An Algorithm for Weighted Positive Influence Dominating Set Based on Learning Automata

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1 4 th Internatonal Conference on Knowledge-Based Engneerng and Innovaton (KBEI-2017) Dec. 22 th, 2017 (Iran Unversty of Scence and Technology) Tehran, Iran An Algorthm for Weghted Postve Influence Domnatng Set Based on Learnng Automata Mohammad Mehd Dalr Khomam Soft computng laboratory, Department of Computer Engneerng and Informaton Technology Amrkabr Unversty of Technology (Tehran Polytechnc), Tehran, Iran Maryam Amr Haer Department of Computer Engneerng & IT, Amrkabr Unversty of Technology (Tehran Polytechnc) Tehran, Iran Mohammad Reza Meybod Soft computng laboratory, Department of Computer Engneerng and Informaton Technology, Amrkabr Unversty of Technology (Tehran Polytechnc), Tehran, Iran Al Mohammad Saghr Soft computng laboratory, Department of Computer Engneerng and Informaton Technology, Amrkabr Unversty of Technology (Tehran Polytechnc), Tehran, Iran Abstract The problem of Influence Maxmzaton (IM) n socal network refers to determnng a set of nodes that can maxmze the spread of nfluence. IM problem has been appled n many domans such as marketng, advertsng and publc opnon montorng. In recent years, dfferent type of algorthms reported n the lterature. A type of algorthms for ths problem s based on domnatng set. In these algorthms, the IM problem s consdered as a verson of domnatng set problem. A lmtaton of most of these algorthms, the graph of the socal network s assumed to be a un-weghted graph whch s not a realstc assumpton. The other drawback of these algorthms s that they generally omt the crucal characterstcs of the socal networks such as dfferent level of nfluence and dynamc nteracton among ndvduals. In order to solve these problems and based on the realstc nature of the socal network, t seems that the Mnmum Weghted Postve Influence Domnatng Set (MWPIDS) problem can support the mentoned characterstcs because t consder the weght of the graph. In ths paper, a learnng automaton based algorthm s proposed to reveal MWPIDS n the socal network graphs. In the proposed algorthm, each vertex of the socal network graph s equpped wth a learnng automaton that determnes the beeng canddate or non-canddate of the correspondng vertex to be n WPIDS or not. Owng to adaptve decson makng characterstcs of learnng automata, the proposed algorthm sgnfcantly reduces the number of canddate soluton. The proposed algorthm, based on learnng automata, teratvely decreases the weght of the obtaned postve nfluence domnatng. In order to evaluate the proposed algorthm, several experments have been conducted on real socal network datasets whch compared to the state-of-the art methods. Expermental results show the superorty of the proposed algorthm over the prevous algorthms. Keywords Learnng Automata; Socal Network Analyss; Domnatng Set; Weghted Postve Influence Domnatng Set I. INTRODUCTION Due to the popularty of onlne socal networks such as Facebook, Myspace and Lnkdn provde companes new settng for enablng large-scale markng through the powerful word-of-mouth effect. Recently, extensve researches around socal network analyss have been nvestgated. They have shown that onlne users tend to share a new nformaton or trust to a news whch s ganed from ther close socal groups such as frends and acquantances nstead of trust to comments of ordnary people or news from publc meda. Many researchers on socal network analyss have been focused on sgnfcant characterstc of socal networks. In most of the studes, a socal network can usually be represented as a network wth a set of nodes corresponds to onlne users and edges representng a relatonshp between users. One of the sgnfcant characterstc of socal networks s the vral markng that ndcates the tendency of ganng largest nfluence on people through adoptng the product. In [1] s shown that the vral markng s effcent than other marketng polces. From the algorthmc aspect the term of vral marketng s transformed nto the Influence Maxmzaton (IM) that s known as an optmzaton problem. The man goal of the nfluence maxmzaton n a socal network s to fnd small k vertces (referred to as ntal seeds) n the networks, such that under the certan dffuson model, the expected number of vertces nfluenced by k s maxmzed. In ths paper, we propose a learnng automata based algorthm for solvng Mnmum Weghted Postve Influence Domnatng Set (MWPIDS) consderng the characterstcs n the socal networks. On the other hand, the theory of learnng automata has been shown ts potental as an adaptve decson makng unt n the dynamc and complex envronment such as sensor networks[2], peer to peer networks[3] and socal networks[4]. 1 Unversal Scentfc Organzaton KBEI-2016

2 The rest of the paper s organzed as follows: In secton 2 revews related work. In secton 3 the theory of learnng automata s ntroduced brefly. In secton 4 the proposed learnng automata s descrbed. The performance of the proposed algorthm s evaluated through the smulaton experments n secton 5. We make concluson and future works n secton 6. II. RELATED WORKS Clark et al. [5] proved that fndng the Mnmum Domnatng Set and ts varatons are NP-hard and proved exponental tme complexty. Several real practcal applcatons of ths problem are ntroduced n socal networks. For example, n terms of nfluence maxmzaton due to the polcy of product adopton, one person can change the decson by acceptng or refusng an offer. There are many researchs have been focused on Postve Influence Domnatng Set (PIDS), however, there are only a lttle works done on Weghted Postve Influence Domnatng Set. In the followng, we wll revewed the outcomes of these studes n a nutshell: The mlestone work for mnmum postve nfluence domnatng set problem s presented by Wang et al. [6]. They proposed an approxmate greedy algorthm and proved that the problem s APX-hard. Moreover, the author s studed the PIDS(Postve Influence Domnatng Set) problem under the determnstc lnear threshold model, n whch the nfluence from a par of nodes s computed by total ncomng nfluence from ts neghbors exceeds a pre-determned threshold. Recently, Rae et al. [7] proposed a new greedy algorthm whch mproves the tme complexty of the prmtve algorthm for PIDS. Although the greedy algorthm acheves many promsng results, but n some cases t s lmted to fnd proper approxmaton and produce unfavorable results. Zhang et al[8] studed PIDS n power-law graph and proved the greedy algorthm wth constant factor. Dneh et al. [9] studed the PIDS from dfferent ponts, and proposed the Total Postve Influence Domnatng Set (TPIDS) nstead of PIDS and consdered fracton of neghbors nstead of half. In addton, they proved an approxmaton factor for TPIDS and showed that n power-law networks whch follow power-law degree dstrbuton, both PIDS and TPIDS admt constant factor approxmaton. Basuchowdhur et al. [10] nvestgated a more generc problem whch s smlar to fndng mnmum domnatng set, but t s dfferent from the nave algorthm n terms of the number of hops needed to reach all nodes n the network and known as Mnmum k- Hop Domnatng Set. Intutvely, by ncreasng the parameter k correspondng to the number of hops, t ncreases the number of nodes whch s trggered by the seeds. In [11] nvestgated partal postve nfluence domnatng set and appled general threshold model for all nodes n networks. Threshold functon s determned as a value of certan fracton of neghbors whch are n actve state. A remarkable applcaton of PIDS s maxmzng the nfluence among partcpant n networks. Gven a socal network, the problem of nfluence maxmzaton refers to select a set of nodes such that the spread of nfluence based on a gven model s maxmzed. In order to modelng the mcro behavors of the ndvduals when they are exposed to specfc nformaton n networks, spreadng models have been proposed. Two well-known models ncludng ndependent cascadng model (IC) and lnear threshold model (LT) are presented n [12]. Based on spreadng models, Kempe et al. [13] nvestgated the nfluence maxmzaton problem n a socal network for fndng a set of ntal nodes that the spread of nformaton