang Zhu, Youguo Wang
PII: DOI: Reference:
S0378-4371(18)30132-8 https://doi.org/10.1016/j.physa.2018.02.060 PHYSA 19180
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Physica A
Received date : 26 June 2017 Revised date : 21 November 2017 Please cite this article as: L. Zhu, Y. Wang, Rumor diffusion model with spatio-temporal diffusion and uncertainty of behavior decision in complex social networks, Physica A (2018), https://doi.org/10.1016/j.physa.2018.02.060 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)
Rumor diffusion model with uncertainty of behavior decision. Diffusion threshold of spatio-temporal diffusion dominated by network topology. Heterogeneity of degree distribution and distance distribution. Theoretical analysis of a revised PDEs model.
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Rumor diffusion model with spatio-temporal diffusion and uncertainty of behavior decision in complex social networks Liang Zhu1, Youguo Wang2 1
College of Telecommunications and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210003, PR China 2 National Engineering Research Center of Communications and Networking, Nanjing University of Posts and Telecommunications, Nanjing 210003, PR China 2 College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210003, PR China 1 [email protected] 2 [email protected] Abstract: In this paper, a rumor diffusion model with uncertainty of human behavior under spatio-temporal diffusion framework is established. Take physical significance of spatial diffusion into account, a diffusion threshold is set under which the rumor is not a trend topic and only spreads along determined physical connections. Heterogeneity of degree distribution and distance distribution has also been considered in theoretical model at the same time. The global existence and uniqueness of classical solution are proved with a Lyapunov function and an approximate classical solution in form of infinite series is constructed with a system of eigenfunction. Simulations and numerical solutions both on Watts–Strogatz (WS) network and Barabási–Albert (BA) network display the variation of density of infected connections from spatial and temporal dimensions. Relevant results show that the density of infected connections is dominated by network topology and uncertainty of human behavior at threshold time. With increase of social capability, rumor diffuses to the steady state in a higher speed. And the variation trends of diffusion size with uncertainty are diverse on different artificial networks. Keywords: rumor diffusion model; spatio-temporal diffusion framework; diffusion threshold; complex social networks 1. Introduction Rumor diffusion, on the basis of online social networks, has produced serious damage on personal reputation and social stability. The evolving theory of telecommunication and developing network technique facilitate communications while promote rumor spreading in our daily lives. Unlike traditional rumor, rumor has characteristics of latent, persistent and explosive in online social networks which put great pressure on both network platforms and governments [1]. Therefore, rumor has drawn wide attention of researchers. Daley and Kendall first established D-K model to analyze the difference between disease diffusion and rumor dissemination [2] based on the mechanism of epidemic spreading. Furthermore, SIS (Susceptible –Infected-Susceptible) model [3] and SIR (Susceptible-Infected-Removed) model [4] on complex networks have been established to explore the transmission mechanism [5-8]. With the improvement of online social network, information could be obtained through multiple channels which challenges classic models based on ordinary differential equations (ODEs). In [9], a spatio-temporal framework on
basis of partial differential equations (PDEs) was first proposed:
I 2 I l f ( x, t , I ), xmin x xmax , t 0 t 2 x I I ( xmin , t ) ( xmax , t ) 0, t 0 x x I ( x, 0) ( x), xmin x xmax
(1)
In above model, authors divided information diffusion process into two separate processes: growth process and social process, meanwhile, used friendship hop as distance. In [10], heterogeneity of user distribution among different distance has been taken into consideration and growth process was described with a liner function. Accuracy of models has been proved with a Digg data set as well. And in [11], a summary of information diffusion in online social networks with PDEs models is proposed. In addition to the topology of network, the user plays a crucial role during the diffusion process as well. Unlike disease transmission, rumor diffusion in online social networks is conducted by rational users who would make strategic choices instead of being randomly infected with some probability [12]. Zinoviev and Duong indicated information diffusion involves all of users considering their knowledge, trust and popularity, which shape their publishing and commenting strategies [13]. Ellwardtet al. made an examination on interplay between person’s individual preference and social influence to suggest personal preference plays a more important role in human behavior [14]. Since users’ behavioral decision is driven by personal interests; game theory has been widely applied to explore human social behavior. Dunia described interaction between neighboring agents as a 2 2 coordination game and showed there is a threshold below which contagion could occur [15]. Jackson and Yariv considered such a diffusion process that agent makes the best response to behavior in last period [16]. Further, impacts of the distribution of costs, network topology and payoff structure on equilibrium points have also been confirmed. Though analyzing random disturbance is difficult in complex networks, the research of it is still important and necessary due to the great influence of random disturbance. In our previous work [17], the variation of connectivity was discussed with a stochastic differential equation (SDE) model and environmental noise was described as a Brownian motion. Zhao et al. held the opinion environmental uncertainty exists in the opinion formation which could be modeled as Gaussian stochastic process [18]. The fuzzy theory was also applied to illustrate imprecise and subjective information with a discrete evacuation model [19]. Cordoni and Persio considered a reaction-diffusion equation on a network and its boundary condition contains time delayed behavior which allows for multiplicative Gaussian noise perturbations [20]. In this paper, we consider the uncertainty of human behavior with random utility theory [12] on the basis of a spatio-temporal framework characterized by equation (1). Different from previous work, users with the same degree at different distances no longer take the same statistical meaning due to spatial diffusion among different distances. And the heterogeneity of degree distribution and distance distribution should be been taken into consideration at the same time in our model. Moreover, spatial diffusion mechanism stems from the function like trend topics and popular recommendation which asks for a certain number of disseminators. And the research object of equation (1) is news event from Digg which has a transmission base generally. As a
result, a spatial diffusion threshold is proposed in this paper below which spatial diffusion would not occur and means the rumor is not a hot message. Meanwhile, several researches have proved the latent nature [21-22] of network rumors which indicates rumor may not attract the attention of departments and keep concealed till spreads to a wide range or causes a great impact. So, it is reasonable to explore the transmission mechanism of rumor with a certain disseminators in the spatio-temporal framework before departments take actions like refuting a rumor. Both numerical analysis and simulations on WS and BA networks have been carried out to research spatio-temporal diffusion with uncertainty of human behavior in homogeneous and heterogenous networks. 2. Theoretical model Several online social networks, like Weibo, Twitter and Facebook, have concentrated on these ‘hot topics’ which owns a high click amount to make optimizations of website functions. Like the Headline function of Weibo, the most popular information would be selected through transmission of mass users and pushed to a new user who has not participated in discussion on these hot topics. While in Twitter, the function of trend topic recommends information with consideration of users’ interests and their friend information at the same time. It is obviously that the function of information push demands popularity which could be mirrored in a high number of forward information or a large amount of information audience. Besides, information transmission through website push is independent of users’ connections which could be interpreted as a mechanism of ‘random walk’. As a result, it is meaningful to consider such a diffusion threshold above which the information could be considered as a hot message and acquired without direct physical connections among users though the determination and size of threshold are different among varied online social networks. In this paper, we explore rumor spreading process under above diffusion framework with consideration of uncertainty of human behavior. Rumor diffuses along determined physical connections among users and is dominated by users’ average stochastic utilities when diffusion size is below threshold while spatial and temporal diffusions are in progress simultaneously above threshold. 2.1 Ordinary differential equation model Inspired by the previous work in [12], we consider users make a choice to spread rumor or not on basis of their average stochastic utilities which are influenced by uncertainty of human behavior. In this section, we will establish an ordinary differential equation model based on random utility theory and make relevant theoretical analysis to complement previous work during the initial stage of rumor diffusion which means rumor is not a hot message and still spreads along determined physical connections. The probability of a user with degree k chooses to spread the rumor could be derived in [12] as follow:
pk (t ) 1 e
ak k
1 bk kP ( k ) I k (t )
(2)
k
where I k (t ) represents the density of infected nodes with node degree k at time t , is a scale parameter to measure the uncertainty in human behavior. P ( k ) describes probability distribution of nodes with node degree k and
k
means the average node degree of networks.
