Identifying Top-k Most Influential Nodes by using the Topological Diffusion Models in the Complex Networks

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1 (IJACSA) Internatonal Journal of Advanced Computer Scence and Applcatons, Vol. 8, No., 07 Identfyng Top-k Most Influental Nodes by usng the Topologcal Dffuson Models n the Complex Networks Maryam Padar, Sarkhosh Seddgh Chaharborj,, Al Harounabad Department of Computer Software, Islamc Azad Unversty, Bushehr branch, Bushehr, Iran School of Mathematcs and Statstcs, Carleton Unversty, Ottawa, Canada Department of Computer Software, Islamc Azad Unversty Central Tehran Branch, Iran Abstract Socal networks are sub-set of complex networks, where users are defned as nodes, and the connectons between users are edges. One of the mportant ssues concernng socal network analyss s dentfyng nfluental and penetrable nodes. Centralty s an mportant method among many others practced for dentfcaton of nfluental nodes. Centralty crtera nclude degree centralty, betweenness centralty, closeness centralty, and Egenvector centralty; all of whch are used n dentfyng those nfluental nodes n weghted and weghtless networks. TOPSIS s another basc and mult-crtera method whch employs four crtera of centralty smultaneously to dentfy nfluental nodes; a fact that makes t more accurate than the above crtera. Another method used for dentfyng nfluental or top-k nfluental nodes n complex socal networks s Heat Dffuson Kernel: As one of the Topologcal Dffuson Models; ths model dentfes nodes based on heat dffuson. In the present paper, to use the topologcal dffuson model, the socal network graph s drawn up by the nteractve and non-nteractve actvtes; then, based on the dffuson, the dynamc equatons of the graph are modeled. Ths was followed by usng mproved heat dffuson kernels to mprove the accuracy of nfluental nodes dentfcaton. After several re-admnstratons of the topologcal dffuson models, those users who dffused more heat were chosen as the most nfluental nodes n the concerned socal network. Fnally, to evaluate the model, the current method was compared wth Technque for Order Preferences by Smlarty to Ideal Soluton (TOPSIS). Keywords Topologcal Dffuson; TOPSIS; Socal Network; Complex Network; Interactve and Non-nteractve Actvtes; Heat Dffuson Kernel I. INTRODUCTION Most networks exstng around us are of complex type. These nclude neural networks, socal networks, organzatonal networks, computer networks, etc. []. Today, socal networks have drawn attenton more than the others. Every socal network s composed of two elements of users and relatonshps: Users are defned as any entty partcpatng n a relatonshp and are called Nodes; relatonshps are the connectons between enttes and are called Edges. Dfferent types of relatonshps (work, famly, frends, etc.) can exst between nodes []. Development of socal networks accelerates the spread of dfferent types of nformaton, ncludng rumors, news, deas, advertsements, etc. People s decsons to refuse or accept nformaton depends on the dffuson or spread method of the nformaton []. Therefore, choosng the ndvduals ntended for spreadng the nformaton gans much mportance. So far, several models have been proposed n socal networks for dentfcaton of those ndvduals who have socal nfluence among people. Many exstng socal nfluence models for the defnton of nfluence dffuson are based solely on the topologcal relatonshp of socal networks nodes. The deas of topologcal dffuson models can be used n the process of dffuson and spread of nfluence, and t can be evaluated through topologcal relatonshps among nodes n a socal network [4], [5]. In ths paper, t s attempted to examne the socal nfluence regardng the heat dffuson kernel phenomenon,.e., the dynamcal equatons are modeled based on heat dffuson. In fact, the heat dffuson process fnds the nfluental nodes