The mpact of network characterstcs on the dffuson of nnovatons Renana Peres School of Busness Admnstraton Hebrew Unversty of Jerusalem, Jerusalem, Israel 91905 peresren@huj.ac.l August 2013 1
The mpact of network characterstcs on the dffuson of nnovatons Abstract: Ths paper studes the nfluence of network topology on the speed and reach of new product dffuson. Whle prevous research has focused on comparng network types, ths paper explores explctly the relatonshp between topology and measurements of dffuson effectveness. We study smultaneously the effect of three network metrcs: the average degree, the relatve degree of socal hubs (.e., the rato of the average degree of hghly-connected ndvduals to the average degree of the entre populaton), and the clusterng coeffcent. A novel network-generaton procedure based on random graphs wth a planted partton s used to generate 160 networks wth a wde range of values for these topologcal metrcs. Usng an agent-based model, we smulate dffuson on these networks and check the dependence of the net present value (NPV) of the number of adopters over tme on the network metrcs. We fnd that the network metrc that most nfluences dffuson s the relatve degree of socal hubs. Ths result emphaszes the mportance of strong hubs. The effect of the average degree s postve but weaker. The clusterng coeffcent has a negatve mpact on dffuson, a fndng that contrbutes to the ongong controversy on the benefts and dsadvantages of transtvty. These results hold for both monopolstc and duopolstc markets. Keywords: clusterng, average degree, degree dstrbuton, agent-based models, dffuson of nnovatons, word of mouth, dffuson of nnovatons. 2
1 Introducton The socal nfluence processes that take place n a gven network are shaped and affected by the network's topologcal characterstcs. In ths paper, we study how the topologcal or structural characterstcs of a socal network nfluence new-product dffuson n that network, n terms of speed of dffuson and the number of adopters. Classcal works n dffuson, focusng on the flow of nformaton among ndvduals, assumed a fully connected market [1]. More recently, lterature has begun to acknowledge the role of network topology n socal nfluence processes, explorng dffuson n topologes such as small world networks [2] and scale-free networks [3]. Emprcal studes have explored how dffuson s nfluenced by aspects of network structure, ncludng weak, long-dstance tes [4] and the exstence of socal hubs [5], and have examned how network structure affects the performance of marketng strateges such as new-product seedng [6,7]. Despte ths nterest, to our knowledge, the drect mpact of topologcal network metrcs on dffuson has not been studed. Specfcally, there has not been a systematc assessment, carred out across multple networks, testng the smultaneous and drect mpact of multple topologcal metrcs on the magntude and speed of dffuson. Although some comparatve studes have been conducted, most of them compare network types (e.g.[8]), referrng to the comprehensve set of propertes characterzng each network rather than solatng the specfc role of each structural dmenson. In ths paper, we conduct a methodologcal nvestgaton of the mpact of network topology on the dffuson of a new product to the market. Specfcally, we focus on the followng structural metrcs: the average degree, the relatve degree of socal hubs (.e., the rato between the average degree of the most-connected nodes and the overall average degree), and the clusterng coeffcent. We apply a graph-theory procedure called random graphs wth a planted partton, whch has so far not been used n network research, and use t to generate 160 networks, wth a large range of values of the nvestgated metrcs. We conduct agent-based smulatons of new product dffuson n these networks, both n a monopoly and under 3