In addition, the process of a user chose to accept the rumor is modeled as a multiple game process which could be divided to several two player games. In equation (2), parameters
a (T R P S ) / ( Rkmax ) and b ( P S ) / ( Rkmax ) contains information of a two player game and T , R, P, S are parameters of utility matrix with a ranking R T P S under the coordination game. For a heterogeneous network, we differentiate nodes with their degree and assume nodes with the same degree take the same statistical characteristics. Then, the threshold of density of infected users in the whole network is taken into account in this paper since it is convenient to observe and count the number of infected users. From this, the diffusion process could be characterized with such an ODE while density of infected users is under threshold: dI k (t ) pk (t )(1 I k (t )) (1 pk (t )) I k (t ) pk (t ) I k (t ) (3) dt where density of infected users satisfies
I (t ) P(k ) I k (t ) I * and k
I * represents
threshold of density of infected users. The front part of equation (3) denotes the increase of influenced individuals who decide to spread a rumor. On the contrary, the latter part of equation (3) means influenced users may also give up spreading rumor at the same moment. It is not difficult to find the state of a user, determined by his/her behavior choice which is up to average stochastic utility, is not constant in this model. Therefore, equation (3) describes a SIS model. For the convenience of analysis, we consider the variable of density of infected connections in the whole network at time t and denote it as y (t ) which takes the following form:
y(t )
1 k
kP(k ) I
k
(t )
(4)
k
Combining with equation (4) and making a variable substitution, equation (3) could be rewritten as:
dy(t ) 1 1 ( kP(k ) ) y(t ) ( aky ( t ) b )/ dt k k 1 e
(5)
Since the discussion of nodes and connections is a pair of dual problem in networks, we focus on the density of infected connections instead of infected nodes in the back of our work. And the relationship of these two variables would also be analyzed in different networks. For the diffusion process described by equation (3), it is important to determine existential conditions and amount of solution in steady state. If inequality I () I * sets up, rumor would not infect a large number of users and spatio-temporal diffusion does not occur as well. Therefore, we discuss the existence of steady solution in equation (5) first. From observation, steady solution of equation (5) y () is not equal to zero otherwise
dy () / dt kP(k )(1 eb / ) k
k 0 which means rumor would continue to diffuse. As a
result, the steady solution y () is a nonzero value which satisfies:
1 k
kP(k ) 1 e k
1 ( aky ( ) bk )/
y()
(6)
The existential condition of steady solution in equation (5) is equivalent to prove the existence of zero point of equation (6). Consider the following function:
f ( y())
1 k
kP(k ) 1 e
1 ( aky ( ) bk )/
(7)
k
Which is consist of multiple S-shaped functions (with value of y () increasing, the function is convex first and then become concave). Obviously, if (df ( y ()) dt ) | y ( ) 0 1 , there exists a zero point of equation (5) which has been proved in [12]. Furthermore, a nonzero steady solution of equation (5) could be determined when uncertainty satisfies
aeb / (1 ebkmax / )2 1 . However, we have to state it is only a sufficient condition of existence of steady solution. When we acquire the prior knowledge of threshold, the threshold time at which rumor becomes a hot message and spatio-temporal diffusion occurs is worth to be researched in forecasting work. For the given value of threshold y * , the Taylor expansion is applied to determine the threshold time y(t * ) y* in reasonable agreements. For equation (5), an approximate linear differential equation could be derived from the Taylor expansion:
dy 1 dt k The
right
part
of
1 akebk / kP ( k ) 1 y (t ) k bk / bk / 2 1 e (1 e ) approximate
equality
is
positive
with
parameters
(8) satisfy
aeb / (1 ebkmax / )2 1 and the threshold time could be derived with an error range o( y (t )) :
t*
f (k ) k f (k ) * ln 1 2 y C1 f1 (k ) k k
(9)
Where
k k P(k ) bk / 1 e 1 ebk / k
(10)
akebk / ak 2ebk / k P ( k ) k 1 bk / 2 bk / 2 (1 e ) k (1 e )
(11)
f1 (k )
f 2 (k )
C1
f (k ) k f (k ) ln 1 2 y (0) f 2 (k ) k k
(12)
In a single source situation, y (0) represents number of neighbors of the source node. For the threshold time described in equation (9), it has a high accuracy while value of threshold is close to zero. Furthermore, with increase of index of Taylor expansion, the truncation error tends to zero exponentially. While the form of threshold time becomes more complicated and not even
exists at cost. In this paper, we only consider the first order expansion to derive a heuristic conclusion. 2.2 Spatio-temporal diffusion model When rumor infects a certain amount of users and diffusion size researches the threshold, spatial and temporal diffusion occur at the same time due to the push service of online social networks. Although [9-11] have established classical spatio-temporal equations, there still remains some work to be improved and perfected. First, heterogeneity of node degree is not taken into consideration in previous models which means these sptaio-temporal equations are established on basis of uniform networks while scale-free features are ubiquitous in fact. Further, mechanism of epidemic infection has been also applied to describe growth process while the difference between epidemic and rumor is ignored. In this paper, we extend the equation (2) in a spatio-temporal framework based on random utility theory and a PDEs model is established while heterogeneity of degree distribution and distance distribution are taken into account. Due to the spatial diffusion among different distances, it is reasonable to assume users with the same number of neighbors acquire different average stochastic utility. Compared with [12], rumor spreads in a more complex way. According to the spatio-temporal equations described in equation (1), a PDEs model with consideration of uncertainty in human behavior is established as follows:
I k ( x, t ) 2 I k ( x, t ) d pk, x (t )(1 I k ( x, t )) (1 pk, x (t )) I k ( x, t ), l x L, t t * 2 x t * I ( x , t ) ( x ) k I (l , t ) I ( L, t ) k k 0 t t
(14)
where above equation describes diffusion process after threshold time t * . In equation (14), distance x means length of shortest path to source node which is characterized as friendship hops intuitively in online social networks and I k ( x, t ) represents density of infected users with degree k in distance x at time t (means the probability to select an infected node with degree k from all the nodes at distance x ). d represents the social capability measuring how fast the information travels across distances. The probability of a user with degree k in distance x at time t chooses to spread the rumor is: 1 pk, x (t ) aky ( x ,t )/ bk / (15) 1 e where
y( x, t ) kP(k ) I k ( x, t ) k
kP(k | x) k
(16)
is the probability to select a connection at the end of which is an infected node in distance x from all the connections linking to nodes in distance x . And P (k | x ) represents the probability to choose a node with degree k from all the nodes in distance x . Considering the physical meaning of real social networks, there is only a source node at the minimum distance which stays in infected state all the time in a single source situation. And for the maximum distance, rumor hardly researches due to a far distance. Hence, variations of density
of infected nodes at distance l and L are fixed which are not in our analysis. Additionally, applying cubic splines interpolation [9] to density of infected nodes at threshold time, a continuous and smooth initial function ( x) could be constructed. For equation (14), assume the threshold time as initial time and make a variable substitution like equation (4), a PDEs model describing variation of infected connections could be derived as follows:
y ( x, t ) 2 y ( x, t ) 1 1 d k kP(k ) 1 e( aky ( x,t )bk )/ y( x, t ), 0 x L, t 0 t 2 x kP ( k | x ) k y ( x, 0) ( x) y (0, t ) y ( L, t ) 0 x (17) where all the parameters take the same meanings as mentioned above and ( x ) is a continuous and smooth initial function to describe density of infected connections among different distances at threshold time. Then, we will discuss the global existence and uniqueness of the solution in equation (17) and derive an approximate classical solution in the form of infinite series. In addition, a numerical method is also constructed through backward difference. To verify the rationality of model and explore the transmission mechanism in artificial networks, simulations and numerical calculations on homogeneous and heterogeneous networks would be carried on to make a comparison in the next section. First, we put forward a theorem to construct a classical solution of equation (17). Theorem 2.2.1: For equation (17), the global existence and uniqueness of solution is proved 1
and a power system is determined in Sobolev Space H 0 (definition of norm in Sobolev Space:
f
H 01
f
L2
f
L2
) in the solution domain (0, L) [0, ) .
