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MM-SIS: Model for multiple information spreading in multiplex network
Accepted Manuscript MM-SIS: Model for multiple information spreading in multiplex network Yunpeng Xiao, Li Zhang, Qian Li, Ling Liu
PII: DOI: Reference:
S0378-4371(18)31118-X https://doi.org/10.1016/j.physa.2018.08.169 PHYSA 20055
To appear in:
Physica A
Received date : 27 May 2018 Revised date : 14 August 2018 Please cite this article as: Y. Xiao, et al., MM-SIS: Model for multiple information spreading in multiplex network, Physica A (2018), https://doi.org/10.1016/j.physa.2018.08.169 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
Highlights: (1) MM-SIS models the interactive diffusion of multiple information in multiplex network (2) The influence factor is introduced to describe interactions between information (3) Epidemic threshold is analyzed and the correctness is verified via experiments (4) The simulation results reveal complex interaction between information
*Manuscript Click here to view linked References
MM-SIS: Model for multiple information spreading in multiplex network Yunpeng Xiaoa,b,∗, Li Zhanga , Qian Lia , Ling Liub a Chongqing
Engineering Laboratory of Internet and Information Security, Chongqing University of Posts and Telecommunications, Chongqing 400065, China b Computer Science, Georgia Institute of Technology, Atlanta, Georgia, 30332, United States
Abstract In social networks, relationships between information are complexly intertwined. However, prior research mainly focuses on studying the independent information in an isolated network, ignoring two factors: the interaction between different information and the diversity of propagation paths. In view of the above problems, this paper proposes the MM-SIS(Multiple information and Multiplex network-SIS)model to explore the detailed processes and characteristics of multiple information in multiplex networks. At the same time, the concept of influence factor is introduced to describe the complex interaction between different information. Moreover, we use the Microscopic Markov Chain method to set dynamic equations, theoretically analyze the epidemic threshold and verify its correctness through experiments. In general, this paper mainly explores the effects of influence factor and the interrelation of network layers on information spreading process. The experimental results show that the above mentioned two factors have obvious impacts on the information outbreak scale and the epidemic threshold. Keywords: Information diffusion, Multiplex network, Multiple information, Markov chain method, SIS model
∗ Corresponding
author Email address: [email protected] (Yunpeng Xiao)
Preprint submitted to Journal of Physica A
August 14, 2018
1. Introduction With the development of Internet, online social networks are prevalent in people’s lives. Many new social media such as micro-blog, blogs and forums have profoundly influenced and changed the way of information dissemination. 5
Online social networks are centered on users, people rely on the friend relationships to interact with others and disseminate information. People use social networks to exchange a lot of information, including hot news, religious beliefs, fashion trends and even emotions [1, 2]. Using these data to analyze and predict users’ behavior and psychology [3] is of great importance to public opinion
10
monitoring [4] and service recommendation. There is a complex interaction between information when they are spreading in the network simultaneously. The widespread popularity of online social networks leads to the explosive growth of information, but these pieces of information are not independent in the network, and they will cooperate or compete
15
with each other in the spreading process [5]. The interaction between information will directly affect the dynamic progress of information, and thus should be taken into consideration. In our real life, many complex systems can be modeled as complex networks to analyze, such as the spread of diseases [6], the propagation of computer viruses [7], the dissemination of innovative products
20
[8] etc. Complex network analysis methods can be used in community detection [9], influence maximization [10], and important nodes identification [11].As one of the complex networks, it is also important to study the properties of online social networks [12], the most commonly used models are SIS and SIR epidemic models [13–15]. Most of these researches focus on single layer networks, but sin-
25
gle layer networks have limitations in modeling some scenarios. Therefore, some researchers have shifted from single, isolated networks to multilayer, overlapped networks, and found that information propagates over multiplex networks exhibit some characteristics that are different from those of independent networks. Although many significant research achievements have been made in the field
30
of information dissemination, there are still some challenges:
2
On the one hand, a comprehensive model is lacked to explore the propagation process of multiple information in multiplex network. At present, most of the previous studies mainly focus on the single information propagates in isolated networks [13, 16, 17], and following scholars focus on the interaction of 35
different information [18–23] or focus on the multiplex networks [24–31], only some studies take both into consideration [32–35]. On the other hand, the relationship between information is complex and diverse. Most studies are not fully considered when exploring the interaction between information. Many prior works have focused on competitive informa-
40
tion, especially on full competition, ignored that it is possible for a user to propagate multiple information simultaneously in real life [32, 36]. The example about diseases in Ref. [37] provides new grounds to the understanding of information spreading in social networks. Along the idea of Mills et al, we realize that the factors driving people to establish links in social network
45
are also complex, that is to say, the nature of the relationship represented by links is different. For instance, friend relationships are established based on the same interests and hobbies, colleague relationships are established based on business contact. Therefore, the potential propagation routes of different information will be different and interact with each other. In this paper, we build
50
a multiplex network based on the specific problems mentioned above and refer to the SIS mechanism to propose an information propagation dynamic model, that is MM-SIS (Multiple information and Multiplex network-SIS) model. Our contribution can be summarized as follows: (1) The MM-SIS model is proposed based on multiple information and multiplex
55
network. This paper focuses on the interaction between information and the diversity of propagation paths when studying the information spreading process. At the same time, we use the SIS epidemic model for reference to explore the propagation mechanism of multiple information in multiplex network.
60
(2) The concept of influence factor is introduced to describe the relationship
3
between information and the cooperative and competitive relationships are integrated into a unified model. Based on this, the concepts of asymmetric influence, bidirectional suppression and bidirectional promotion are defined to represent the complex and diverse interaction between information. 65
(3) We establish dynamic equations of our model by using the Microscopic Markov Chain method. Then, we theoretically analyze the epidemic threshold and verify its correctness through experiments. The rest of this paper is organized as follows. Section 2 introduces the related work. Section 3 formulates the problems and gives the necessary definitions. In
70
Section 4, we describe the proposed model in detail. Section 5 presents and analyzes the experimental results of the model. Finally, Section 6 concludes the paper.
2. Related works In this section, we will review related work from the following aspects. 75
Single network topology for multiple information: [15, 18–20] use real data to prove that there is a real interaction between information. Myers et al. [18] use real data from Twitter to suggest that interactions cause a relative change in the spreading probability of a contagion by 71% on the average. Su et al.[19] compare information to organisms and employ game theory to study whether
80
a contagion can be promoted or suppressed by others in the diffusion process. Zhang et al.[20] propose IDA model to study the contagion adoption behavior under a set of interactions. In the case of full competition (a node can transmit at most one information at any given time), [21] shows that winner takes all, that is to say, the stronger virus completely wipes-out the weaker one. However, [22]
85
uses mean field theory to show that if the cross-immunity satisfies a threshold condition, viruses can coexist in the network. Trpevski et al.[23] extend the SIS model to study competitive rumors spreading over networks. Multiple network topology for single information: A system composed of interconnected sub-networks is often called a network of networks (NON). NON has 4
90
a variety of titles in different papers[24] including interdependent networks[25], multiplex networks[26] and multidimensional networks[27], etc. Zhuang et al.[26] adopt the content-dependent linear threshold model to study the effect of clustering coefficient in multiplex networks. Li et al.[28] use bond percolation and cascading failure to show that interaction between layers can greatly enhance
95
the information diffusion process. Cozzo et al.[29] propose a contact-based information spreading model to investigate the critical point of the multiplex system. Wang et al.[30] establish a collaboration network to perform the trustworthy service selection in web service social network environment where the concept of the collaboration reputation is first proposed. What’s more, targeted
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immunization strategy for epidemic spreading over a multilayer network is less efficient than in isolated networks [31]. Multiple network topology for multiple information: The above two scenarios inspired scholars to study the propagation of multiple information spreading over multiplex networks, which is the most similar scenario to this paper. Sahneh
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et al.[32] address the problem of two competitive viruses propagating in a host population where each virus has a distinct contact network for propagation. Wu et al.[33] use the mean-field approach to investigate the impact of awareness on epidemic spreading process. Watkins et al.[34] study the problem about optimal resource allocation for competitive spreading processes on bilayer network. Zhao
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et al.[35] present the concept of state-dependent infectious rate and propose a unified framework to study the cases of two pathogens in two-layered networks. What’s more, the spread of epidemics and information can be regarded as a Markov process. S. G´omez et al.[38] study the disease in complex network based on discrete-time Markov chain approach. Granell et al.[39] present a model in
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which awareness and epidemic spread on separate layers and employ Microscopic Markov Chain Approach, they find that awareness can suppress the propagation of epidemic. Wei et al.[36] develop the SI1 I2 S model to study competing memes in a composite network and show that the first eigenvalue of system matrix is a critical metric to identify epidemic behavior. The abovementioned researches
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are based on Markov Chain Approach, Microscopic Markov Chain Approach 5
help us quantify the microscopic dynamics at the individual level.
3. Problem definition 3.1. Related definitions In general, the multiplex network can be represented as G(V, E1 , E2 , ..., El ), 125
where V = {V1 , V2 , . . . . . . , VN } represents the set of nodes, the cardinality |V | =N is used to denote the total number of whole network users and Ei represents the set of links in spcacei .If there is a connection between node i and node j, then we stipulate that the information can spread from node i to node j. Inf = {Inf1 , Inf2 , ..., InfM } is used to represent the set of multiple
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information. Since this paper uses the SIS communication mechanism for reference, the user is in Ii state when the user spreads information Infi , otherwise it is in Si state. Table. 1 shows the symbols we have used in the paper. The basic concepts and symbol representations in this paper are as follows: Influence factor: symbol δ is used to represent impact factor. Specifically, δi
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represents the influence factor of Infi . We specify that information Infi is suppressed by others when 0 ≤ δi < 1 , and the smaller δi means the greater the suppression strength, especially when δi = 0, it means information Infi and others are completely mutual exclusive; information Infi is promoted by others when δi > 1, and the larger δi means the greater the promotion strength;
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information Infi is independent with others, when δi = 1. Infection rate: symbol β is used to represent infection rate. Specifically, βi represents the infection rate of Infi . The infection rate indicates the probability that if a node does not spread the information and its neighbor transmits the information, then he is influenced by a neighbor user to spread the message.
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Cure rate: symbol θ is used to represent cure rate. Specifically, θi represents the cure rate of Infi . When the node is in I state, it will return to S state spontaneously with θ probability. In this paper, if node i in Ii state loses interest in Infi and forgets it, the node will changes from Ii state to Si state. This parameter describes the duration of information. The larger θ means the
6
150
shorter duration of information. If Infi is not attractive enough, it is quickly forgotten and the value of θi is large. Table 1. Symbols and definitions Symbols
Definitions
Inf
information
δi
influence factor of Infi
βi
infection rate of Infi
θi
cure rate of Infi
A(B)
the adjacency matrix of spaceA(spaceB)
λA (λB )
the largest eigenvalue of matrix A(matrix B)
ρ(G)
correlation coefficient of multiplex work G
3.2. Problem formulation To formally formulate the problem of our research, let G(V, E1 , E2 , ..., El ) be the whole multiplex network, and let T = {(β1 , θ1 , δ1 ), (β2 , θ2 , δ2 ), ..., (βM ,θM ,δM )} 155
represents the basic attributes of all information. We use the above attributes to study the spread of information, that is, to calculate the state of each node We use PiX (t) to represent the probability of the node i in
at a certain time.
the state X at time t. Specifically, the problem is defined as follows: G(V, E1 , E2 , ..., El ) ⇒ f : (G, T ) → PiX (t) T = {(β , θ , δ ), (β , θ , δ ), ..., (β ,θ ,δ )} 1
160
1
1
2
2
2
M
M
M
3.2.1. Problem input
Given the related definitions, the input to this problem can be defined as follows: 1. Whole network topology G(V, E1 , E2 , ..., El ). 2. The basic characteristics of the information to be studied and the inter165
action between the information T = {(β1 , θ1 , δ1 ), (β2 , θ2 , δ2 ), ..., (βM ,θM ,δM )}
7
3.2.2. Problem output Based on the above description, the problems to be solved are as follows: How to explore the influence of interaction between information and interrelation between network layers on information dissemination? Firstly, we initial170
ize the whole multiplex network G(V, E1 , E2 , ..., El ), that is, defining the number of nodes and the network topology of each layer. Then we initialize the basic characteristics of information T = {(β1 , θ1 , δ1 ), (β2 , θ2 , δ2 ), ..., (βM ,θM ,δM )}, and let different information propagates on different layers. We set the interrelation into positive interrelation, negative interrelation and neutral interrelation,
175
and the interactions are classfied into promotion, suppression and unrelated. Under different conditions, PiX (t) can be got through experiments, thus the overall information propagation situation is obtained. Further analysis can yield the impacts of interactions and interrelations on information dissemination.
