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1 Abtelung für Stadt und Regonalentwcklung Department of Urban and Regonal Development Gunther Maer, Alexander Kaufmann The Development of Computer Networks Frst Results from a Mcroeconomc Model SREDscusson
2 The Development of ComputerNetworks: Frst Results from a Mcroeconomc Model Gunther Maer, Alexander Kaufmann Insttute for Urban and Regonal Studes, Venna Unversty of Economcs and Busness Admnstraton Augasse 26 A1090 Venna Austra Abstract Computer networks lke the Internet are ganng mportance n socal and economc lfe. The acceleratng pace of the adopton of network technologes for busness purposes s a rather recent phenomenon. Many applcatons are stll n the early, sometmes even expermental phase. Nevertheless, t seems to be certan that networks wll change the socoeconomc structures we know today. Ths s the background for our specal nterest n the development of networks, n the role of spatal factors nfluencng the formaton of networks and consequences of networks on spatal structures, and n the role of externaltes. Ths paper dscusses a smple economc model based on a mcroeconomc calculus that ncorporates the man factors that generate the growth of computer networks. The paper provdes frst analytc results about the generaton of computer networks. The paper dscusses (1) under what condtons economc factors wll ntate the process of network formaton, (2) the relatonshp between ndvdual and socal evaluaton, and (3) the effcency of a network that s generated based on economc mechansms.
3 1. Introducton The development of computer technologes and ts applcatons has been extremely rapd n the eghtes and nnetes. Computer networks belong to the most sgnfcant felds of techncal development. Ths s a consequence of two development paths: 1. The growng mportance of PCs due to the rapdly ncreasng computng capacty. 2. The organzatonal and communcatve advantages of electronc networks have led to the wdespread use of prvate networks, not only wthn companes and ts subsdares but also n customersupplerrelatons. The man mpulse came from a publc network  the Internet. It s generally beleved that computer networks lke the Internet wll fundamentally change economy and socety (see e.g., Castells, 1996, MacLuhan, 1992, Gllespe, 1991). Castells (1996) even compares ths new development wth the Industral Revoluton. In order to understand what consequences computer networks wll have on socety and economy t seems necessary to frst understand why and how such networks develop. In ths paper we try to take a step n ths drecton. We attempt to fnd key economc factors that may have drven the rapd nternatonal dffuson of networkng over the past years, and to dentfy the condtons that may stmulate or hamper ths development. We wll do ths by developng a conceptual framework and a smple economc model that captures the key factors. In ths paper we apply a mcroeconomc perspectve. Ratonal agents are assumed to decde about whether to connect to another agent through a network lnk or not based on a comparson of ther costs and benefts of ths step. In order to keep the analyss manageable we use relatvely smple concepts of costs and benefts n ths context. Most mportantly, we assume that all the costs and benefts materalze n the current perod so that we don t have to take nto account dscountng and expectatons about the future development of the network. 2
4 Of course, n realty there are dfferent components of the costs of a network lnk: nstallaton costs, fxed and varable telecommuncaton costs, provder fees, etc. In our analyss we assume only the costs of establshng the network lnk. A smlar argument holds for benefts as well. Here we assume that the man effect of a computer network lnk les n the reducton of dstance frcton. Ths, n turn, allows for an ncreased level of nteracton between actors. The actors are assumed to derve utlty from ths nteracton. A maor economc element of any network nfrastructure les n the exstence of network externaltes (Capello, 1994). In our smple model network externaltes orgnate from the fact that when an actor connects to another actor who s already on a network, dstance frcton between all actors already on the network and the new member s reduced. As we wll see, these network externaltes are the man drvng force behnd network development. They also rase the queston of optmalty and ndvdual vs. socal valuaton. 2. The Model In order to gan some nsght nto the development of computer networks, we now develop a model that allows us to talk about the above mentoned mechansms and relatonshps n more precse terms. We start by descrbng the ntal condton of the problem,.e. the stuaton wthout any computer network at all, by a complete graph G = ( V, E) (1) wth V beng a set of n vertces and E the set of all n( n 1) 2 possble connectons between these vertces. The weghts of the graph are gven by a weght or dstance matrx D. Economc actors are located at the vertces of the graph G. Ths weght matrx represents the general dstances between each par of actors. 3