Proof. First, it is obvious that the growth function
g ( y ( x, t ))
1 1 kP(k ) y ( x, t ) aky ( x ,t )/ bk / 1 e kP(k | x) k
(18)
k
is local Lipschitz continuous. Thus, local existence and uniqueness of the solution of equation (17) could be obtained. Afterwards, the priori estimates of solution would be analyzed with a Lyapunov function and characteristics of solution would be promoted to the global domain. Assume y ( x, t ; ) is the solution of equation (17) which satisfies the initial condition. It is experiential to formulate a Lyapinov function in the following form: L 1 y ( x ,t ; ) V ( y ( x, t ; )) y x2 ( x, t ; ) g ( ) d dx, y ( x, t ; ) H 01 0 0 2
L
0
1 2 1 1 2 1 ( aky ( x ,t: ) bk )/ kP(k ) ln(1 e ) y ( x, t ; ) C2 dx yx ( x, t ; ) y ( x, t ; ) 2 2 kP ( k | x ) ak k k (19)
where C2 is a constant. According to definition of Lyapinov function, we have:
V ( y ( x, t ; )) V ( y ( x, 0)) V ( ( x))
(20)
Meanwhile, function ln(1 eaky ( x ,t ; )/ bk / ) ak increases monotonically with increase of y ( x, t ; ) which leads to:
b (21) ak ak a Applying equation (21) to equation (18), the following inequality could be derived combined with equation (20): L 1 1 2 2 (22) 0 ( 2 yx ( x, t; ) 2 y ( x, t; ) C3 )dx V ( y( x, t; )) V ( ( x)) Since ( x ) is a continuous polynomial function defined in a bound interval, its Lyapunov ln(1 e( aky ( x ,t ; ) bk )/ )
lim
y ( x ,t ; )
ln(1 e( aky ( x ,t ; ) bk )/ ) y ( x, t ; )
function is bounded. Applying Poincaré inequality to equation (22), we have
yx ( x, t; )
L2
y( x, t; )
L2
and
are bounded which supports that a power system is determined by equation (17) in
domain (0, L) [0, ) . Meanwhile, on the basis of priori estimates and prolongation theorem, the global existence and uniqueness of solution are set up. Then, we will discuss the form of classical solution of equation (17) through Taylor expansion. Based on the global existence and uniqueness of solution, an approximate form of classical solution could be derived through the following theorem. Theorem 2.2.2: For PDEs model described in equation (17), there exists a classical solution in the form of
y( x, t ) y (1) ( x, t ) y (2) ( x, t ) near the initial value y ( x, 0) where
y (1) ( x, t ) , y (2) ( x, t ) are determined by equation (24) and (25). Proof. First, equation (16) is a non-homogeneous nonlinear PDE whose solution is difficult to derive normally. As a result, we consider the Taylor expansion of the first part of equation (18) at threshold value y ( x, 0) and such an approximate PDE model is derived:
y ( x, t ) 2 y ( x, t ) d C5 y ( x, t ) C6 , 0 x L, t 0 t 2 x y ( x,0) ( x) y (0, t ) y ( L, t ) 0 x which could be divided into two problems as follows: y (1) ( x, t ) 2 y (1) ( x, t ) d C5 y (1) ( x, t ), 0 x L, t 0 t 2 x (1) y ( x,0) ( x) y (1) (0, t ) y (1) ( L, t ) 0 x and
(23)
(24)
y (2) ( x, t ) 2 y (2) ( x, t ) d C5 y (2) ( x, t ) C6 , 0 x L, t 0 t 2 x (2) y ( x,0) 0 y (2) (0, t ) y (2) ( L, t ) 0 x
(25)
where
C5
1 1 y( x, 0) akeaky ( x ,0)/ kP(k ) kP ( k ) 1 eaky ( x ,0)/ bk / kP(k | x) k (1 eaky ( x,0)/ bk / )2 (26) kP(k | x) k k
k
C6 1
a keaky ( x ,0)/ kP ( k ) kP(k | x) k (1 eaky ( x ,0)/ bk / )2
(27)
k
are constants and y ( x, 0) represents density of infected connection at distance x in threshold time. Since the PDE described in equation (24) is a homogeneous linear problem, its solution
y (1) ( x, t ) could be derived directly by separation of variable:
(2n 1) 2 L [C5 y ( x, t ) ( x) cos xdx e 0 2L n 0 L
(1)
(2 n 1)2 2 d 4 L2
]t
cos
(2n 1) x 2L
(28)
For equation (25), we expanse C6 , y (2) ( x, t ) as series according to the system of eigenfunctions cos (2n 1) x : 2L
4C6 (1) n 1 (2n 1) x C6 2n 1 cos 2L n 0 y (2) ( x, t ) v (t ) cos (2n 1) x n 2L n 0
(29)
Substitute equation (29) into equation (25) and combined with initial condition, the following equations are set up due to coefficients of series on each side of the equation are consistent.
' 4C6 (1) n 1 (2n 1) 2 2 d v ( t ) C v ( t ) vn (t ) n 5 n 4 L2 2n 1 v (0) 0 n
(30)
Thereout, we have
vn (t )
16C6 L2
(2 n 1) (1) n [C5 4 L2 e (2n 1)[4C5 L2 (2n 1) 2 2 d ] 2
2
d
]t
1
(31)
and
y ( x, t ) (2)
n 0
16C6 L2
(2 n 1) (1) n [C5 4 L2 e (2n 1)[4C5 L2 (2n 1) 2 2 d ] 2
2
d
]t
(2n 1) 1 cos x 2L
(32) Thus, y( x, t ) y (1) ( x, t ) y (2) ( x, t ) is an approximate classical solution near the initial value y ( x, 0) . It is easy to prove the uniform convergence of series y ( x, t ) , meanwhile, its first order partial derivative with respect to time t and second order partial derivative with respect to distance x are uniformly convergent as well. Consequently, it is meaningful to formulate a classical solution in the series form. From this, the proof is completed. Normally, classical solution is analyzed by theoretical deduction and it is difficult to apply the classical solution to calculation which is especially in the form of infinite series. So, a numerical method is established to approximate the analytic solution with a truncation error. In the next section, specific parameters would be determined to give relevant calculation results while we only introduce a framework of numerical approximation in this part. For the solution domain in equation (17), we determine the maximum time T and divide solution domain into grids where spatial and temporal interval are set as h and respectively. Thus, the number of grids is LT / ( h ) and the divided domain could be denoted as
x x j , j 0,1,..., J L / h , t tn , n 0,1,..., N T / . In equation (17), various iterative equations are formulated on basis of multiple difference methods. Forward difference method is most commonly used due to its explicit iterative equation while the solution is not always stable under any initial conditions. Take a comprehensive consideration on solution accuracy and computational complexity; we apply backward difference to formulate an implicit iterative equation. Assume y ( x, t ) is the solution of equation (16), following relational expression could be derived based on Taylor expansion: y ( x j , ti 1 ) y ( x j , ti ) y( x j 1 , ti ) 2 y( x j , ti ) y( x j 1 , ti ) d g ( y ( x j , ti )) o( h 2 ) (33) h2 For these sufficiently small h and which tend to zero in limit of theoretical analysis, the truncation
error
tends
to
zero.
Denoting
Y (i ) ( y ( x1 , ti ) y ( x2 , ti ) ... y ( xJ 1 , ti ))T ,
G (i ) ( g ( y ( x1 , ti )) g ( y ( x2 , ti )) ... g ( y ( xJ 1 , ti )))T , equation (33) could be described in form of matrix: QY (i ) Y (i 1) G (i 1)
(34)
where
1 2d / h 2 d / h 2 0 2 2 1 2d / h d / h 2 d / h Q 0 d / h 2 1 2d / h 2 0 0 0
0 0 1 2d / h 2 0
(35)
Obviously, Q is symmetrical and diagonally dominant which implies that a LU (Lower-Upper) decomposition exists. And then, equation could be solved gradually. Though, compared with forward difference method, truncation error of backward difference method is
O( h2 ) as well, stability of solution does not depend on division of domain. In addition,
central difference method and Crank-Nicholson format are also suitable with higher accuracy and complexity while we won’t make an introduction in details. 3. Simulations: In this section, numerical solutions of equation (17) based on above iterative matrix under determined parameters would be calculated. And then, simulations on artificial networks are carried out to investigate the dynamics of PDEs model. The parameters of utility matrix are set as R 4, S 1, T 0, P 0 to indicate a coordination game analyzed in literature [12]. Spatial interval h takes value 0.05 while temporal interval is 0.01. Meanwhile, artificial networks including WS network [23] and BA network [24] are constructed as follows. The WS network with the total number of nodes N 5000 is generated with a rewiring probability 0.3 and average degree of WS network is set as 15. While BA network with the same scale of WS network is built with two processes. In growth process, 30 initial nodes construct a global coupled network. In preferential attachment process, the other nodes (with 15 edges) connect to nodes already existing in the network with a probability which is proportional to the degree of nodes. Parameters of simulations have been researched in previous literatures [9-12]. Additionally, to reduce the random error, all the simulations are average over 100 independent runs and each simulation is performed with a randomly chosen source node. In addition, maximums of network distance ought to be same for all the source nodes to ensure that rumor diffuses in similar spatial dimensions. In [12], it has been illustrated that uncertainty has great influence on diffusion size for the case where temporal diffusion is taken into account only. Since density of infected connections is analyzed instead of infected nodes in this paper, we first research variation of density of infected connections under different values of uncertainty and threshold I * at threshold time while spatial diffusion has not been considered.
