Measuring Homophily in Social Network: Identification of Flow of Inspiring Influence under New Vistas of Evolutionary Dynamics

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1 (IJACSA) Internatonal Journal of Advanced Computer Scence and Applcatons, Measurng Homophly n Socal Network: Identfcaton of Flow of Insprng Influence under New Vstas of Evolutonary Dynamcs Hameed Al-Qaher Department of Quanttatve Methods and Informaton Systems Kuwat Unverst, Safat, Kuwat Abstract Interacton wth dfferent person leads to dfferent knds of deas and sharng or some nourshng effects whch mght nfluence others to beleve or trust or even jon some assocaton and subsequently become the member of that communty. Ths wll facltate to enjoy all knds of socal prvleges. These concepts of groupng smlar objects can be experenced as well as could be mplemented on any Socal Networks. The concept of homophly could assst to desgn the afflaton graph (of smlar and close smlar enttes) of every member of any socal network thus dentfyng the most popular communty. In ths paper we propose and dscuss three ter datamnng algorthms) of a socal network and evolutonary dynamcs from graph propertes perspectve (embeddedness, betweenness and graph occupancy). A novel contrbuton s made n the proposal ncorporatng the prncple of evolutonary dynamcs to nvestgate the graph propertes. The work also has been extended towards certan specfc ntrospecton about the dstrbuton of the mpact, and ncentves of evolutonary algorthm for socal network based events. The experments demonstrate the nterplay between on-lne strateges and socal network occupancy to maxmze ther ndvdual proft levels. Keywords Homophly; Afflaton; Embeddedness; Betweenness; Graph occupancy; Evolutonary dynamcs I. INTRODUCTION Dfferent propertes of socal network have demonstrated potental nterplay between events, partcpants and socal network tself. There are numbers of nstances, where, attrbute of socal network could drve the applcaton area of the network tself [21]. Hence, certan propertes of socal network have an emergng mpact [27] and homophly lke behavor s defntely one of them. Pror research demonstrates mpressve role of such behavor on the applcaton and analyss of socal network [ 28]. Homophly [1] [5] [6] [20], the tendency of ndvduals to form assocaton wth ndvduals of smlar soco-cultural background, becomes the basc governng structural component of any socal network and t has been the focus of many socal network studes [2] [7]. Socal network studes reveal that socal networks are homogeneous wth regard to many soco-demographc, behavoral and nterpersonal characterstcs [3]. In any exstng socal network such as Facebook or Twtter there les some common functonal attrbutes such as postng photos, sendng messages, lkes, Soumya Banerjee Department of Computer Scence Brla Insttute of Technology, Mesra, Inda dslkes, etc.. These knds of actvtes lead to the concept of afflaton towards a communty [20]. From nfluence of socal propagaton, Facebook and Twtter are dedcated to dssemnatng the nformaton and thus the concept of Twtter follower graph and cascadng of nfluence also renforces the hypothess of dfferent nfluence measurement model [4]. Consderng the broader defnton of the problem, ths paper fnds a close smlarty between graph theory and a socal network homophlc structure and explores the emprcal sgnfcance of nfluence propagaton or a popular communty rankng and detecton. The paper valdates the exstng graph postulates wth a proposed mnng algorthm and smulaton appled a Facebook data set. Investgaton yelds certan sgnfcant results wth regards to popular communty structure and rankng based on dfferent classcal graph theory propertes lke path traversal, Betweenness and Embeddedness of socal network nodes. After ntal valdaton through graph smulaton, an ntatve has been solcted wth a selforganzng and evolutonary prncple, whch could dynamcally trace the varants of socal network. The role of evolutonary dynamcs [20] s also consderably sgnfcant as t s defned as a study of the mathematcal prncples accordng to whch lfe has evolved and contnues to evolve. The evoluton s also vsble n the formaton of socal graph. Renforcement and valdaton of graph propertes has been demonstrated through the algorthmc strateges coned from evolutonary dynamcs [13] [14].The remanng part of the paper s organzed as follows: Secton 2 elaborates the statement of the problem wth the parameters of graph theory followed by exstng methodologes, examples of socal and afflaton graph n secton 2A and motvaton of the analyss has been dscussed n secton 2B. Secton 3 descrbes mathematcal treatments responsble for proposed algorthm, presented n secton 4. Secton 5 dscusses the data set for experments and ther mplcaton on the graph propertes of socal network. Secton 5.1 ntroduced the role of evolutonary dynamcs to valdate the smple graph propertes for socal network nstances, whch may mplcate n the mnng of graph related nferences. Fnally secton 6 gves concluson and mentons further scope of relevance research on the paradgm. II. STATEMENT OF THE PROBLEM A communty s formed n order to propagate or transfer or share dfferent knowledge across the network. And to dsperse 14 P a g e