4. Proposed model 180
In this section, the MM-SIS model is introduced in detail, which aims to reveal the co-evolution rule of multiple information on top of multiplex networks. The details of our model are shown in Fig. 1.
Figure 1: Model framework
8
4.1. Multiple information and multiplex network In this paper, we consider the diversity of propagation paths to build a 185
multiplex network. The nature of the relationship represented by the links is different, so different information will spread along different network topologies. Fig. 2 shows the above process more vividly. To be brief, this paper divides the links in the network into two types, which are represented by spaceA and spaceB, respectively. The nodes in spaceA and
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spaceB correspond randomly and each pair of nodes represents the same user. Information1 spreads on spaceA while information 2 spreads on spaceB. Note that our model can also be easily extended to more layers. We assume that the number of nodes in the network is N ,V = {V1 , V2 , . . . , VN } is the set of nodes, EA is the set of links in spaceA, EB is the set of links in spaceB. Therefore, we
195
use a triple G(V, EA , EB ) to represent the whole multiplex network. Meanwhile, we define the adjacency matrix A = [aij ]N ×N , where aij = 1 if there is an edge between node i and node j in spaceA, otherwise aij = 0. Similarly, we define the adjacency matrix B of SpaceB, B = [bij ]N ×N .
Figure 2: Schematic figure of multiplex network. Information 1 spreads on space A while information 2 spreads on space B, the nodes contacted by dotted line represent the identical user. δ1 and δ2 are influence factors. Inf is the abbreviation of information
4.2. Modeling 200
In this paper, the information spreading process refers to the SIS mechanism. If there are M pieces of information in the network and they are not completely
9
mutually exclusive (a node can be infected by multiple information at the same time), there will be 2M kinds of node states in the network. In order to facilitate the expression of the model and simplify the theoretical expression, this paper 205
uses two pieces of information(i.e. Inf = {Inf1 , Inf2 })as an example to study, and the research principle of more than two information is similar to this one. So each node can be in one of four different states X, X ∈ {I1 S2 , S1 I2 , I1 I2 , S1 S2 }. I1 S2 represents a user only transmits information 1, S1 I2 represents a user only transmits information 2, I1 I2 represents a user transmits information 1
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and information 2 simultaneously and S1 S2 represents a user neither transmits information 1 nor information 2. We regulate that when a node is in S1 S2 state, it may be infected by information 1 at a rate of β1 , then assume that the node’s neighbors are independent of each other, it will become infected with probability: fi1 = 1 −
∏
j∈i′ neighbors in space A
(1 − β1 PjI1 (t))
(1)
in which PjI1 (t) = PjI1 S2 (t) + PjI1 I2 (t), where PjX (t) represents the probability that node j is in X state at time t. However, if a node has already propagated information 2 and failed to recover from information 2 (i.e. in S1 I2 state), its probability of being infected by information 1 becomes δ1 β1 , and its probability of being infected will be: u1i = 1 −
∏
j∈i′ neighbors in space A
(1 − δ1 β1 PjI1 (t))
(2)
(1 − β2 PjI2 (t))
(3)
(1 − δ2 β2 PjI2 (t))
(4)
With the same principle, we can obtain: fi2 = 1 −
u2i = 1 −
∏
∏
j∈i′ neighbors in space B
j∈i′ neighbors in space B
And we assume that θ1 (θ2 ) is the cure rate of information 1 (information 2). The transition between four states of a specific node is indicated in Fig. 3. In this paper, we assume that information 1 is the preferred information, 215
that is to say, users give priority to information 1 when faced with two pieces 10
Figure 3: The transition between four states of a specific node, S represents the susceptible state,I represents the infected state. of information [23]. The following discrete time microscopic Markov equations is presented to describe the co-evolution process of multiple information. PiI1 S2 (t + 1) = PiS1 S2 (t)fi1 (1 − u2i ) + PiS1 I2 (t)u1i θ2 + PiI1 S2 (t)(1 − θ1 )(1 − u2i ) + PiI1 I2 (t)(1 − θ1 )θ2 P S1 I2 (t + 1) = P S1 S2 (t)(1 − f 1 )f 2 + P S1 I2 (t)(1− i i i i i I1 I2 I1 S2 2 1 (t)θ1 fi + Pi (t)θ1 (1 − θ2 ) ui )(1 − θ2 ) + Pi PiI1 I2 (t + 1) = PiS1 S2 (t)fi1 u2i + PiS1 I2 (t)u1i (1 − θ2 )+ P I1 S2 (t)(1 − θ )u2 + P I1 I2 (t)(1 − θ )(1 − θ ) i
1
i
i
1
(5)
2
Note that in this paper we focus on the evolution process of the nodes in I state, and it is easy to get: PiS1 S2 (t + 1) = 1 − PiI1 S2 (t + 1) − PiS1 I2 (t + 1) − PiI1 I2 (t + 1)
(6)
so do not need an extra equation to represent PiS1 S2 (t + 1). 4.3. Theoretical analysis 220
Epidemic threshold is a critical point of the information outbreak. Put another way, we will study the conditions under which the information will not spread in the MM-SIS model. Lemma 1 (Hirsch and Smale,1974 [40]) : The system is asymptotically stable → − − → − → at P = 0 if the eigenvalues of ∇g( 0 ) are less than 1 in absolute value, where 11
225
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[ − →] m → − → − Jacobian matrix J = ∇g( 0 ) = ∂g ∂Pn | P = 0 . m,n − → − → For convenience, we use P (t + 1) = g( P (t)) to represent eqs(1). Vector −−−→ −−−→ −−−→ −−−→ − → P (t) = (P I1 S2 (t), P S1 I2 (t), P I1 I2 (t)), where P I1 S2 (t) = (P1I1 S2 (t), P2I1 S2 (t), ..., PNI1 S2 (t)). −−S−→ −−−→ P 1 I2 (t) and P I1 I2 (t) are represented similarly. To study the epidemic threshold, we analyze the Jacobian matrix of system −−−→ −−−→ −−−→ − → − → − → − → − → P (t + 1) = g( P (t)). When (P I1 S2 (t), P S1 I2 (t), P I1 I2 (t)) = ( 0 , 0 , 0 ), the Jacobian matrix is shown as follows:
S1
− → − → − = Z J(→ 0,0,0) Z
Z S2 Z
S4
S5 S3
Where Z is a N × N zero matrix, matrixes S1 -S5 are as follows: S1 = (1 − θ1 )E + β1 A S2 = (1 − θ2 )E + β2 B
235
S3 = (1 − θ1 )(1 − θ2 )E S4 = (1 − θ1 )θ2 E + β1 A S5 = θ1 (1 − θ2 )E + β2 B Where E is an identity matrix. This Jacobian matrix is an upper triangular matrix, so its eigenvalues are the same as those of S1 , S2 and S3 . According to −−−→ −−−→ −−−→ − → − → − → Lemma 1, if the system is stable at (P I1 S2 (t), P S1 I2 (t), P I1 I2 (t)) = ( 0 , 0 , 0 ), − → − → − is less than 1. That is, the the absolute value of the eigenvalue of J(→ 0,0,0)
following conditions need to be met: max{|(1 − θ1 )(1 − θ2 )|, |(1 − θ1 ) + β1 λA |, |(1 − θ2 ) + β2 λB |} < 1
(7)
Where λA and λB are the largest eigenvalues of adjacency matrix A and 240
adjacency matrix B respectively. The system is asymptotically stable at point −−−→ −−−→ −−−→ − → − → − → (P I1 S2 (t), P S1 I2 (t), P I1 I2 (t)) = ( 0 , 0 , 0 ), it means initial infected nodes will die out in the network exponentially and none information can spread in the network. Since 0 < θ1 (θ2 ) ≤ 1, 0 ≤ (1−θ1 )(1−θ2 ) < 1 is obviously satisfied. Matrix A
245
and matrix B are nonnegative, irreducible, symmetric, square matrixes(because
12
spaceA and spaceB are undirected networks), so the largest eigenvalues (λA , λB ) are positive real number according to Perron-Frobenius Theorem. 0 < θ1 (θ2 ) ≤ 1 and 0 < β1 (β2 ) ≤ 1,thus, (1 − θ1 ) + β1 λA > 0 and (1 − θ2 ) + β2 λB > 0 are satisfied and condition(7) can be rewritten as follows:
250
(1 − θ ) + β λ < 1 1 1 A (1 − θ ) + β λ < 1 2 2 B
namely
β1 /θ1 < 1/λA
namely
β2 /θ2 < 1/λB
(8)
So we include that when β1 /θ1 < 1/λA and β2 /θ2 < 1/λB , there is no information spreading in the network. If the above-mentioned two information expected to survive, they should at least satisfy β1 /θ1 ≥ 1/λA or β2 /θ2 ≥ 1/λB . In the following sections we will verify the correctness of this theoretical analysis and explore the impact of influence factor and layer interrelation on the process
255
of information dissemination.
5. Results and analysis 5.1. Experimental datasets In this paper, we simulate the proposed model on the real network and the algorithm generated network. The specific descriptions are as follows. 260
Real world network: The real dataset used in this paper is to describe the relationship between employees in a company, including two types of relationship1 . According to the requirements of our experiments, we select the colleague relationships between nodes to establish spaceA and select the friend relationships between nodes to establish spaceB. The statistical characteristics of the
265
dataset are shown in Table. 2. Generated network: Both spaceA and spaceB are scale-free networks. we use the Barab´asi-Albert model with N = 2000 nodes, m = 3, in which, m represents the number of links when a new node is added. The degree of node in spaceA is 2-96, in spaceB is 2-104. 1 http://deim.urv.cat/
manlio.dedomenico/data.php
13
Table 2. Statistics of dataset Dataset
270
Space A
Space B
Users
71
71
Edges
892
575
Degree
0-30
0-25
5.2. Experiments and analysis In this section, we simulate the proposed model and further analyze the experimental results. Firstly, we verify the correctness of the theoretical analysis. Then exploring the influence of interaction between multiple information and the influence of interrelation between network layers respectively. Finally, we comprehensively explore how multiple information and multiplex network affect the epidemic threshold.
the number of infected nodes
5.2.1. Theoretical analysis verification the number of infected nodes
275
250 PI1I2 200
I1
P
PI2 150
100
50
0 0
20
40
(a) t
60
80
100
250 PI1I2 I
200
P
1
PI2
150
100
50
0 0
20
40
60
80
100
(b) t
(a)
(b)
Figure 4: Spreading process of information over time, (a) and (b) show that information 1 and information 2 both can spread in the network.(a) β1 /θ1 ≥ 1/λA , β2 /θ2 < 1/λB ;(b) β1 /θ1 < 1/λA , β2 /θ2 ≥ 1/λB .