(a) (b) Fig .1 Variation of density of infected connections under different parameters: (a) density of infected connections varies with uncertainty on both WS and BA networks. (b) density of infected connections varies with threshold I * on BA network. Fig.1 (a) illustrates density of infected connections changes with uncertainty while threshold is fixed as 0.1 and social capability d 0.1 in both WS and BA networks. Rumor hardly diffuses with small values of uncertainty. With increase of , rumor diffuses till the threshold time and density of infected connections takes the same value of threshold approximately in WS network. While in BA network, the density increases quickly first and then tend to be stable with an increasing uncertainty. Fig.1 (a) shows density of infected connections
under fixed threshold is dominated by network topology and uncertainty. Homogeneity of WS network in degree weakens the difference of connection among nodes. Especially, in a regular network, density of nodes is even equivalent to density of sides with the consideration of numerical calculation. While in BA network, nodes obey a long-tail distribution due to the property of sacle-free and there exists significant differences in connection among different nodes. In Fig.1 (b), we fix uncertainty as 0.01 and adjust threshold from 0.25 to 0.3 to explore the variation of density of infected connections in BA network. The red line in Fig.1 (b) represents a proportional function with gradient is 1. Note that, density of infected connections is below threshold for these small values of threshold which means nodes with small degree make up the main part of infected nodes in threshold time. With increase of threshold, density of infected connections grows rapidly and flattens out around I * 0.3 . In this situation, hub nodes with large number of connections are infected at the threshold time. Compared to nodes with fewer connections, hub nodes take a higher probability to touch with infected nodes while density of nodes is large enough in the network.
(a) (b) Fig.2 The evolution of rumor diffusion on both WS and BA networks: (a) numerical solutions. (b) simulations. Fig.2 shows the evolution of rumor diffusion through numerical and simulation values on both WS and BA networks. In Fig.2 (a), we set threshold I * and social capability d as 0.1. Meanwhile, uncertainty is 0.1 on WS network while takes value of 0.01 on BA network. We make the following explanation for the selection of uncertainty on artificial networks: First, values of uncertainty should be selected with consideration of network topology to ensure rumor would diffuse in the network. Then, we hope diffusion sizes on both artificial networks tend to be consistent through adjusting values of uncertainty in case produce a phenomenon that rumor always seems to spread to a larger range on a network than another. In fact, the influence of uncertainty on diffusion size of networks is significantly different on WS and BA networks. The relevant conclusions would be proposed to make an analysis in the latter. In addition, for a fixed threshold, simulations without spatial diffusion have been carried on to generate initial density of infected connections among different distances and cubic splines interpolation is applied to construct an initial function ( x ) with these discrete points. From observation of Fig.2 (a) and (b), we find the speed of rumor diffusion rises firstly and then decreases to 0 finally from numerical solution and simulation results. In the early time, susceptible users construct principle part of the whole network and the choice not to spread a rumor would gain higher returns for most users which lead to a slow growth rate of rumor diffusion. With increasing of infected users, the
behavior choice of spreading rumor would offer a high returns that result in an increasing probability of accepting the rumor. While only a small number of susceptible users existing, the growing speed slows down and total density of infected connections researches a maximum gradually.
(a)
(b)
(c) (d) Fig.3 The evolution of rumor diffusion in different distances: (a) numerical solutions on WS network. (b) simulations on WS network. (c) numerical solutions on BA network. (d) simulations on BA network. Fig.3 illustrates the evolutions of rumor diffusion at different distances while a total variation of infected connections has been proposed. Fig.3 (a) and (b) display rumor diffusion on both spatial and temporal dimensions with uncertainty 0.1 , social capability d 0.1 and threshold I * 0.1 on WS network. As shown in Fig.3 (a) and (b), density of infected connections keeps a similar variation trend at different distances till the whole network is infected. Due to the homogeneity of connection distributions and initial values at different distances, variation of density of infected connections shows a homogeneous characteristic approximately at different distances on WS network. In Fig.3 (c) and (d), uncertainty is set as 0.01 while other parameters remain unchanged. On BA network, nodes are mainly distributed at distance 3 where initial infected nodes are agminate as well. Since the existence of hub nodes at distance 3, rumor diffuses at a higher rapid compared with other distances. However, in the adjacent distance of source node, rumor spreads slowly since few initial infected user exists. After that, though similar variation trend of density of infected connections exists in numerical solutions and simulations, we still have to explain the difference between them. In simulation, only distance and round of rumor diffusion in integer have practical significance while domain could be divided into sufficiently small grids to acquire a satisfactory result in numerical solution. That is why the
image of numerical solution looks smoother than simulation results. In addition, it is not difficult to find the diffusion size is larger in simulation than in numerical solution which is dominated by the following factor. Spatial diffusion stems from the difference of density of infected connections in adjacent distances. In numerical solution, spatial diffusion between two adjacent integer distances would generate loss due to these middle discrete points. However, excessive spatial interval would produce a huge truncation error which is a universal shortcoming in differential method.
(a)
(b)
(c) Fig.4 Effective rounds of rumor diffusion and density of infected connections in steady state with different parameters: (a) effective rounds on both WS and BA networks. (b) density of infected connections in steady state on WS network. (c) density of infected connections in steady state on BA network. In order to explore the influence of social capability d and uncertainty on rumor diffusion, relevant numerical solutions and simulations have been shown in Fig.4. The uncertainty is fixed as 0.1 on WS network while takes value of 0.01 on BA network to ensure rumor would diffuse to the threshold 0.1 in Fig.4 (a). Assuming steady state means variation of density of infected connections is less than 104 in two successive diffusions, we define effective rounds of rumor diffusion as the minimum value of diffusion rounds from threshold state to steady state. Through Fig.4 (a), effective rounds of rumor diffusion decreases with social capability increases in total which implies a rising value of social capability accelerates rumor diffusion among different distances. As shown in Fig.4 (b) and (c), where social capability is fixed as 0.1 and threshold is 0.1, we find that the density of infected connections in steady state grows up to a peak, then, decreases and tends to be stable with uncertainty increasing. The above phenomenon implies that uncertainty is not always harmful to system performance which could be interpreted as stochastic benefits in nonlinear systems. On both WS network and BA network, rumor hardly diffuses with small value
of uncertainty and spatial diffusion would not occur. Since for these small values of uncertainty, users almost make a choice on basis of deterministic utilities while the optimal choice is refusing rumors contributed by few influenced users in initial time. With increase of uncertainty, probability of a user chooses to spread a rumor tends to 0.5 no matter how much neighbors he/she has. It means users’ choice is completely random and is not related to utility which has been proved in [12]. Comparing curves on both artificial networks, we find the value of uncertainty to cause rumor diffusion is smaller on BA network which indicates an easier way of rumor spreading in a heterogeneous environment. 4. Conclusion In this paper, a PDEs model with uncertainty of human behavior is established under a spatio-temporal framework. With the consideration of physical meaning of spatial diffusion, a threshold is proposed below which the rumor would not become a hot message and only diffuses along determined physical connections. We also derive an approximate threshold time when threshold is reached. The distribution of infected connections at different distances constructs the initial function for the PDEs model at threshold time. Different from previous works, the heterogeneity of degree distribution and distance distribution has been taken into account at the same time. In theoretical analysis part of the PDEs model, the global existence and uniqueness of solutions have been proved with a Lyapinov function. After that, a classical solution around threshold is derived in the form of infinite series with a system of eigenfunction. In order to acquire an approximate numerical solution, an iterative matrix based on backward method is proposed with a truncation error. Simulations have also been carried out to explore rumor diffusion on artificial networks. We find the heterogeneity of degree distribution leads to different densities of infected connections under the same threshold which is also influenced by uncertainty on BA network. In addition, diffusion speeds at different distances are diverse from each other which dominated by distance distribution and initial infected users distribution on BA network while rumor diffuses at similar rapids at different distances on WS network. Is has also been explored that an increasing social capability would accelerate rumor diffusion to reach a steady state. Compared with WS network, rumor diffuses more easily with relatively small values of uncertainty while rumor declines at a faster speed with uncertainty increasing in BA network. Conflict of interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments Thank all reviewers for their precious comments. This research was supported by the National Nature Science Foundation of China (No.61179027) and the QingLan Project of Jiangsu Province of China (No. QL 06212006) . Reference: [1] X. Zhao, J. Wang, Dynamical Behaviors of Rumor, Spreading Model with Control Measures, Abstract and Applied Analysis, Hindawi Publishing Corporation, 2014, 2014. [2] D.J. Delay, D.G. Kendall, Epidemics and rumours, Nature 204 (1964) 1118.
