Adaptive Influence Maximization in Microblog under the Competitive Independent Cascade Model

Transcription

1 International Jornal of Knowledge Engineering, Vol. 1, No. 2, September 215 Adaptie Inflence Maximization in Microblog nder the Competitie Independent Cascade Model Zheng Ding, Kai Ni, and Zhiqiang He Abstract With the rapid deelopment of social media technology, many different pieces of information, ideas, prodcts, and innoations are propagating widely in online social networks. Information diffsion has been frther researched by many scientists and experts in the past few decades. In this paper, we stdy the competitie inflence propagation in sina microblog nder the competitie independent cascade model, which extends the classical independent cascade model. his paper pays attention to the problem that adaptiely selecting a certain nmber of seeds to maximize its inflence benefit (IB nder a competitie diffsion model. We call this problem the adaptie inflence maximization (AIM problem. he traditional Monte-Carlo greedy algorithm can select a specified nmber of seeds, and has a ery high complexity. A new efficient algorithm called M-based algorithm is designed to select seeds adaptiely, and faster than the Monte-Carlo greedy algorithm. Index erms Adaptie inflence maximization, competitie independent cascade model, microblog. I. INRODUCION As we moe frther into the information age, we can commnicate with each other in online social media more conenient and faster than eer before. So mch information srronds s daily and we are free to choose which pieces of media message to adopt and repost, and which pieces to ignore. his process is called information propagation. he information diffsion model throgh a social network has been stdied by many scholars. For example, ssceptible infectie Ssceptible (SIS model and ssceptible infectie remoal (SIR model [1] are first proposed to describe the information diffsion process. It is in both cases that state of nodes transfer from one of S (ssceptible, I (infectie, R (remoal to one of the two other states with certain probabilities. Simple differential eqations are sed to describe the percentage of nmber of nodes in each state. Kempe et al. [2] proposed the independent cascade (IC model and the linear threshold (L model, which characterize how information are disseminated throghot the network starting from the initial seeds nodes, which are based on a graph strctre and are widely sed today. All the models aboe only stdy the propagation of a piece Manscript receied March 3, 215; reised Jne 2, 215. his work is spported by National High echnology Research and Deelopment Program of China (863 Program SS215AA1133, and National Natral Science Fondation of China ( he athors are with the Key Laboratory of Uniersal Wireless Commnication, Ministry of Edcation, Beijing Uniersity of Posts and elecommnications, Beijing 1876, China ( dzsd213@163.com, nikai@bpt.ed.cn, hezq@bpt.ed.cn. of information in a social network. We call this single-item diffsion. In fact it is often the case that different information are propagating concrrently in a social network. Information competition is omnipresent in social network, sch as the sales competition of similar goods, the campaign between the parties in social media, and the competitie spread of rmors and trth in pblic. Competitie inflence diffsion model has been stdied by a lot of papers and books, Chen et al. [3] propose the competitie independent cascade (CIC models and the competitie linear threshold (CL model where the different kinds of information diffsion process proceed iteratiely in a synchronos way along a discrete time-axis, starting from sets of initial actiated nodes. In addition, broecheler et al. [4] pt forward the weighted generalized annotated program (wgap framework for expressing competitie diffsion models. Borodin et al. [5] extend the well-stdied linear-threshold model in three different ways to model competitie inflence diffsion. he competitie inflence maximization problem corresponds to the isse of competitie information diffsion, which aims to expand the inflence coerage of a piece of information at the same time that its competitie information is spreading. he literatre on the competitie inflence maximization problem is abndant. CLDAG algorithm is designed to sole the inflence blocking maximization problem nder the competitie linear threshold model in the et al. [6]. Carnes et al. [7] propose two models for competitie information diffsion, and gie an efficient algorithm for the inflence maximization problem from the follower s perspectie. In this paper, we make a contribtion to the competitie inflence propagation in microblog nder the competitie independent cascade model. Competitie inflence maximization problem is redefined as adaptie inflence maximization problem which takes the cost of initial seeds into consideration and select the initial seeds adaptiely. he method for adaptie inflence maximization is worth frther researching and widely sed in the commercial field. First of all, we introdce the competitie independent cascade model in sina microblog in Section II. Adaptie inflence maximization problem is proposed in Section III, while applying the Monte-Carlo greedy algorithm to try to sole the AIM problem. A more efficient algorithm called M-based algorithm is designed to sole AIM problem in Section IV. At last, we make a comparison between these two methods and draw a conclsion abot the whole paper. II. COMPEIIVE INDEPENDEN CASCADE MODEL IN SINA MICROBLOG Chen et al. [3] proposed the competitie independent DOI: /IJKE.215.V