14
I
P
1
1.5
1
0.5
0 0
20
40
60
80
100
the number of infected nodes
the number of infected nodes
2
2 I
P
1
1.5
1
0.5
0 0
(a) t
20
40
60
80
100
(b) t
Figure 5: The infected nodes of information 1 over time with β1 /θ1 < 1/λA , β2 /θ2 < 1/λB . (a) and (b) shows that information 1 die out exponentially in the multiplex network. (a) δ1 = 10; (b) δ1 = 30. From the above analysis we conclude that if the two pieces of information studied in this paper can spread in the network, the condition β1 /θ1 ≥ 1/λA 280
and condition β2 /θ2 ≥ 1/λB should be reached at least one. In Fig. 4, (a) shows the spreading process of information 1 and information 2 when β1 /θ1 ≥ 1/λA , β2 /θ2 < 1/λB and (b)shows the spreading process of information 1 and information 2 when β1 /θ1 < 1/λA , β2 /θ2 ≥ 1/λB . In figure (a) and figure (b) we can see that P I1 > 0 and P I2 > 0 at stable state (small fluctuations of
285
its infected nodes), that is to say, both of the information can spread in the network. These results preliminarily prove the correctness of our conclusion. Alex Beutel et al. explored the propagation of two viruses on an isolated network, they found that even both viruses are independently too weak to survive on their own when β1 /θ1 < 1/λA , β2 /θ2 < 1/λB , with enough cooperation they
290
can [22]. But in this multiplex network we find different results. We study information 1 as an example (information 2 is same with information 1). In Fig. 5, (a) depicts the evolution process of the number of infected nodes of information 1 with δ1 = 10, β1 /θ1 < 1/λA and β2 /θ2 < 1/λB . We can see that information 1 die out exponentially. In order to rule out such a situation caused by
295
insufficient cooperation, we increase the strength of cooperation. However, (b)
15
presents the same result with (a) when δ1 = 30. The above results show that as long as β1 /θ1 < 1/λA and β2 /θ2 < 1/λB no matter how strong the cooperation is, information will die out exponentially in this model. These results further N ∑ illustrate the correctness of our theoretical analysis. P I1 (t) = (PiI1 (t)) is i=1
used to represent the total number of nodes that propagate information 1 at N N ∑ ∑ time t. Similarly, P I2 (t) = (PiI2 (t)) , P I1 I2 (t) = (PiI1 I2 (t)). i=1
i=1
600
600
δ1<1,δ2<1
400
δ >1,δ >1
200
δ1=1,δ2=1
1
40
(a) t
60
80
δ1>1,δ2>1
400
2
δ1=1,δ2=1
200
δ1>1,δ2<1 20
δ1<1,δ2<1
100
δ >1,δ <1 1
0 0
20
40
2
60
80
100
the number of I1I2,
the number of I2,
the number of I1,
800
800
700 600
1000
1000
0 0
PI1I2
1200
1200
PI2
PI1
5.2.2. Multiple information analysis
500 400 δ1<1,δ2<1
300 200
δ >1,δ >1
100
δ1=1,δ2=1
0 0
(b) t
1
2
δ1>1,δ2<1 20
40
(c) t
60
80
100
Figure 6: The impact of influence factor on spreading process in generated network, with β1 = β2 = 0.14, θ1 = θ2 = 0.24. (a) represents the change curve of P I1 over time; (b) represents the change curve of P I2 over time; (c) represents
PI2
40
the number of I2,
50 40 δ1<1,δ2<1
30
δ >1,δ >1 1
2
δ =1,δ =1
20
1
2
δ >1,δ <1 1
10 0
50
10
20
30
40
2
50
30 δ <1,δ <1
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2
δ1=1,δ2=1
10
δ >1,δ <1 1
0 0
(a) t
10
20
(b) t
30
40
2
50
the number of I1I2,
PI1
60
PI1I2
the change curve of P I1 I2 over time.
the number of I1,
300
40
30
20
δ <1,δ <1 1
2
δ1>1,δ2>1
10
δ =1,δ =1 1
2
δ1>1,δ2<1 0 0
10
20
(c) t
30
40
50
Figure 7: The impact of influence factor on spreading process in real network, with β1 = β2 = 0.14, θ1 = θ2 = 0.24. (a) represents the change curve of P I1 over time; (b) represents the change curve of P I2 over time; (c) represents the change curve of P I1 I2 over time. In this section, we will explore the impact of influence factor on informa16
tion evolution process. First of all, we need to introduce several concepts. If 305
information 1 promotes the propagation of information 2 while information 2 suppresses the propagation of information 1 (or information 1 suppresses the propagation of information 2 while information 2 promotes the propagation of information 1), the relationship between information 1 and information 2 is defined as asymmetrical influence; if information 1 and information 2 promote
310
each other, the relationship between them is defined as bidirectional promotion; if information 1 and information 2 suppress each other, the relationship between them is defined as bidirectional suppression. Fig. 6 and Fig. 7 illustrate the simulation results in generated network and real network respectively. In Fig. 6 and Fig. 7, (a) plots the number of nodes
315
infected by information 1 over time, (b) plots the number of nodes infected by information 2 over time, (c) plots the number of nodes infected by both information 1 and information 2 over time. From (a) and (b) we can see that cooperation can promote the propagation of information, that to say, more nodes transmit information 1 or information 2; competition can suppress the
320
propagation of information, that to say, less nodes transmit information 1 or information 2. (c) shows that cooperative relationship enables more users to accept the two information simultaneously, while competitive relationship leads to a significant reduction in the number of users who accept both information. Here, we should pay attention to the case that relationship between information
325
is asymmetrical influence(i.e., δ1 > 1, δ2 < 1). From (a) we can see that, for information 1, the promotion strength of asymmetrical influence is not as stronger as bidirectional promotion (i.e., δ1 > 1, δ2 > 1); from (b) we can see that, for information 2, the suppression strength of asymmetrical influence is more stronger than bidirectional suppression (i.e., δ1 < 1, δ2 < 1). Fig. 8 further
330
explains the number of infected nodes at stable state as a function of δ1 and δ2 . P∗I1 represents the number of infected nodes by information 1 at stable state and indicates the prevalence of information 1. We define P∗I2 and P∗I1 I2 , similarly.
17
Figure 8: The number of infected nodes at stable state as a function of δ1 and δ2 with β1 = β2 = 0.14, θ1 = θ2 = 0.24. (a) plots the number of I1 at stable state; (b) plots the number of I2 at stable state; (c) plots the number of I1 I2 at stable state. 5.2.3. Multiplex network analysis In order to capture the impact of the layers interrelation on the informa335
tion dissemination process, the correlation coefficient is used to measure the interrelation between network layers.
ρ(G) = √ ∑
∑
(dA,i − dA )(dB,i − dB ) √ ∑ 2 2 (dA,i − dA ) (dB,i − dB )
(9)
In Eq. 9, dA,i represents the degree of node i in spaceA and dA represents the average degree in spaceA. Similarly, dB,i represents the degree of node i in spaceB and dB represents the average degree in spaceB. 340
The effect of interrelation on the preferred information 1 is shown in Fig. 9. (a) shows that in the case of cooperation, the positive correlation (ρ(G) > 0) promotes the propagation of information 1 while the negative correlation(ρ(G) < 0) suppresses the propagation of information 1. However, it is just the reverse in the case of competition .In other words, the positive correlation suppresses the
345
propagation while the negative correlation promotes the propagation when they are competitive. Fig. 10 shows that for information 2, positive correlation promotes the propagation while negative correlation suppresses the propagation in both cooperation and competition cases. What’s more, in the case of competition, information 2 is more sensitive to the interlayer relationship, that is, the 18
350
positive and negative correlation has a great difference in the number of nodes infected by information 2. Combining Fig. 9 and Fig. 10, we can see that interrelations between layers have different effects on the preferred information and the secondary information in the case of competition. Cooperative
Competitive
PI1
1000
1000
800
the number of I1,
the number of I1,
PI1
1200
800 600 400
ρ(G)>0 ρ(G)<0 ρ(G)=0
200 0 0
50
100
150
200
600
400
0 0
250
(a) t
ρ(G)>0 ρ(G)<0 ρ(G)=0
200
50
100
(b) t
150
200
250
Figure 9: The influence of the interrelation between network layers on the spreading process of information 1 in the case of cooperation or competition. Specifically (a) represents in the case of cooperation and shows that positive correlation promotes the propagation while negative correlation suppresses the propagation, δ1 = 1.5; (b) represents in the case of competition and shows that positive correlation suppresses the propagation while negative correlation promotes the propagation, δ1 = 0.5.
5.2.4. MM-comprehensive analysis 355
Finally, we comprehensively study the effects of influence factor and layers interrelation on epidemic threshold. The simulation results of information 1 and information 2 are shown in Fig. 11 and Fig. 12 respectively. We use A to mark the position of β1 /θ1 = 1/λA in Fig. 11 and use B to mark the position of β2 /θ2 = 1/λB in Fig. 12. For clarity, we use the black arrows to point the
360
position of A or B, and the red arrows to point the position of the epidemic threshold. In Fig. 11 and Fig. 12, (a), (b) and (c) plots the stable-state infection fraction curves as a function of β when space A and space B are positively
19
Competitive
Cooperative 800
I
I
P2
P2
1200 1000
700
the number of I2,
the number of I2,
600
800 600 400 ρ(G)>0 ρ(G)<0
200
ρ(G)=0 0 0
50
100
(a) t
150
200
500 400 300 200
0 0
250
ρ(G)>0 ρ(G)<0 ρ(G)=0
100 50
100
150
200
250
(b) t
Figure 10: The influence of the interrelation between network layers on the spreading process of information 2 in the case of cooperation or competition. specifically (a) represents in the case of cooperation and shows that that positive correlation promotes the propagation while negative correlation suppresses the propagation δ2 = 1.5; (b) represents in the case of competition and shows that that positive correlation promotes the propagation while negative correlation suppresses the propagation δ2 = 0.5. interrelated, neutrally interrelated and negatively interrelated respectively. We analyze the preferred information 1 at first, Fig. 11 shows that as the correlation 365
coefficient ρ(G) decreases the epidemic threshold gradually increases(the red arrow points a larger number),but its not very obvious. It is worth noting that the position where the coves of δ1 > 1, δ1 = 1 and δ1 < 1 begin to become larger than zero is coincident whether in Fig. 11(a), Fig. 11(b) or Fig. 11(c), which indicates the epidemic threshold is equal when δ1 > 1 , δ1 = 1 or δ1 < 1. In
370
general, for the preferred information, influence factor only affects the fraction of infected nodes at stable state but has no significant effects on its epidemic threshold, regardless of the layers interrelation. From Fig. 12, we can see that as the correlation coefficient ρ(G) decreases the epidemic threshold of information 2 also gradually increases, but the result
375
is more obvious than that of information 1. Different from information 1, the influence factor not only affects the fraction of infected nodes at stable state but
20
1200
600 400 200 0 0
A0.1
0.2
0.3
0.4
(a) β1
0.5
0.6
800 600 400 200
A0.1
δ1>1 1000
δ1<1
0 0
0.7
PI*1
I
δ1<1
δ1=1
the number of I1,
800
1000
δ1=1
the number of I1,
the number of I1,
1000
δ1>1
1200 δ1>1
P*1
*
PI1
1200
0.2
0.3
0.4
(b) β1
0.5
0.6
800
δ1=1
600 400 200 0 0
0.7
δ1=1
A0.1
0.2
0.3
0.4
(c) β1
0.5
0.6
0.7
Figure 11: The number of nodes infected by information 1 at stable state as a function of β1 for different values of δ1 .(a) layers are positively interrelated, ρ(G) > 0;(b) layers are neutrally interrelated,ρ(G) = 0;(c) layers are negatively
1200
I
δ2<1
600 400 200 0 0
B0.1
0.2
2
1000
δ =1
0.3
(a) β2
0.4
0.5
0.6
0.7
800
δ2<1
600 400 200 0 0
B0.1
0.2
δ2>1 δ =1 2
2
the numbet of I2,
the number of I2,
2
800
1200
δ >1 1000
δ =1
PI*2
1200
δ2>1
the number of I2,
1000
P*2
*
PI2
interrelated,ρ(G) < 0.A represents the position of β1 /θ1 = 1/λA .