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Received date : 26 June 2017 Revised date : 21 November 2017 Please cite this article as: L. Zhu, Y. Wang, Rumor diffusion model with spatio-temporal diffusion and uncertainty of behavior decision in complex social networks, Physica A (2018), https://doi.org/10.1016/j.physa.2018.02.060 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)
Rumor diffusion model with uncertainty of behavior decision. Diffusion threshold of spatio-temporal diffusion dominated by network topology. Heterogeneity of degree distribution and distance distribution. Theoretical analysis of a revised PDEs model.
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Rumor diffusion model with spatio-temporal diffusion and uncertainty of behavior decision in complex social networks Liang Zhu1, Youguo Wang2 1
College of Telecommunications and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210003, PR China 2 National Engineering Research Center of Communications and Networking, Nanjing University of Posts and Telecommunications, Nanjing 210003, PR China 2 College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210003, PR China 1 [email protected] 2 [email protected] Abstract: In this paper, a rumor diffusion model with uncertainty of human behavior under spatio-temporal diffusion framework is established. Take physical significance of spatial diffusion into account, a diffusion threshold is set under which the rumor is not a trend topic and only spreads along determined physical connections. Heterogeneity of degree distribution and distance distribution has also been considered in theoretical model at the same time. The global existence and uniqueness of classical solution are proved with a Lyapunov function and an approximate classical solution in form of infinite series is constructed with a system of eigenfunction. Simulations and numerical solutions both on Watts–Strogatz (WS) network and Barabási–Albert (BA) network display the variation of density of infected connections from spatial and temporal dimensions. Relevant results show that the density of infected connections is dominated by network topology and uncertainty of human behavior at threshold time. With increase of social capability, rumor diffuses to the steady state in a higher speed. And the variation trends of diffusion size with uncertainty are diverse on different artificial networks. Keywords: rumor diffusion model; spatio-temporal diffusion framework; diffusion threshold; complex social networks 1. Introduction Rumor diffusion, on the basis of online social networks, has produced serious damage on personal reputation and social stability. The evolving theory of telecommunication and developing network technique facilitate communications while promote rumor spreading in our daily lives. Unlike traditional rumor, rumor has characteristics of latent, persistent and explosive in online social networks which put great pressure on both network platforms and governments [1]. Therefore, rumor has drawn wide attention of researchers. Daley and Kendall first established D-K model to analyze the difference between disease diffusion and rumor dissemination [2] based on the mechanism of epidemic spreading. Furthermore, SIS (Susceptible –Infected-Susceptible) model [3] and SIR (Susceptible-Infected-Removed) model [4] on complex networks have been established to explore the transmission mechanism [5-8]. With the improvement of online social network, information could be obtained through multiple channels which challenges classic models based on ordinary differential equations (ODEs). In [9], a spatio-temporal framework on
basis of partial differential equations (PDEs) was first proposed:
I 2 I l f ( x, t , I ), xmin x xmax , t 0 t 2 x I I ( xmin , t ) ( xmax , t ) 0, t 0 x x I ( x, 0) ( x), xmin x xmax
(1)
In above model, authors divided information diffusion process into two separate processes: growth process and social process, meanwhile, used friendship hop as distance. In [10], heterogeneity of user distribution among different distance has been taken into consideration and growth process was described with a liner function. Accuracy of models has been proved with a Digg data set as well. And in [11], a summary of information diffusion in online social networks with PDEs models is proposed. In addition to the topology of network, the user plays a crucial role during the diffusion process as well. Unlike disease transmission, rumor diffusion in online social networks is conducted by rational users who would make strategic choices instead of being randomly infected with some probability [12]. Zinoviev and Duong indicated information diffusion involves all of users considering their knowledge, trust and popularity, which shape their publishing and commenting strategies [13]. Ellwardtet al. made an examination on interplay between person’s individual preference and social influence to suggest personal preference plays a more important role in human behavior [14]. Since users’ behavioral decision is driven by personal interests; game theory has been widely applied to explore human social behavior. Dunia described interaction between neighboring agents as a 2 2 coordination game and showed there is a threshold below which contagion could occur [15]. Jackson and Yariv considered such a diffusion process that agent makes the best response to behavior in last period [16]. Further, impacts of the distribution of costs, network topology and payoff structure on equilibrium points have also been confirmed. Though analyzing random disturbance is difficult in complex networks, the research of it is still important and necessary due to the great influence of random disturbance. In our previous work [17], the variation of connectivity was discussed with a stochastic differential equation (SDE) model and environmental noise was described as a Brownian motion. Zhao et al. held the opinion environmental uncertainty exists in the opinion formation which could be modeled as Gaussian stochastic process [18]. The fuzzy theory was also applied to illustrate imprecise and subjective information with a discrete evacuation model [19]. Cordoni and Persio considered a reaction-diffusion equation on a network and its boundary condition contains time delayed behavior which allows for multiplicative Gaussian noise perturbations [20]. In this paper, we consider the uncertainty of human behavior with random utility theory [12] on the basis of a spatio-temporal framework characterized by equation (1). Different from previous work, users with the same degree at different distances no longer take the same statistical meaning due to spatial diffusion among different distances. And the heterogeneity of degree distribution and distance distribution should be been taken into consideration at the same time in our model. Moreover, spatial diffusion mechanism stems from the function like trend topics and popular recommendation which asks for a certain number of disseminators. And the research object of equation (1) is news event from Digg which has a transmission base generally. As a
result, a spatial diffusion threshold is proposed in this paper below which spatial diffusion would not occur and means the rumor is not a hot message. Meanwhile, several researches have proved the latent nature [21-22] of network rumors which indicates rumor may not attract the attention of departments and keep concealed till spreads to a wide range or causes a great impact. So, it is reasonable to explore the transmission mechanism of rumor with a certain disseminators in the spatio-temporal framework before departments take actions like refuting a rumor. Both numerical analysis and simulations on WS and BA networks have been carried out to research spatio-temporal diffusion with uncertainty of human behavior in homogeneous and heterogenous networks. 2. Theoretical model Several online social networks, like Weibo, Twitter and Facebook, have concentrated on these ‘hot topics’ which owns a high click amount to make optimizations of website functions. Like the Headline function of Weibo, the most popular information would be selected through transmission of mass users and pushed to a new user who has not participated in discussion on these hot topics. While in Twitter, the function of trend topic recommends information with consideration of users’ interests and their friend information at the same time. It is obviously that the function of information push demands popularity which could be mirrored in a high number of forward information or a large amount of information audience. Besides, information transmission through website push is independent of users’ connections which could be interpreted as a mechanism of ‘random walk’. As a result, it is meaningful to consider such a diffusion threshold above which the information could be considered as a hot message and acquired without direct physical connections among users though the determination and size of threshold are different among varied online social networks. In this paper, we explore rumor spreading process under above diffusion framework with consideration of uncertainty of human behavior. Rumor diffuses along determined physical connections among users and is dominated by users’ average stochastic utilities when diffusion size is below threshold while spatial and temporal diffusions are in progress simultaneously above threshold. 2.1 Ordinary differential equation model Inspired by the previous work in [12], we consider users make a choice to spread rumor or not on basis of their average stochastic utilities which are influenced by uncertainty of human behavior. In this section, we will establish an ordinary differential equation model based on random utility theory and make relevant theoretical analysis to complement previous work during the initial stage of rumor diffusion which means rumor is not a hot message and still spreads along determined physical connections. The probability of a user with degree k chooses to spread the rumor could be derived in [12] as follow:
pk (t ) 1 e
ak k
1 bk kP ( k ) I k (t )
(2)
k
where I k (t ) represents the density of infected nodes with node degree k at time t , is a scale parameter to measure the uncertainty in human behavior. P ( k ) describes probability distribution of nodes with node degree k and
k
means the average node degree of networks.
