Identify Influential Spreaders in Asymmetrically Interacting Multiplex Networks
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Identify Influential Spreaders in AsymmetricallyInteracting Multiplex Networks
Qi Zeng, Ying Liu, Liming Pan, and Ming Tang
Abstract —Identifying the most influential spreaders is important to understand and control the spreading process in a network. Asmany real-world complex systems can be modeled as multilayer networks, the question of identifying important nodes in multilayernetwork has attracted much attention. Existing studies focus on the multilayer network structure, while neglecting how the structuraland dynamical coupling of multiple layers influence the dynamical importance of nodes in the network. Here we investigate on thisquestion in an information-disease coupled spreading dynamics on multiplex networks. Firstly, we explicitly reveal that three interlayercoupling factors, which are the two-layer relative spreading speed, the interlayer coupling strength and the two-layer degree correlation,significantly impact the spreading influence of a node on the contact layer. The suppression effect from the information layer makes thestructural centrality on the contact layer fail to predict the spreading influence of nodes in the multiplex network. Then by mapping thecoevolving spreading dynamics into percolation process and using the message-passing approach, we propose a method to calculatethe size of the disease outbreaks from a single seed node, which can be used to estimate the nodes’ spreading influence in thecoevolving dynamics. Our work provides insights on the importance of nodes in the multiplex network and gives a feasible framework toinvestigate influential spreaders in the asymmetrically coevolving dynamics.
Index Terms —multiplex network, influential spreader, asymmetrically interacting dynamics, centrality measure. ✦ NTRODUCTION M ANY activities in society can be described as spread-ing processes on networks, such as the spreadingof epidemic disease through contacts between human be-ings, information dissemination through email and mobilephone, and the diffusion of ideas among friends and com-munity members. Identifying the most influential spreadersis an important step to control the spreading processes, e.g. to hinder epidemic outbreaks [1], conduct successful ad-vertisement for new products [2] and protect key membersin ecosystem [3]. A commonly accepted way to rank andidentify important nodes in a network is to use the central-ity measures, such as degree, betweenness [4], eigenvectorcentrality [5], PageRank [6], nonbacktracking centrality [7],and k-shell index [8]. Based on the idea of centrality, thereare a lot of achievements in the identification of importantnodes in single layer networks [9], [10], [11], which help usto better understand the network structure and function.However in the real-world, from city infrastructure tohuman interaction patterns, many complex systems areinterconnected and are better described by the multilayernetwork [12]. For example, the air transportation networkcan be described as a multilayer network where nodesrepresent airports and each commercial airline corresponds • Q. Zeng and Y. Liu (corresponding author) are with the School ofComputer Science, Southwest Petroleum University, Chengdu 610500,China. Y. Liu is also with Big Data Research Center, University ofElectronic Science and Technology of China, Chengdu 611731, China.E-mail: [email protected] • L. Pan is with the School of Computer Science and Technology, NanjingNormal University, Nanjing 210023, China. • M. Tang (corresponding author) is with School of Physics and ElectronicScience, East China Normal University, Shanghai 200241, China ; Shang-hai Key Laboratory of Multidimensional Information Processing, EastChina Normal University, Shanghai 200241, China.E-mail: [email protected] to a different layer [13]. In social networks, individualsinteract in different ways like being friends, colleagues,schoolmates, or interacting on different online social plat-forms [14], where each type of connections is representedby a layer. The multiplex network is a particular type ofmultilayer network, in which a set of nodes represent thesame individuals in all layers, and their edges in differentlayers represent various friendship patterns, such as thesocial networks.It is thus natural to use the multilayer formalism tostudy the scenarios where different dynamical processes in-terplay, such as the cooperative contagion processes spread-ing in a host population [15], and the competitive epi-demic spreading or opinion spreading on multilayer net-works [16], [17], [18]. A special case attracting much atten-tion is the information-disease asymmetrically interactingprocesses [19], [20]. When epidemic disease outbreaks in adistrict, the information on it is swiftly transmitted throughthe online social media, telephone, mass media, et al. Thespreading of information suppresses the disease spreadingbecause of the awareness people arise after receiving theinformation [21], and the spreading of disease promotesthe diffusion of information, which are two asymmetricallyinteracting processes.While many real-world complex systems can be modeledas multilayer networks, neglecting the multiple relation-ships between nodes or simply aggregating them into a sin-gle network alters the structural and dynamical propertiesof the system, leading to inaccurate identification of impor-tant nodes [22], [23]. In recent years, there are some progressin identifying the critical nodes on multilayer network,which are in general extend the centrality measures fromsingle network to multilayer network and the focus is onthe multilayer structure alone. Examples are the Multiplex