0.3
(b) β2
0.4
0.5
0.6
0.7
800
δ2<1
600 400 200 0 0
B0.1
0.2
0.3
0.4
(c) β2
0.5
0.6
0.7
Figure 12: The number of nodes infected by information 2 at stable state as a function of β2 for different values of δ2 . (a) layers are positively interrelated, ρ(G) > 0; (b) layers are neutrally interrelated,ρ(G) = 0;(c) layers are negatively interrelated,ρ(G) < 0. B represents the position of β2 /θ2 = 1/λB . also has significant effects on the epidemic threshold of information 2. In conclusion, for the secondary information the cooperative relationship increases the number of infected nodes and reduces the epidemic threshold while competitive 380
relationship reduces the number of infected nodes and increases the epidemic threshold.
21
6. Conclusion In summary, this paper proposes an information propagation dynamic model based on multiple information and multiplex network (MM-SIS model) and sets 385
a group of discrete time Microscopic Markov equations to describe the information spreading process. Firstly, this paper theoretically analyzes the epidemic threshold in the MM-SIS model, which correctness has been verified through experiments. Through a series of experiments, we find that: (1) The asymmetric influence between information (i.e., δ1 > 1, δ2 < 1 or δ1 < 1, δ2 > 1)
390
causes stronger suppression effect than bidirectional suppression (i.e., δ1 < 1, δ2 < 1) and weaker promotion effect than bidirectional promotion (i.e., δ1 > 1, δ2 > 1). (2) The epidemic threshold of information increases as the correlation coefficient between the network layers decreases. (3) For the preferred information, positive correlation promotes the dissemination while negative correlation
395
suppresses the dissemination in the case of cooperation, whereas, it is opposite in the case of competition. For the secondary considered information, positive correlation promotes the dissemination and negative correlation suppresses the dissemination in both cooperation and competition cases. (4) For the preferred information, influence factor can affect the final outbreak scale of information
400
but has no obvious effects on the epidemic threshold. For the secondary information, influence factor affects both final outbreak scale of the information and the epidemic threshold.
Acknowledgment This paper is partially supported by the National Natural Science Foun405
dation of China (Grant No. 61772098); the China Scholarship Council (Grant No.201707845009); Chongqing Science and Technology Commission Project (Grant No. cstc2017jcyjAX0099) and Chongqing key research and development project (No.cstc2017zdcy-zdyf0299, cstc2017zdcy-zdyf0436) and Chongqing Graduate Education Teaching Reform Project(No.yjg183081); And Ling Liu’s research
22
410
is partially support by the National Science Foundation under Grants NSF 1547102, SaTC 1564097 and an IBM faculty award.
References [1] X. Xiong, Y. Y. Li, S. J. Qiao, An emotional contagion model for heterogeneous social media with multiple behaviors, Physi415
ca A: Statistical Mechanics and its Applications 490 (2018) 185–202. doi:10.1016/j.physa.2017.08.025. [2] J. Goldenberg, B. Libai, E. Muller, Talk of the network: a complex systems look at the underlying process of word-of-mouth, Marketing Letters 12 (2001) 211–223. doi:10.1023/A:1011122126881.
420
[3] H. H. Wang, A. A. Raza, Y. B. Lin, R. Rosenfeld, Behavior analysis of low-literate users of a viral speech-based telephone service, in: Proceedings of the 4th Annual Symposium on Computing for Development, ACM, 2013, pp. 1–9. doi:10.1145/2537052.2537062. [4] N. Masuda, Opinion control in complex networks, New Journal of Physics
425
17 (2015) 33031–33041. doi:10.1088/1367-2630/17/3/033031. [5] L. Weng, A. Flammini, A. Vespignani, F. Menczer, Competition among memes in a world with limited attention, Scientific Reports 2 (2012) 335. doi:10.1038/srep00335. [6] H. F. Zhang, M. Small, X. C. Fu, B. H. Wang, Modeling the influence
430
of information on the coevolution of contact networks and the dynamics of infectious diseases, Physica D: Nonlinear Phenomena 241 (2012) 1512– 1517. doi:10.1016/j.physd.2012.05.011. [7] L. X. Yang, X. F. Yang, J. M. Liu, C. Q. Gan, Epidemics of computer viruses: a complex-network approach, Applied Mathematics and Computation
435
219 (2013) 8705–8717. doi:10.1016/j.amc.2013.02.031.
23
[8] M. Llas, P. M. Gleiser, J. M. L´opez, Nonequilibrium phase transition in a model for the propagation of innovations among economic agents, Phys. Rev. E 68 (2003) 066101. doi:10.1103/PhysRevE.68.066101. [9] M. 440
E.
J.
Newman,
M.
Girvan,
munity structure in networks,
Finding
and
evaluating
com-
Phys. Rev. E 69 (2004) 026113.
doi:10.1103/PhysRevE.69.026113. [10] S. Pei, L. Muchnik, J. S. Andrade, Jr, Z. Zheng, H. A. Makes, Searching for superspreaders of information in real-world social media, Scientific Reports 4 (2014) 5547. doi:10.1038/srep05547. 445
[11] D. Chen, L. L¨ u, M. S. Shang, Y. C. Zhang, T. Zhou, Identifying influential nodes in complex networks, Physica A: Statistical Mechanics and its Applications 391 (2012) 1777–1787. doi:10.1016/j.physa.2011.09.017. [12] H. Hu, X. Wang, Evolution of a large online social network, Physics Letters A 373 (2009) 1105–1110. doi:10.1016/j.physleta.2009.02.004.
450
[13] Z. Qian, S. Tang, X. Zhang, Z. Zheng, The independent spreaders involved sir rumor model in complex networks, Physica A: Statistical Mechanics and its Applications 429 (2015) 95–102. doi:10.1016/j.physa.2015.02.022. [14] R. Pastor-Satorras, C. Castellano, P. V. Mieghem, A. Vespignani, Epidemic processes in complex networks, Review of Modern Physics 87 (2015) 120–
455
131. doi:10.1103/RevModPhys.87.925. [15] Y. Liu, B. Wang, B. Wu, S. Shang, Y. Zhang, C. Shi, Characterizing superspreading in microblog: an epidemic-based information propagation model, Physica A: Statistical Mechanics and its Applications 463 (2016) 202–218. doi:10.1016/j.physa.2016.07.022.
460
[16] J. Leskovec, M. Mcglohon, C. Faloutsos, N. Glance, M. Hurst, Patterns of cascading behavior in large blog graphs, in: Proceedings of the 2007 SIAM International Conference on Data Mining, SIAM, 2007, pp. 551–556. doi:10.1137/1.9781611972771.60. 24
[17] C. Nowzari, V. M. Preciado, G. J. Pappas, Analysis and control of epi465
demics: a survey of spreading processes on complex networks, IEEE Control Systems 36 (2016) 26–46. doi:10.1109/MCS.2015.2495000. [18] S. A. Myers, J. Leskovec, Clash of the contagions: cooperation and competition in information diffusion, in: Proceedings of the 12th IEEE International Conference on Data Mining, IEEE, 2012, pp. 539–548.
470
doi:10.1109/ICDM.2012.159. [19] Y. Su, X. Zhang, L. Liu, S. Song, B. Fang, Understanding information interactions in diffusion: perspective,
Frontiers
of
Computer
an evolutionary game-theoretic Science
10
(2016)
518–531.
doi:10.1007/s11704-015-5008-y. 475
[20] X. Zhang, Y. Su, S. Qiu, S. Xie, B. Fang, IAD: interaction-aware diffusion framework in social networks, IEEE Transactions on Knowledge and Data Engineering (2018) 1–1.doi:10.1109/TKDE.2018.2857492. [21] B. A. Prakash, A. Beutel, R. Rosenfeld, C. Faloutsos, Winner takes all: competing viruses or ideas on fair-play networks, in: Proceedings of the
480
21st international conference on World Wide Web, ACM, 2012, pp. 1037– 1046. doi:10.1145/2187836.2187975. [22] A. Beutel, B. A. Prakash, R. Rosenfeld, C. Faloutsos, Interacting viruses in networks: can both survive?, in: Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining, ACM,
485
2012, pp. 426–434. doi:10.1145/2339530.2339601. [23] D.
Trpevski,
spreading
over
W.
K.
Tang,
networks,
L.
Phys.
Kocarev, Rev.
E
Model 81
for
(2010)
rumor 056102.
doi:10.1103/PhysRevE.81.056102. [24] Z. Wang, L. Wang, A. Szolnoki, M. Perc, Evolutionary games on multi490
layer networks: a colloquium, European Physical Journal B 88 (2015) 124. doi:10.1140/epjb/e2015-60270-7. 25
[25] G. Dong, L. Tian, D. Zhou, R. Du, J. Xiao, H. E. Stanley, Robustness of n interdependent networks with partial support-dependence relationship, EPL 102 (2013) 68004. doi:10.1209/0295-5075/102/68004. 495
[26] Y. Zhuang, A. Arenas, O. Yaˇgan, Clustering determines the dynamics of complex contagions in multiplex networks, Phys. Rev. E 95 (2017) 012312. doi:10.1103/PhysRevE.95.012312. [27] M. Berlingerio, M. Coscia, F. Giannotti, A. Monreale, D. Pedreschi, Multidimensional networks: foundations of structural analysis, World Wide Web
500
16 (2013) 567–593. doi:10.1007/s11280-012-0190-4. [28] W. Li, S. Tang, W. Fang, Q. Guo, X. Zhang, Z. Zheng, How multiple social networks affect user awareness:
the information diffu-
sion process in multiplex networks, Phys. Rev. E 92 (2015) 042810. doi:10.1103/PhysRevE.92.042810. 505
[29] E. Cozzo, R. A. Ba˜ nos, S. Meloni, Y. Moreno, Contact-based social contagion in multiplex networks, Phys. Rev. E 88 (2013) 050801. doi:10.1103/PhysRevE.88.050801. [30] S. Wang, L. Huang, C. H. Hsu, F. Yang, Collaboration reputation for trustworthy web service selection in social networks, Journal of Computer and
510
System Sciences 82 (2016) 130–143. doi:10.1016/j.jcss.2015.06.009. [31] C.
Buono,
L.
A.
Braunstein,
Immunization
demic spreading on multilayer networks,
strategy
for
epi-
EPL 109 (2015) 26001.
doi:10.1209/0295-5075/109/26001. [32] F. D. Sahneh, 515
C. Scoglio,
arbitrary multilayer networks,
Competitive epidemic spreading over Phys. Rev. E 89 (2014) 062817.
doi:10.1103/PhysRevE.89.062817. [33] Q. Wu, X. Fu, M. Small, X. J. Xu, The impact of awareness on epidemic spreading in networks, Chaos 22 (2012) 013101. doi:10.1063/1.3673573.
26
[34] N. J. Watkins, C. Nowzari, V. M. Preciado, G. J. Pappas, Optimal re520
source allocation for competitive spreading processes on bilayer networks, IEEE Transactions on Control of Network Systems 5 (2015) 298–307. doi:10.1109/TCNS.2016.2607838. [35] Y. Zhao, M. Zheng, Z. Liu, A unified framework of mutual influence between two pathogens in multiplex networks, Chaos 24 (2014) 043129.
525
doi:10.1063/1.4902254. [36] X. Wei, N. C. Valler, B. A. Prakash, I. Neamtiu, M. Faloutsos, C. Faloutsos, Competing memes propagation on networks: a network science perspective, IEEE Journal on Selected Areas in Communications 31 (2013) 1049–1060. doi:10.1109/JSAC.2013.130607.
530
[37] H. L. Mills, pled networks:
A. Ganesh,
C. Colijn,
Pathogen spread on cou-
effect of host and network properties on transmis-
sion thresholds, Journal of Theoretical Biology 320 (2013) 47–57. doi:10.1016/j.jtbi.2012.12.006. [38] S. G´omez, A. Arenas, J. Borge-Holthoefer, S. Meloni, Y. Moreno, Discrete535
time markov chain approach to contact-based disease spreading in complex networks, EPL 89 (2010) 38009. doi:10.1209/0295-5075/89/38009. [39] C. Granell, S. G´omez, A. Arenas, Dynamical interplay between awareness and epidemic spreading in multiplex networks, Phys. Rev. Lett. 111 (2013) 128701. doi:10.1103/PhysRevLett.111.128701.
540
[40] M. W. Hirsch, S. Smale, Differential Equations, Dynamic Systems, and Linear Algebra, Academic Press, New York, 1974.