n complex networks usng heat transfer laws based on nteractve and non-nteractve actvtes. Therefore, users who receve heat more than others and have a greater ncrease n temperature curve are dentfed as the most nfluental nodes n a socal network. Then, through defnng the modfed heat dffuson kernels, the accuracy of the dentfcaton of the nfluental nodes can be ncreased. II. RELATED WORK Doo & Lu n [] proposed a model of socal nfluence based on actvtes. Actvty-based socal nfluence s very effectve n fndng nodes n the socal networks. In ther paper, three types of topologcal dffuson,.e., the Lnear Threshold models (LT), Independent Cascade model (IC) and heat dffuson model have been fully expressed. The authors n [] carred out n 05 for the frst tme the problem of nfluence maxmzaton as a combnaton optmzaton problem. They consdered two nfluences spread models,.e. ndependent cascade and the lnear threshold and evaluated these models extensvely. But authors n [7] focused on the lnear threshold model and presented a standard greedy algorthm n whch the selecton of a node wth the maxmum edge s ncreased repeatedly. The authors n [8] dscussed mostly the tme-crtcal nfluence maxmzaton, where each node wants to reach maxmze nfluence spread wthn a gven deadlne. 0 P a g e

2 (IJACSA) Internatonal Journal of Advanced Computer Scence and Applcatons, Vol. 8, No., 07 The authors n [] presented three dffuson models together wth three algorthms for selecton of the best people. Ther paper presented a new approach for analyzng socal networks; subsequently, complexty analyss shows that the proposed model s also scalable to great socal networks. The authors n [] desgned a two-stage greedy algorthm (GAUP ) to fnd the most effectve nodes n a network; GAUP ntally computes the preferences of the user to a pattern that has latent feature model based on SVD or a model based on vector space, then to fnd top-k nodes n the second stage, t utlzes a greedy algorthm. A new technque [] known as Technque for Order Preferences was presented by Smlarty to Ideal Soluton (TOPSIS weghted) to mprove the rankng of node spread. Wth ths method, the authors n [] not only consdered dfferent centralty measures as the mult-attrbute to the network but also proposed a new algorthm to calculate the weght of each attrbute; and to evaluate ts performance n four real networks they used the Susceptble Infected Recovered (SIR) model to do the smulaton. Hu, et al. s experments on four real networks showed that the proposed method could rank the spreadng ablty of nodes more accurately than the orgnal method. III. METHOD Socal network models can be descrbed by mathematcal tools, such as graph and matrx. The most mportant property of graphs s ther topologcal capablty n whch a vertex s created for each user, and f two users are frends wth each other, the two vertces are connected. If a drecton s attrbuted to each edge, the sad network s called drected, n whch the order of vertces n an edge s mportant. Each row n the matrx corresponds to a vertex, and each column n the matrx also corresponds to a vertex. If there s a relatonshp/edge between two vertces of an edge, number s used; otherwse, number 0 s nserted []. Users can send text, photo or vdeo and lke ther frends, or wrte comments for them; hence, the users actvtes can be dvded nto two categores: the nteractve actvtes,.e., those user actvtes that nclude nodes other than themself, and the non-nteractve actvtes, whch nclude only hmself and not someone else. As an example, permsson for commentng on the profle pcture of another person s an nteractve actvty because t s mplemented between two nodes and the pcture n the profle page s a non-nteractve actvty because t ncludes only the node tself. If s taken as a row and j as a column of a matrx, assume that IA j represents some nteractve actvtes from node v