duopolstc competton. We test the relatve nfluence of each structural metrc on the effectveness of dffuson, measured as the Net Present Value (NPV) of the number of adopters. Our man fndngs are: 1. Among the nvestgated metrcs, the relatve degree of hubs has the strongest postve mpact on dffuson. Ths result s nterestng n lght of the controversy on the contrbuton of socal hubs [9,10]. Our results ndcate that the effect of hubs s stronger than the effect of the overall average degree, whch s also sgnfcant and postve, but weaker. 2. The clusterng coeffcent has a negatve mpact on dffuson. Ths result s n lne wth prevous works comparng network types; however, t solates the role of clusterng from that of other topologcal network metrcs. In addton, ths fndng contrbutes to an ongong dscusson on the benefts and dsadvantages of transtvty (that s, the lkelhood that the other nodes connected wth a node are also connected to one another), of whch clusterng s a measure [11], mplyng the possble drawbacks of transtvty n the context of dffuson. Ths paper offers three man contrbutons: Frst, t measures the drect mpact of structural parameters on dffuson. We vary three major network metrcs, and run a multvarate regresson to explore smultaneously ther relatve roles. Second, we use a network generaton method that has not been used so far n network research, to create a set of networks wth a wde range of values for the varous metrcs we examne, wthout the need to use networks of dfferent types. Thrd, the agent-based smulaton enables us to represent real-lfe dffuson scenaros by () consderng both a monopoly and a compettve market; () usng a cascade agent actvaton model [12]; ths s the ndvdual-level analog to the Bass dffuson model [1], whch s the standard model used to descrbe dffuson processes; and () evaluatng the effectveness of the dffuson process by measurng the NPV of the number of adopters, whch reflects both the reach of the dffuson process and ts speed. Prevous studes used such smulatons, but dd not focus on solatng the roles of specfc topologcal metrcs n the dffuson process. The rest of ths paper s organzed as follows: In secton 2 we revew the lterature and descrbe the topologcal metrcs we use and ther antcpated nfluence. Secton 3 descrbes the network generaton procedure. Secton 4 descrbes the agent-based model. The results and conclusons are presented n sectons 5 and 6, respectvely. 4
2 Network structure and dffuson propagaton Socal nfluence processes are strongly affected by the topology of the network n whch they take place. Thus far, most studes focusng on the relatonshps between network topologes and socal nfluence have been non-comparatve n nature, wth each paper focusng on a sngle network type. To the extent that comparson was done, t focused on comparng performance across dfferent types of networks, and was mostly theoretc wthout measurng actual dffuson or nformaton flow ( see [13,14]; see [8] for an excepton of an emprcal study). For example, n small world networks, whch are "rewred" lattce graphs n whch several lattce tes are replaced wth random connectons, nformaton flow and nfluence processes are expected to be more rapd than n regular lattce graphs, due to the "shortcuts" between nodes 1 [14 15]. Lkewse, nformaton flow n fully random graphs s expected to be rapd [16,17]. These studes provde mportant nsghts wth regard to the nfluence of network structure on nformaton flow. However, a specfc network type usually bnds several topologcal dmensons: For example, small-world networks combne both hgh clusterng and short path length; most random graphs have a Possonan degree dstrbuton (wth some exceptons; see [16]). In scale-free networks the clusterng depends on the network sze [13]. Thus, t s hard to solate the relatve role of each structural dmenson on the bass of an overall comparson of dfferent types of graphs. For example, f dffuson n a lattce graph s slower than n a random graph, s ths because of the lower clusterng of the random graph, or perhaps the varablty n the degree of the nodes? Do the hubs n scale-free networks generate faster dffuson n comparson wth other graphs, although the average degree n scale-free networks s very low? These questons are stll unanswered. 1 The network lterature uses a varety of terms to descrbe network members and ther connectons. We use the term "network member" when talkng about the real ndvduals n the network, "node" to descrbe ths person n the theoretcal context, and "agent" when speakng about the agent-based model. In all contexts, we refer to the connectons among network members, nodes, or agents as "tes". 5