s maxmzed. We Chen et al. [14] nvestgate the IM, and modfed the algorthms to apply n the context of large-scale data mnng n real socal networks. Chen et al. [15] proposed a heurstc based algorthm for nfluence maxmzaton n arborescence whch s called (MIA) to change the scalablty of the algorthm usng the maxmum nfluence path of each par of nodes. Km et al. [16] proposed scalable and parallelzable nfluence approxmaton algorthm by calculatng ndependent paths n networks. Yang et al. [17] extended the nfluence maxmzaton problem and proposed a coordnate descent method to solve the transmsson cost problem. Goyal et al. [18]used the smple path between neghbor nodes to estmate the nfluence spread n networks. Due to the dynamc and complex nature of socal networks, all works whch were revewed n ths paper suffer from two major challenges that descrbed as below. Postve and negatve nteractons through communcaton should be fully captured. Note that varyng the behavor of the users reflected by postve and negatve nteractons affect the decsons of the algorthm whch analyses the socal network. Snce the propertes of users are dfferent from each other, they reflect dfferent levels of nfluence durng the communcaton whch IM gnores them. None of exstng algorthms reported for PIDS and ts varatons consders the menton characterstcs. Our man contrbutons are lsted as follows: We use a realstc model whch s superor to model ude by the pror Influence Maxmzaton algorthm. We desgn an effcent learnng automata based algorthm to solve WPIDS n socal networks. Our proposed algorthm s the frst learnng based algorthm for solvng the WPIDS We expermentally proved that the obtaned results are comparable and compettve wth exstng stateof-the art algorthms. III. STATEMENT OF THE PROBLEM The socal networks can be represented by a graph G ( V, E, W ) where V V, V,..., V } { 1 2 n s a set of nodes, 2

3 E V V s a set of edge and W ndcates the weght functon such that F : v. Mnmum Weghted Postve Influence Domnatng Set (MWPIDS) s a subset of S V such that each node v V s domnated by at least n 2 w v nodes n S wth mnmum weght where n s the degree of node v. The topology of the underlay learnng automata network ' ' ' can be represent by a graph G ( V, E ) n whch ' V { 2 LA1, LA,..., LA n } E LA ' s V underlyng network and connectng the networks, each node s a set of learnng automata n ' ' ' V V s a set of lnks n the underlay network. In socal s mapped to an LA ' n learnng automata network based on one to one functon ' H : V V. A prmary goal of MWPIDS s to learn the select optmal acton among a lst of canddate ones n whch s guded toward the optmal soluton. Fg 1 demonstrates a snapshot of the graph of the socal network. Node w Node Nodej Node k Node n Node Node l m Fg. 1. A Snapshot of a graph of the socal network IV. LEARNING AUTOMATA Learnng automaton (LA) [12, 14] s an abstract model of fnte state machne so that t can perform fnte actons whch randomly chosen and evaluated by a stochastc envronment and response s gven as a reward or penalty to LA. LA uses feedback of envronment and select the next acton. Durng ths process, LA learns how to choose the best acton from a set of allowed actons. Fgure 2 llustrates the relatonshp between the LA and the t s random envronment. LA wth varable structure can be descrbed wth quadruple denotes the fnte set {,, P, T} where { 1,..., r } of actons and,..., } { 1 m s the set of nput values that can be taken by renforcement sgnal and P { p 1,..., p r } P( n 1) T[ ( n), ( n), p( n)} s the probablty vector of each acton. s learnng algorthm where s updated n each teraton. Equatons (1) and (2) llustrate changes n probablty vector accordng to performance evaluaton of α n each step. Automaton updates ts acton probablty set based on equaton (1) for favorable responses as: p ( n 1) p ( n) a[1 p ( n)] (1) p ( n 1) (1 a) p ( n) j j j, And equaton (2) for unfavorable ones: p ( n 1) (1 a) p ( n) b p j ( n 1) ( ) (1 b) p j ( n) r 1 P(n j j, j Where ) s the acton probablty vector at nstant n. r s the number of actons that can