to the neghborng node v j and NIA are some non-nteractve actvtes at node v. Upon combnaton of nteractve and non-nteractve actvtes, there are several methods for spreadng heat dffuson kernel. Interactve and non-nteractve actvtes are defned as IA j / IA jk and NA / MAX ( NA), respectvely. k:( v, ) E j v k The MAX(NA) s defned as the largest number of non- Greedy Algorthm based on Users Preferences (GAUP) Sngular Value Decomposton (SVD) nteractve actvty n V []. Fg. s an example of the topologcal structure of Twtter socal network dsplayng top users. The postve nteger numbers on the edges represent the number of nteractve actvtes, and postve nteger numbers wth underlne at each node represent the number of non-nteractve actvtes mplemented by that node. v v v v v 5 Fg.. A sample of socal network (nteractve and non-nteractve actvtes). Topologcal dffuson, as one of the processes of nfluence dffuson n socal networks, exhbts the spread of nfluence through topologcal relatonshps between nodes. Heat dffuson model s one of such topologcal dffuson models. In ths model, t s assumed that at the ntal tme t 0, n a socal network of n nodes, all heat nodes except the node v has prmary heat of zero. Node v whch has some heat, s selected as a heat source and at tme t, v dffuses the heat equally between all ts neghbors. At tme t, nodes wth non-zero heat dffuser ther heat to all ther neghbors. Wth the repetton of ths process for a perod t for all n nodes, the nfluence of each node s found. Suppose that G = (V, E) represents a drectonal graph of socal network, V = {v, v,..., v n } s a set of vertces representng the number of users, and E = {(u,v) u,v V} s a set of edges representng the frendshp relatonshp between users [4]. The heat at vertex v at tme t s defned by the functon H (t); heat flows from a hgh-temperature node to a low-temperature node followng the edges between vertces. In the drectonal graph, at tme nterval t, vertex v dffuses the heat to the amount of DH (t) through output edges to next nodes. At the same tme, the vertex v j, receves the heat RH (t) through nput edges. Heat varatons at vertex V v between the tme nterval t and t+t s defned wth the sum of the dfferences between the heat that receved, and the heat dffuses to, all ts neghbors. H ( t t ) H ( t ) RH ( t ) DH ( t ). () 70 v 5 5 DH (t) and RH (t) are defned as () and (): IA j RH ( t ) t H j( t ). j: ( v, ) E j v IA jk k: ( v, ) E jv k NA DH ( t ) t H ( t ). MAX ( NA) v v v 7 7 v 8 7 () () P a g e

3 (IJACSA) Internatonal Journal of Advanced Computer Scence and Applcatons, Vol. 8, No., 07 Parameter α, called heat dffuson coeffcent, controls the rate of heat transfer. Value of α s a real number between 0 and. If α tends to 0, heat transfer would be dffcult, and the heat wll not spread everywhere. If α tends to, wthout heat loss, all heat wll be dstrbuted among all neghbors,.e., f α s bg, the heat s dffused rapdly from one node to another. In ths paper, the value of α s taken as 0.5, whch means that the heat s transferred wth a lttle loss between nodes. β n () s used as a weght for non-nteractng actvtes and takes a real number between 0 and. If β s consdered 0, t means that some of the non-nteractng actvtes are gnored and have no effect on the heat dffuson process. But f β s set to, n heat dffuson process, node v loses ts heat wth a lower rate; n ths paper, n all curves, the value of β s taken to be 0.5. Equaton () and () can be combned wth equaton of heat varaton () and obtan (4). ( jk ) ( ) t k : (, ) E j k v v j: (, ) E j. H t t H t v v H j() t IA j IA NA H ( t ) MAX ( NA) In general, for n nodes, t can be wrtten as (5): (4) H ( t t ) H ( t ) tkh ( t ). (5) In (5), K s an n n matrx representng the heat dffuson kernel from graph G, and H(t) s a column vector representng heat dstrbuton at tme t n graph G, whch s defned for prmary heat source