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PageRank and eigenvector-based centrality in multiplex net-work. But how the interplay between the multiple processeson the multilayer network impact the functional importanceof nodes is still unknown, and new ranking methods thatconsidering the structural and dynamical interplay betweenlayers are still lacking.In modern times, the epidemic infectious diseasespreads more easily due to the growing connectivityamong metropolitan centers in the world urbanizationprogress [24]. Such as the ongoing coronavirus disease 2019(COVID-19) has caused more than 20 millions confirmedcases and 700 thousands deaths in just a few months,impacting politically and economically on the daily livesof people. Timely and accurately identify the influentialspreaders is thus of great importance to make optimal useof available resource to suppress epidemic disease [25]. Inthis manuscript, based on an asymmetrically interactinginformation-disease spreading model on multiplex network,we work on explicitly revealing how the structural anddynamical coupling of multiple layers influence the func-tional importance of nodes and then step further to pro-pose a new method to rank and identify critical nodes inthe asymmetrically interacting multiplex network. In ourstudy, the multiplex network consists of two layers, wherethe spreading of information on one layer suppresses thespreading of disease on the other layer, while the spreadingof disease promotes the information diffusion. As in realapplications controlling the spreading of disease is usuallythe fundamental purpose, we concentrate on the influenceof nodes in disease spreading.Firstly, by taking the degree and eigenvector centralityof nodes on the physical contact layer as the benchmarkmeasure, we study on how the two-layer coupling factorsimpact their accuracy to predict the spreading influence ofnodes. As the spreading influence of nodes in the contactlayer is suppressed by information spreading, the accuracyof centralities is impacted. The application of discoveringthe performance of centralities under different structuraland dynamical parameters is two-fold. One the one hand,due to the difficulty in collecting network data, when weonly have the contact layer data and obtain the centralityof nodes from the contact layer alone, we can evaluate howaccurate the obtained centrality can predict the influentialspreaders. On the other hand and more importantly, basedon the network data of both layers, we should define newmeasure to accurately identify the most influential spreadersin the asymmetrically interacting dynamics.Then we propose an effective framework to rank thenode influence in the asymmetrically interacting multiplexnetworks. Specifically speaking, by mapping the coevolv-ing spreading dynamics into bond percolation and usingthe message-passing approach, we calculate the spreadingoutbreak size for each node as seed, which can be usedto rank the influence of disease spreaders in the multiplexnetwork. The accurate identification of disease spreaders isvery applicable in real-world epidemic control.To the best of our knowledge, this work is the first stepin explicitly studying how the two-layer coupling factorsimpact the spreading influence and the centrality of nodesin identifying influential spreaders, and ranking the nodeinfluence by considering the coupling factors. Our main contributions in this paper are as follows: • We discover how three coupling factors, which arethe relative spreading speed of the two layers, thecoupling strength and the inter-layer degree correla-tion, impact the accuracy of centrality measures inpredicting spreading influence in multiplex network. • By mapping the coevolving spreading dynamics intopercolation, we propose a method to accurately rankthe spreading influence of nodes in the multiplexnetwork. • We numerically evaluate our proposed method inmultiplex networks and show its superiority overthe benchmark centralities in identifying influentialspreaders.The rest of the paper is organized as follows. In section2, we briefly introduce related works. In section 3, theinformation-disease spreading model on multiplex networkis described. In section 4, we demonstrate the impact ofinterlayer coupling factors on the accuracy of centralities. Insection 5, we map the coevolving spreading dynamics intopercolation and propose a new ranking method. Finally insection 7, we give the conclusions.