2 International Jornal of Knowledge Engineering, Vol. 1, No. 2, September 215 cascade model to describe the mlti-item diffsion, we apply CIC model to sina microblog and only consider two kinds of competitie information which we refer to as the positie information and the negatie information. In this model, a social network is considered as a directed graph G ( V, E, where V is the set of nodes respecting sers in sina microblog and E is the set of directed edges respecting inflence relationships between sers. Each node has three possible states: inactie, positie, negatie. he positie and the negatie inflence probabilities of each edge are the same, that is the propagating probability p(. A node may change from the inactie state to the positie state or negatie state, bt cannot change from actie state to inactie state, and the state of actiated nodes is no longer changes. he propagation process starts from initial seed sets S and S, and the reslt is to obtain the positie and negatie actie set S and S while the information to complete the diffse process. he diffsion rle [3] is as follows: 1 Gien the initial seed sets S and S with S S at time step t. At eery time step t 1, nodes in actiated seed sets S and t 1 S try to actiate their t1 neighbors. 2 Positie actiate attempt: eery actie node S try t1 in to actiate their neighbors N ( \ ( S S, carries ot a positie actiation attempt with sccessfl probability p(,. If the attempt is sccessfl, is added into a positie sccessfl attempt set for node, if not, no longer be actiated. 3 Negatie actiate attempt is in the same way with the aboe process of positie actiate attempt. In the case that a node is both positiely actiated and negatiely actiated in the same time, the state of the node is set to the positie. 4 he competitie diffsion ends with the final actie sets S and t S ntil the sets of positie and negatie nodes t will no longer change, that is, St S and t1 St S. t1 In or model, the diffsion probability p(, of node to node is proportional to the similar degree and familiar degree of node and node. he similar degree is represented by the microblog text similarity S(, of ser and ser, and the familiar degree is represented by the ratio of common friends F(,. p(, S(, F(,, 1 (1 where the parameters α and β indicate the inflence weight on diffsion probability of the similar degree and the familiar degree. he math definition of similar degree and familiar degree are as follows, S(,, S(, [,1] where and are the microblog text s key words of ser and ser, represents the key words nmber of text (2. where N N N F(,, F(, [,1] N N N, and ser, F (3 N represent the neighbor friends of ser,, represents the nmber of common friends of ser N represents the friend nmber of ser. 1 Fig. 1. A directed social graph in which nodes hae two attribtes: Microblog text and friends F. III. ADAPIVE INFLUENCE MAXIMIZAION PROBLEM In this section, we first gie the definition of the adaptie inflence maximization (AIM problem, then introdce the Monte-Carlo algorithm which is sed to sole the AIM problem and analyze its performance. A. Problem Definition he adaptie inflence maximization problem is a optimization problem in which gien a social graph G ( V, E, the aboe described competitie independent cascade model, a negatie seed set S, the goal is to find a positie seed set S V \ S, sch that the positie inflence benefit B is maximized. he positie inflence benefit B is the positie coerage ( S, S mins the cost of selected positie seeds CS (. CS ( is proportional to the size of positie seeds S. he mathematical representation is: * S V \ S S arg max ( S, S C( S (4 C( S S (5 Compared with the classical inflence maximization problem, there is no limit for the nmber of the reqired seed set in AIM problem. Besides, we take the cost of seeds into consideration so that make the inflence maximization problem hae a optimal soltion. 2 2 F