2 (IJACSA) Internatonal Journal of Advanced Computer Scence and Applcatons, the utlty of one s communty, t s requred to apply some technques, such as votng or pollng, on whch knowledge about the partcular communty has to be shared. When formng communty, certan basc parameters along wth the lnks among the members of the communty are to be consdered. Parameters and lnks are as follows: Frend lst Communty lnks Graph occupancy perod: Intal tme and fnal tme Paths and connectvty Path traversed Embeddedness Betweenness of nodes Afflaton Co-evaluaton of socal and afflaton network A. Frend Lst Frendshp s developed n a socal ste based on some common factor, lke as, members belongng to same school, workng area, communty and so on. Smlar type of characterstc people can be blocked nto a common structure and even a few from the block can also belong to some other structure based upon ther choce. [8] The homophly test of frendshp n a socal ste can be nterpreted wth the help of the followng example. Let there be a network where m fracton of all ndvduals are male and f fracton of all ndvduals are female. Consderng a gven edge n ths network, f we ndependently assgn each node the gender male wth probablty m and the gender female wth probablty f then the both ends of the edge wll be male wth probablty m 2 and smlarly both ends wll be female wth probablty f 2. But f the frst end of the edge s male and the second end s female or vce versa then there exst cross-gender edge. Ths condton wll take place wth a probablty 2mf. Thus the test for homophly accordng to gender can be summarzed as f the fracton of cross-gender edges s less than 2mf then there s a presence of homophly [8]. B. Communty Lnks Communty can be created by a group of members by selecton and socal nfluence method. The tendency of people to form frendshps wth others who are lke them are termed as selecton [8]. The selecton crtera are manly race or ethncty or smlar characterstcs. People may also modfy ther behavors to brng them more closely nto algnment wth the behavor of ther frends. Ths process s vvdly descrbed as socalzaton and socal nfluence [8]. The ndvdual smlar characterstcs drve the formaton of lnks but socal nfluence s a mechansm by whch the exstng lnks n the network serve to share people s characterstcs. C. Graph occupancy perod: Intal tme and fnal tme It s the amounts of tme spend on vstng a node. The duraton of remanng n a focus s calculated by checkng the dfference n fnal tme and ntal tme. Ths ndcates the graph occupancy perod. D. Paths and connectvty Accordng to the socal scentsts John Barnes defnes graph theory as Termnologcal jungle, n whch any newcomer may plant a tree. [8] A path s defned to be as a sequence of nodes wth the property that each consecutve par n the sequence s connected by an edge. The paths can also be analyzed as not just the nodes but also the sequence of edges lnkng these nodes. Connectvty can be descrbed by sayng a graph s connected f for every par of nodes, there s a path between them. If a graph s not connected, then t s separated nto a set of connected peces. Connected components of a graph are a subset of the nodes such that the followng two propertes hold [8]. 1) Every node n the subset has a path to every other [8] 2) The subset s not part of some larger set wth the property that every node can reach every other [8]. E. Path Traversed The path traversal s mportant n spread of mportant nformaton. It s requred to examne whether somethng flowng through a network has to travel just a few hops or more. The Length of a path s the number of steps t contans from begnnng to end that s the number of edges n the sequence that comprses t. The path traversal technque used over here s the Breadth - Frst - Search. The method of the traversal s one just need to keep dscoverng nodes layer-by-layer, buldng each new layer from the nodes that are connected to at least one node n the prevous layer. Snce t searches the graph outward from a startng node, reachng the closest node frst, t s named as breadth-frst-search [8]. F. Embeddedness The number of common neghbors the two end ponts n a network has referred to as embeddedness of an edge. Ths s llustrated wth the help of a schematc dagram n schema1. G A H Schema 1 An afflaton network Case B C Here, embeddedness for two node A and node B has two common neghbors node G and node H. Thus the concept of embeddedness provdes nformaton that f two ndvduals are D E F 15 P a g e