27
PII: DOI: Reference:
S0378-4371(18)31118-X https://doi.org/10.1016/j.physa.2018.08.169 PHYSA 20055
To appear in:
Physica A
Received date : 27 May 2018 Revised date : 14 August 2018 Please cite this article as: Y. Xiao, et al., MM-SIS: Model for multiple information spreading in multiplex network, Physica A (2018), https://doi.org/10.1016/j.physa.2018.08.169 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
Highlights: (1) MM-SIS models the interactive diffusion of multiple information in multiplex network (2) The influence factor is introduced to describe interactions between information (3) Epidemic threshold is analyzed and the correctness is verified via experiments (4) The simulation results reveal complex interaction between information
*Manuscript Click here to view linked References
MM-SIS: Model for multiple information spreading in multiplex network Yunpeng Xiaoa,b,∗, Li Zhanga , Qian Lia , Ling Liub a Chongqing
Engineering Laboratory of Internet and Information Security, Chongqing University of Posts and Telecommunications, Chongqing 400065, China b Computer Science, Georgia Institute of Technology, Atlanta, Georgia, 30332, United States
Abstract In social networks, relationships between information are complexly intertwined. However, prior research mainly focuses on studying the independent information in an isolated network, ignoring two factors: the interaction between different information and the diversity of propagation paths. In view of the above problems, this paper proposes the MM-SIS(Multiple information and Multiplex network-SIS)model to explore the detailed processes and characteristics of multiple information in multiplex networks. At the same time, the concept of influence factor is introduced to describe the complex interaction between different information. Moreover, we use the Microscopic Markov Chain method to set dynamic equations, theoretically analyze the epidemic threshold and verify its correctness through experiments. In general, this paper mainly explores the effects of influence factor and the interrelation of network layers on information spreading process. The experimental results show that the above mentioned two factors have obvious impacts on the information outbreak scale and the epidemic threshold. Keywords: Information diffusion, Multiplex network, Multiple information, Markov chain method, SIS model
∗ Corresponding
author Email address: [email protected] (Yunpeng Xiao)
Preprint submitted to Journal of Physica A
August 14, 2018
1. Introduction With the development of Internet, online social networks are prevalent in people’s lives. Many new social media such as micro-blog, blogs and forums have profoundly influenced and changed the way of information dissemination. 5
Online social networks are centered on users, people rely on the friend relationships to interact with others and disseminate information. People use social networks to exchange a lot of information, including hot news, religious beliefs, fashion trends and even emotions [1, 2]. Using these data to analyze and predict users’ behavior and psychology [3] is of great importance to public opinion
10
monitoring [4] and service recommendation. There is a complex interaction between information when they are spreading in the network simultaneously. The widespread popularity of online social networks leads to the explosive growth of information, but these pieces of information are not independent in the network, and they will cooperate or compete
15
with each other in the spreading process [5]. The interaction between information will directly affect the dynamic progress of information, and thus should be taken into consideration. In our real life, many complex systems can be modeled as complex networks to analyze, such as the spread of diseases [6], the propagation of computer viruses [7], the dissemination of innovative products
20
[8] etc. Complex network analysis methods can be used in community detection [9], influence maximization [10], and important nodes identification [11].As one of the complex networks, it is also important to study the properties of online social networks [12], the most commonly used models are SIS and SIR epidemic models [13–15]. Most of these researches focus on single layer networks, but sin-
25
gle layer networks have limitations in modeling some scenarios. Therefore, some researchers have shifted from single, isolated networks to multilayer, overlapped networks, and found that information propagates over multiplex networks exhibit some characteristics that are different from those of independent networks. Although many significant research achievements have been made in the field
30
of information dissemination, there are still some challenges:
2
On the one hand, a comprehensive model is lacked to explore the propagation process of multiple information in multiplex network. At present, most of the previous studies mainly focus on the single information propagates in isolated networks [13, 16, 17], and following scholars focus on the interaction of 35
different information [18–23] or focus on the multiplex networks [24–31], only some studies take both into consideration [32–35]. On the other hand, the relationship between information is complex and diverse. Most studies are not fully considered when exploring the interaction between information. Many prior works have focused on competitive informa-
40
tion, especially on full competition, ignored that it is possible for a user to propagate multiple information simultaneously in real life [32, 36]. The example about diseases in Ref. [37] provides new grounds to the understanding of information spreading in social networks. Along the idea of Mills et al, we realize that the factors driving people to establish links in social network
45
are also complex, that is to say, the nature of the relationship represented by links is different. For instance, friend relationships are established based on the same interests and hobbies, colleague relationships are established based on business contact. Therefore, the potential propagation routes of different information will be different and interact with each other. In this paper, we build
50
a multiplex network based on the specific problems mentioned above and refer to the SIS mechanism to propose an information propagation dynamic model, that is MM-SIS (Multiple information and Multiplex network-SIS) model. Our contribution can be summarized as follows: (1) The MM-SIS model is proposed based on multiple information and multiplex
55
network. This paper focuses on the interaction between information and the diversity of propagation paths when studying the information spreading process. At the same time, we use the SIS epidemic model for reference to explore the propagation mechanism of multiple information in multiplex network.
60
(2) The concept of influence factor is introduced to describe the relationship
3
between information and the cooperative and competitive relationships are integrated into a unified model. Based on this, the concepts of asymmetric influence, bidirectional suppression and bidirectional promotion are defined to represent the complex and diverse interaction between information. 65
(3) We establish dynamic equations of our model by using the Microscopic Markov Chain method. Then, we theoretically analyze the epidemic threshold and verify its correctness through experiments. The rest of this paper is organized as follows. Section 2 introduces the related work. Section 3 formulates the problems and gives the necessary definitions. In
70
Section 4, we describe the proposed model in detail. Section 5 presents and analyzes the experimental results of the model. Finally, Section 6 concludes the paper.
2. Related works In this section, we will review related work from the following aspects. 75
Single network topology for multiple information: [15, 18–20] use real data to prove that there is a real interaction between information. Myers et al. [18] use real data from Twitter to suggest that interactions cause a relative change in the spreading probability of a contagion by 71% on the average. Su et al.[19] compare information to organisms and employ game theory to study whether
80
a contagion can be promoted or suppressed by others in the diffusion process. Zhang et al.[20] propose IDA model to study the contagion adoption behavior under a set of interactions. In the case of full competition (a node can transmit at most one information at any given time), [21] shows that winner takes all, that is to say, the stronger virus completely wipes-out the weaker one. However, [22]
85
uses mean field theory to show that if the cross-immunity satisfies a threshold condition, viruses can coexist in the network. Trpevski et al.[23] extend the SIS model to study competitive rumors spreading over networks. Multiple network topology for single information: A system composed of interconnected sub-networks is often called a network of networks (NON). NON has 4
90
a variety of titles in different papers[24] including interdependent networks[25], multiplex networks[26] and multidimensional networks[27], etc. Zhuang et al.[26] adopt the content-dependent linear threshold model to study the effect of clustering coefficient in multiplex networks. Li et al.[28] use bond percolation and cascading failure to show that interaction between layers can greatly enhance
95
the information diffusion process. Cozzo et al.[29] propose a contact-based information spreading model to investigate the critical point of the multiplex system. Wang et al.[30] establish a collaboration network to perform the trustworthy service selection in web service social network environment where the concept of the collaboration reputation is first proposed. What’s more, targeted
100
immunization strategy for epidemic spreading over a multilayer network is less efficient than in isolated networks [31]. Multiple network topology for multiple information: The above two scenarios inspired scholars to study the propagation of multiple information spreading over multiplex networks, which is the most similar scenario to this paper. Sahneh
105
et al.[32] address the problem of two competitive viruses propagating in a host population where each virus has a distinct contact network for propagation. Wu et al.[33] use the mean-field approach to investigate the impact of awareness on epidemic spreading process. Watkins et al.[34] study the problem about optimal resource allocation for competitive spreading processes on bilayer network. Zhao
110
et al.[35] present the concept of state-dependent infectious rate and propose a unified framework to study the cases of two pathogens in two-layered networks. What’s more, the spread of epidemics and information can be regarded as a Markov process. S. G´omez et al.[38] study the disease in complex network based on discrete-time Markov chain approach. Granell et al.[39] present a model in
115
which awareness and epidemic spread on separate layers and employ Microscopic Markov Chain Approach, they find that awareness can suppress the propagation of epidemic. Wei et al.[36] develop the SI1 I2 S model to study competing memes in a composite network and show that the first eigenvalue of system matrix is a critical metric to identify epidemic behavior. The abovementioned researches
120
are based on Markov Chain Approach, Microscopic Markov Chain Approach 5
help us quantify the microscopic dynamics at the individual level.
3. Problem definition 3.1. Related definitions In general, the multiplex network can be represented as G(V, E1 , E2 , ..., El ), 125
where V = {V1 , V2 , . . . . . . , VN } represents the set of nodes, the cardinality |V | =N is used to denote the total number of whole network users and Ei represents the set of links in spcacei .If there is a connection between node i and node j, then we stipulate that the information can spread from node i to node j. Inf = {Inf1 , Inf2 , ..., InfM } is used to represent the set of multiple
130
information. Since this paper uses the SIS communication mechanism for reference, the user is in Ii state when the user spreads information Infi , otherwise it is in Si state. Table. 1 shows the symbols we have used in the paper. The basic concepts and symbol representations in this paper are as follows: Influence factor: symbol δ is used to represent impact factor. Specifically, δi
135
represents the influence factor of Infi . We specify that information Infi is suppressed by others when 0 ≤ δi < 1 , and the smaller δi means the greater the suppression strength, especially when δi = 0, it means information Infi and others are completely mutual exclusive; information Infi is promoted by others when δi > 1, and the larger δi means the greater the promotion strength;
140
information Infi is independent with others, when δi = 1. Infection rate: symbol β is used to represent infection rate. Specifically, βi represents the infection rate of Infi . The infection rate indicates the probability that if a node does not spread the information and its neighbor transmits the information, then he is influenced by a neighbor user to spread the message.
145
Cure rate: symbol θ is used to represent cure rate. Specifically, θi represents the cure rate of Infi . When the node is in I state, it will return to S state spontaneously with θ probability. In this paper, if node i in Ii state loses interest in Infi and forgets it, the node will changes from Ii state to Si state. This parameter describes the duration of information. The larger θ means the
6
150
shorter duration of information. If Infi is not attractive enough, it is quickly forgotten and the value of θi is large. Table 1. Symbols and definitions Symbols
Definitions
Inf
information
δi
influence factor of Infi
βi
infection rate of Infi
θi
cure rate of Infi
A(B)
the adjacency matrix of spaceA(spaceB)
λA (λB )
the largest eigenvalue of matrix A(matrix B)
ρ(G)
correlation coefficient of multiplex work G
3.2. Problem formulation To formally formulate the problem of our research, let G(V, E1 , E2 , ..., El ) be the whole multiplex network, and let T = {(β1 , θ1 , δ1 ), (β2 , θ2 , δ2 ), ..., (βM ,θM ,δM )} 155
represents the basic attributes of all information. We use the above attributes to study the spread of information, that is, to calculate the state of each node We use PiX (t) to represent the probability of the node i in
at a certain time.
the state X at time t. Specifically, the problem is defined as follows: G(V, E1 , E2 , ..., El ) ⇒ f : (G, T ) → PiX (t) T = {(β , θ , δ ), (β , θ , δ ), ..., (β ,θ ,δ )} 1
160
1
1
2
2
2
M
M
M
3.2.1. Problem input
Given the related definitions, the input to this problem can be defined as follows: 1. Whole network topology G(V, E1 , E2 , ..., El ). 2. The basic characteristics of the information to be studied and the inter165
action between the information T = {(β1 , θ1 , δ1 ), (β2 , θ2 , δ2 ), ..., (βM ,θM ,δM )}
7
3.2.2. Problem output Based on the above description, the problems to be solved are as follows: How to explore the influence of interaction between information and interrelation between network layers on information dissemination? Firstly, we initial170
ize the whole multiplex network G(V, E1 , E2 , ..., El ), that is, defining the number of nodes and the network topology of each layer. Then we initialize the basic characteristics of information T = {(β1 , θ1 , δ1 ), (β2 , θ2 , δ2 ), ..., (βM ,θM ,δM )}, and let different information propagates on different layers. We set the interrelation into positive interrelation, negative interrelation and neutral interrelation,
175
and the interactions are classfied into promotion, suppression and unrelated. Under different conditions, PiX (t) can be got through experiments, thus the overall information propagation situation is obtained. Further analysis can yield the impacts of interactions and interrelations on information dissemination.