3 International Jornal of Knowledge Engineering, Vol. 1, No. 2, September 215 he inflence benefit fnction is sbmodlar bt not monotonos in or model. We say that a set fnction f :2 V R is sbmodlar if for any sbsets S V and anyelement V \, f ( S { } f ( S f ( { } f (. First of all, we proe the sbmodlarity of the inflence benefit fnction in here. For any sbsets S V and any element V \, to get ( ( S { }, S S { } ( (, S S S S ( ( { }, S { } ( (, (6 is eqal to get ( S { }, S ( S, S ( { }, S (, S (7 becase the formla (7 is eqialent to the formla (6 with both sides of the former ineqality sbtract a constant *. he aboe CIC model in sina microblog is homogeneos becase the propagation probabilities of each edge are the same for positie and negatie inflence, and follows B-FP(1 reglation. he positie inflence coerage fnction is sbmodlar that is proed in chapter 4 of Wei Chen et al. [3]. So the ineqality (7 is tre and ineqality (6 is proed. herefore, he inflence benefit fnction ( S, S is sb-modlar. It is obios that inflence benefit is not monotonos so that AIM problem can be soled by Monte-Carlo greedy algorithm sing the sbmodlarity and monotonicity of inflence coerage. Next sbsection describes the Monte-Carlo greedy algorithm for the NP-hard problem in detail. B. Monte-Carlo Greedy Algorithm Monte-Carlo Greedy Algorithm is sed to sole the inflence maximization in single information diffsion process in [3], we apply it to sole AIM problem in competitie information diffsion process. he main idea of Monte-Carlo greedy algorithm is adding the node which makes the inflence coerage maximm into the positie seed set ntil the nmber of seed set reach the specified nmber. his algorithm tilize the sbmodlarity and monotonicity property of the inflence coerage. Gien seed set S, we simlate the randomized diffsion process for R times to get the aerage of the nmber of the positie actie nodes. We take the aerage of these conts in order R to get high accracy in or estimate. In the competitie inflence diffsion model, the inflence spread is a piece of information s actiated nodes nmber while two pieces of information propagate. In this Algorithm, we first determine the nmber of selected positie seeds k ranging from 1 to the size of the graph N. hen we find the optimal seeds set to maximize the positie inflence benefit. Monte-Carlo Greedy Algorithm can sole the AIM problem, howeer, the greedy algorithm reqires a large nmber of traersal to choose the optimal node, which cannot be done efficiently. It takes abot O(n 2 logn time to sole the AIM problem. When dealing with larger networks, the time consmption is nacceptable. In next section, we propose a faster algorithm to select seed nodes adaptiely. he Monte-Carlo greedy algorithm scheme can be obtained in Algorithm I ia the following recrsion procedres: Algorithm I Monte-Carlo greedy algorithm for adaptie inflence maximization. Inpt: G : social graph inclding arc weights for CIC model; S : the negatie seed set Otpt: S : the selected positie seed set Initialize S ; for k 1: N for j 1: k arg max ( S, S, G S S {} end S arg max B S end retrn S ; wv \( S S R IV. M-BASED ALGORIHM FOR HE AIM PROBLEM A. M-Based Algorithm he positie and negatie inflence coerage is not only sbmodlar bt monotonos. While a node is added in positie seed set, the positie inflence coerage may remain the same or increase, the positie inflence benefit whose change is continos will decrease or increase accordingly. It is easy to know the inflence benefit fnction is conex. So the positie benefit has a maximm ale, the node which can enhance the positie inflence benefit will be selected as the positie seeds by browsing the whole nodes except negatie seeds. his idea is based on the maximm ale property of the inflence spread, so we call this idea M-based algorithm. he mathematical description of M-based algorithm is as follows: gien a social network considered as a directed graph G ( V, E and a negatie seed set S, we want to select a positie seed set S to maximize the positie inflence benefit IB( S. First of all, S is set to an empty set. If V \ S can make the positie inflence benefit IB( S increasing, is added in positie seed set S, ntil there is no node can enhance the positie inflence benefit. Becase the diffsion process is random, IB( S is a estimated ale which is the aerage of positie inflence benefit in M times competitie diffsion. he nmber of the positie seeds do not need to specify, we obtain the adaptie seed set in this algorithm. In terms of complexity it takes abot O(n times to sole the AIM problem. he M-based algorithm process can be obtained in Algorithm II: Algorithm II M-based algorithm for adaptie inflence maximization. Inpt: G : social graph inclding arc weights for CIC model; S : the negatie seed set 131