The goal of ths paper s to study systematcally the nfluence of structural factors on the speed of dffuson. We focus on three structural dmensons: average degree, relatve degree of socal hubs, and clusterng coeffcent. 2.1 Average degree The degree of a node s the number of tes t has wth other nodes. The average degree of a network s the average of the degrees of all the nodes n the network, and can be consdered as a metrc of the network's level of connectvty. Emprcal comparsons of the average degrees of real-lfe networks ndcate values rangng from ~7 n some neural networks to <113 n a networks of flm actors [18,19]. For theoretcal networks, the average degree can often be derved from the networks' creaton procedure: For example, for a regular lattce or for a Watts- Strogatz network, where all nodes have an equal number of k tes, the average degree s k, and a random Erdos-Reny graph of N nodes wth a te probablty p wll have an average degree of. 2.2 Relatve degree of socal hubs Socal hubs are nodes wth a hgh degree n relaton to the degrees of other nodes. Not all networks have socal hubs. A regular lattce does not have any. In random graphs and small world networks, the degree dstrbuton s Possonan [13], and socal hubs do not dffer much from the other members of the network. Socal hubs are most prevalent n scale-free networks, where degree dstrbuton follows a power law, wth most of the nodes havng a small number of tes, and a small number of nodes havng an extremely hgh degree [13,20]. The contrbuton of socal hubs to dffuson processes s a subject of ongong debate. Some studes argue that hubs enhance the spread of nformaton [21] and the speed of dffuson [6], and hence t s benefcal to target these ndvduals when attemptng to ntroduce new products or deas nto a network [7]. Other studes clam that the role of hubs s complcated and depends on the level of contagon n the system [22], as well as on these ndvduals' nherent propensty to adopt [5]. Watts and Dodds (2007) suggest that n many cases, the large mass of less-connected unts n a network determnes the speed and magntude of socal nfluence [9]. 6
Here, we contrbute to ths dscusson by systematcally varyng the level of the relatve degree of hubs across the networks we generate, and testng the nfluence of ths varable, whle controllng for other topologcal metrcs. We measure the relatve degree of socal hubs as the degree rato of the average degree of the top 10% most connected nodes to the overall average degree. 2.3 Clusterng Coeffcent The clusterng coeffcent serves as a measure of a network's transtvty, that s, the lkelhood that f nodes A and B are connected to each other, and nodes B and C are connected to each other, then nodes A and C are also connected. In other words, the clusterng coeffcent ndcates the lkelhood that a person n a gven network s frends wth the frends of hs or her frends. Clusterng can be measured ether locally for each node countng the number of ts tes, and seeng how many of them are connected to each other or globally, countng the numbers of "trangles" relatve to the number of "open trples", where an open trple s a sngle node wth tes to two other nodes. For example, f nodes a, b and c are all connected to each other, they form a sngle trangle, and three open trples: abc, bac, acb. Heren we use Newman's (2003) defnton of a global clusterng coeffcent [19]: (1). Clusterng s strongly dependent on network type, and can also vary to some extent across networks of the same type. For a one-dmensonal lattce arranged n form of a rng, where each node s connected to ts k nearest neghbors, the clusterng coeffcent s, whch approaches for large values of k. In a random graph, clusterng tends to be much lower, gven by the approxmaton of k/n, where N s the network sze, and k s the average degree of the network. The clusterng coeffcent of a small-world Watts-Strogatz network s between these values [14] and depends on the value of p, the rewrng probablty of the regular lattce [23]. In scale-free networks generated by the Barabas-Albert procedure, clusterng s hgher than that n a random graph, but lower than that n a small world graph, and 7
s gven approxmately by [13]. The strong dependence of clusterng on network type can be an obstacle for systematcally testng how clusterng affects dffuson, snce t s hard to determne whether dfferences result from the clusterng or from other characterstcs that are typcal of specfc network types, such as level of randomness, path length, etc. Network theory s splt wth regard to whether transtvty s benefcal for network processes. In some contexts, a node, that s, an ndvdual n the network, s better off not closng a trad, but rather formng tes wth a new node; n other contexts, such as when redundancy and multple ponts of nfluence are needed, ndvduals gan from closng a trad and ncreasng the network's transtvty (see [11] for revew). Here, we shed some lght as to the role of transtvty n the context of dffuson, by varyng the clusterng coeffcent, and testng drectly ts effect on dffuson, controllng for other topologcal metrcs. Network lterature suggests addtonal network metrcs that mght contrbute to dffuson processes, ncludng average path length, dameter, and densty. We focus on the three dscussed above because they represent ndependent topologcal network dmensons, and snce they are global and can be pre-determned as nput for the network generaton procedure suggested below. 