3 (IJACSA) Internatonal Journal of Advanced Computer Scence and Applcatons, connected by an embedded edge then ths makes t easer for them to develop a trust level and generate confdence for transferrng vtal nformaton or nteractng wth each other [8]. G. Betweenness Betweenness of a node s explaned as the total amount of flow that t carres when there exsts a unt of flow between each par of nodes s dvded up evenly over shortest path. s wth hgh betweenness occupy crtcal roles n the network structure. To compute betweenness effcently we use the notaton of breadth-frst-search. For a gve graphcal structure the calculaton of betweenness s done on the perspectve of tme. For each gven node the total flow from that node to all other s dstrbuted over the edges. Ths technque s appled on every node n order to smply add up the flow from all of them to get the betwenness on every edge [8](shown n schema 2). Schema 3 Afflaton through bpartte graph. Ths knd of formaton can lead to a knd of co-evaluaton whch mght ndcate the selecton choce of each ndvdual and there socal nfluence. For example f two people belong to a same focus then there s a probablty that they become frends and can nfluence each other wth ther communty they belong. Accordng to the graph theoretc representaton nodes are used as both people and foc but the dfference s created by dstnct type of edges. Frstly, an edge n a socal network, t connects two people and ndcates frendshp. Secondly, an edge n an afflaton network, usually known as socal-afflaton network. Ths edge connects a person to a focus and desgnates the operaton of the person n the focus. These two parameters can be resembled n the followng schema 3. Person Schema 2: Local betweenness(the local betweenness of actor 1 s 2 H. Afflaton Afflaton, a concept that s assocated wth homophly graph, [8] [9] can be used to represent the partcpants,.e. a set of people, n a set of foc (representng some knd of communty). For example, node A, representng a person could partcpate n focus X through an edge. These knds of graph are sad to be afflaton network, snce t represents the afflaton of people (on left) wth foc (on rght). Afflaton network s one of the examples of the bpartte graph. Bpartte graph: A graph s sad to be bpartte f ts nodes can be dvded nto two sets n such a way that every edge connects a node n one set to a node n the other set [8]. Scheme 3 s an example showng nodes A and B representng people partcpatng n and Lterature Club and Soft Computng ) foc. I. Co-evoluton of socal and afflaton networks New frend lnks are formed and people become assocated wth new foc over the perod of tme. Schema 3 (a) When A, B, C are three dfferent persons. Schema 3 (b) A and B represent people but C denotes focus. III. RECENT TRENDS AND MOTIVATION Whle explorng the ncentve generaton, the basc economcs model of ncentve dstrbuton has become crucal trend to be studed. Research already revealed the mpact of ncentves on worker self-selecton n a controlled and restrcted laboratory experment. Subjects face the choce between a fxed and a varable payment scheme. Consderng the status of the treatment, the varable payment s a pece rate, a tournament, or a revenue-sharng scheme [22] [23]. 16 P a g e

4 (IJACSA) Internatonal Journal of Advanced Computer Scence and Applcatons, The extenson of applyng the graph propertes of socal network s broadly nspred by Jula Poncela Casasnovas s research [24]. She addressed the study of the evoluton of cooperaton on complex networks, usng among the dfferent socal dlemmas. Emphaszng manly on the Prsoner s Dlemma game as a metaphor of the problem, her research analysed possble outcomes of the dynamcs, dependng on the underlyng topology. Very recently, n 2013, t has been ponted out that the topology not only hghlghts homophly or other assocated propertes but also t leads toward role dscovery problem. role dscovery problem [25] fnds groups of nodes that share smlar topologcal structure n the graph. But s t only topology or economes of ncentve to quantfy the dstrbuton of pay off under network? We nvestgated from sgnfcant work on evolutonary games of Nowak et. al. [26] and found that even payoff