4 International Jornal of Knowledge Engineering, Vol. 1, No. 2, September 215 Otpt: S : the selected positie seed set Initialize S ; V \ S ^ if IB( S { } IB( S S ( ^ IB S ; IB( S M M ^ Fig. 3 shows the positie inflence benefit s change sing M-based algorithm. Fig. 4 shows the program rnning times sing Monte-Carlo Greedy algorithm and M-based algorithm. From the reslt we can see M-based algorithm faster than Monte-Carlo Greedy algorithm, bt the inflence benefit is slightly smaller. Becase the diffsion has random property, the inflence benefit may float. And the experiment shows selected seed set is not niqe. B. Experiments and Reslts In this section, we se a dataset contains the directed ser network and their microblog content extracted from sina API in 214. Each node in the network represents an ser, and each directed edge connecting two nodes indicates that the two sers hae following relationship. he dataset is consist of 592 sers. able I reports the basic statistics of the dataset. ABLE I: SAISICS OF HE DAASE Nmber of nodes 592 Nmber of edges Aerage degree 5 Maximal degree 327 Eery node in the dataset is nmbered, the node nmber ranges from to 591. he negatie seeds are selected randomly in this experiment, their nmber are 22, 12, 233, 389, 42, 44, 449, 53, 553. In or experiments, the seed cost parameter γ is set to 1, Monte-Carlo Greedy algorithm with R = 1 nder CIC model, M-based algorithm with M = 1 nder CIC model, and the reslt is the aerage of many times simlation. At the same time, the reslt cre is smoothed by Origin Pro with the parameter points of windows is 2. Simlation compter configration is Intel CPU with 3.4GHZ and RAM with 5.82G. Fig. 3. Experiment reslt of m-based algorithm. Fig. 4. Rnning time of Monte-Carlo greedy algorithm and M-based algorithm. V. CONCLUSIONS In this paper, we inestigate the adaptie inflence maximization in competitie diffsion of social network. We design M-based algorithm to oercome the slowness of the Monte Carlo greedy algorithm. Howeer, the problem is still worth contining research. Fig. 2. Experiment reslt of Monte-Carlo greedy algorithm. We se aboe two algorithms to select positie seeds, 72 positie seeds which reach the maximm positie inflence benefit are selected by Monte-Carlo Greedy algorithm, 7positie seeds which reach the maximm positie inflence benefit are selected by M-based algorithm. he program of selecting optimal positie seeds sing Monte Carlo Greedy algorithm takes hors and the program sing M-based algorithm takes.853 hors. Fig. 2 depicts the positie inflence benefit s change with the increase of the positie seeds nmber sing Monte-Carlo Greedy algorithm. REFERENCES [1] M. E. Newman, he strctre and fnction of complex networks, SIAM Reiew, ol. 45, no. 2, pp , 23. [2] D. Kempe, J. Kleinberg, and E. ardos, Maximizing the spread of inflence throgh a social network, in Proc. the Ninth ACM SIGKDD International Conference on Knowledge Discoery and Data Mining, ACM, 23, pp [3] W. Chen, L. V. Lakshmanan, and C. Castillo, Information and inflence propagation in social networks, Synthesis Lectres on Data Management, ol. 5, no. 4, pp , 213. [4] M. Broecheler, P. Shakarian, and V. Sbrahmanian, A scalable framework for modeling competitie diffsion in social networks, in Proc. 21 IEEE Second International Conference on Social Compting (SocialCom, IEEE, 21, pp [5] A. Borodin, Y. Films, and J. Oren, hreshold models for competitie inflence in social networks, Internet and Network Economics, Springer, 21, pp

5 International Jornal of Knowledge Engineering, Vol. 1, No. 2, September 215 [6] X. He, G. Song, W. Chen, and Q. Jiang, Inflence blocking maximization in social networks nder the competitie linear threshold model. SDM, SIAM, 212, pp [7]. Carnes, C. Nagarajan, S. M. Wild, and A. V. Zylen, Maximizing inflence in a competitie social network: a follower s perspectie, in Proc. the Ninth International Conference on Electronic Commerce, ACM, 27, pp Zheng Ding receied a B.E. degree in commnication engineering from Chongqing Uniersity of Posts and elecommnications, Chongqing, China, in 212. Crrently she is working towards the M.S. degree in information and commnication engineering in Beijing Uniersity of Posts and elecommnications, Beijing, China. Her research interests are in the area of data mining and the Complex network dynamics. commnication, particlarly on the complex network theory. Zhiqiang He receied the B.E. and Ph.D. degrees (with distinction in signal and information processing from Beijing Uniersity of Posts and elecommnications, China, in 1999 and 24, respectiely. Since Jly 24, he has been with the School of Information and Commnication Engineering, Beijing Uniersity of Posts and elecommnications, where he is crrently an associate professor and the director of the Center of Information heory and echnology. His research interests inclde signal and information processing in wireless commnications, data mining in social network, and nderwater acost. Kai Ni receied a B.S. degree in information engineering and a Ph.D. in signal and information processing from Beijing Uniersity of Posts and elecommnications (BUP, Beijing, China, in 1998 in 23, respectiely. Crrently he is a professor in the School of Information and Commnication Engineering of BUP. His research interests are in the area of data mining and broadband wireless 133