3 Generatng the networks To test how network metrcs nfluence dffuson, t s necessary to generate a large number of networks, wth a wde range of values for each metrc under nvestgaton. Snce the standard network types (random, lattce, small-world, etc.) largely dctate dependences among the network metrcs, we ntroduce here a new network generaton method, based on mergng several random graphs, whch can generate networks wth a wde range of values for the average degree, relatve degree of socal hubs, and clusterng coeffcent. Ths method, termed "random graphs wth a planted partton", s drawn from graph theory [24,25] 2 and to the best of our knowledge has not yet been used for smulatng socal networks. 2 We thank Mchael Krvelevch for several useful dscussons on ths topc. 8
Assume a market of sze N. For smplcty, tes are symmetrc,.e., f node a s connected to b, then b s also connected to a. The N nodes are organzed nto three separate bns (denoted bns 1, 2 and 3) of szes N1, N2, and N3, respectvely. The probablty that a customer from bn and a customer from bn j are connected s p j. For three bns, there are sx probabltes: p 11, p 22, p 33, p 12, p 13, p 23, as llustrated n Fgure 1. If all these probabltes are the same, the graph s a standard random graph. Manpulatng the probabltes allows one to create random networks wth dfferent values for the average degree, degree rato and clusterng as follows: The average degree of the nodes n bn s gven by, where,j,k=1,2,3 and. The term s the average number of tes that the node has wth other nodes n bn (assumng a node cannot connect to tself), and the two other terms are the number of tes that the node has wth nodes n bns j and k, respectvely. Hence, the overall average degree s. If we arbtrarly defne bn 1 as the bn of hubs and set p11 to be hgher than the other probabltes, then we can defne the relatve degree of hubs as D1/D. Fgure 1: A random graph wth a planted partton To calculate the clusterng coeffcent, we need to calculate the number of trangles and the number of open trples. The number of trangles s gven by the followng expresson: (2) 9
The frst sum s the number of trangles n whch all nodes belong to the same bn. The second sum s the number of trangles n whch two nodes are n the same bn and one s n a dfferent bn: If = 1, for example, we frst choose a par of nodes n bn 1 (there are such pars); ther probablty to be connected s p11. Takng randomly a node n bn 2 (there are N 2 such nodes), the probablty that both nodes of bn 1's par wll be connected to t s ; the same logc apples for bn 3. The thrd term s the number of trangles n whch each node s n a dfferent bn. The number of open trples s calculated n a smlar way. A unt n bn can be n an open trple wth ether two other unts n, two unts n j, or two unts n k, or wth one n and one n j, one n and one n k, or one n j and one n k. Snce there are N unts n bn, the number of open trples that contan a node from bn s gven by: (3) The clusterng coeffcent s then calculated as. Gven values of the average degree, relatve degree of hubs, and clusterng coeffcent, we can calculate the probabltes that satsfy these values and then use a random number generator to create the networks correspondng to these probabltes. Note that ths procedure does not guarantee a soluton for every degree/rato/clusterng combnaton, or that an obtaned soluton s unque (snce we have 6 probabltes and 3 equatons). However, snce the goal s to generate networks wth a desred structure and not to obtan a specfc network, ths procedure s adequate for our purposes. Ths procedure was used to generate 160 networks, usng the followng values: N = 1000, N 1 = 100, N 2 = 450, and N 3 = 450. The dstrbuton of parameters across the networks s descrbed n Fgure 2. As Fgure 2 llustrates, the average degree ranges from 2 to 49 (average s 24.5), the clusterng coeffcent ranges from 0.01 to 0.48 wth an average of 0.21, and the relatve degree of 10