H(0). Now the lmt of (5) s taken on t when t approaches zero and gves (). H ( t t ) H ( t ) lm lm KH ( t ), t0 t t0 dh () t KH ( t ), () dt Then we take ntegral of both sdes of (): dh () t Kdt. (7) dt Now f we take the ln of both sdes of (7), the heat equaton s defned as (8): Kt H ( t ) H (0). (8) Then, usng heat change equatons RH and DH, two types of heat dffuson kernel are provded. Now, through MATLAB software, curves are drawn and wth the help of curves, the most nfluental node can easly be found. The frst defned kernel s called heat dffuson kernel based on nteractve and non-nteractve actvtes. Matrx K s defned as: e IA j / IA jk ; ( v j, v ) E k: ( v, ) j v E k K (, j ) NA / MAX ( NA) ; j and d 0 0 ; otherwse. Fg. (a) llustrates the topologcal structure of heat dffuson based on nteractve and non-nteractve actvtes. Ths graph has been calculated and plotted usng () and Fg.. As shown n the fgure node v neghbors sx nodes v, v, v 4, v, v 8 and v. Two nodes v 4 and v have four neghborng nodes; nodes v, v, v 5 and v neghbor node v 4 and nodes v, v 4, v 5 and v 7 are four neghbors of node v. Each of nodes v, v 5, v 7, v 8 and v s drectly related to two nodes. Fnally the nodes v and v have one neghbor, t means that the node v s the only neghbor of node v and node v s the only neghbor of node v. Fg. (b) s the dagram of heat dffuson n whch, as an example, node v 7 has been chosen as the heat source and s plotted by usng (8) the dagram of heat dffuson. X-axs s the tme lne and Y-axs s the amount of heat at each node. The heat dffuson n ths dagram s such that ntally (at tme zero), the heat source has a lot of heat and the rest of the nodes are wthout heat. At tme, v 7 dffused heat to ts neghbors (v and v 8 ), and these two nodes gan heat. But node v receves more heat and goes up. In ths manner, the heat source reduces wth tme due to dffuson of heat to other nodes and the other nodes also receve heat from ther neghbors and ncrease. Node v receves more heat because t s drectly lnked wth sx nodes and ncrease more than all nodes; so t s obvous that t s ascendng n the dagram. But nodes v and v wth just one neghbor wll be the lowest lmt of the dagram. A pont worthy to note n ths dagram s that n the tme nterval, two nodes v and v 4 are hgher than node v ; the reason for that s that these two nodes receve more heat than node v ; hence n the dagram, nodes v and v 4 have hgher ncreases. The degree of each node s one of the nfluental factors for nformaton dffuson. If d j represents the degree of each node, the weght on the edges connected to node v j s calculated as (): v v v Weght v 0.77 (a) v v j (). () d j v v Topologcal structure v 7 v v P a g e

4 (IJACSA) Internatonal Journal of Advanced Computer Scence and Applcatons, Vol. 8, No., 07 (a) Topologcal structure (b) Dagram of heat dffuson wth heat source v 7 Fg.. Heat dffuson based on nteractve and non-nteractve actvtes. Therefore, RH (t) can be changed nto (): H j(t) RH ( t) t. () j:( v j, v ) V d j Now a new kernel called heat dffuson kernel can defne through () and () based on the non-nteractve actvtes. H j(t) RH ( t) t. () j:( v j, v ) V d j NA DH ( t ) t H ( t ). MAX ( NA) Matrx K changes as (): / d j ; ( v j, v ) E K (, j ) NA / MAX ( NA) ; j and d 0 0 ; otherwse. () () Fg. (a) s drawn usng () and Fg.. As t was mentoned above, d j represents the degree of each node meanng that f the degree of each node s calculated based on (), weght of the edge between nodes s obtaned. As an example, vertces v and v 8 are two neghbors of node v 7 ; therefore, the weght of 0.5 s assgned to E(v 7, v ) and E(v 7, v 8 ), respectvely, whch means half of the heat of v 7 s transferred to v and the other half to v 8. Node v has just one frend, whch s vertex v. Thus, the weght of E(v, v ) s equal to. It means that the whole heat of v s dffused to v. Smlarly, the edge