In addition, the process of a user chose to accept the rumor is modeled as a multiple game process which could be divided to several two player games. In equation (2), parameters
a (T R P S ) / ( Rkmax ) and b ( P S ) / ( Rkmax ) contains information of a two player game and T , R, P, S are parameters of utility matrix with a ranking R T P S under the coordination game. For a heterogeneous network, we differentiate nodes with their degree and assume nodes with the same degree take the same statistical characteristics. Then, the threshold of density of infected users in the whole network is taken into account in this paper since it is convenient to observe and count the number of infected users. From this, the diffusion process could be characterized with such an ODE while density of infected users is under threshold: dI k (t ) pk (t )(1 I k (t )) (1 pk (t )) I k (t ) pk (t ) I k (t ) (3) dt where density of infected users satisfies
I (t ) P(k ) I k (t ) I * and k
I * represents
threshold of density of infected users. The front part of equation (3) denotes the increase of influenced individuals who decide to spread a rumor. On the contrary, the latter part of equation (3) means influenced users may also give up spreading rumor at the same moment. It is not difficult to find the state of a user, determined by his/her behavior choice which is up to average stochastic utility, is not constant in this model. Therefore, equation (3) describes a SIS model. For the convenience of analysis, we consider the variable of density of infected connections in the whole network at time t and denote it as y (t ) which takes the following form:
y(t )
1 k
kP(k ) I
k
(t )
(4)
k
Combining with equation (4) and making a variable substitution, equation (3) could be rewritten as:
dy(t ) 1 1 ( kP(k ) ) y(t ) ( aky ( t ) b )/ dt k k 1 e
(5)
Since the discussion of nodes and connections is a pair of dual problem in networks, we focus on the density of infected connections instead of infected nodes in the back of our work. And the relationship of these two variables would also be analyzed in different networks. For the diffusion process described by equation (3), it is important to determine existential conditions and amount of solution in steady state. If inequality I () I * sets up, rumor would not infect a large number of users and spatio-temporal diffusion does not occur as well. Therefore, we discuss the existence of steady solution in equation (5) first. From observation, steady solution of equation (5) y () is not equal to zero otherwise
dy () / dt kP(k )(1 eb / ) k
k 0 which means rumor would continue to diffuse. As a
result, the steady solution y () is a nonzero value which satisfies:
1 k
kP(k ) 1 e k
1 ( aky ( ) bk )/
y()
(6)
The existential condition of steady solution in equation (5) is equivalent to prove the existence of zero point of equation (6). Consider the following function:
f ( y())
1 k
kP(k ) 1 e
1 ( aky ( ) bk )/
(7)
k
Which is consist of multiple S-shaped functions (with value of y () increasing, the function is convex first and then become concave). Obviously, if (df ( y ()) dt ) | y ( ) 0 1 , there exists a zero point of equation (5) which has been proved in [12]. Furthermore, a nonzero steady solution of equation (5) could be determined when uncertainty satisfies
aeb / (1 ebkmax / )2 1 . However, we have to state it is only a sufficient condition of existence of steady solution. When we acquire the prior knowledge of threshold, the threshold time at which rumor becomes a hot message and spatio-temporal diffusion occurs is worth to be researched in forecasting work. For the given value of threshold y * , the Taylor expansion is applied to determine the threshold time y(t * ) y* in reasonable agreements. For equation (5), an approximate linear differential equation could be derived from the Taylor expansion:
dy 1 dt k The
right
part
of
1 akebk / kP ( k ) 1 y (t ) k bk / bk / 2 1 e (1 e ) approximate
equality
is
positive
with
parameters
(8) satisfy
aeb / (1 ebkmax / )2 1 and the threshold time could be derived with an error range o( y (t )) :
t*
f (k ) k f (k ) * ln 1 2 y C1 f1 (k ) k k
(9)
Where
k k P(k ) bk / 1 e 1 ebk / k
(10)
akebk / ak 2ebk / k P ( k ) k 1 bk / 2 bk / 2 (1 e ) k (1 e )
(11)
f1 (k )
f 2 (k )
C1
f (k ) k f (k ) ln 1 2 y (0) f 2 (k ) k k
(12)
In a single source situation, y (0) represents number of neighbors of the source node. For the threshold time described in equation (9), it has a high accuracy while value of threshold is close to zero. Furthermore, with increase of index of Taylor expansion, the truncation error tends to zero exponentially. While the form of threshold time becomes more complicated and not even
exists at cost. In this paper, we only consider the first order expansion to derive a heuristic conclusion. 2.2 Spatio-temporal diffusion model When rumor infects a certain amount of users and diffusion size researches the threshold, spatial and temporal diffusion occur at the same time due to the push service of online social networks. Although [9-11] have established classical spatio-temporal equations, there still remains some work to be improved and perfected. First, heterogeneity of node degree is not taken into consideration in previous models which means these sptaio-temporal equations are established on basis of uniform networks while scale-free features are ubiquitous in fact. Further, mechanism of epidemic infection has been also applied to describe growth process while the difference between epidemic and rumor is ignored. In this paper, we extend the equation (2) in a spatio-temporal framework based on random utility theory and a PDEs model is established while heterogeneity of degree distribution and distance distribution are taken into account. Due to the spatial diffusion among different distances, it is reasonable to assume users with the same number of neighbors acquire different average stochastic utility. Compared with [12], rumor spreads in a more complex way. According to the spatio-temporal equations described in equation (1), a PDEs model with consideration of uncertainty in human behavior is established as follows:
I k ( x, t ) 2 I k ( x, t ) d pk, x (t )(1 I k ( x, t )) (1 pk, x (t )) I k ( x, t ), l x L, t t * 2 x t * I ( x , t ) ( x ) k I (l , t ) I ( L, t ) k k 0 t t
(14)
where above equation describes diffusion process after threshold time t * . In equation (14), distance x means length of shortest path to source node which is characterized as friendship hops intuitively in online social networks and I k ( x, t ) represents density of infected users with degree k in distance x at time t (means the probability to select an infected node with degree k from all the nodes at distance x ). d represents the social capability measuring how fast the information travels across distances. The probability of a user with degree k in distance x at time t chooses to spread the rumor is: 1 pk, x (t ) aky ( x ,t )/ bk / (15) 1 e where
y( x, t ) kP(k ) I k ( x, t ) k
kP(k | x) k
(16)
is the probability to select a connection at the end of which is an infected node in distance x from all the connections linking to nodes in distance x . And P (k | x ) represents the probability to choose a node with degree k from all the nodes in distance x . Considering the physical meaning of real social networks, there is only a source node at the minimum distance which stays in infected state all the time in a single source situation. And for the maximum distance, rumor hardly researches due to a far distance. Hence, variations of density
of infected nodes at distance l and L are fixed which are not in our analysis. Additionally, applying cubic splines interpolation [9] to density of infected nodes at threshold time, a continuous and smooth initial function ( x) could be constructed. For equation (14), assume the threshold time as initial time and make a variable substitution like equation (4), a PDEs model describing variation of infected connections could be derived as follows:
y ( x, t ) 2 y ( x, t ) 1 1 d k kP(k ) 1 e( aky ( x,t )bk )/ y( x, t ), 0 x L, t 0 t 2 x kP ( k | x ) k y ( x, 0) ( x) y (0, t ) y ( L, t ) 0 x (17) where all the parameters take the same meanings as mentioned above and ( x ) is a continuous and smooth initial function to describe density of infected connections among different distances at threshold time. Then, we will discuss the global existence and uniqueness of the solution in equation (17) and derive an approximate classical solution in the form of infinite series. In addition, a numerical method is also constructed through backward difference. To verify the rationality of model and explore the transmission mechanism in artificial networks, simulations and numerical calculations on homogeneous and heterogeneous networks would be carried on to make a comparison in the next section. First, we put forward a theorem to construct a classical solution of equation (17). Theorem 2.2.1: For equation (17), the global existence and uniqueness of solution is proved 1
and a power system is determined in Sobolev Space H 0 (definition of norm in Sobolev Space:
f
H 01
f
L2
f
L2
) in the solution domain (0, L) [0, ) .