hubs (.e., the rato between the average degree of the 10% most connected nodes and the overall average degree) ranges from 1.09 to 6.7 wth an average of 3.96. Fgure 3 llustrates the network structure of a network wth an average degree of 7.5, relatve degree of hubs of 6.67, and a clusterng coeffcent of 0.45. The probabltes are: p 11 = 0.49, p 22 = 0.0001, p 33 = 0.012, p 12 = 0.013, p 13 = 0.013, p 23 = 0. Fgure 2: The dstrbuton of parameters across the 160 networks Fgure 3: An example of a random graph wth a planted partton. Average degree = 7.5, relatve degree of hubs = 6.67, clusterng coeffcent 0.45. p 11 = 0.49, p 22 = 0.0001, p 33 = 0.012, p 12 = 0.013, p 13 = 0.013, p 23 = 0. We next use these networks as the bass for an agent-based dffuson model. 4 An agent-based model for smulatng dffuson An agent-based model s used here to descrbe the dffuson process, n lne wth the agentbased econophyscs approach [26]. Agent-based models are ncreasngly used for descrbng complex economc systems, and have been extensvely used to descrbe dffuson of nnovatons [27,28]. Here, a network wth a pre-specfed structure s created through the random-graphwth-a-planted-partton generaton process descrbed above, and an agent-based model s used to smulate the process of new product dffuson n ths network. The model s composed of 1000 agents, and s used to descrbe the dffuson of both a monopolstc frm wth no competton and a duopoly wth two competng frms. 11
4.1 Adopton Probabltes For each network, we smulate the dffuson of the adopton of a new product. Assume frst a monopolstc case. Tme s dscrete, the market starts wth all agents at state "0", and when an agent adopts, ts state changes to "1". We assume that adopton s un-drectonal, that s, an agent can convert from 0 to 1, but cannot ds-adopt and convert from 1 to 0. The transton rule s based on classcal dffuson theory, whch suggests that the adopton decson s a result of the combned nfluence of two factors: external nfluence, represented by the probablty that an agent wll be nfluenced by sales people, advertsng, promotons, and other marketng efforts 3 ; and nternal nfluence, whch refers to the nfluence of all means of socal nteracton such as word of mouth, or mtaton. We denote by q the susceptblty of agent to the nternal nfluence of a sngle other agent,.e., the probablty that a gven agent n a gven tme perod wll convnce agent to adopt. The agent actvaton rule used here s based on a competng rsk, or a cascade, approach, where at tme t, each pror adopter connected to agent ndependently tres to convnce to adopt. Thus, the dscrete-tme hazard of to adopt s 1 mnus the probablty that all these adopters, as well as the advertsng efforts, faled the task: P ( t) S ( t ) 1 (1 )(1 q ), where S (t) s the number of adopters n s personal socal network [29]. Note that ths formulaton s not the only means of descrbng contagon n agent-based models. Other models, such as those based on the Isng model analogue, have used a thresholdbased approach, where the agent changes states when a certan threshold of utlty s reached [6]. However, the competng-rsk formulaton s more approprate for modelng new product dffuson, snce t converges to the Bass model as the dscrete tme nterval reduces to zero [30]. In addton, ths approach consders the overall nfluence from all agents connected to the potental adopter, unlke other models, whch choose randomly the agent that nfluences [22]. Lba, Muller and Peres [18] extended ths monopolstc model to descrbe adopton n a compettve scenaro, as follows: f two frms, A and B, compete n the market, then an agent can be n one of three states "0", "A" or "B". Each of the competng frms s assgned ts own values for external nfluence, A and B, and for the nternal nfluence of a sngle agent, q A and q B. 3 In the dffuson lterature s often denoted as p. Here, p s used to represent probabltes. 12
Adopters of A and B each ndependently nfluence a potental adopter to adopt ther respectve frms. The probablty of beng successfully convnced by at least one adopter of A or B s gven by: P A ( t) A S ( t ) 1 (1 A )(1 qa ) ; P B ( t) B S ( t ) 1 (1 B )(1 qb ), Where A S and B S denote all agents n s personal socal network who have adopted ether A or B. Now, n a dscrete tme perod t, there could be one of three scenaros: a) Agent s convnced to adopt from A but s not convnced to adopt from B. The probablty of ths A B happenng s P (1 P ). b) Agent s convnced to adopt from B only. The probablty of ths B A s P (1 P ). c) Agent s persuaded to adopt from both A and B, but as per the model A B assumptons has to choose one of the two. The probablty of ths happenng s P P, and the agent adopts from A rather than B accordng to the rato of the