weght between vertces s calculated. For example, node v 7 s consdered as a heat source and the heat dffuson dagram s plotted n Fg. (b) usng (8). The heat dffuson process n ths dagram s as follows: at tme nterval zero, the heat source has hgh heat, and the rest of the nodes are wthout heat. (b) Dagram of heat dffuson wth heat source v7 Fg.. Heat dffuson based on non-nteractve actvtes. Wth the passage of tme, heat of the source reduces due to heat dffuson to other nodes; accordngly, heat ncreases n two nodes v and v 8, whch are drectly related to heat source (v 7 ). But as shown, fnally, t s node v, wth more lnks to other nodes, whch fnally receves more heat and goes hgher than node v 8. From tme nterval 4 onwards, node v ncreases and surpasses all other nodes, even the heat source. Node v, wth lnks to more nodes, receves hgher heat. Therefore, expectedly, t follows an ascendng trend. A comparson of Fg. (b) and Fg. (b) dagrams ndcates that n both dagrams, node v s hgher than all others; therefore, t s selected as the most nfluental node. But, as prevously mentoned, the y-axs represents the recept of the amount of heat; where each node recevng more heat goes hgher. Now, a comparson of node v n both dagrams reveals that node v n dagram Fg. (b) s hgher than n dagram Fg. (b); therefore, t can be stated that dagram Fg. (b) has receved more heat. It s thereupon mpled that the heat dffuson kernel based on nteractve and non-nteractve actvtes () acts better than the heat dffuson kernel based on non-nteractve actvty (); because the nodes n heat dffuson kernel based on nteractve and non-nteractve actvty have receved more heat. It s now attempted to obtan a modfed kernel by combnaton of kernels n such a way that the most nfluental node receves more heat (.e., t s hgher n the y-axs). Ths core s expressed as (): P a g e

5 (IJACSA) Internatonal Journal of Advanced Computer Scence and Applcatons, Vol. 8, No., 07 IA j / IA jk ; ( v j, v ) E k: ( v, ) j v E k d j K (, j ) NA / MAX ( NA ) ; j and d 0 0 ; otherwse. () of nteractve and non-nteractve method s shown as a crcle. As seen n the dagram of Fg. 5, the trangular curve s hgher than the crcle curve,.e. node v n the proposed method receves more heat, and therefore trangular curve has hgher ncrease. So the proposed kernel acts better than the heat dffuson kernel based on the nteractve and non-nteractve actvtes. Fg. 4(a), s the topologcal structure of the proposed kernel drawn usng Fg. and (). As can be seen, the weght on the edges vares when () s appled. Heat dffuson dagram wth heat source v 7 usng (8) s ndcated n Fg. 4(b). In ths dagram also the heat source has a descendng trend over tme due to heat dffuson to other nodes; and stll, node v wth more lnks has hgher heat and ncreases more than other nodes. Accordngly, node v s selected as the most nfluental node n the proposed kernel smlar to the prevously presented kernels. (a) The topologcal structure of proposed kernel Fg. 5. Comparng best kernel wth proposed kernel. IV. EVALUATION In ths secton, two real data sets, Revje.paj and EIES.paj, are selected; then usng Pajek software, the data s analyzed and mplemented usng MATLAB software. After that, TOPSIS method, whch s a multple crteron decson-makng method, and s defned based on postve deal soluton and negatve deal soluton, s used for evaluaton. TOPSIS model evaluaton and prortzaton procedure wll be as follows [4]: The frst step s development of a decson matrx; that s, usng m crtera, n ndexes wll be evaluated. The second step s normalzaton of the decson matrx. To that end, (4) s used, whch s a vector method. r j x j ;,, m ; j,, n. m x j j (4) The thrd step s makng weghted normalzed decson matrx where the weght of crtera s calculated by (5). v j w j r j ;,, m ; j,, n. (5) (b) Dagram of heat dffuson wth heat source v7 Fg. 4. Heat dffuson of proposed