Proof. First, it is obvious that the growth function
g ( y ( x, t ))
1 1 kP(k ) y ( x, t ) aky ( x ,t )/ bk / 1 e kP(k | x) k
(18)
k
is local Lipschitz continuous. Thus, local existence and uniqueness of the solution of equation (17) could be obtained. Afterwards, the priori estimates of solution would be analyzed with a Lyapunov function and characteristics of solution would be promoted to the global domain. Assume y ( x, t ; ) is the solution of equation (17) which satisfies the initial condition. It is experiential to formulate a Lyapinov function in the following form: L 1 y ( x ,t ; ) V ( y ( x, t ; )) y x2 ( x, t ; ) g ( ) d dx, y ( x, t ; ) H 01 0 0 2
L
0
1 2 1 1 2 1 ( aky ( x ,t: ) bk )/ kP(k ) ln(1 e ) y ( x, t ; ) C2 dx yx ( x, t ; ) y ( x, t ; ) 2 2 kP ( k | x ) ak k k (19)
where C2 is a constant. According to definition of Lyapinov function, we have:
V ( y ( x, t ; )) V ( y ( x, 0)) V ( ( x))
(20)
Meanwhile, function ln(1 eaky ( x ,t ; )/ bk / ) ak increases monotonically with increase of y ( x, t ; ) which leads to:
b (21) ak ak a Applying equation (21) to equation (18), the following inequality could be derived combined with equation (20): L 1 1 2 2 (22) 0 ( 2 yx ( x, t; ) 2 y ( x, t; ) C3 )dx V ( y( x, t; )) V ( ( x)) Since ( x ) is a continuous polynomial function defined in a bound interval, its Lyapunov ln(1 e( aky ( x ,t ; ) bk )/ )
lim
y ( x ,t ; )
ln(1 e( aky ( x ,t ; ) bk )/ ) y ( x, t ; )
function is bounded. Applying Poincaré inequality to equation (22), we have
yx ( x, t; )
L2
y( x, t; )
L2
and
are bounded which supports that a power system is determined by equation (17) in
domain (0, L) [0, ) . Meanwhile, on the basis of priori estimates and prolongation theorem, the global existence and uniqueness of solution are set up. Then, we will discuss the form of classical solution of equation (17) through Taylor expansion. Based on the global existence and uniqueness of solution, an approximate form of classical solution could be derived through the following theorem. Theorem 2.2.2: For PDEs model described in equation (17), there exists a classical solution in the form of
y( x, t ) y (1) ( x, t ) y (2) ( x, t ) near the initial value y ( x, 0) where
y (1) ( x, t ) , y (2) ( x, t ) are determined by equation (24) and (25). Proof. First, equation (16) is a non-homogeneous nonlinear PDE whose solution is difficult to derive normally. As a result, we consider the Taylor expansion of the first part of equation (18) at threshold value y ( x, 0) and such an approximate PDE model is derived:
y ( x, t ) 2 y ( x, t ) d C5 y ( x, t ) C6 , 0 x L, t 0 t 2 x y ( x,0) ( x) y (0, t ) y ( L, t ) 0 x which could be divided into two problems as follows: y (1) ( x, t ) 2 y (1) ( x, t ) d C5 y (1) ( x, t ), 0 x L, t 0 t 2 x (1) y ( x,0) ( x) y (1) (0, t ) y (1) ( L, t ) 0 x and
(23)
(24)
y (2) ( x, t ) 2 y (2) ( x, t ) d C5 y (2) ( x, t ) C6 , 0 x L, t 0 t 2 x (2) y ( x,0) 0 y (2) (0, t ) y (2) ( L, t ) 0 x
(25)
where
C5
1 1 y( x, 0) akeaky ( x ,0)/ kP(k ) kP ( k ) 1 eaky ( x ,0)/ bk / kP(k | x) k (1 eaky ( x,0)/ bk / )2 (26) kP(k | x) k k
k
C6 1
a keaky ( x ,0)/ kP ( k ) kP(k | x) k (1 eaky ( x ,0)/ bk / )2
(27)
k
are constants and y ( x, 0) represents density of infected connection at distance x in threshold time. Since the PDE described in equation (24) is a homogeneous linear problem, its solution
y (1) ( x, t ) could be derived directly by separation of variable:
(2n 1) 2 L [C5 y ( x, t ) ( x) cos xdx e 0 2L n 0 L
(1)
(2 n 1)2 2 d 4 L2
]t
cos
(2n 1) x 2L
(28)
For equation (25), we expanse C6 , y (2) ( x, t ) as series according to the system of eigenfunctions cos (2n 1) x : 2L
4C6 (1) n 1 (2n 1) x C6 2n 1 cos 2L n 0 y (2) ( x, t ) v (t ) cos (2n 1) x n 2L n 0
(29)
Substitute equation (29) into equation (25) and combined with initial condition, the following equations are set up due to coefficients of series on each side of the equation are consistent.
' 4C6 (1) n 1 (2n 1) 2 2 d v ( t ) C v ( t ) vn (t ) n 5 n 4 L2 2n 1 v (0) 0 n
(30)
Thereout, we have
vn (t )
16C6 L2
(2 n 1) (1) n [C5 4 L2 e (2n 1)[4C5 L2 (2n 1) 2 2 d ] 2
2
d
]t
1
(31)
and
y ( x, t ) (2)
n 0
16C6 L2
(2 n 1) (1) n [C5 4 L2 e (2n 1)[4C5 L2 (2n 1) 2 2 d ] 2
2
d
]t
(2n 1) 1 cos x 2L
(32) Thus, y( x, t ) y (1) ( x, t ) y (2) ( x, t ) is an approximate classical solution near the initial value y ( x, 0) . It is easy to prove the uniform convergence of series y ( x, t ) , meanwhile, its first order partial derivative with respect to time t and second order partial derivative with respect to distance x are uniformly convergent as well. Consequently, it is meaningful to formulate a classical solution in the series form. From this, the proof is completed. Normally, classical solution is analyzed by theoretical deduction and it is difficult to apply the classical solution to calculation which is especially in the form of infinite series. So, a numerical method is established to approximate the analytic solution with a truncation error. In the next section, specific parameters would be determined to give relevant calculation results while we only introduce a framework of numerical approximation in this part. For the solution domain in equation (17), we determine the maximum time T and divide solution domain into grids where spatial and temporal interval are set as h and respectively. Thus, the number of grids is LT / ( h ) and the divided domain could be denoted as
x x j , j 0,1,..., J L / h , t tn , n 0,1,..., N T / . In equation (17), various iterative equations are formulated on basis of multiple difference methods. Forward difference method is most commonly used due to its explicit iterative equation while the solution is not always stable under any initial conditions. Take a comprehensive consideration on solution accuracy and computational complexity; we apply backward difference to formulate an implicit iterative equation. Assume y ( x, t ) is the solution of equation (16), following relational expression could be derived based on Taylor expansion: y ( x j , ti 1 ) y ( x j , ti ) y( x j 1 , ti ) 2 y( x j , ti ) y( x j 1 , ti ) d g ( y ( x j , ti )) o( h 2 ) (33) h2 For these sufficiently small h and which tend to zero in limit of theoretical analysis, the truncation
error
tends
to
zero.
Denoting
Y (i ) ( y ( x1 , ti ) y ( x2 , ti ) ... y ( xJ 1 , ti ))T ,
G (i ) ( g ( y ( x1 , ti )) g ( y ( x2 , ti )) ... g ( y ( xJ 1 , ti )))T , equation (33) could be described in form of matrix: QY (i ) Y (i 1) G (i 1)
(34)
where
1 2d / h 2 d / h 2 0 2 2 1 2d / h d / h 2 d / h Q 0 d / h 2 1 2d / h 2 0 0 0
0 0 1 2d / h 2 0
(35)
Obviously, Q is symmetrical and diagonally dominant which implies that a LU (Lower-Upper) decomposition exists. And then, equation could be solved gradually. Though, compared with forward difference method, truncation error of backward difference method is
O( h2 ) as well, stability of solution does not depend on division of domain. In addition,
central difference method and Crank-Nicholson format are also suitable with higher accuracy and complexity while we won’t make an introduction in details. 3. Simulations: In this section, numerical solutions of equation (17) based on above iterative matrix under determined parameters would be calculated. And then, simulations on artificial networks are carried out to investigate the dynamics of PDEs model. The parameters of utility matrix are set as R 4, S 1, T 0, P 0 to indicate a coordination game analyzed in literature [12]. Spatial interval h takes value 0.05 while temporal interval is 0.01. Meanwhile, artificial networks including WS network [23] and BA network [24] are constructed as follows. The WS network with the total number of nodes N 5000 is generated with a rewiring probability 0.3 and average degree of WS network is set as 15. While BA network with the same scale of WS network is built with two processes. In growth process, 30 initial nodes construct a global coupled network. In preferential attachment process, the other nodes (with 15 edges) connect to nodes already existing in the network with a probability which is proportional to the degree of nodes. Parameters of simulations have been researched in previous literatures [9-12]. Additionally, to reduce the random error, all the simulations are average over 100 independent runs and each simulation is performed with a randomly chosen source node. In addition, maximums of network distance ought to be same for all the source nodes to ensure that rumor diffuses in similar spatial dimensions. In [12], it has been illustrated that uncertainty has great influence on diffusion size for the case where temporal diffusion is taken into account only. Since density of infected connections is analyzed instead of infected nodes in this paper, we first research variation of density of infected connections under different values of uncertainty and threshold I * at threshold time while spatial diffusion has not been considered.