probabltes,. The probabltes of actually adoptng from frm A, adoptng from frm B, or not adoptng from ether are, respectvely: (4) P ( adopt A) P (1 P ) P A B P A B A A B P ( adopt B) P (1 P ) P P P ( adopt none) (1 P )(1 P ) B A B A where, 1 B A A In the smulaton, for each agent n each perod, the adopton probablty s realzed by drawng a random number from a unform dstrbuton and comparng t to adopton probabltes P (adopt A) and P (adopt B). If ths number s between 0 and P (adopt A), A s adopted; f t s between P (adopt A) and P (adopt A) + P (adopt B), B s adopted; and f t s between P (adopt A) + P (adopt B) and 1 there s no adopton (snce the number s randomly drawn, the order A, B does not matter). A sngle run of the smulaton (namely, an adopton process startng from zero wth a gven network structure, and q) ends after 30 tme perods, whch s consstent wth common practce 13
n smlar models [29]. Gven the parameter values used here, the 30 tme perods are such that most of the market has adopted by that tme. For smplcty, A and B are assumed to be equal for frms A and B, and dentcal across agents. To take nto account the possble heterogeneous nature of customer propensty to be affected by others, the value of q s assumed to be normally dstrbuted throughout the network. For robustness, we also examned cases n whch q was dstrbuted n a power law dstrbuton wth the power-law exponent parameter smulated n the commonly used range of 2-3. We also looked at a unform dstrbuton n whch the range was plus mnus the standard devaton used n the Normal dstrbuton analyss. The results reported next are robust to the specfcaton of q. The values of and q were chosen to be consstent wth prevous research regardng the ranges of these parameter values n dffuson models and n agent-based models (e.g.[19]). The parameter was assgned the followng values: = 0.001, 0.005, 0.01, 0.05, 0.1. The values of q dffer across networks: In networks wth hgh average degrees, our prelmnary smulatons show that the nterestng dynamcs occur for lower values of q, (snce the combnaton of hgh q values and hgh degree generates almost nstantaneous dffuson). Therefore, two sets of q values were used: For the 80 networks wth lower average degrees, we used a dstrbuton wth an average q of 0.005, 0.01, 0.02, 0.03, 0.04, and for the 80 networks wth hgher average degrees, we used dstrbutons wth an average q of 0.08, 0.1, 0.12, 0.16. The expermental desgn was a full factoral experment, where for each network we tested all combnatons of the dfferent values of and q (from the approprate set). For each of the 160 networks, we ran the smulaton 6000 tmes: we ran the smulaton for each combnaton of and q as elaborated above, for both monopolstc and duopoly scenaros, and, to control for random effects, for each parameter set we ran 120 smulatons. Smulatons were conducted usng C++ code. The overall runtme on an Intel 3.1 GHz 5-2400 core processor, 16GB RAM was 47 days. 4.2 The tme value of dffuson An effectve dffuson process s quck, and t affects a large number of network members. To measure the effectveness of dffuson, we evaluated the NPV of the number of adopters, that 14
s, the dscounted sum of the number of adopters. Thus, for a gven frm the NPV s, where S (t) s the number of agents who adopted from frm ( =A,B) at tme perod t. Note that n our smulatons, the frms are symmetrc, so S (t)=s (t). T s the total tme horzon, and d s the dscount factor. In the smulaton, a standard dscount factor of 10% s used. 5 Results To estmate the mpact of network topology on dffuson, we examned how the NPV of the number of adopters s dependent on the varous topologcal metrcs and dffuson parameters. Fgure 4 comprses three graphs, llustratng, respectvely, the dependence of the ln_npv on each of the three topologcal metrcs we used. For llustraton purposes we present values for =0.005 and q=002, averagng the values of each remanng varable across all smulaton runs and across all the networks. The ln was used to enable a comparson to be made between the coeffcents of the monopoly and duopoly cases. To evaluate the relatve role of each of the topologcal metrcs n the dffuson process, we ran a multvarate regresson, where ln_npv was regressed smultaneously aganst the topologcal metrcs and dffuson parameters. The regresson data were pooled over the 160 networks. The explanatory varables were the average degree (Degree), the relatve degree of hubs (Rel_hubs), and the clusterng coeffcent (Cluster). The dffuson parameters and q were mean-centralzed. The resultant regresson equaton was the followng: (5) ln_ NPV 0 1Degree 2 Rel _ hubs 3Cluster 4 5q. Each data pont n the regresson was the average over the 120 runs wth the same parameter combnaton. The estmaton was performed separately for the monopoly and duopoly condtons. For the duopoly case the NPV of one of the frms (frm A) was used; snce both frms are dentcal, the choce of A or B s equvalent. 15