ELATED WORKS
To identify important nodes in the multilayer network, alot of centrality measures have been proposed, which arein general extended from centrality in single networks. Forexample, the Multiplex PageRank is a natural extension ofPageRank in multiplex network, which considers the cen-trality of a node in one network is affected by the centralityof the node in another network [26]. Another FunctionalMultiple PageRank is defined on the weight of multilinks(connections in different layers) where the link overlapbetween layers is considered [27]. The eigenvector centralityin multiplex networks takes into account the mutual in-fluence of layers [28], and a supracentrality matrix whichcouples the centrality matrices of the individual layers isused [29]. The tensor framework mathematically describesthe intralayer and interlayer relationships [30], and tensordecomposition is then used to identify critical nodes [31].In the tensorial formalism, Domenico et al. generalizesthe eigenvector centrality, PageRank and betweenness tomultilayer network and define the versatility to identifythe most important nodes [32]. A family of multilayer PCI(Power Community Index) measures generalized from h-index considers the density of a node’s intra and interconnections and can be computed from the local structureof the network [33].Although there are such progresses on identifying criti-cal nodes on multilayer network, the focus is on the struc-ture of the network, while neglecting how the structural anddynamical coupling between multiple layers influence thedynamical importance of nodes in the network. In multi-layer network, the interplay between the network structureand spreading dynamics on top of it will largely influencethe role of nodes in the network [34]. Identifying the criticalnodes must take both the structural and dynamical char-acteristics of the network into consideration. A few workssummed up the centrality of nodes in both layers and
OURNAL OF L A TEX CLASS FILES, VOL. , NO. , 3 synthesized the dynamical parameters to form a measureto quantify the node influence in symmetrically interactinginterconnected networks, and the results showed that tak-ing both layers’ structure and dynamical parameters intoconsideration leaded to a better ranking result of the criticalnodes [35]. How the coevolving dynamics impacts on thefunctional importance of nodes and accurate identificationof important nodes remains an open question.
HE INFORMATION - DISEASE SPREADING MODELON MULTIPLEX NETWORK
We use an asymmetrically interacting model to describe thecoevolving dynamics of disease and information spread-ing [36], [37] on multiplex network as shown in Fig. 1.Consider a multiplex network consisting of two layers. Theupper layer represents the information communication net-work labeled as layer A and the bottom layer represents thephysical contact network labeled as layer B. In the commu-nication layer (layer A), the classical susceptible-infected-recovered (SIR) model is used to describe the spreading ofinformation on disease. In the SIR model, node can be in oneof the three states: (1) susceptible (S), in which the individualhas not received any information about the disease, (2)infected (or informed for information transmission I), inwhich the individual is aware of the disease and is able totransmit the information, or (3) recovered (R) in which theindividual has received the information but is not willingto transmit the information to others. The informed nodetries to transmit information to its neighbors at rate β A . Theinformed node recovers with rate µ A in the next time step.Once a node becomes recovered, it will remain in the statein the subsequent time steps.The spreading of disease on the physical contact layer Bis described by the SIRV model [39], where a vaccinated(V)state is introduced. A vaccinated node will not be infectedby any node. The SIR dynamics is the same as layer A,with the infection rate and recovery rate denoted as β B and µ B respectively. The dynamical coupling of the twolayers is as follows. For an informed individual in layerA, if its counterpart in layer B is susceptible, then thecounterpart node changes to vaccination state with rate λ AB . The parameter λ AB ranges in [0 , to represent theextent to which people take care of the information and arewilling to take vaccination. For a susceptible node in layerB, if its counterpart in layer A is in susceptible state, thenin the next time step it may become either vaccinated withrate λ AB if its counterpart in layer A is getting informed orinfected by its infected neighbors in layer B. In this case,vaccination and infection will compete for the