4. Proposed model 180
In this section, the MM-SIS model is introduced in detail, which aims to reveal the co-evolution rule of multiple information on top of multiplex networks. The details of our model are shown in Fig. 1.
Figure 1: Model framework
8
4.1. Multiple information and multiplex network In this paper, we consider the diversity of propagation paths to build a 185
multiplex network. The nature of the relationship represented by the links is different, so different information will spread along different network topologies. Fig. 2 shows the above process more vividly. To be brief, this paper divides the links in the network into two types, which are represented by spaceA and spaceB, respectively. The nodes in spaceA and
190
spaceB correspond randomly and each pair of nodes represents the same user. Information1 spreads on spaceA while information 2 spreads on spaceB. Note that our model can also be easily extended to more layers. We assume that the number of nodes in the network is N ,V = {V1 , V2 , . . . , VN } is the set of nodes, EA is the set of links in spaceA, EB is the set of links in spaceB. Therefore, we
195
use a triple G(V, EA , EB ) to represent the whole multiplex network. Meanwhile, we define the adjacency matrix A = [aij ]N ×N , where aij = 1 if there is an edge between node i and node j in spaceA, otherwise aij = 0. Similarly, we define the adjacency matrix B of SpaceB, B = [bij ]N ×N .
Figure 2: Schematic figure of multiplex network. Information 1 spreads on space A while information 2 spreads on space B, the nodes contacted by dotted line represent the identical user. δ1 and δ2 are influence factors. Inf is the abbreviation of information
4.2. Modeling 200
In this paper, the information spreading process refers to the SIS mechanism. If there are M pieces of information in the network and they are not completely
9
mutually exclusive (a node can be infected by multiple information at the same time), there will be 2M kinds of node states in the network. In order to facilitate the expression of the model and simplify the theoretical expression, this paper 205
uses two pieces of information(i.e. Inf = {Inf1 , Inf2 })as an example to study, and the research principle of more than two information is similar to this one. So each node can be in one of four different states X, X ∈ {I1 S2 , S1 I2 , I1 I2 , S1 S2 }. I1 S2 represents a user only transmits information 1, S1 I2 represents a user only transmits information 2, I1 I2 represents a user transmits information 1
210
and information 2 simultaneously and S1 S2 represents a user neither transmits information 1 nor information 2. We regulate that when a node is in S1 S2 state, it may be infected by information 1 at a rate of β1 , then assume that the node’s neighbors are independent of each other, it will become infected with probability: fi1 = 1 −
∏
j∈i′ neighbors in space A
(1 − β1 PjI1 (t))
(1)
in which PjI1 (t) = PjI1 S2 (t) + PjI1 I2 (t), where PjX (t) represents the probability that node j is in X state at time t. However, if a node has already propagated information 2 and failed to recover from information 2 (i.e. in S1 I2 state), its probability of being infected by information 1 becomes δ1 β1 , and its probability of being infected will be: u1i = 1 −
∏
j∈i′ neighbors in space A
(1 − δ1 β1 PjI1 (t))
(2)
(1 − β2 PjI2 (t))
(3)
(1 − δ2 β2 PjI2 (t))
(4)
With the same principle, we can obtain: fi2 = 1 −
u2i = 1 −
∏
∏
j∈i′ neighbors in space B
j∈i′ neighbors in space B
And we assume that θ1 (θ2 ) is the cure rate of information 1 (information 2). The transition between four states of a specific node is indicated in Fig. 3. In this paper, we assume that information 1 is the preferred information, 215
that is to say, users give priority to information 1 when faced with two pieces 10
Figure 3: The transition between four states of a specific node, S represents the susceptible state,I represents the infected state. of information [23]. The following discrete time microscopic Markov equations is presented to describe the co-evolution process of multiple information. PiI1 S2 (t + 1) = PiS1 S2 (t)fi1 (1 − u2i ) + PiS1 I2 (t)u1i θ2 + PiI1 S2 (t)(1 − θ1 )(1 − u2i ) + PiI1 I2 (t)(1 − θ1 )θ2 P S1 I2 (t + 1) = P S1 S2 (t)(1 − f 1 )f 2 + P S1 I2 (t)(1− i i i i i I1 I2 I1 S2 2 1 (t)θ1 fi + Pi (t)θ1 (1 − θ2 ) ui )(1 − θ2 ) + Pi PiI1 I2 (t + 1) = PiS1 S2 (t)fi1 u2i + PiS1 I2 (t)u1i (1 − θ2 )+ P I1 S2 (t)(1 − θ )u2 + P I1 I2 (t)(1 − θ )(1 − θ ) i
1
i
i
1
(5)
2
Note that in this paper we focus on the evolution process of the nodes in I state, and it is easy to get: PiS1 S2 (t + 1) = 1 − PiI1 S2 (t + 1) − PiS1 I2 (t + 1) − PiI1 I2 (t + 1)
(6)
so do not need an extra equation to represent PiS1 S2 (t + 1). 4.3. Theoretical analysis 220
Epidemic threshold is a critical point of the information outbreak. Put another way, we will study the conditions under which the information will not spread in the MM-SIS model. Lemma 1 (Hirsch and Smale,1974 [40]) : The system is asymptotically stable → − − → − → at P = 0 if the eigenvalues of ∇g( 0 ) are less than 1 in absolute value, where 11
225
230
[ − →] m → − → − Jacobian matrix J = ∇g( 0 ) = ∂g ∂Pn | P = 0 . m,n − → − → For convenience, we use P (t + 1) = g( P (t)) to represent eqs(1). Vector −−−→ −−−→ −−−→ −−−→ − → P (t) = (P I1 S2 (t), P S1 I2 (t), P I1 I2 (t)), where P I1 S2 (t) = (P1I1 S2 (t), P2I1 S2 (t), ..., PNI1 S2 (t)). −−S−→ −−−→ P 1 I2 (t) and P I1 I2 (t) are represented similarly. To study the epidemic threshold, we analyze the Jacobian matrix of system −−−→ −−−→ −−−→ − → − → − → − → − → P (t + 1) = g( P (t)). When (P I1 S2 (t), P S1 I2 (t), P I1 I2 (t)) = ( 0 , 0 , 0 ), the Jacobian matrix is shown as follows:
S1
− → − → − = Z J(→ 0,0,0) Z
Z S2 Z
S4
S5 S3
Where Z is a N × N zero matrix, matrixes S1 -S5 are as follows: S1 = (1 − θ1 )E + β1 A S2 = (1 − θ2 )E + β2 B
235
S3 = (1 − θ1 )(1 − θ2 )E S4 = (1 − θ1 )θ2 E + β1 A S5 = θ1 (1 − θ2 )E + β2 B Where E is an identity matrix. This Jacobian matrix is an upper triangular matrix, so its eigenvalues are the same as those of S1 , S2 and S3 . According to −−−→ −−−→ −−−→ − → − → − → Lemma 1, if the system is stable at (P I1 S2 (t), P S1 I2 (t), P I1 I2 (t)) = ( 0 , 0 , 0 ), − → − → − is less than 1. That is, the the absolute value of the eigenvalue of J(→ 0,0,0)
following conditions need to be met: max{|(1 − θ1 )(1 − θ2 )|, |(1 − θ1 ) + β1 λA |, |(1 − θ2 ) + β2 λB |} < 1
(7)
Where λA and λB are the largest eigenvalues of adjacency matrix A and 240
adjacency matrix B respectively. The system is asymptotically stable at point −−−→ −−−→ −−−→ − → − → − → (P I1 S2 (t), P S1 I2 (t), P I1 I2 (t)) = ( 0 , 0 , 0 ), it means initial infected nodes will die out in the network exponentially and none information can spread in the network. Since 0 < θ1 (θ2 ) ≤ 1, 0 ≤ (1−θ1 )(1−θ2 ) < 1 is obviously satisfied. Matrix A
245
and matrix B are nonnegative, irreducible, symmetric, square matrixes(because
12
spaceA and spaceB are undirected networks), so the largest eigenvalues (λA , λB ) are positive real number according to Perron-Frobenius Theorem. 0 < θ1 (θ2 ) ≤ 1 and 0 < β1 (β2 ) ≤ 1,thus, (1 − θ1 ) + β1 λA > 0 and (1 − θ2 ) + β2 λB > 0 are satisfied and condition(7) can be rewritten as follows:
250
(1 − θ ) + β λ < 1 1 1 A (1 − θ ) + β λ < 1 2 2 B
namely
β1 /θ1 < 1/λA
namely
β2 /θ2 < 1/λB
(8)
So we include that when β1 /θ1 < 1/λA and β2 /θ2 < 1/λB , there is no information spreading in the network. If the above-mentioned two information expected to survive, they should at least satisfy β1 /θ1 ≥ 1/λA or β2 /θ2 ≥ 1/λB . In the following sections we will verify the correctness of this theoretical analysis and explore the impact of influence factor and layer interrelation on the process
255
of information dissemination.
5. Results and analysis 5.1. Experimental datasets In this paper, we simulate the proposed model on the real network and the algorithm generated network. The specific descriptions are as follows. 260
Real world network: The real dataset used in this paper is to describe the relationship between employees in a company, including two types of relationship1 . According to the requirements of our experiments, we select the colleague relationships between nodes to establish spaceA and select the friend relationships between nodes to establish spaceB. The statistical characteristics of the
265
dataset are shown in Table. 2. Generated network: Both spaceA and spaceB are scale-free networks. we use the Barab´asi-Albert model with N = 2000 nodes, m = 3, in which, m represents the number of links when a new node is added. The degree of node in spaceA is 2-96, in spaceB is 2-104. 1 http://deim.urv.cat/
manlio.dedomenico/data.php
13
Table 2. Statistics of dataset Dataset
270
Space A
Space B
Users
71
71
Edges
892
575
Degree
0-30
0-25
5.2. Experiments and analysis In this section, we simulate the proposed model and further analyze the experimental results. Firstly, we verify the correctness of the theoretical analysis. Then exploring the influence of interaction between multiple information and the influence of interrelation between network layers respectively. Finally, we comprehensively explore how multiple information and multiplex network affect the epidemic threshold.
the number of infected nodes
5.2.1. Theoretical analysis verification the number of infected nodes
275
250 PI1I2 200
I1
P
PI2 150
100
50
0 0
20
40
(a) t
60
80
100
250 PI1I2 I
200
P
1
PI2
150
100
50
0 0
20
40
60
80
100
(b) t
(a)
(b)
Figure 4: Spreading process of information over time, (a) and (b) show that information 1 and information 2 both can spread in the network.(a) β1 /θ1 ≥ 1/λA , β2 /θ2 < 1/λB ;(b) β1 /θ1 < 1/λA , β2 /θ2 ≥ 1/λB .