Fgure 4: Network performance as a functon of topologcal metrcs, for monopoly and duopoly. = 0.005, q = 0.02, averaged on all other parameters and on all networks. The results are dsplayed n Table 1. The table presents the regresson coeffcents for the monopoly and duopoly cases. In both cases, the topologcal metrc that has the strongest mpact on dffuson s the relatve degree of hubs. That s, the hgher the rato between the average degree of the top 10% most connected agents and the overall average degree, the more effectve the dffuson process. The effect of the average degree s also postve, but smaller. Ths result contrbutes to the controversy as to the mportance of socal hubs to the dffuson process: When controllng for dffuson parameters and other network metrcs, the mpact of the average degree of socal hubs s postve and strong, even more than that of the overall average degree of the network. The effect of the clusterng coeffcent s negatve, ndcatng that global clusterng has a negatve effect on dffuson. Ths fndng contrbutes to the dscusson on the role of transtvty n network processes [11]: Whle transtvty mght have benefts, n the context of dffuson ts mpact s negatve. By varyng the clusterng coeffcent ndependently of other network metrcs, and testng t smultaneously wth the other metrcs, we have managed to solate ts effect and assess ts relatve role n dffuson. Interestngly, these results are robust to the compettve market structure and are the same for both monopolstc and duopolstc markets. All results are sgnfcant wth p < 0.001. Table 1: Estmaton results n monopolstc and duopolstc markets 6 Dscusson and Conclusons Ths paper focuses on the effects of average degree, relatve degree of hubs and clusterng coeffcent on the dffuson of a new product. We ntroduce a network generaton procedure, based on random graphs wth a planted partton, to generate 160 networks encompassng a large 16
range of values for the parameters under nvestgaton. Usng agent-based models, we smulate dffuson processes on these networks, for monopolstc and duopolstc markets. By drectly manpulatng the topologcal metrcs and measurng ther mpact smultaneously, we can solate the relatve nfluence of each metrc. We fnd that the relatve degree of hubs, as well as the average degree, have strong and postve effects on dffuson. The clusterng coeffcent, however, has a negatve mpact on dffuson: the hgher the level of global clusterng, the weaker the dffuson process. These fndngs shed lght on the ways n whch underlyng network topologes nfluence dffuson, and they contrbute to two ongong dscussons n the network lterature. The frst result emphaszes the mportance of hubs,.e., nodes whose degrees are relatvely hgh compared wth the degrees of the rest of the populaton. Specfcally, our analyss suggests that a degree dstrbuton that tends towards a power-law s lkely to be assocated wth a more effectve dffuson process (n terms of NPV) compared wth a more unform degree dstrbuton. Ths result demonstrates that the role of socal hubs n dffuson processes s ntrcate and depends on the aspect of dffuson that s beng tested: Whle the relatve degree of hubs mght be less mportant when measurng the length of ndvdual cascades of nfluence (the length of a sngle nformaton, or nfluence chan) [9], t s mportant to the overall effectveness of dffuson. Our fndng regardng the negatve mpact of clusterng contrbutes two nterestng nsghts to the ongong dscusson on the pros and cons of network transtvty [11]: Frst, the result sheds lght on the role of transtvty, n solaton from other network metrcs. Whle most other studes have compared network types, thereby consderng clusterng n combnaton wth other network metrcs, here we measure the drect effect of clusterng on dffuson. Second, our result demonstrates the drawbacks of transtvty n the context of dffuson. It seems that the redundances generated by hgh clusterng mpede dffuson. Note that although ths paper studes a wde range of parameter values, ts fndngs cannot automatcally be generalzed to all network types. Snce the network creaton procedure s based on random graphs, the degree dstrbuton s a Possonan mx functon, and s dfferent from some degree dstrbutons found n real networks, such as the power law dstrbuton. The results are not expected to be fundamentally dfferent for such networks; however, further research s needed to verfy ths emprcally. 17