determnes reproductve rate and successful ndvduals have a hgher payoff and produce more offsprng. Stll t cannot be assured that the payoff also could be sgnfed by the carryng capacty of ndvdual partcpants n networked games. Fnally, ths work adopts the emergng strateges of evolutonary game based ncentve dstrbuton for any nstance of socal network under test. IV. EXPLORING MATHEMATICAL TREATMENTS The most nfluental communty (focus) can be determned by the frequency of the clcks made by the ndvdual nodes. However, vstng a focus and beng a follower and then a member of that focus are two dfferent aspects. Vstng a node could mean only collect nformaton whle beng a member means makng the communty well know. As such, to fnd the most nfluental focus we need to fnd all possble paths from the source to the destnaton and fndng the shortest path by computng the Betweenness values. Consderng the graph one node at a tme, the dstrbuton of the total flow over the edge from that node to the other nodes could be computed. And hence, the betweenness of the every node could be calculated as follows: Betweenness of every node = flows from dfferent nodes (1) The shortest path traversed from the source to the destnaton can be found by the Breadth-Frst Search algorthm. Hence the number of shortest path to each node should be the sum of the number of shortest path to all nodes drectly above t n the breadth-frst search [8]. Valuaton of the members of the communty or focus can be denoted by the tme spent on t by a partcular node. Here les the concept of Graph Occupancy, whch can be calculated by calculatng the tme dfference between Fnal tme of leavng the focus and Intal tme of enterng the focus. Mathematcally, ths can be expressed as follows: Where t f b /* frequency s non-lnear relaton wth t={1,2,..m} betweenness. (2) Snce we concentrate on each node for calculatng the betweenness and thus fndng ther frequency of partcpants n makng a focus famous thus we have the followng set of Betweenness (B ) and Frequency (F ).Ths s expressed as: b b and F f f.. B,..., 1 2 b n f n 1, 2,... Equaton 2, descrbes the frequency of acceptance of a focus by dfferent nodes. Based on a number of choces made by each ndvdual, we can select out the most benefcal or famous (leader) communtes, n terms of the populaton focus. Ths entre logc can be expressed mathematcally as: g n p B F snp C 1 where, p s the popularty of a node and C s a constant for each delay unt process. In equaton [3], the number of paths to be followed to reach the destnaton s obtaned by the betweenness calculaton denoted as and frequency of the partcular node can also be obtaned and denoted as F. The summaton of these two values yelds the mnmum possble path to be traversed to pont the popularty of the communty. The maxmum popularty of a communty could be found by the product of ths result, n other words, by applyng the concept of Max-Mn functon, the maxmum popularty of a focus can be obtaned for the mnmum path travelled from the source to the destnaton. V. PROPOSED ALGORITHM -I Fndng the most nfluental communty could be acheved through the followng algorthm whch has dstnct blocks to evaluate the popularty and nfluence of node(s). The algorthm contans certan unconventonal graph propertes such as betweenness and embeddedness. (3) 1. Begn 2. Intalzaton of lnks present n between the nodes and focus 3. Intalzaton of varables Betweeness=, = {. } Graph frequency= ( ), = {. } Popularty P 4. Fndng value = {t=1, 2, 3.n} 5. Calculatng the betweenness: Betweenness of every node= flows from dfferent nodes 6. Values form the betweenness leads to fnd the shortest path 7. Calculatng the shortest path based on Breadth frst search 8. Select a random node whch has vsted a partcular communty node at least once 17 P a g e

5 (IJACSA) Internatonal Journal of Advanced Computer Scence and Applcatons, 9. whle do Calculate the Graph frequency f then F= end f Calculate g(p)= + +C) 10. end whle 11. end begn VI. IMPLEMENTATION AND ANALYSIS The data set for valdatng the proposed model was collected from Facebook communty network. The reason to choose Facebook was ts close resemblance, n terms of nodes, betweenness and edges, to the classcal graph theory models. A set of nne homophlc nodes, at a partcular tme, were selected for ntal valdaton. In these sets, each node s connected to ts nfluental nodes, whch s denoted by the lnks and ther weghts. There are communtes that belong to the nodes and each of these nodes tres to promote ther own communtes. To denote the most nfluental focus or communty, we need to fnd the shortest path (whch also motvates the