14
I
P
1
1.5
1
0.5
0 0
20
40
60
80
100
the number of infected nodes
the number of infected nodes
2
2 I
P
1
1.5
1
0.5
0 0
(a) t
20
40
60
80
100
(b) t
Figure 5: The infected nodes of information 1 over time with β1 /θ1 < 1/λA , β2 /θ2 < 1/λB . (a) and (b) shows that information 1 die out exponentially in the multiplex network. (a) δ1 = 10; (b) δ1 = 30. From the above analysis we conclude that if the two pieces of information studied in this paper can spread in the network, the condition β1 /θ1 ≥ 1/λA 280
and condition β2 /θ2 ≥ 1/λB should be reached at least one. In Fig. 4, (a) shows the spreading process of information 1 and information 2 when β1 /θ1 ≥ 1/λA , β2 /θ2 < 1/λB and (b)shows the spreading process of information 1 and information 2 when β1 /θ1 < 1/λA , β2 /θ2 ≥ 1/λB . In figure (a) and figure (b) we can see that P I1 > 0 and P I2 > 0 at stable state (small fluctuations of
285
its infected nodes), that is to say, both of the information can spread in the network. These results preliminarily prove the correctness of our conclusion. Alex Beutel et al. explored the propagation of two viruses on an isolated network, they found that even both viruses are independently too weak to survive on their own when β1 /θ1 < 1/λA , β2 /θ2 < 1/λB , with enough cooperation they
290
can [22]. But in this multiplex network we find different results. We study information 1 as an example (information 2 is same with information 1). In Fig. 5, (a) depicts the evolution process of the number of infected nodes of information 1 with δ1 = 10, β1 /θ1 < 1/λA and β2 /θ2 < 1/λB . We can see that information 1 die out exponentially. In order to rule out such a situation caused by
295
insufficient cooperation, we increase the strength of cooperation. However, (b)
15
presents the same result with (a) when δ1 = 30. The above results show that as long as β1 /θ1 < 1/λA and β2 /θ2 < 1/λB no matter how strong the cooperation is, information will die out exponentially in this model. These results further N ∑ illustrate the correctness of our theoretical analysis. P I1 (t) = (PiI1 (t)) is i=1
used to represent the total number of nodes that propagate information 1 at N N ∑ ∑ time t. Similarly, P I2 (t) = (PiI2 (t)) , P I1 I2 (t) = (PiI1 I2 (t)). i=1
i=1
600
600
δ1<1,δ2<1
400
δ >1,δ >1
200
δ1=1,δ2=1
1
40
(a) t
60
80
δ1>1,δ2>1
400
2
δ1=1,δ2=1
200
δ1>1,δ2<1 20
δ1<1,δ2<1
100
δ >1,δ <1 1
0 0
20
40
2
60
80
100
the number of I1I2,
the number of I2,
the number of I1,
800
800
700 600
1000
1000
0 0
PI1I2
1200
1200
PI2
PI1
5.2.2. Multiple information analysis
500 400 δ1<1,δ2<1
300 200
δ >1,δ >1
100
δ1=1,δ2=1
0 0
(b) t
1
2
δ1>1,δ2<1 20
40
(c) t
60
80
100
Figure 6: The impact of influence factor on spreading process in generated network, with β1 = β2 = 0.14, θ1 = θ2 = 0.24. (a) represents the change curve of P I1 over time; (b) represents the change curve of P I2 over time; (c) represents
PI2
40
the number of I2,
50 40 δ1<1,δ2<1
30
δ >1,δ >1 1
2
δ =1,δ =1
20
1
2
δ >1,δ <1 1
10 0
50
10
20
30
40
2
50
30 δ <1,δ <1
20
1
2
δ >1,δ >1 1
2
δ1=1,δ2=1
10
δ >1,δ <1 1
0 0
(a) t
10
20
(b) t
30
40
2
50
the number of I1I2,
PI1
60
PI1I2
the change curve of P I1 I2 over time.
the number of I1,
300
40
30
20
δ <1,δ <1 1
2
δ1>1,δ2>1
10
δ =1,δ =1 1
2
δ1>1,δ2<1 0 0
10
20
(c) t
30
40
50
Figure 7: The impact of influence factor on spreading process in real network, with β1 = β2 = 0.14, θ1 = θ2 = 0.24. (a) represents the change curve of P I1 over time; (b) represents the change curve of P I2 over time; (c) represents the change curve of P I1 I2 over time. In this section, we will explore the impact of influence factor on informa16
tion evolution process. First of all, we need to introduce several concepts. If 305
information 1 promotes the propagation of information 2 while information 2 suppresses the propagation of information 1 (or information 1 suppresses the propagation of information 2 while information 2 promotes the propagation of information 1), the relationship between information 1 and information 2 is defined as asymmetrical influence; if information 1 and information 2 promote
310
each other, the relationship between them is defined as bidirectional promotion; if information 1 and information 2 suppress each other, the relationship between them is defined as bidirectional suppression. Fig. 6 and Fig. 7 illustrate the simulation results in generated network and real network respectively. In Fig. 6 and Fig. 7, (a) plots the number of nodes
315
infected by information 1 over time, (b) plots the number of nodes infected by information 2 over time, (c) plots the number of nodes infected by both information 1 and information 2 over time. From (a) and (b) we can see that cooperation can promote the propagation of information, that to say, more nodes transmit information 1 or information 2; competition can suppress the
320
propagation of information, that to say, less nodes transmit information 1 or information 2. (c) shows that cooperative relationship enables more users to accept the two information simultaneously, while competitive relationship leads to a significant reduction in the number of users who accept both information. Here, we should pay attention to the case that relationship between information
325
is asymmetrical influence(i.e., δ1 > 1, δ2 < 1). From (a) we can see that, for information 1, the promotion strength of asymmetrical influence is not as stronger as bidirectional promotion (i.e., δ1 > 1, δ2 > 1); from (b) we can see that, for information 2, the suppression strength of asymmetrical influence is more stronger than bidirectional suppression (i.e., δ1 < 1, δ2 < 1). Fig. 8 further
330
explains the number of infected nodes at stable state as a function of δ1 and δ2 . P∗I1 represents the number of infected nodes by information 1 at stable state and indicates the prevalence of information 1. We define P∗I2 and P∗I1 I2 , similarly.
17
Figure 8: The number of infected nodes at stable state as a function of δ1 and δ2 with β1 = β2 = 0.14, θ1 = θ2 = 0.24. (a) plots the number of I1 at stable state; (b) plots the number of I2 at stable state; (c) plots the number of I1 I2 at stable state. 5.2.3. Multiplex network analysis In order to capture the impact of the layers interrelation on the informa335
tion dissemination process, the correlation coefficient is used to measure the interrelation between network layers.
ρ(G) = √ ∑
∑
(dA,i − dA )(dB,i − dB ) √ ∑ 2 2 (dA,i − dA ) (dB,i − dB )
(9)
In Eq. 9, dA,i represents the degree of node i in spaceA and dA represents the average degree in spaceA. Similarly, dB,i represents the degree of node i in spaceB and dB represents the average degree in spaceB. 340
The effect of interrelation on the preferred information 1 is shown in Fig. 9. (a) shows that in the case of cooperation, the positive correlation (ρ(G) > 0) promotes the propagation of information 1 while the negative correlation(ρ(G) < 0) suppresses the propagation of information 1. However, it is just the reverse in the case of competition .In other words, the positive correlation suppresses the
345
propagation while the negative correlation promotes the propagation when they are competitive. Fig. 10 shows that for information 2, positive correlation promotes the propagation while negative correlation suppresses the propagation in both cooperation and competition cases. What’s more, in the case of competition, information 2 is more sensitive to the interlayer relationship, that is, the 18
350
positive and negative correlation has a great difference in the number of nodes infected by information 2. Combining Fig. 9 and Fig. 10, we can see that interrelations between layers have different effects on the preferred information and the secondary information in the case of competition. Cooperative
Competitive
PI1
1000
1000
800
the number of I1,
the number of I1,
PI1
1200
800 600 400
ρ(G)>0 ρ(G)<0 ρ(G)=0
200 0 0
50
100
150
200
600
400
0 0
250
(a) t
ρ(G)>0 ρ(G)<0 ρ(G)=0
200
50
100
(b) t
150
200
250
Figure 9: The influence of the interrelation between network layers on the spreading process of information 1 in the case of cooperation or competition. Specifically (a) represents in the case of cooperation and shows that positive correlation promotes the propagation while negative correlation suppresses the propagation, δ1 = 1.5; (b) represents in the case of competition and shows that positive correlation suppresses the propagation while negative correlation promotes the propagation, δ1 = 0.5.
5.2.4. MM-comprehensive analysis 355
Finally, we comprehensively study the effects of influence factor and layers interrelation on epidemic threshold. The simulation results of information 1 and information 2 are shown in Fig. 11 and Fig. 12 respectively. We use A to mark the position of β1 /θ1 = 1/λA in Fig. 11 and use B to mark the position of β2 /θ2 = 1/λB in Fig. 12. For clarity, we use the black arrows to point the
360
position of A or B, and the red arrows to point the position of the epidemic threshold. In Fig. 11 and Fig. 12, (a), (b) and (c) plots the stable-state infection fraction curves as a function of β when space A and space B are positively
19
Competitive
Cooperative 800
I
I
P2
P2
1200 1000
700
the number of I2,
the number of I2,
600
800 600 400 ρ(G)>0 ρ(G)<0
200
ρ(G)=0 0 0
50
100
(a) t
150
200
500 400 300 200
0 0
250
ρ(G)>0 ρ(G)<0 ρ(G)=0
100 50
100
150
200
250
(b) t
Figure 10: The influence of the interrelation between network layers on the spreading process of information 2 in the case of cooperation or competition. specifically (a) represents in the case of cooperation and shows that that positive correlation promotes the propagation while negative correlation suppresses the propagation δ2 = 1.5; (b) represents in the case of competition and shows that that positive correlation promotes the propagation while negative correlation suppresses the propagation δ2 = 0.5. interrelated, neutrally interrelated and negatively interrelated respectively. We analyze the preferred information 1 at first, Fig. 11 shows that as the correlation 365
coefficient ρ(G) decreases the epidemic threshold gradually increases(the red arrow points a larger number),but its not very obvious. It is worth noting that the position where the coves of δ1 > 1, δ1 = 1 and δ1 < 1 begin to become larger than zero is coincident whether in Fig. 11(a), Fig. 11(b) or Fig. 11(c), which indicates the epidemic threshold is equal when δ1 > 1 , δ1 = 1 or δ1 < 1. In
370
general, for the preferred information, influence factor only affects the fraction of infected nodes at stable state but has no significant effects on its epidemic threshold, regardless of the layers interrelation. From Fig. 12, we can see that as the correlation coefficient ρ(G) decreases the epidemic threshold of information 2 also gradually increases, but the result
375
is more obvious than that of information 1. Different from information 1, the influence factor not only affects the fraction of infected nodes at stable state but
20
1200
600 400 200 0 0
A0.1
0.2
0.3
0.4
(a) β1
0.5
0.6
800 600 400 200
A0.1
δ1>1 1000
δ1<1
0 0
0.7
PI*1
I
δ1<1
δ1=1
the number of I1,
800
1000
δ1=1
the number of I1,
the number of I1,
1000
δ1>1
1200 δ1>1
P*1
*
PI1
1200
0.2
0.3
0.4
(b) β1
0.5
0.6
800
δ1=1
600 400 200 0 0
0.7
δ1=1
A0.1
0.2
0.3
0.4
(c) β1
0.5
0.6
0.7
Figure 11: The number of nodes infected by information 1 at stable state as a function of β1 for different values of δ1 .(a) layers are positively interrelated, ρ(G) > 0;(b) layers are neutrally interrelated,ρ(G) = 0;(c) layers are negatively
1200
I
δ2<1
600 400 200 0 0
B0.1
0.2
2
1000
δ =1
0.3
(a) β2
0.4
0.5
0.6
0.7
800
δ2<1
600 400 200 0 0
B0.1
0.2
δ2>1 δ =1 2
2
the numbet of I2,
the number of I2,
2
800
1200
δ >1 1000
δ =1
PI*2
1200
δ2>1
the number of I2,
1000
P*2
*
PI2
interrelated,ρ(G) < 0.A represents the position of β1 /θ1 = 1/λA .
0.3
(b) β2
0.4
0.5
0.6
0.7
800
δ2<1
600 400 200 0 0
B0.1
0.2
0.3
0.4
(c) β2
0.5
0.6
0.7
Figure 12: The number of nodes infected by information 2 at stable state as a function of β2 for different values of δ2 . (a) layers are positively interrelated, ρ(G) > 0; (b) layers are neutrally interrelated,ρ(G) = 0;(c) layers are negatively interrelated,ρ(G) < 0. B represents the position of β2 /θ2 = 1/λB . also has significant effects on the epidemic threshold of information 2. In conclusion, for the secondary information the cooperative relationship increases the number of infected nodes and reduces the epidemic threshold while competitive 380
relationship reduces the number of infected nodes and increases the epidemic threshold.