kernel. Heat dffuson kernel based on nteractve and nonnteractve actvtes s compared wth the proposed kernel n two dagrams Fg. (b) and 4(b) n Fg. 5. In ths dagram, too, node v 7 s selected as a heat source; contnuous lne represents the heat source of proposed method, and a dotted lne represents the heat source of the method of nteractve and nonnteractve actvtes. Node v (as the most nfluental node) for the proposed method s ndcated as trangular and for the case In the fourth step, the postve and negatve deals are calculated. The hghest performance of each ndex (postve deal) and the lowest performance of each ndex (negatve deal) are represented by A + and A -, respectvely. These two ndexes are calculated n () and (7). max v j j bmn v j j c A K K. () mn v j j bmax v j j c A K K. (7) 4 P a g e

6 (IJACSA) Internatonal Journal of Advanced Computer Scence and Applcatons, Vol. 8, No., 07 In the ffth step, Eucldean dstances of each crteron from the postve deal and negatve deal are calculated by (8) and (), respectvely. ;,, m ; j,, n. (8) Node v4 whch has the hghest prorty n TOPSIS model s consdered as the heat source. Top nodes wth the heat source v4 for Revje data set are shown n Table, n the order of prorty. ;,, m ; j,, n. TABLE. I. S v j v j n j S v j v j n j () The sxth step s the calculaton of deal soluton,.e., the relatve closeness of each crteron to the deal soluton, whch s obtaned by (0). C nodes v, v, v4, v, v7, v4, v and v and the lowest degree s 8, whch belongs to node v50. Ten hghest nodes prortzed by TOPSIS model are shown n Table. S S S ;,, m. (0) Prortzaton s based on the value of C where ths value can be 0 C. When ths value s closer to one, t ndcates the hghest rank, and when ths value s close to zero t ndcates the lowest rank. A. Revje Data Set Revje s a data set of Slovenan magaznes and journals publshed n and 000 [5]. 4 dfferent magaznes and journals were lsted, and over 0,000 people were asked to read magaznes and journals. A typcal network s created from ths data set n whch the magaznes are the vertces. In ths network, edges are drected and weghted; the rng at vertces corresponds to the readers of the magazne. If a reader has read two or more magaznes, then those magaznes are lnked together, and the weght on that ndcates the number of readers of two magaznes. The topologcal structure of ths data set s shown n Fg.. THE ARRANGEMENT OF SUPERIOR NODES BY THE TOPSIS MODEL FOR REVIJE DATA SET Prorty Model Model TOPSIS TABLE. II v4 v v v v v7 v v v v THE ARRANGEMENT OF SUPERIOR NODES WITH THE HEAT SOURCE V4 FOR REVIJE DATA SET Prorty Method Heat dffuson kernel based on nteractve and nonnteractve actvty Heat dffuson kernel based on the nonnteractve actvty Proposed heat dffuson kernel v4 v v v4 v 4 v v v7 v v v4 5 v 7 v v 0 v v v v v7 8 v v v v v7 0 v v 0 v v v v By comparng Tables and, t can be seen that n all methods, node v4 s the prorty, but other prortes are dfferent. For example, the second prorty n TOPSIS method s node v and n the heat dffuson method, node v. Hence, the heat dagrams of Revje dataset for two nodes v and v are drawn n Fg. 7 and from that, the most nfluental node wll be determned. For readablty of the dagrams, the heat source (node v4) s drawn only for the proposed heat dffuson kernel; heat dffuson kernel based on nteractve and non-nteractve actvtes are shown wth, heat dffuson kernel based on non-nteractve actvty s shown by, and the proposed heat dffuson kernel s shown by φ. Fg.. The topologcal structure of Revje data set. The data set of magaznes and journals have nteractve and non-nteractve actvtes and nclude 4 vertces and 8 edges. The hghest degree of nodes s, whch s related to As shown n Fg. 7, nodes v and v are compared for the above-mentoned heat dffuson models, n both dagrams (a) and (b). Fg. 7(b) whch s related to the node v, receves more heat compared wth dagram Fg. 7(a) and thus has a hgher ascendng trend. So node v s more nfluental than node v and has a hgher prorty. Hence, for advertsement and news dffuson, frstly node v and then node v are selected. 5 P a g e