other nodes to jon the communty and ncrease the popularty). Embeddedness of the nodes are also consdered to dentfy the common neghbors among them. Table 1 shows the parameters to be consdered for the proposed algorthm. The output value n the table was computed after mplementng the algorthm (usng MATLAB verson (R2010 a)). The detaled explanatons of the parameters are as follows: Frend Lst: 9 dfferent s represent 9 dfferent frends wth ther communtes Lnk Present: edges among the nodes wth weghts. Embeddness: A B edge havng common neghbors. There may be present or mght not be present. If not present, then denoted by NIL. If present then node number s gven. Betweenness: nvolves reasonng about the set of all shortest paths, between pars of nodes. Graph occupancy: amount of tme spent on a node. Leadng to the further popularty of a communty and ncreasng the node lnkage. The correspondng outputs are gven n the followng Table 1. TABLE I. LIST OF NODES WITH THEIR CONNECTIVITY, FLOW OF INFORMATION AND FURTHER INCREASE IN POPULARITY Fren d lst Lnks present Embedd edness (4,5,6) 4 (3,5,6) 3 Parameters Betweenness (1,5) (1,6) (2,3) (2,5) (4,5,6,7) 4 (3,4) (1,3,5,6,8) NIL (1,2,3,4) NIL (1,2,3,6) 4 (3) NIL (9) NIL (4,1) (4,6) (5,3) (5,4) (6,2) (6,3) (7,3) (8,9) Graph occupancy (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) (7,2) (7,3) (7,4) (7,5) (8,2) (8,3) (8,4) (8,5) (4,8) NIL (9,4) (9,1) (9,3) (9,4) (9,5) (9,6) P a g e

6 (IJACSA) Internatonal Journal of Advanced Computer Scence and Applcatons, Fg. 2. Enhancements n new connectons wth the exstng nvokng nodes Fg. 1. s wth ther lnks and weghts Fgure 1 represents nne dfferent nfluental nodes wth ther lnks represented n the drected graph format. Each of the edges has ther weghts marked on t. Here, the shortest path has been consdered from a source node say 1 and the destnaton node, 6.Tme duraton, that s the amount of tme spent by the nodes n ther communtes, can be an nvokng factor to jon ther communty. Usng equaton (3), Fgure 2demonstrates that the most nfluencng or nsprng node enhances the number of functonally actve node. In ths fgure, t s shown that the nspred nodes have added on more of the connectvty wth other dfferent nodes. To crawl Facebook, we mplemented a dstrbuted, multthreaded crawler usng Python wth support for remote method nvocaton (RMI) [11]. Facebook provdes a feature to show 10 randomly selected users from a gven regonal network; we performed repeated queres to ths servce to gather 50 user IDs to seed our breadth-frst searches of socal lnks on each network 1 [11]. A. Role of Evolutonary Dynamcs Based on the ntal smulaton, t has been demonstrated that there exst a strong cohesve drectons wth graph theory and socal network n the context of homophlc communty detecton. 1 We need further nvestgaton for the specfc attrbutes n the context of more homophly dentfcaton from the perspectve of ether drected or undrected graph. The extenson of the algorthm wll be sgnfcant to quantfy the applcaton specfc nvestgaton of homophly. The extenson also could revaldate the correlaton of graph propertes under socal network. Inspred by the phenomenal contrbuton n evolutonary game theory by Nowak and hs colleagues, several non lnear characterstcs have been started adoptng the concept of evolutonary dynamcs [12 ] [13 ]. It s evdent that evolutonary dynamcs are defned by nonlnear dfferental equatons and therefore can be mported for revaldatng complex graph and network wth the growth of a socal graph as mentoned n the frst part of the algorthm. An evolutonary dynamc assgns each populaton game F an ordnary F dfferental equaton [14] x V (x) on the smplex x. One smple and general way to defne an evolutonary dynamc s va a growth rate functon: g : R n n x R (4) Here, g represents the (absolute) growth rate of strategy and t wll as a functon of the current payoff toreward the strategy. The prevous algorthm only consdered the shortest path, breadth frst search and graph frequency. As Evolutonary Dynamcs can also retrospect the growth aspects, propagaton and mutaton of message under any state of graph, hence therefore the conventonal graph propertes have been revaldated usng the extended algorthm. It should be mentoned that proposed algorthm tres to ncorporate potental strength of evolutonary dynamcs for smulatng 19 P a g e