(a) (b) Fig .1 Variation of density of infected connections under different parameters: (a) density of infected connections varies with uncertainty on both WS and BA networks. (b) density of infected connections varies with threshold I * on BA network. Fig.1 (a) illustrates density of infected connections changes with uncertainty while threshold is fixed as 0.1 and social capability d 0.1 in both WS and BA networks. Rumor hardly diffuses with small values of uncertainty. With increase of , rumor diffuses till the threshold time and density of infected connections takes the same value of threshold approximately in WS network. While in BA network, the density increases quickly first and then tend to be stable with an increasing uncertainty. Fig.1 (a) shows density of infected connections
under fixed threshold is dominated by network topology and uncertainty. Homogeneity of WS network in degree weakens the difference of connection among nodes. Especially, in a regular network, density of nodes is even equivalent to density of sides with the consideration of numerical calculation. While in BA network, nodes obey a long-tail distribution due to the property of sacle-free and there exists significant differences in connection among different nodes. In Fig.1 (b), we fix uncertainty as 0.01 and adjust threshold from 0.25 to 0.3 to explore the variation of density of infected connections in BA network. The red line in Fig.1 (b) represents a proportional function with gradient is 1. Note that, density of infected connections is below threshold for these small values of threshold which means nodes with small degree make up the main part of infected nodes in threshold time. With increase of threshold, density of infected connections grows rapidly and flattens out around I * 0.3 . In this situation, hub nodes with large number of connections are infected at the threshold time. Compared to nodes with fewer connections, hub nodes take a higher probability to touch with infected nodes while density of nodes is large enough in the network.
(a) (b) Fig.2 The evolution of rumor diffusion on both WS and BA networks: (a) numerical solutions. (b) simulations. Fig.2 shows the evolution of rumor diffusion through numerical and simulation values on both WS and BA networks. In Fig.2 (a), we set threshold I * and social capability d as 0.1. Meanwhile, uncertainty is 0.1 on WS network while takes value of 0.01 on BA network. We make the following explanation for the selection of uncertainty on artificial networks: First, values of uncertainty should be selected with consideration of network topology to ensure rumor would diffuse in the network. Then, we hope diffusion sizes on both artificial networks tend to be consistent through adjusting values of uncertainty in case produce a phenomenon that rumor always seems to spread to a larger range on a network than another. In fact, the influence of uncertainty on diffusion size of networks is significantly different on WS and BA networks. The relevant conclusions would be proposed to make an analysis in the latter. In addition, for a fixed threshold, simulations without spatial diffusion have been carried on to generate initial density of infected connections among different distances and cubic splines interpolation is applied to construct an initial function ( x ) with these discrete points. From observation of Fig.2 (a) and (b), we find the speed of rumor diffusion rises firstly and then decreases to 0 finally from numerical solution and simulation results. In the early time, susceptible users construct principle part of the whole network and the choice not to spread a rumor would gain higher returns for most users which lead to a slow growth rate of rumor diffusion. With increasing of infected users, the
behavior choice of spreading rumor would offer a high returns that result in an increasing probability of accepting the rumor. While only a small number of susceptible users existing, the growing speed slows down and total density of infected connections researches a maximum gradually.
(a)
(b)
(c) (d) Fig.3 The evolution of rumor diffusion in different distances: (a) numerical solutions on WS network. (b) simulations on WS network. (c) numerical solutions on BA network. (d) simulations on BA network. Fig.3 illustrates the evolutions of rumor diffusion at different distances while a total variation of infected connections has been proposed. Fig.3 (a) and (b) display rumor diffusion on both spatial and temporal dimensions with uncertainty 0.1 , social capability d 0.1 and threshold I * 0.1 on WS network. As shown in Fig.3 (a) and (b), density of infected connections keeps a similar variation trend at different distances till the whole network is infected. Due to the homogeneity of connection distributions and initial values at different distances, variation of density of infected connections shows a homogeneous characteristic approximately at different distances on WS network. In Fig.3 (c) and (d), uncertainty is set as 0.01 while other parameters remain unchanged. On BA network, nodes are mainly distributed at distance 3 where initial infected nodes are agminate as well. Since the existence of hub nodes at distance 3, rumor diffuses at a higher rapid compared with other distances. However, in the adjacent distance of source node, rumor spreads slowly since few initial infected user exists. After that, though similar variation trend of density of infected connections exists in numerical solutions and simulations, we still have to explain the difference between them. In simulation, only distance and round of rumor diffusion in integer have practical significance while domain could be divided into sufficiently small grids to acquire a satisfactory result in numerical solution. That is why the
image of numerical solution looks smoother than simulation results. In addition, it is not difficult to find the diffusion size is larger in simulation than in numerical solution which is dominated by the following factor. Spatial diffusion stems from the difference of density of infected connections in adjacent distances. In numerical solution, spatial diffusion between two adjacent integer distances would generate loss due to these middle discrete points. However, excessive spatial interval would produce a huge truncation error which is a universal shortcoming in differential method.
(a)
(b)
(c) Fig.4 Effective rounds of rumor diffusion and density of infected connections in steady state with different parameters: (a) effective rounds on both WS and BA networks. (b) density of infected connections in steady state on WS network. (c) density of infected connections in steady state on BA network. In order to explore the influence of social capability d and uncertainty on rumor diffusion, relevant numerical solutions and simulations have been shown in Fig.4. The uncertainty is fixed as 0.1 on WS network while takes value of 0.01 on BA network to ensure rumor would diffuse to the threshold 0.1 in Fig.4 (a). Assuming steady state means variation of density of infected connections is less than 104 in two successive diffusions, we define effective rounds of rumor diffusion as the minimum value of diffusion rounds from threshold state to steady state. Through Fig.4 (a), effective rounds of rumor diffusion decreases with social capability increases in total which implies a rising value of social capability accelerates rumor diffusion among different distances. As shown in Fig.4 (b) and (c), where social capability is fixed as 0.1 and threshold is 0.1, we find that the density of infected connections in steady state grows up to a peak, then, decreases and tends to be stable with uncertainty increasing. The above phenomenon implies that uncertainty is not always harmful to system performance which could be interpreted as stochastic benefits in nonlinear systems. On both WS network and BA network, rumor hardly diffuses with small value
of uncertainty and spatial diffusion would not occur. Since for these small values of uncertainty, users almost make a choice on basis of deterministic utilities while the optimal choice is refusing rumors contributed by few influenced users in initial time. With increase of uncertainty, probability of a user chooses to spread a rumor tends to 0.5 no matter how much neighbors he/she has. It means users’ choice is completely random and is not related to utility which has been proved in [12]. Comparing curves on both artificial networks, we find the value of uncertainty to cause rumor diffusion is smaller on BA network which indicates an easier way of rumor spreading in a heterogeneous environment. 4. Conclusion In this paper, a PDEs model with uncertainty of human behavior is established under a spatio-temporal framework. With the consideration of physical meaning of spatial diffusion, a threshold is proposed below which the rumor would not become a hot message and only diffuses along determined physical connections. We also derive an approximate threshold time when threshold is reached. The distribution of infected connections at different distances constructs the initial function for the PDEs model at threshold time. Different from previous works, the heterogeneity of degree distribution and distance distribution has been taken into account at the same time. In theoretical analysis part of the PDEs model, the global existence and uniqueness of solutions have been proved with a Lyapinov function. After that, a classical solution around threshold is derived in the form of infinite series with a system of eigenfunction. In order to acquire an approximate numerical solution, an iterative matrix based on backward method is proposed with a truncation error. Simulations have also been carried out to explore rumor diffusion on artificial networks. We find the heterogeneity of degree distribution leads to different densities of infected connections under the same threshold which is also influenced by uncertainty on BA network. In addition, diffusion speeds at different distances are diverse from each other which dominated by distance distribution and initial infected users distribution on BA network while rumor diffuses at similar rapids at different distances on WS network. Is has also been explored that an increasing social capability would accelerate rumor diffusion to reach a steady state. Compared with WS network, rumor diffuses more easily with relatively small values of uncertainty while rumor declines at a faster speed with uncertainty increasing in BA network. Conflict of interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments Thank all reviewers for their precious comments. This research was supported by the National Nature Science Foundation of China (No.61179027) and the QingLan Project of Jiangsu Province of China (No. QL 06212006) . Reference: [1] X. Zhao, J. Wang, Dynamical Behaviors of Rumor, Spreading Model with Control Measures, Abstract and Applied Analysis, Hindawi Publishing Corporation, 2014, 2014. [2] D.J. Delay, D.G. Kendall, Epidemics and rumours, Nature 204 (1964) 1118.
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