chance toaffect such susceptible node in layer B. Take p A and p B as the probability that the vaccination or infection winsrespectively, then p A = 1 − (1 − β A ) n Ai [1 − (1 − β A ) n Ai ] + [1 − (1 − β B ) n Bi ] (1)and p B = 1 − (1 − β B ) n Bi [1 − (1 − β A ) n Ai ] + [1 − (1 − β B ) n Bi ] , (2) where n Ai is the number of informed neighbors of the coun-terpart node in layer A, and n Bi is the number of infectedneighbors of the considered node in layer B. If vaccinationwins out, the node in layer A is informed with probability − (1 − β A ) n Ai and then the couterpart node in layer B isvaccinated with rate λ AB . Else, if the infection wins out, thenode is infected in layer B with probability − (1 − β B ) n Bi and its counterpart node changes to the informed state withrate λ BA , which ranges in [0 , . This represents the extentto which an infected individual is aware of and willing totransmit the information of the epidemic disease. EVEALING THE IMPACT OF INTERLAYER COU - PLING FACTORS
It is pointed out that the asymmetrically interacting dy-namics will alter the activities of nodes on the multiplexnetworks [19]. Now we study on how three dynamicaland structural two-layer coupling factors, which are the therelative spreading speed of the two layers, the dynamicalcoupling strength and the interlayer degree correlation,affect the spreading influence of nodes and thus changethe accuracy of centrality in predicting their influence inthe physical contact layer, which is our focus. The relativespreading speed of the two layers is defined as γ λAB = λ A /λ B , (3)where λ A = β A /µ A and λ B = β B /µ B are the effectivetransmission rate in layer A and B respectively. For simplic-ity, we set µ A = µ B = 1 . The dynamical coupling strengthof the two layers is represented by the two parameters λ AB and λ BA , which are the vaccination rate and informed raterespectively. The interlayer degree correlation m s is quanti-fied by the Spearman rank correlation coefficient, which isdefined as m s = 1 − P Ni =1 ∆ i N ( N − , (4)where N is the network size and ∆ i is the rank differenceof node i in the degree sorting list in each layer. In simulations, we use the uncorrelated configuration model(UCM) to generate each layer of the multiplex network,which follows a power law degree distribution p ( k ) ∼ k − γ .We first construct layer A with N = 10000 , the power expo-nent γ = 2 . , and average degree < k > = 6 . The minimaldegree k min = 3 , and the maximal degree k max = √ N .Then we copy the nodes of layer A, randomly exchangetheir degree sequence and generate the edges to form layerB. Each node in layer A has a counterpart in layer B. Atthe beginning of the interacting dynamical processes, allnodes are set to be susceptible in both layers except theseed node. The seed node is infected in layer B and itscounterpart in layer A is informed, which will initiate thedisease spreading on layer B and information spreading onlayer A respectively. The spreading processes stop until onboth layers there is no infected (informed) nodes. We recordthe final proportion of recovered nodes in layer B as thespreading influence of the seed node, which is averagedover 100 independent realizations. OURNAL OF L A TEX CLASS FILES, VOL. , NO. , 4
Fig. 1. The information-disease spreading processes on a communication-contact multiplex network. (a) The multiplex network consists of acommunication layer A with SIR dynamics and a physical contact layer B with SIRV dynamics. (b) An infected node in layer B promotes theinformation spreading by changing its counterpart into informed state with rate λ BA . (c) An informed node in layer A suppresses the diseasespreading by changing its counterpart into vaccinated state with rate λ AB . Suppose an undirected network is represented as G(V,E), where V = { v , v , ..., v n } is the set of nodes and E = { e , e , ..., e m } is the set of edges. The adjacency matrixof the graph G is A n ∗ n = a ij , where a ij = 1 means thereis an edge between node i and j , otherwise a ij = 0 . Thedegree k i = P n a ij of node i is defined as the numberof its direct neighbors, which is a simple but effective wayto quantify the potential influence of nodes in the network.The larger the degree, the more neighbors the node is able toinfluence directly. The time complexity of calculating degreeis O ( N ) , where N is the size of the network.The idea of eigenvector centrality is that not all neigh-bors are equivalent. The node is important if it connectsto many neighbors which are themselves important. Theeigenvector centrality of node i is defined as e i = λ − N X n =1 a ij e j , (5)where λ is the largest eigenvalue of the