be taken by the LA. The learnng rates a and b denote the reward and penalty parameters and determne the amount of ncreases and decreases of the acton probabltes, respectvely. If a=b L learnng algorthm s called lnear reward penalty ( R P ), f b<<a gven learnng algorthm s called lnear reward-ε penalty ( ), and fnally, f b=0 t s called lnear reward LR p nacton ( L ) [24-26]. R I V. THE PROPOSED ALGORITHM In ths secton, a learnng automata based algorthm for fndng MWPIDS problem wll be proposed. In ths algorthm, each node of the socal network s equpped wth a learnng automaton whch consst of two actons canddate and non-canddate. The network of the learnng automata resded n the nodes of the socal network has been utlzed to fnd a proper WPIDS. For the sake of more clarfcaton about the relatonshp between learnng automata and the graph of the WPIDS Fgure 2 s gven. The detaled descrptons of the proposed algorthm are gven n the rest of ths secton. (2) Random Envronment Acton Response Learnng Automata Fg.2. The relatonshp between the LA and ts random envronment 3

4 Node w Node n LA o LA n Node m Node Node l LA Nodej LA j Node k LA k Mean probablty nsoluton 1 n Max( P ) soluton Where Max P ) corresponds to the maxmum probablty ( of learnng Automata and n soluton (3) demonstrated as the number of canddate LA whch s present n soluton. The general structure of proposed algorthm s shown n the followng dagram as Fgure 3. Thus LA learns how to choose the best acton wth for WPIDS n successve teratons. Ths process wll be contnued untl the algorthm reaches to a near optmal soluton. In the next secton the experment smulatons are demonstrated n order to evaluate the proposed algorthm. LA m LA l Fg2. A snapshot of mappng the network of the learnng automata The proposed algorthm conssts of four steps: 1. Intalzaton, 2. Actvaton and acton selecton, 3. Verfcaton and calculatng renforcement sgnal and, 4. Updatng probablty vector. These steps are descrbed as follows: In the ntalzaton, each node n the networks s equpped wth learnng automata that has two actons canddate and non-canddate and ntally are dsabled. At frst all learnng automata are actvated and choosng ts acton randomly. Because of probabltes of all acton n frst teraton are equal, hence each acton s selected wth equal probablty. After an acton selecton step s done the verfcaton and calculatng renforcement sgnal s stars. In ths step, the set of learnng automata whch selects the canddate actons are checked n terms of correctness, whch means that the obtaned set s WPIDS or not. Then ther weghts are calculated. If the weght of WPIDS s less than the former WPIDS then the selected acton corresponds to current learnng automata whch s presented n the set are rewarded else the selects acton are penalzed. Ths process s contnued untl the weght of MPIDS s equal to pre-defned threshold. Fgure 3 denotes the flowchart of the proposed algorthm. Note that, at frst, learnng automata chooses ts acton wth equal probablty and wll gradually converge to an approprate value durng teratons of the learnng process. One of the most mportant parts n all learnng algorthms s how the algorthm s termnated. In ths way, there are varous solutons are suggested. One of the common way s usng a lmtaton on the number of teratons. If the number of teratons exceeds from a predefned threshold, then the algorthm stops. Another approach s computng the mean of the probablty vector. After some teratons, f the mean of the probablty vector s fxed then the algorthm stops. The mean of probabltes s expressed n equaton (3) Intalzaton of the network of LAs Weghted Postve Influence Domnatng Set (WPIDS) Formaton No Start WPIDS Verfcaton and Envronment Evaluaton Termnaton Condton End Yes Fg3. Flow Chart of Proposed Algorthm based on Learnng Automata for WPIDS. VI. SIMULATION RESULTS In order to evaluate the proposed algorthm, DIMACS [19] benchmark s used n experments. DIMACS s a wellknown benchmark. Ths benchmark contans dfferent networks whch are descrbed n Table 1. For the proposed algorthm, all experments have been mplemented usng L R- I learnng algorthm for the learnng automata. Snce the proposed algorthm operate on networks wth vertex weght and there s no dataset for the socal networks wth vertex 4