21
6. Conclusion In summary, this paper proposes an information propagation dynamic model based on multiple information and multiplex network (MM-SIS model) and sets 385
a group of discrete time Microscopic Markov equations to describe the information spreading process. Firstly, this paper theoretically analyzes the epidemic threshold in the MM-SIS model, which correctness has been verified through experiments. Through a series of experiments, we find that: (1) The asymmetric influence between information (i.e., δ1 > 1, δ2 < 1 or δ1 < 1, δ2 > 1)
390
causes stronger suppression effect than bidirectional suppression (i.e., δ1 < 1, δ2 < 1) and weaker promotion effect than bidirectional promotion (i.e., δ1 > 1, δ2 > 1). (2) The epidemic threshold of information increases as the correlation coefficient between the network layers decreases. (3) For the preferred information, positive correlation promotes the dissemination while negative correlation
395
suppresses the dissemination in the case of cooperation, whereas, it is opposite in the case of competition. For the secondary considered information, positive correlation promotes the dissemination and negative correlation suppresses the dissemination in both cooperation and competition cases. (4) For the preferred information, influence factor can affect the final outbreak scale of information
400
but has no obvious effects on the epidemic threshold. For the secondary information, influence factor affects both final outbreak scale of the information and the epidemic threshold.
Acknowledgment This paper is partially supported by the National Natural Science Foun405
dation of China (Grant No. 61772098); the China Scholarship Council (Grant No.201707845009); Chongqing Science and Technology Commission Project (Grant No. cstc2017jcyjAX0099) and Chongqing key research and development project (No.cstc2017zdcy-zdyf0299, cstc2017zdcy-zdyf0436) and Chongqing Graduate Education Teaching Reform Project(No.yjg183081); And Ling Liu’s research
22
410
is partially support by the National Science Foundation under Grants NSF 1547102, SaTC 1564097 and an IBM faculty award.
References [1] X. Xiong, Y. Y. Li, S. J. Qiao, An emotional contagion model for heterogeneous social media with multiple behaviors, Physi415
ca A: Statistical Mechanics and its Applications 490 (2018) 185–202. doi:10.1016/j.physa.2017.08.025. [2] J. Goldenberg, B. Libai, E. Muller, Talk of the network: a complex systems look at the underlying process of word-of-mouth, Marketing Letters 12 (2001) 211–223. doi:10.1023/A:1011122126881.
420
[3] H. H. Wang, A. A. Raza, Y. B. Lin, R. Rosenfeld, Behavior analysis of low-literate users of a viral speech-based telephone service, in: Proceedings of the 4th Annual Symposium on Computing for Development, ACM, 2013, pp. 1–9. doi:10.1145/2537052.2537062. [4] N. Masuda, Opinion control in complex networks, New Journal of Physics
425
17 (2015) 33031–33041. doi:10.1088/1367-2630/17/3/033031. [5] L. Weng, A. Flammini, A. Vespignani, F. Menczer, Competition among memes in a world with limited attention, Scientific Reports 2 (2012) 335. doi:10.1038/srep00335. [6] H. F. Zhang, M. Small, X. C. Fu, B. H. Wang, Modeling the influence
430
of information on the coevolution of contact networks and the dynamics of infectious diseases, Physica D: Nonlinear Phenomena 241 (2012) 1512– 1517. doi:10.1016/j.physd.2012.05.011. [7] L. X. Yang, X. F. Yang, J. M. Liu, C. Q. Gan, Epidemics of computer viruses: a complex-network approach, Applied Mathematics and Computation
435
219 (2013) 8705–8717. doi:10.1016/j.amc.2013.02.031.
23
[8] M. Llas, P. M. Gleiser, J. M. L´opez, Nonequilibrium phase transition in a model for the propagation of innovations among economic agents, Phys. Rev. E 68 (2003) 066101. doi:10.1103/PhysRevE.68.066101. [9] M. 440
E.
J.
Newman,
M.
Girvan,
munity structure in networks,
Finding
and
evaluating
com-
Phys. Rev. E 69 (2004) 026113.
doi:10.1103/PhysRevE.69.026113. [10] S. Pei, L. Muchnik, J. S. Andrade, Jr, Z. Zheng, H. A. Makes, Searching for superspreaders of information in real-world social media, Scientific Reports 4 (2014) 5547. doi:10.1038/srep05547. 445
[11] D. Chen, L. L¨ u, M. S. Shang, Y. C. Zhang, T. Zhou, Identifying influential nodes in complex networks, Physica A: Statistical Mechanics and its Applications 391 (2012) 1777–1787. doi:10.1016/j.physa.2011.09.017. [12] H. Hu, X. Wang, Evolution of a large online social network, Physics Letters A 373 (2009) 1105–1110. doi:10.1016/j.physleta.2009.02.004.
450
[13] Z. Qian, S. Tang, X. Zhang, Z. Zheng, The independent spreaders involved sir rumor model in complex networks, Physica A: Statistical Mechanics and its Applications 429 (2015) 95–102. doi:10.1016/j.physa.2015.02.022. [14] R. Pastor-Satorras, C. Castellano, P. V. Mieghem, A. Vespignani, Epidemic processes in complex networks, Review of Modern Physics 87 (2015) 120–
455
131. doi:10.1103/RevModPhys.87.925. [15] Y. Liu, B. Wang, B. Wu, S. Shang, Y. Zhang, C. Shi, Characterizing superspreading in microblog: an epidemic-based information propagation model, Physica A: Statistical Mechanics and its Applications 463 (2016) 202–218. doi:10.1016/j.physa.2016.07.022.
460
[16] J. Leskovec, M. Mcglohon, C. Faloutsos, N. Glance, M. Hurst, Patterns of cascading behavior in large blog graphs, in: Proceedings of the 2007 SIAM International Conference on Data Mining, SIAM, 2007, pp. 551–556. doi:10.1137/1.9781611972771.60. 24
[17] C. Nowzari, V. M. Preciado, G. J. Pappas, Analysis and control of epi465
demics: a survey of spreading processes on complex networks, IEEE Control Systems 36 (2016) 26–46. doi:10.1109/MCS.2015.2495000. [18] S. A. Myers, J. Leskovec, Clash of the contagions: cooperation and competition in information diffusion, in: Proceedings of the 12th IEEE International Conference on Data Mining, IEEE, 2012, pp. 539–548.
470
doi:10.1109/ICDM.2012.159. [19] Y. Su, X. Zhang, L. Liu, S. Song, B. Fang, Understanding information interactions in diffusion: perspective,
Frontiers
of
Computer
an evolutionary game-theoretic Science
10
(2016)
518–531.
doi:10.1007/s11704-015-5008-y. 475
[20] X. Zhang, Y. Su, S. Qiu, S. Xie, B. Fang, IAD: interaction-aware diffusion framework in social networks, IEEE Transactions on Knowledge and Data Engineering (2018) 1–1.doi:10.1109/TKDE.2018.2857492. [21] B. A. Prakash, A. Beutel, R. Rosenfeld, C. Faloutsos, Winner takes all: competing viruses or ideas on fair-play networks, in: Proceedings of the
480
21st international conference on World Wide Web, ACM, 2012, pp. 1037– 1046. doi:10.1145/2187836.2187975. [22] A. Beutel, B. A. Prakash, R. Rosenfeld, C. Faloutsos, Interacting viruses in networks: can both survive?, in: Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining, ACM,
485
2012, pp. 426–434. doi:10.1145/2339530.2339601. [23] D.
Trpevski,
spreading
over
W.
K.
Tang,
networks,
L.
Phys.
Kocarev, Rev.
E
Model 81
for
(2010)
rumor 056102.
doi:10.1103/PhysRevE.81.056102. [24] Z. Wang, L. Wang, A. Szolnoki, M. Perc, Evolutionary games on multi490
layer networks: a colloquium, European Physical Journal B 88 (2015) 124. doi:10.1140/epjb/e2015-60270-7. 25
[25] G. Dong, L. Tian, D. Zhou, R. Du, J. Xiao, H. E. Stanley, Robustness of n interdependent networks with partial support-dependence relationship, EPL 102 (2013) 68004. doi:10.1209/0295-5075/102/68004. 495
[26] Y. Zhuang, A. Arenas, O. Yaˇgan, Clustering determines the dynamics of complex contagions in multiplex networks, Phys. Rev. E 95 (2017) 012312. doi:10.1103/PhysRevE.95.012312. [27] M. Berlingerio, M. Coscia, F. Giannotti, A. Monreale, D. Pedreschi, Multidimensional networks: foundations of structural analysis, World Wide Web
500
16 (2013) 567–593. doi:10.1007/s11280-012-0190-4. [28] W. Li, S. Tang, W. Fang, Q. Guo, X. Zhang, Z. Zheng, How multiple social networks affect user awareness:
the information diffu-
sion process in multiplex networks, Phys. Rev. E 92 (2015) 042810. doi:10.1103/PhysRevE.92.042810. 505
[29] E. Cozzo, R. A. Ba˜ nos, S. Meloni, Y. Moreno, Contact-based social contagion in multiplex networks, Phys. Rev. E 88 (2013) 050801. doi:10.1103/PhysRevE.88.050801. [30] S. Wang, L. Huang, C. H. Hsu, F. Yang, Collaboration reputation for trustworthy web service selection in social networks, Journal of Computer and
510
System Sciences 82 (2016) 130–143. doi:10.1016/j.jcss.2015.06.009. [31] C.
Buono,
L.
A.
Braunstein,
Immunization
demic spreading on multilayer networks,
strategy
for
epi-
EPL 109 (2015) 26001.
doi:10.1209/0295-5075/109/26001. [32] F. D. Sahneh, 515
C. Scoglio,
arbitrary multilayer networks,
Competitive epidemic spreading over Phys. Rev. E 89 (2014) 062817.
doi:10.1103/PhysRevE.89.062817. [33] Q. Wu, X. Fu, M. Small, X. J. Xu, The impact of awareness on epidemic spreading in networks, Chaos 22 (2012) 013101. doi:10.1063/1.3673573.
26
[34] N. J. Watkins, C. Nowzari, V. M. Preciado, G. J. Pappas, Optimal re520
source allocation for competitive spreading processes on bilayer networks, IEEE Transactions on Control of Network Systems 5 (2015) 298–307. doi:10.1109/TCNS.2016.2607838. [35] Y. Zhao, M. Zheng, Z. Liu, A unified framework of mutual influence between two pathogens in multiplex networks, Chaos 24 (2014) 043129.
525
doi:10.1063/1.4902254. [36] X. Wei, N. C. Valler, B. A. Prakash, I. Neamtiu, M. Faloutsos, C. Faloutsos, Competing memes propagation on networks: a network science perspective, IEEE Journal on Selected Areas in Communications 31 (2013) 1049–1060. doi:10.1109/JSAC.2013.130607.
530
[37] H. L. Mills, pled networks:
A. Ganesh,
C. Colijn,
Pathogen spread on cou-
effect of host and network properties on transmis-
sion thresholds, Journal of Theoretical Biology 320 (2013) 47–57. doi:10.1016/j.jtbi.2012.12.006. [38] S. G´omez, A. Arenas, J. Borge-Holthoefer, S. Meloni, Y. Moreno, Discrete535
time markov chain approach to contact-based disease spreading in complex networks, EPL 89 (2010) 38009. doi:10.1209/0295-5075/89/38009. [39] C. Granell, S. G´omez, A. Arenas, Dynamical interplay between awareness and epidemic spreading in multiplex networks, Phys. Rev. Lett. 111 (2013) 128701. doi:10.1103/PhysRevLett.111.128701.
540
[40] M. W. Hirsch, S. Smale, Differential Equations, Dynamic Systems, and Linear Algebra, Academic Press, New York, 1974.
27