adjacency matrixA, e = { e , e , ..., e n } T is the eigenvector of matrix Acorresponding to the largest eigenvalue λ . If writing in theform of matrix, then it is λe = Ae . In our work, the degreeand eigenvector centrality of nodes in layer B are used asbenchmark methods to identify the most influential nodesin disease spreading. We use the Kendall’s tau correlation coefficient [38] andthe imprecision function [8] to quantify how accurate theconsidered measures can predict the disease-spreading in-fluence of nodes in the asymmetrically interacting processes. The Kendall’s tau correlation coefficient quantifies the con-sistency of two ranking lists for a set of objects. It is definedas τ = P i
Firstly, we study on the impact of relative spreading speedof two layers on the accuracy of centrality in layer B.The accuracy of centrality in predicting node influence isquantified by the Kendall’s τ correlation of centrality andspreading influence as shown in Fig. 2. To concentrate onthe relative spreading speed γ λAB , we fix other couplingparameters and demonstrate the results. The impact of theseparameters when they vary will be discussed in later part.At γ λAB = 0 , it corresponds to the case when there is onlydisease spreading on layer B. The change of correlation τ is due to the change of spreading influence of nodes,which is suppressed by information spreading in layer A.It can be seen from Fig. 2 that τ decreases with γ λAB . InFig. 2 (a) when γ λAB < . , the τ e B and τ k B are relativelystable. This means when the information spreads slowly,or even slower than the disease, it has little impact on thedisease spreading. When γ λAB increases to the range [1.0,3.0], τ e B and τ k B largely decrease. This is because whenthe information spreads faster than the disease, more nodesin layer B will be vaccinated. The spreading influence ofnodes in layer B is suppressed by the information spreading.The faster the information spreads, the more nodes getvaccinated, making the e B and k B less accurate. Considerthe real-world scenario that the epidemics control agencywants to identify the most influential disease spreaders andquarantine them. When the information spreads slowly, itis workable to identify the influential spreaders from thestructure of contact network. But when information spreadsfast, using only the contact data is not adequate any more.As for γ λAB > . , where the information spreads evenfaster, the value of τ becomes stable. This is because whenthe information spreads very fast, the number of vaccinatednodes achieves its upper limit, and the spreading influenceof nodes impacted by the vaccination will not change. Thechange of other three parameters λ AB , λ BA and m s , do notinfluence the decreasing trend of τ , as demonstrated in Fig. 2(b)-(d).Next, we work on the interlayer coupling strength. Thereare two parameters λ AB and λ BA reflecting the couplingstrength between two layers. From Fig. 3(a), we can see thatwith the increase of λ AB , τ k B decreases significantly, whichimplies that the simple degree centrality in layer B becomesworse to rank the node influence. As for τ e B , it first increasesa little and then decreases. In general, τ decreases withthe increase of λ AB . This is because when λ AB increases,the effect of layer A on B are getting more strong, thusthe centrality on layer B are becoming less accurate underthe asymmetrically interacting processes. The Fig. 3 (b)and (c) displays similar trends as (a). In Fig. 3(b), it canbe seen that with the increase of λ BA , τ increases. When λ BA is small, the amount of informed nodes in layer Acaused by the notification from layer B is small. So thespreading of information relies more on the structure andcentrality of nodes in layer A. In this case, the suppressionof disease is more dependent on the structure of layer A,leading to the relatively low accuracy of centrality in layerB to predict the spreading influence of nodes. When λ BA becomes large, more informed nodes in layer A are causedby the notification from their counterparts in layer B, so the the distribution of informed nodes is more random and thesuppression of disease then depends less on the centralityof nodes in layer A. When the number of informed nodesis large enough, the distribution of them can be consideredas uniform in layer A. In this case, the spreading influenceof nodes in layer B are reduced proportionally to theirdegree. The larger λ BA is, the stronger such effect is. Thusthe τ increases as λ BA increases. The Fig. 3 (e) and (f)displays similar trends as (d).In real-world scenarios, if theinfected individuals can timely report their heathy status,corresponding to