7 (IJACSA) Internatonal Journal of Advanced Computer Scence and Applcatons, the behavor and growth of socal graph structure. Seres of more precse plots have been accomplshed n post smulaton of extended algorthm (Fgure 4-8). Persstence across the network also ndcates the out degree and also any average number of dstnct tags of groups and of tag assgnments of users havng k out neghbours could be evaluated from socal network as shown. The users, who have more contacts n the socal network, tend also to be more actve n terms of tags and groups. Average number of dstrbutve tags denoted as n t. group tags on the specfc message as n g and subsequently n w represents the lst of predefned choces. Correlatons between the actvty of partcpants and ther number of declared frends and neghbours can be dentfed: here also the k out neghbours have been taken nto consderaton. The data has been log-bnned (Fgure 3): by defnton a bn of constant logarthmc wdth sgnfes that the logarthm of the upper edge of a bn (x +1) s equal to the logarthm of the lower edge of that bn (x ) plus the bn wdth (b) [19]. Here, the symbols ndcate the average, and the error bars wth near optmal 25 and 75 percentles for each bn. The algorthm deploys the Python 2.7.3, whch was released on Aprl The present mplementaton also allows the feature of Automatc numberng of felds n the str.format() method, whch have been reflected n the post mplementaton stage of algorthm. Algorthm II: Graph Pattern (G, P (t), ) 1. Defne the ntal state of the graph,.e. defne G = /* s the 2 Kronecker symbol*/ 2. Solve for P(t), whch provdes functons P(t) and (t) /*P(t) : Probablty of sample occupancy tme for t m Evaluate: dp dy dt dt P (5) /* The probablty Y(t) that at least one mutaton has occurred whle the system was at state before tme t*/ 3. Intalze the system wth N classes of socal network nstances at tme t = 0, wth ts ntalstart of connecton, k out : out-degree of graph, occupancy on graph, and termnaton nstance: evaluate: (t m P (t m ) ) P (t m ) (6) /* where: probablty ( t m ) that the socal network s at state, P(t): sample occupancy, represents probable mutaton rate of messages across partcpants accordng to the proporton of occupaton*/ 4. Sample the next mutaton tme accordng to the cumulatve probablty P(t). Ths can be done va the 2 It s smply a functon on two varables, and j whch are ntegers, when for each socal network nstances cardnalty of the varables s large, then t could assst to approxmate nference based wth a constraned, lower complexty, adaptvely szed sum for the target cardnal value [10 ]. nverson method, such that the next tme t m = P 1(r), where r s a unform random varable between 0 and Add t m to the current tme of occupancy of partcpants and betweenness. 6. Choose the specfc score and plot accordng to ther respectve nore transton of the network and update the state of the system as per Step Remove extnct and redundant classes from the lst and reduce the number N of classes accordngly. 8. Return to Step 1 untl fnshed. Fg. 3. Dstrbuton of taggng wth out degree (k out) Fg. 4. Degree Correlaton and dstrbuton of Message taggng In case of large scale network lke Facebook, the most conventonally nvestgated mxng pattern nvolves the degree (number of neghbours) of nodes. Ths type of mxng mprovses the lkelhood; leadng users wth a gven number of neghbours connect wth users of smlar degree. Ths property s emphaszed by computng mult-pont degree correlaton functons. Complementary cumulatve condtonal dstrbutons as mentoned wth group taggng, specfc message taggng and pre-defned choce taggng, compared wth the global cumulatve dstrbutons denoted by black lnes (Fgure 4). Even among the subset of users wth a gven k ou t, a strong dsparty s stll observed n the amount of actvty and also around a specfc communty. Subsequently, the 20 P a g e

8 occupancy on a specfc graph nstance results n the followng plot: Log-log plot of the dstrbuton of the contact duratons and of the cumulated duraton of all the contacts two ndvduals m and n have over a day (w mn ). An nterestng nference could be drawn that out of 88% of the total contacts sustaned less than 1 mnute on a specfc tag or comments, but more than 0.2% perssted more than 5 mnutes aganst a specfc topc of nterest. For the cumulated duratons, 64% of the total duraton of contacts between two ndvduals durng one day last less than 2 mnutes, but 9% last more than 10 mnutes and 0.38% more than 1 hour. The small symbols recprocate to the actual dstrbutons, and the large symbols to the log-bnned dstrbutons [18] (Fgure 5). (IJACSA) Internatonal Journal of Advanced Computer Scence and Applcatons, Fg. 6. Evolutonary Dynamcs plot wth varyng mutaton and Probablty of occupancy Fg. 5. Occupancy Graph for Socal Network wth Mutaton Probablty of Messages Degree dstrbutons of evolutonary dynamcs and the emprcal data: o represents the data ponts from the smulated data