7 (IJACSA) Internatonal Journal of Advanced Computer Scence and Applcatons, Vol. 8, No., 07 TABLE. III. Prorty Model Model TOPSIS THE ARRANGEMENT OF SUPERIOR NODES BY THE TOPSIS MODEL FOR EIES DATA SET v v v 8 v v v v v 4 v v 4 TABLE. IV. THE ARRANGEMENT OF SUPERIOR NODES WITH THE HEAT SOURCE V 4 FOR EIES DATA SET (a) Comparng node v for all three dffuson models (b) Comparng node v for all three dffuson models Fg. 7. Comparng the second prorty for models presented n Revje. B. EIES Data Set EIES data set 4 [5] of Wasserman and Faust data collecton s the second set of real data that has been consdered n ths paper. Ths data set also has nteractve and non-nteractve actvtes; ts communcaton network s drectonal, weghted and has nodes and 40 edges. Node v wth the degree and node v 5 wth the degree are hghest and lowest degree n ths data set, respectvely. Fg.8 shows the topologcal structure of the data set. Fg. 8. The topologcal structure of EIES data set. Prorty Method Heat dffuson kernel based on nteractve and nonnteractve actvty Heat dffuson kernel based on the non-nteractve actvty Proposed heat dffuson kernel v v v v 8 v v v v 4 v 7 v v v v 4 v v v v 5 v 8 v v v v v v 8 v v v v 4 v 7 v In Table, the top nodes based on the TOPSIS model are shown n the order of prorty. Accordng to Table, node v n TOPSIS model has the hghest prorty; accordngly, n Table 4, the order of prortes of superor nodes for data set EIES s ndcated usng v as the heat source. Comparng Tables and 4, t can be seen that the prorty n all methods belongs to node v, but other prortes vary. For example, the second prorty n some methods s v whle n some other ones, t s v. Thus, the dagram of heat dffuson of data set EIES for two nodes v and v s drawn n Fg.. Then, the most nfluental node s found from the dagrams. For readablty of dagrams, the heat source (node v ) s drawn just for the proposed heat dffuson kernel; heat dffuson kernel based on nteractve and non-nteractve actvtes are shown by, the heat dffuson kernel based on non-nteractve actvty s shown by and proposed heat dffuson kernel s shown by φ. By comparng the two dagrams (a) and (b) n Fg., t s clear that the two dagrams have lttle dfference. Now, f ths s compared to dagram (c), whch s related to node v 8 (thrd prorty of TOPSIS model); for data set EIES, t can be stated that node v 8 has lower prorty compared wth node v. Therefore, nodes v, v and v have the best condton for dffuson n socal networks and they are more approprate for spreadng deas and news or advertsements P a g e

8 (IJACSA) Internatonal Journal of Advanced Computer Scence and Applcatons, Vol. 8, No., 07 non-nteractve actvtes are shown by, the heat dffuson kernel based on non-nteractve actvty s shown by and proposed heat dffuson kernel s shown by φ. In Fg. (a), the dagram wth the heat source for node v 4 n Revje data set s shown, where the dffuson kernel related to φ, whch s the proposed dffuson kernel, s hgher; n other words, snce the proposed kernel has hgher heat compared to the other two kernels, t has an ascendng trend. In ths dagram, heat dffuson kernel based on the non-nteractng actvty s lower than the other kernels. (a) Comparng node v for all three dffuson models In Fg. (b), the dagram wth the heat source for node v n data set EIES s drawn n whch agan the proposed dffuson kernel (φ ) has ncreased more compared to the other two kernels,.e., n EIES data set, the proposed kernel has hgher heat. On the other hand, n ths dagram also the heat dffuson kernel based on non-nteractve actvty s lower than the other kernels. (b) Comparng node v for all three dffuson models (a) The heat source of node v4 for