large λ BA , then it is easier to identifycorrectly the influential disease spreaders. Otherwise, asthe infected ones are hidden, it becomes more difficultto identify the influential spreaders under the interplaybetween information spreading and disease spreading. Thiswill prevent effective epidemic control from the healthyagencies.Finally, we discuss the effect of degree correlation m s between layers. The spearman rank correlation coefficientis used to quantify the degree correlation of nodes in twolayers. As shown in Fig. 4, the τ decreases with the increaseof degree correlation m s . Remember that at the initial stepof interacting spreading, a seed node in layer B is infectedand its counterpart in layer A is informed. The affectedrange of disease spreading from the seed node is determinedby the centrality of seed in both layer A and B. From therespect of degree correlation, all seeds can be divided intofour cases: (1) nodes with large k A and small k B . In thiscase, because of their small degree in layer B, the spreadinginfluence of such nodes is relatively small. Although theinformation spreading is large due to their large k A , thesuppression effect is less obvious. Thus the performance ofcentrality is not largely affected. (2) nodes with small k A and large k B . In this case, due to the centrality of seed,the disease spreading is relatively large and the informationspreading is small, thus there will be a small number ofvaccinated nodes and the suppression effect is as well small.(3) nodes with large k A and large k B . For these nodesas initial spreaders in both layers, the suppression effectfor disease spreading is obvious, making the centrality inpredicting the disease spreading less accurate. The largerdegree correlation m s is, the more such kind of nodes arein the network, corresponding to the largely reduced τ inFig. 4. (4) nodes with small k A and small k B . In this case,neither the disease nor information will break out and thesenodes are ranked low in the list. In all, when the two-layer degree correlation is large, the suppression effect onthe disease spreading is the largest, and the τ of centralityand spreading influence is impacted the most. The aboveresults implies that in real-world epidemic control wherethe information spreading and disease spreading interplays,using the contact network data is not adequate to accuratelyidentify the critical spreaders, especially when the informa-tion spreads fast, the vaccination willingness of people isstrong and the degree correlation is large. So to identify theinfluential spreaders more accurately in multiplex network,we need new framework and method which is our work inthe next part. OURNAL OF L A TEX CLASS FILES, VOL. , NO. , 6 γ λ AB τ (a) e B k B γ λ AB τ (b) γ λ AB τ (c) γ λ AB τ (d) Fig. 2. The τ e B and τ k B as a function of γ λAB . The other parameters are set as (a) λ AB = 0 . , λ BA = 0 . , m s = 0 . , (b) λ AB = 1 . , λ BA = 0 . , m s = 0 . , (c) λ AB = 1 . , λ BA = 0 . , m s = 0 . and (d) λ AB = 1 . , λ BA = 0 . , m s = 0 . . λ AB τ (a) e B k B λ AB τ (b) λ AB τ (c) λ BA τ (d) λ BA τ (e) λ BA τ (f) Fig. 3. The τ e B and τ k B as a function of λ AB and λ BA respectively. The other parameters are set as (a) γ λAB = 2 . , m s = 0 . , λ BA = 0 . , (b) γ λAB = 2 . , m s = 0 . , λ BA = 0 . , (c) γ λAB = 2 . , m s = 0 . , λ BA = 0 . ,(d) γ λAB = 2 . , m s = 0 . , λ AB = 1 . , (e) γ λAB = 2 . , m s = 0 . , λ AB = 1 . and (f) γ λAB = 2 . , m s = 0 . , λ AB = 1 . . APPING TO BOND PERCOLATION
In this part, we map the coevolving spreading dynamics intobond percolation and calculate analytically the prevalencewhen the epidemic is originated from a single seed i on themultiplex network by using the message passage method.The message passing approach is an inference method thatcan provide exact predictions or good approximations ana-lytically in many problems in network sciences, such as pre-dicting the size of the giant component in percolation [40].In the SIR dynamics, the ultimate outbreak size correspondsto the size of the giant component in percolation, which isour interests.We first introduce the percolation on a single network. Inthe percolation process, edges are occupied with probability T p called transmissibility, and a giant component appears if T p is sufficient high. The mapping of percolation to theSIR model is straight forward: edges are occupied withprobability T p , equal to the time-integrated probability T that an infection occurs on the edges. Here T = 1 − e − βt is the probability that a neighbor of an infected node isinfected before it recovers, where β is the disease-causinginfection probability and t is the time the