and x represents the data ponts from the emprcal data new project The results of the degree dstrbutons of the extended algorthm wth dynamc values of mutaton are shown n Fgure 6. The upper fgures are the comparson of developer and project degree dstrbutons n lnear coordnates. The lower fgures are the comparson of developer and project degree dstrbutons n log-log coordnates. The R 2 of developer degree dstrbuton from the smulated data n lower fgure s and the R 2 of project degree dstrbuton from the smulated data n lower fgure s Also the largest project sze of the smulated data s just We can further lower ths value by tunng the mutaton parameter [17] and P (t),and graph functon of the extended algorthm. H log represents homophly log evaluated from message exchanged towards any specfc and common nterest [17]. Fg. 7. Recursve smulaton on test functon towards Homophly from Betweenness In order to quantfy the complex large cardnal network, we ncorporate classcal test functon lke DeJong 3 (The conventonal De Jong s functons s the so-called sphere functon, ths functon s unmodal and convex by nature)[ 16].Dmenson s represented n the formulas by varable D, so as can be observed that t becomes smple to calculate selected functons for an arbtrary dmenson for arbtrary number of partcpants. Smlarly, the result of homphly structure from ntal betweeness and other asscoted propertes s safely turned out from local mmma due to the sze of ts popluaton and avaable best soluton.the extended part of the algorthm demonstrates the frquencey of nterplay and ncentve dstrbuton betwwen dfferent communtes played n socal network. 3 represents Dejong Functon 21 P a g e

9 Algorthm III Dstrbuton n Populaton graph The workng strategy Varables: N./*Homogeneous populaton of sze*/ t /* Tme Step t*/ r /* relatve ftness */ (IJACSA) Internatonal Journal of Advanced Computer Scence and Applcatons, /*temporal event could be Bnomal tme step 4 (equaton 8) of propagaton of base event under socal network, f randomly chosen ndvdual from SN then fxaton of probablty of new mutant μ 1 represents probable mutaton rate of messages across partcpants accordng to the proporton of occupaton*/ μ 11/r 11/r μ 1 11/r 11/r 2 4 n 1 [ ][t t N 1 t 2( 1)... t 2!/N N (7) (8) The dstrbuton of trend on populaton graph of the new attrbutes concernng the people refers category of homophly structures (shown n red lne) where the majorty of people n the populaton have lower levels of the attrbute. Gradually, as tme elapses, the most nfluental message/ broadcast confguraton propagates across the populaton (ndcated by green lne) but subsequently; evoluton dynamcs pushes the hgher level of the attrbute wth certan ncentve. Ths s natural recpent of ncentves under socal network for those persons who are partcpatng and nteractng. In addton to, there are drfts of ncentve dstrbuton under dfferent settngs. Hence, eventually after a consderable perod of evolutonary dynamc, drft ncreases at the hgh end of the attrbute for homophlc structure and the dstrbuton approaches reverts back agan to normal. Fg. 9. Smulaton of Pay off dstrbuton At ths stage of smulaton, t was not evdent how the approprate nteracton could ental hgher range of ncentve among the partcpants. Therefore, a specfc data set was chosen to exhbt dfferent nterplay among the socal partcpants. The observatons show that the frequency of cooperators (ndcated by blue), devators (ndcated byred) and loners (represented as yellow) under smooth partcpaton. Ths s also measured as the multplcaton factor accordng to trust of dscusson and nteracton. Indvduals are arranged on random regular graphs where each node has eght neghbors and they nteract n randomly formed groups of sze N = 5. For small multplcaton factors,ndvduals domnate. The reason s smple: n ths case even n a group of cooperators, the payoffs do not exceed the ncentve of ndvdual. All of the proposed components of three algorthms mprovse dynamc equlbrum consderng the populaton of socal network. The range of multplcaton wth trust ncreases the pay off a very small dscrmnaton ofr.and above the threshold devators exst.. Only for much larger value of r~4.056 cooperators reappear and co-exst wth devators. Snce ndvduals are absent, the dynamcs agan retan voluntary partcpaton nto compulsory nteracton. Fnally, for r= 4.6 cooperators take over and manage to dsplace devators (see Fgure 9 and table 1). Fg. 8. Tme versus messages Table 1. Confgurng desgn of Incentves on SN 4 Refer to the Appendx for Proof The proposed desgn nvolves 4 stages and 4 groups descrbed below and summarzed n the followng Table: 22 P a g e