5 weght, the benchmark s modfed usng method reported by Pullan et al. [20] whch s descrbed as follows. Based on ths method, the weght corresponds to node v can be computed by ( mod 200) +1. Ths method was also used n[21, 22]. Table1. Descrpton of the datasets used for experments. Networks #nodes #Edges D max D avg Brock k C k C k Dsjc k The results are reported n terms of the mean and standard devaton wth the average of n 1000 runs for the proposed algorthm. Note that n denotes the number of nodes n the network. The executon tme of the proposed algorthm s reported n terms of mnute. The metrcs used for the comparson are descrbed as below. Dynamc Threshold (DT) : ths metrc computes the average weght of the PIDS. Mean-Value: Ths metrc s computed usng equaton (3). Number of actvated nodes: Ths metrc computes the number of actvated nodes of the network. The desgn of the experments s gven as follows. The frst experment s conducted to study the mpact of learnng rate on the executon tme and DT. The second experment s desgned to study the behavor of the proposed algorthm wth respect to DT and Mean value durng the process of fndng the soluton for WIPIDS n networks. The thrd experment s conducted to study the mpact of selectng nfluental nodes on the performance of the nfluence maxmzaton algorthms wth respect to the number of actvated nodes. Experment 1 Ths experment s carred out to study the mpact of rate a on the performance of the proposed algorthm performance of the proposed algorthms. For ths purpose, the WPIDS algorthm s evaluated for fve dfferent learnng parameter {0.001, 0.01, 0.0.5, 0.1, 0.5}. As can be seen n Fgure 5 and Fgure 6, the dynamc threshold and executon tme versus dfferent learnng rate are plotted respectvely. The results show that the proposed Algorthm for whch learnng rate s small, works better than when the learnng rate s large due to avodng local mnmum. But choosng a small learnng rate may ncrease the learnng tme. For ths problem, we have shown that expermentally the learnng rate 0.05 can create a trade-off between executon tme of the learnng algorthm and the accuracy of the obtaned soluton s determned by the learnng algorthm. Hence, for the rest of the experment, we select 0.05 as the learnng rate for the learnng algorthm. Dynamc Threashold Executon Tme(Mnute) Brock200 C250 C500 Dsjc Learnng rate (a) Fg 4. Brock200 C250 C500 Dsjc Learnng Rate (a) Fg. 5. Comparson of proposed algorthm n terms executon tme for dfferent learnng rate. Experment 2 Ths experment s nvestgated to study the behavor of the proposed algorthm durng the process of fndng the soluton for WIPIDS n networks. In order to perform ths experment, we plot both DT and Mean-Value versus teratons. We note that DT s scaled between (0, 1) by dvdng DT by the weght of maxmum WPIDS for each network. The results of ths experment for dfferent test graphs are gven n Fg. 6. As t s shown, DT gradually converges to the weght of mnmum postve nfluence domnatng set and at the same tme Mean-Value converges to 1. Ths means that the algorthm gradually tends to fnd the WPIDS wth mnmum. 5

6 2) a nodes. Based on these experment our algorthm s compared wth four well-known algorthm whch s reported for nfluence maxmzaton ncludng:,hgh-degree [12], PageRank[23], DegGreedy[24]and random generaton algorthm on the all datasets. In Hgh-degree algorthm k nodes s obtaned based on the maxmum degree. For PageRank algorthm, we have chosen the k hghest ranked nodes as ntal seed. The Page-rank algorthm computes and ranked nodes based on the sgnfcance. The DegGreedy algorthm try to maxmze the nfluence based on node neghborhoods greedly and proved effcency of the algorthm n terms of scalablty and hgher spread of nfluence. In random nodes are selected unformly at random. For ths experment, the learnng rates of the LA s are set to In addton, for all networks, we assume that the threshold 0. 5 based the Lnear Threshold model. In addton we consdered that all undrected datasets are double weght and drected for nfluence maxmzaton, n whch the weght node s equal to 1/d where d s the degree of node. Fgure 7 shows number of actvated nodes n comparson wth ntal seed sze for all data sets. 6)b 6) c 7)a 6)d Fg 6. The plot of DT and Mean-Value versus teraton number for dfferent Networks 7)b Experment 3 The goal of ths experment s to study the mpact of selectng nfluental nodes on the performance of the nfluence maxmzaton wth respect to number of actvated 6