infected noderemains infective. If using the discrete time rather thancontinuous, which is common in computer simulation, then T = 1 − (1 − β ) t , where t is measured in time steps [41].The giant component appearing in the percolation processcorresponds to the potential epidemic outbreak of diseasewith a non-negligible fraction of the network size.To map the coevolving dynamics on multiplex networkinto percolation process, we need the following assump- OURNAL OF L A TEX CLASS FILES, VOL. , NO. , 7 m s0.20.40.60.8 τ (a) e B k B m s0.20.40.60.8 τ (b) m s00.20.40.60.8 τ (c) m s0.30.40.50.60.7 τ (d) Fig. 4. τ e B and τ k B as a function of m s . The other parameters are set as (a) γ λAB = 2 . , λ AB = 0 . , λ BA = 0 . , (b) γ λAB = 2 . , λ AB = 1 . , λ BA = 0 . , (c) γ λAB = 2 . , λ AB = 1 . , λ BA = 0 . and (d) γ λAB = 2 . , λ AB = 1 . , λ BA = 0 . . tion that the information spreading is much faster that theepidemic spreading [42]. This is reasonable for the Inter-net time, as information is easily transmitted worldwidethrough online social media, telephone, mass media, et al.Like the COVID-19 epidemics, the whole world gets toknow its information soon after it outbreaks in Wuhan,China. In addition, the vaccination in layer B can be re-garded as a type of ”disease” because each node in layer Bcan be in either infected states or vaccinated states [36]. Thedisease spreading and vaccination are then viewed as twocompeting ”diseases” on layer B. In the limit of large net-work size N, when two competing diseases spreads, it canbe considered as if they were spreading non-concurrently,one after the other [42]. So we can treat our coevolving dy-namics as a fast dynamics of information spreading spreadsfirst and a slow dynamics of disease spreading spreadssubsequently. First we consider the fast dynamics, i.e. the informationspreading in layer A. This is a simple SIR process. Let H Ai → j be the probability that node i is not connected to the giantcomponent via node j . Then H Ai → j can be obtained by theself-consistency equations [43] H Ai → j = 1 − T A + T A Y k ∈ ∂j \ i H Aj → k , (8)where T A is the edge occupation probability in percolation,and ∂j \ i is the neighbors of node j except i . This equationrepresents that either the edge connecting i and j is notoccupied, or although it is occupied, j is not connected tothe giant component through any of its neighbors other than i . The probability that i in the giant component is P Ai = 1 − Y j ∈ ∂i H Ai → j , (9) where ∂i is the neighbor set of node i . Mapping to SIRdynamics, T A = 1 − e − β A when t = 1 . H Ai → j is theprobability that node j by following the link from i doesnot trigger out outbreaks with the transmissibility T A . P Ai isthe probability that a node i in layer A triggers the epidemicoutbreak in terms of H Ai → j . Now consider the slow dynamics in layer B . Let H Bi → j bethe probability that node i is not connected to the giantcomponent via node j . This can happen because of (a) thenode j is vaccinated with probability λ AB P Ai ; (b) the node j is not vaccinated, but the edge i → j is not occupied withprobability − β B ; and (c) the node j is not vaccinated andthe edge i → j is occupied, but is not connected to the giantcomponent via any of the neighbors. Conclude the abovescenarios we have H Bi → j = λ AB P Ai + (cid:16) − λ AB P Ai (cid:17) (cid:16) − T B (cid:17) + T B (cid:16) − λ AB P Ai (cid:17) Y k ∈ ∂j \ i H Bj → k , (10)where T B = β B because we simulate the SIR dynamicsin discrete time and the infected nodes recover after onetime step, corresponding to t = 1 in the above mentioneddefinition. The probability that i in the giant component is P Bi = 1 − Y j ∈ ∂i H Bi → j . (11)This P Bi is the probability that a node i in layer B triggersan epidemic outbreak in terms of H Bi → j .Then according to ref. [44], the size of epidemic when itoriginates from a seed i is S Bi = 1 N (1 + N X j =1 ,j = i P Bj ) . (12) OURNAL OF L A TEX CLASS FILES, VOL. , NO. , 8
As the seed i must be included in the epidemic outbreak,it corresponds to in the summation. After obtaining theprobability and the outbreak size, the average prevalencewhen the epidemic is initiated by a seed i is defined as ρ Pi = P Bi ∗ S Bi . (13)Thus we can take ρ P as the indicator of node influencein layer B in the asymmetrically interacting processes onmultiplex network. The calculation of influence is describedin algorithm 1. Algorithm 1
Influence Calculation in Multiplex Network