Revje data set (c) Comparng node v8 for all three dffuson models Fg.. Comparng top nodes wth heat source v for models presented n EIES. V. CONCLUSIONS The most obvous problem n the feld of socal networks s fndng k nfluental nodes n a network of ndvduals so as to beneft from the nfluence of these ndvduals n the entre network and dffuse the news n entre network to the most neghbors n the best and fastest possble way. Earler, n the Methods secton, three dfferent heat dffuson kernels were defned: heat dffuson kernel based on nteractve and nonnteractve actvtes, heat dffuson kernel based on nonnteractng actvty and proposed heat dffuson kernel; ths was followed n Experments and Evaluaton secton by fndng nfluental nodes for two data sets Revje and EIES (more specfcally, node v 4 for Revje data set and node v for EIES data set). Now n ths secton, the best heat dffuson kernel s depcted usng Fg.. As stated prevously, for readablty of dagrams, the heat dffuson kernel based on nteractve and (b) The heat source of node v for EIES data set Fg.. Comparson of kernels to determne the hghest prorty of the kernel. Therefore, proposed heat dffuson kernel n both data set Revje and EIES have hgher ncrease and more heat. Thus, the order of kernels from the lowest to the hghest prorty can be concluded as the followng: ) Heat dffuson kernel based on the non-nteractve actvty. ) Heat dffuson kernel based on nteractve and nonnteractve actvty. ) Proposed heat dffuson kernel. 7 P a g e

9 (IJACSA) Internatonal Journal of Advanced Computer Scence and Applcatons, Vol. 8, No., 07 REFERENCES [] S. Boccalett, V. Latora, Y. Moreno, M. Chavez, and D. U. Hwang, Complex networks: Structure and dynamcs, Physcs Reports, vol. 44, no. 4, pp , Feb. 00. [] M. E. Newman, The structure and functon of complex networks, 00. و. 7-5 pp. SIAM revew, vo. 45, no., [] M. Doo, and L. Lu, Extractng top-k most nfluental nodes by actvty analyss, In Informaton Reuse and Integraton (IRI), 04 IEEE 5th Internatonal Conference on, 04, pp. 7-. [4] R. M. Bond, C. J. Farss, J. J. Jones, A. D. Kramer, C. Marlow, J. E. Settle, and J. H.Fowler, A -mllon-person experment n socal nfluence and poltcal moblzaton, Nature, vol. 48, no. 745, pp. 5-8, September 0. [5] I. Anger, and C. Kttl, Measurng nfluence on Twtter, In Proceedngs of the th Internatonal Conference on Knowledge Management and Knowledge Technologes, ACM, no., September 0. [] D. Kempe, J. Klenberg, and É. Tardos, Maxmzng the spread of nfluence through a socal network, Theory of Computng, vol., no. 4, pp. 5-47, Aprl 05. [7] Z. Lu, L. Fan, W. Wu, B. Thurasngham, and K. Yang, Effcent nfluence spread estmaton for nfluence maxmzaton under the lnear threshold model, Computatonal Socal Networks, vol., no., pp., Dec 04. [8] W. Chen, W. Lu, and N. Zhang, Tme-crtcal nfluence maxmzaton n socal networks wth tme-delayed dffuson process, In AAAI, vol. 0, pp. -5, July 0. [] H. Ma, H. Yang, M. R. Lyu, and I. Kng, Mnng socal networks usng heat dffuson processes for marketng canddates selecton, In Proceedngs of the 7th ACM conference on Informaton and knowledge management, 008, pp. -4. [] J. Zhou, Y. Zhang, and J. Cheng, Preference-based mnng of top-k nfluental nodes n socal networks, Future Generaton Computer Systems, vol., pp , Feb 04. [] Y. Du, C. Gao, Y. Hu, S. Mahadevan, and Y. Deng,. A new method of dentfyng nfluental nodes n complex networks based on TOPSIS, Physca A: Statstcal Mechancs and ts Applcatons, vol., pp. 57-, Aprl 04. [] J. Hu, Y. Du, H. Mo, D. We, and Y. Deng, A modfed weghted TOPSIS to dentfy nfluental nodes n complex networks, Physca A: Statstcal Mechancs and ts Applcatons, vol. 444, pp. 7-85, Feb 0. [] J. Scott, Socal network analyss, Sage, 0. [4] P. Van Meghem, Graph spectra for complex networks, Cambrdge Unversty Press. 0, pp. -7. [5] V. Batagelj and A. Mrvar (00). Pajek datasets. 8 P a g e