10 Group 1: Ths group comprses of more than 65 subjects. The default ntal value of the partcpatng reward s 10 unts and the fnal value could be measured as Ω. Group 2: Ths group represents almost 50 subjects wth ntal ncentve but n subsequent sessons the experence dffers wth dfferent fractons of ncentves dependng on the frequency of nteracton as shown n Fgure 8. Group n: The group enhances ts subject lne > 100, but n addton to the normal ncentve dstrbuton, there wll be dfferent treatments for ncentve ether there should be donaton or addtonal amount the partcpants wll push nto t. Populaton varables Mean Std. Dev. AutoCorr. Homophly St.No. (IJACSA) Internatonal Journal of Advanced Computer Scence and Applcatons, B. Comment on fnal Homophly Sturucure The homphly evlaluton from the socal graph s completely based on the ncrease of posts, tags and other socal network acton artfacts. Keepng n mnd about the non-lnear aspects and growth strategy of socal network, evolutonary dynamcs, comparatvely better vsualzaton of densty of attrbutes has become possble. Especally, betweenness and graph occupancey and mutaton could be the one of those key factors. Recent research of Tom A.B. Snjders and hs colleagues [15] demonstrated the relaton between evolutonary dynamcs and homophly of socal network n some common aspects of frendshp, recommendaton, group study and selecton etc. Fgure 7 exhbts trend of precse vsualzaton of homophly structure as guded by the extended algorthm wth mutaton as prme parameter. The followng table 2 should be consdered to understand the relatonshp shown n Fgure 8: the red crcle sgnfes hgher cluster densty and therefore provdng the remanng count of homophly over the populaton. Record Set trackng From snap.stanford.edu/data/ Male and Female Age Densty of Propagaton Network Varables Out Degree Local Betweeness > VII. CONCLUSION AND FURTHER SCOPE OF RESEARCH Afflaton beng an mportant factor of homophly graph has a relevant role n promotng the focus through the nodes. The dea s how a frend n the Facebook or any socal network can nfluence hs frends to jon the communty he or she belongs to. Ths paper nvestgates such possbltes by explorng the homophly communty n socal network by usng graph property based algorthm and smulaton. Further, ncorporaton of evolutonary dynamcs also contrbuted for nvestgatng homophly property wth better approxmaton. Subsequently, the growth of socal network, temporal behavor and trend of t, can also be nvestgated by augmentng the exstng evolutonary dynamcs algorthm. On the other hand, the rankng algorthm or conventonal classfer model can also be extended usng ntal attrbutes of embeddedness and graph traversal wth graph occupancy tme. Soft computng based (fuzzy and rough set) homophly dentfcaton from graph propertes could be an emergng research on computatonal socal network. Fnally, n the extended analyss part, certan sgnfcant observatons are made n terms of maxmzng beneft or proft, based on ther role at specfc nstances under socal network. Experments have ncorporated certan publc data sources to demonstrate mutual nterplay of socal network partcpants, ther specfc contrbuton towards the network and share of ncentve f any. REFERENCES [1] Vojtek, P., &Belková, M. Homophly of neghborhood n Graph Relatonal Classfer. Pages of: Geffert, V.,Karhumäk, J., Berton, A., Preneel, B., Návrat, P., &Belková, M. (eds), SOFSEM 2010: 36th Conf. on Current Trends n Theory and Practce of Computer Scence, SpndleruvMlyn, Czech Republc, 2010, Proc. LNCS, vol Sprnger. [2] Hall Bsgn, Ntn Agarwal and Xa owexu, Investgatng Homophly n Onlne Socal Network, 2010,IEEE/WIC/ACM Internatonal Conference on Web Intellgence and Intellgent Web Technology, pp [3] M. McPherson, L Smth Lovn and J.M. Cook, Brds of a feather: Homophly n Socal Network, Annual Revew of Socology, 27(1): pp , [4] Eytan Bakshy, Wnter A mason, Jake M.H.Ofman, Duncan J. watts, Everyone s an Influencer : Quantfyng Influence on Twtter, publshed n Proceedngs WSDM 2011, Hongkong, February [5] Reza Zafaran, Wllam D Cole, and Huan Lu. Sentment Propagaton n Socal Networks A Case Study n LveJournal. In Patrca Cha, Sun-K And Salerno, John And Mabry, edtor, SBP Advances n SocalComputng, Lecture Notes n Computer Scence 6007, pp Sprnger Berln, [6] James H. Fowler, Jame E. Settle, and Ncholas A. Chrstaks.Correlated genotypes n frendshp networks.pnas, pp. 108:1993, [7] Cosma Rohlla Shalz and Andrew C. Thomas. Homophly and Contagon Are Genercally Confounded n Observatonal Socal Network Studes. Socologcal Methods and Research (arxv: v3), P a g e

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