Input: the network of layer A G A and layer B G B ; the infor-mation transmission rate λ A , the disease transmissionrate λ B and the vaccination rate λ AB Output: the set of influence of each node i as the initial seed ρ Pi calculate H Ai → j and P Ai from network A calculate H Bi → j and P Bi from network B based on theobtained value of P Ai calculate S Bi from network B based on the obtainedvalue of P Bi calculate ρ Pi based on the obtained value of P Bi and S Bi VALUATION OF THE PROPOSED METHOD INIDENTIFYING INFLUENTIAL SPREADERS ON MULTI - PLEX NETWORK
Now we evaluate the proposed method in identifying influ-ential spreaders on multiplex network. As we have revealedthat the dynamical interplay of the two layers impacts thespreading influence of nodes, we compare the accuracy of ρ P with degree and eigenvector centrality under differentvalues of the parameters.In Fig. 5, we vary the two-layer relative spreading speed γ λAB and set all other parameters as fixed. It can be seenthat the imprecision defined in 4.1 of ρ P is much lower thanthat of the degree and eigenvector centrality. Although at p = 0 . the imprecision of ρ P is a little bit higher thanthat of k B , in most cases, ρ P is the best.In Fig. 6 imprecisions under different λ AB and λ BA aredemonstrated. It can be seen that, under all values of λ AB and λ BA respectively, the imprecision of ρ P is significantlylower than that of k B and e B in almost all the cases.Finally, we demonstrate the imprecisions of the threemethods under different degree correlations m s . From Fig. 7(a) and (b), we can see that the ρ P is the best indicator forspreading influence. When the degree correlation increasesto 0.5 or 0.7, the imprecision of ρ is equal to or slightlyhigher than that of e B . We think this is because whenwe calculate the ρ P , it is based on the assumption thatthe layer A spreads information first. When the degreecorrelation is large, the hub nodes in layer B is probably tobe vaccinated. Then layer B is separated by the vaccinatednodes into several small clusters and a giant component.The calculation of ρ P on layer B is less accurate because thenetwork has been dismantled. To evaluate this, we calculatethe number of components and the size of giant componentwhen m s varies. As shown in Fig. 8, with the increase of degree correlation m s the number of components in layer Bincreases and the size of the giant component decreases. Ingeneral, ρ P is the best way to predict the spreading influenceof nodes under asymmetrically interacting processes. Theseresults imply that in the modern world, where the spreadingof information is very easy and fast to world-wide, thecontrol strategy based on the contact layer alone will lossits effectiveness. To accurately identify the most influentialspreaders and control the spread of epidemic disease, weneed not only the physical contact network data, but alsothe information transmission network data and the couplingparameters. ONCLUSION
In this paper, we study on how the two-layer coupling fac-tors impact the centrality of nodes to predict their spreadinginfluence and propose a method to identify the most influ-ential spreaders in multiplex networks. The results showthat the benchmark centralities like degree and eigenvectorcentrality in one layer alone can not predict the influen-tial spreaders accurately due to the interplay between thespreading of information and disease on multiplex net-works. The relative spreading speed of two layers, the inter-layer dynamical coupling strength and the two-layer degreecorrelation play an important role in affecting the spreadinginfluence of nodes. By mapping the coevolving spread-ing dynamics into bond percolation, we use the message-passing approach to calculate the epidemic outbreak sizewhen spreading is initiated by a single seed. The obtainedmeasure takes both the intralayer and interlayer structuraland dynamical information into account and is thus veryaccurate in identifying the most influential disease spread-ers. Our work provides new ideas for making effectiveepidemic control strategy and gives a feasible frameworkto study the identification of critical nodes on multilayernetworks. Although we study on the identification of criticalnodes in the asymmetrically interacting spreading dynam-ics, the interacting dynamics on the multilayer network canbe interdependent or competitive. How these interactingdynamics influences the functional importance of nodes isthe question in future study. A CKNOWLEDGMENTS
This work is supported by the National Natural ScienceFoundation of China (No. 61802321, 11975099), the SichuanScience and Technology Program (No. 2020YJ0125), and theNatural Science Foundation of Shanghai (No. 18ZR1412200) R EFERENCES [1] L. C. Freeman, ”Centrality in social networks conceptual clarifica-tion,”
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