7 Acknowledgements Ths paper was supported by the Israel Scence Foundaton, The Israel Assocaton for Internet Study, and the Kmart fund. My thanks to Mchael Krvelevch for dscusson on random graphs wth a planted partton. I also thank Zeev Rudnck, Barak Lba and Etan Muller. Ths work s supported by the Israel Internet Assocaton, and the Kmart foundaton. 8 References [1] F.M. Bass, A new product growth model for consumer durables, Manage. Sc. 15 (1969) 215-227. [2] J. Goldenberg, B. Lba, E. Muller, Talk of the network: A complex systems look at the underlyng process of word-of-mouth, Market. Lett. 12 (2001) 211-223. [3] M. Barthelemy, A. Barrat, R. Pastor-Satorras, A. Vespgnan, Velocty and herarchcal spread of epdemc outbreaks n scale-free networks, Phys. Rev. Lett. 92 (2004) 178701-1-4. [4] M. Granovetter, The strength of weak tes, Am. J. Socol, 78 (1973) 1360-1380. [5] J. Goldenberg, S. Han, D. Lehmann, J. Hong, The role of hubs n the adopton process, J. Marketng, 73 (2009) 1-13. [6] C.E Lacana, S.L. Rovere, Isng-lke agent-based technology dffuson model: Adopton patterns vs. seedng strateges, Physca A, 390 (2011) 1139-1149. [7] F. Stonedahl, W. Rand, U. Wlensky, Evolvng vral marketng strateges, n: Proceedngs of the 12th Annual Conference on Genetc and Evolutonary Computaton, ACM, New York, 2010, pp. 1195-1202. [8] M. Uchda, S. Shrayama, Influence of a network structure on the network effect n the communcaton servce market, Physca A, 387 (2008) 5303-5310. [9] D. J. Watts, P.S. Dodds, Influentals, networks, and publc opnon formaton, J. Consum. Res. 34 (2007) 441-458. [10] K. Keller, J. Berry, The Influentals: One Amercan n Ten Tells the Other Nne How to Vote, Where to Eat, and What to Buy, The Free Press, New York, 2003. [11] M.J. Burger, V. Buskens, Socal context and network formaton: An expermental study, Soc. Networks 31 (2009) 63 75. [12] J. Leskovec, L.A. Adamc, B.A. Huberman, The dynamcs of vral marketng, ACM TWEB 1 (2007) 5-51. [13] R. Albert, A.L. Barabás, Statstcal mechancs of complex networks, Rev. Mod. Phys. 74 (2002) 47 97. 18
[14] D. J. Watts, S. H. Strogatz, Collectve dynamcs of small-world networks, Nature 393 (1998) 440-442. [15] M. E. J. Newman, D. J. Watts, Scalng and percolaton n the small-world network model, Phys. Rev. E, 60.6 (1999)7332-7342. [16] M.E.J. Newman, D.J. Watts, S.H. Strogatz, Random graph models of socal networks, Proc. Natl. Acad. Sc. U.S.A 99 Suppl 1(2002) 2566-2572. [17] P. Erdős, A. Rény, On random graphs, Publcatones Mathematcae Debrecen 6 (1959) 290-297. [18] B. Lba, E. Muller, R. Peres, Decomposng the value of word-of-mouth seedng programs: Acceleraton vs. expanson, J. Market. Res. 50 (2013) 161-176. [19] M.E.J. Newman, The structure and functon of complex networks, SIAM R 45 (2003) 167 256. [20] M. Faloutsos, P. Faloutsos, C. Faloutsos, On power-law relatonshps of the nternet topology, ACM SIGCOMM Comp. Comm. R. 29 (1999) 251-262. [21] M.A. Janssen, W. Jager, Smulatng market dynamcs: Interactons between consumer psychology and socal networks, Artf. Lfe 9 (2003) 343-356. [22] L. Deng, Y. Lu, F. Xong, An opnon dffuson model wth clustered early adopters, Physca A 392 (2013) 3546 3554. [23] A. Barrat, M. Wegt, On the propertes of small-world network models, Eur. Phys. J. B 13 (2000) 547-560. [24] S. Fortunato, Communty detecton n graphs, Phys. Rep. 486 (2010) 75-174. [25] A. Condon, R.M. Karp, Algorthms for graph parttonng on the planted partton model, Random Struct. Algor.,18 (2001) 116-140. [26] C. Schnckus, Between complexty of modellng and modellng of complexty: An essay on econophyscs, Physca A 392 (2013) 3654-3665. [27] M. Haenlen, B. Lba, Targetng revenue leaders for new products, J. Marketng 77 (2013) 1-55. [28] R. Garca, Uses of agent-based modelng n nnovaton/new product development research, J. Prod. Innovat. Manag., 22 (2005) 380-398. [29] J. Goldenberg, B. Lba, E. Muller, Usng complex systems analyss to advance marketng theory development: Modelng heterogenety effects on new product growth through stochastc cellular automata, AMS Rev. 9 (2001) 1-18. [30] J. Goldenberg, O. Lowengart, D. Shapra (2009), Zoomng n: Self emergence movements n new-product growth, Market. Sc. 28 (2) 274-292. 19