7 showed that the proposed algorthm leadng to promsng results n terms of the number of actvated nodes for nfluence maxmzaton. Note that, due to the exponental complexty of WMPIDS problem, the algorthm s not supposed to reach to an optmal soluton n reasonable tme, but the results confrm that proposed algorthm produces better solutons than other well-known algorthms n terms of nfluence maxmzaton. 7)C 7)D Fgure 7: Comparng spread of nfluence usng random, hghdegree, PageRank, deggreedy and WPIS algorthms for the datasets As can be seen from fgure 7, t's compared spread of nfluence usng random, hgh-degree, PageRank, DegGreedy and WPIDS algorthms for the all datasets. From the result, we may conclude that, DegGreedy algorthm outperforms than other heurstc algorthms, n addton hgh-degree centralty algorthm whch relyng solely on the structural propertes of a network doesn't make sense for nfluence maxmzaton. On the other hand, we need to also explctly capture the dynamcs of nformaton whch s captured by WPIDS nodes n the networks. VII. CONCLUSION In ths paper, an algorthm for mnmum weghted postve nfluence domnatng set n socal networks usng learnng automata was proposed. Ths algorthm s the frst algorthm for the mnmum weghted postve nfluence domnatng set problem called MWPIDS n the lterature. Based on the proposed algorthm, each node wth the am of a learnng automaton s able to determne tself to the canddate or non-canddate for the MWPIDS. The proposed learnng automata based algorthm, take the advantage of cooperaton of learnng automata wth each other to fnd MWPIDS. In order to evaluate the performance of the proposed algorthm, a number of experments have been conducted on popular networks. Expermental results VIII. REFERENCES [1] Farbank, V., A study nto the effectveness of vral marketng over the nternet. Retreved Aprl, : p [2] Esnaashar, M. and M. Meybod, A cellular learnng automata based clusterng algorthm for wreless sensor networks. Sensor Letters, (5): p [3] Saghr, A.M. and M.R. Meybod, A Self-adaptve Algorthm for Topology Matchng n Unstructured Peer-to-Peer Networks. Journal of Network and Systems Management, (2): p [4] Rezvanan, A., M. Rahmat, and M.R. Meybod, Samplng from complex networks usng dstrbuted learnng automata. Physca A: Statstcal Mechancs and ts Applcatons, : p [5] Clark, B.N., C.J. Colbourn, and D.S. Johnson, Unt dsk graphs. Dscrete mathematcs, (1-3): p [6] Wang, F., et al., On postve nfluence domnatng sets n socal networks. Theoretcal Computer Scence, (3): p [7] Rae, H., N. Yazdan, and M. Asadpour. A new algorthm for postve nfluence domnatng set n socal networks. n Proceedngs of the 2012 Internatonal Conference on Advances n Socal Networks Analyss and Mnng (ASONAM 2012) IEEE Computer Socety. [8] Zhang, W., et al., Postve nfluence domnatng sets n power-law graphs. Socal Network Analyss and Mnng, (1): p [9] Dnh, T.N., et al., On the approxmablty of postve nfluence domnatng set n socal networks. Journal of Combnatoral Optmzaton, (3): p [10] Basuchowdhur, P. and S. Majumder. Fndng nfluental nodes n socal networks usng mnmum k-hop domnatng set. n Internatonal Conference on Appled Algorthms Sprnger. [11] Zhang, Z., et al. Vral marketng wth postve nfluence. n INFOCOM 2017-IEEE Conference on Computer Communcatons, IEEE IEEE. [12] Kempe, D., J. Klenberg, and É. Tardos. Maxmzng the spread of nfluence through a socal network. n Proceedngs of the nnth ACM 7

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