5 The computer network s descrbed by another graph N = ( V, L) wth the same set of vertces as G and a set of edges L that s a subset of E and represents exstng computer lnks. Intally, we set L = {} 0 and denote ths graph as N (, 0 = V L0 ). The graph N may consst of a number of components,.e. the subgraphs formed of connected vertces and the correspondng edges. Snce unconnected vertces are defned as components, the number of components may be between 1 (when all vertces are connected by the same network) and n (when no network connectons exst). We wll denote the components of graph N as where C (N) s the number of components of graph N. c ( N), = 1,, C( N) (2) Wth these defntons, each vertex must belong to exactly one of the components of N. We wll wrte the component that contans vertex V as c. Note that when V and V are drectly or ndrectly connected by a network lnk,.e. by network lnks n L, they belong to the same component and c = c. The actors are assumed to beneft from nteracton. They can ether nteract along the edges of the underlyng graph G, or along the network lnks n N. Of course, they can nteract along N only wth those actors that belong to the same component;.e. those who are on the same physcal network. Network lnks ease nteracton and are therefore benefcal for the actors. We defne the beneft of nteracton as a standard nteracton potental. For N 0, the case wth no network lnks, the beneft for actor s therefore B ( N 0) = exp( α D) (3) Now, what s the effect of a network lnk on the benefts of a certan actor? It can be argued that n today s computer networks dstance frcton s neglgble. Delays n the communcaton over those networks orgnate from other factors lke lmted capacty of a server, lmted capacty of the fnal 4
6 lnk, a general level of congeston, etc. that are not related to dstance. Therefore, t can be argued that dstance frcton s completely elmnated by network lnks. The benefts for an actor on the network would therefore become D c B ( N) = exp( α ) + exp(0) (4) Ths, of course, s an extreme poston that may hold only for pure communcaton. As soon as there are other forms of nteracton nvolved, lke the shpment of products or personal vstatons, dstance frcton s agan mportant. Therefore, a more moderate formulaton would be to allow the network lnks to reduce dstance frcton by a certan factor δ ( 0 δ 1). The benefts for an actor would therefore become c c B ( N) = exp( α D) + exp[ α(1 δ ) D ]. (5) Ths s the most general formulaton. It contans the two others as specal cases (δ = 0 and δ = 1, respectvely). For any δ > 0 the actors nvolved benefts from a network connecton. But, of course, there may also be costs nvolved n establshng and operatng a network lnk. These costs are probably proportonal to the length of the network lnk. If we denote a drect network connecton between vertex and vertex as c L, the costs for ths lnk can be wrtten as C(. (6) L ) = γd Note that wth the establshment of a lnk from to we also establsh a lnk from to. Therefore, the queston arses, who wll bear the costs of ths network lnk. In the worst case, one of the two actors that set up a new network lnk wll have to bear all the costs. Therefore, n hs/her decson makng every actor wll have to assume that he/she wll have to bear all the costs of the network lnk. Consequently, we can wrte for C ( ), the costs the lnk may create for actor as L C ( L ) = C( L ) = γd. (7) 5
7 Smlarly, C (N) s the costs network N generates for actor. The queston of our paper s, how N evolves over tme, when the decsons about whether to establsh a certan network lnk or not are based upon the economc calculus of the actors. For each par of graphs G and N we can compute the benefts and costs for each actor. Ths yelds vectors of benefts and costs Snce G s a constant, we wll use the smplfed notaton B = B(N,G) and C = C(N,G). (8) B = B(N); C = C(N). (9) Gven these benefts and costs, the actors are, of course, nterested n maxmzng ther profts: Π ( N) = B ( N) C ( N) (10) As mentoned above, we assume throughout that all the costs and benefts occur only n the current perod. That means that actors look only at the current stuaton and do not make any nvestmenttype decsons where they trade off current expendtures for future benefts. Also, we assume that there are no capacty constrants. Therefore, the addton of a network connecton cannot lower the beneft of actors already on the network. These two assumptons smplfy the problem consderably, but at the same tme severely lmt the potental value of the analyss. We hope to be able to remove these assumptons n future work. + Let us wrte as N = ( V, L + L ) the graph that we get when addng the network connecton L to a base graph N. Smlarly, N = ( V, L L ) s the graph that we get when we remove the network connecton L from the base graph N. But, snce we do not allow for capacty constrants, network lnks wll only be added n our analyss, never removed. At every tme perod each actor can choose from n dfferent strateges. She can stay wth the current stuaton, drop the network lnk to any of the say k other actors he/she s currently connected to, or establsh a lnk to any one of the nk1 actors she s currently not connected to. 6
8 The margnal profts of these strateges can be derved drectly from the above defntons: Π ( N ) Π ( N) = [ B ( N ) B ( N)] [ C ( N ) C ( N)] Π ( N ) Π ( N) = [ B ( N ) B ( N)] [ C ( N ) C ( N)] (11) (12) The margnal proft of no change at all s zero, of course. At every tme perod a ratonal actor wll try to mplement the strategy that provdes hm/her the maxmum margnal proft. However, mplementng such a strategy has mplcatons for other actors as well. It can well be that addng lnk L yelds the hghest margnal proft for actor, but does not yeld the hghest margnal proft for actor. A number of mechansms are concevable that decde such stuatons. For example, we may assume that only the strategy wth the hghest margnal proft of all actors wll be mplemented. The most meanngful mechansm from a mcroeconomc pont of few s what we call the mutual best strategy. Ths means that only those network lnks wll be establshed that yeld the hghest margnal proft for both actors nvolved. 3. Some Results 3.1. Network Generaton We can derve some nterestng frst results even from ths very smple structure. The frst queston we wll ask s, under what condtons the economc forces wll be suffcent to generate a network. More formally: Under what condtons wll economcally ratonal actors establsh at least one network lnk. A related queston that we can answer wth the frst one s whch one of the possble connectons wll be establshed frst. From (5) we know that the followng holds: + 0 ] exp( αd B ( N ) B ( N0) = exp[ α(1 δ ) D ) (13) The margnal costs for actor are smply a lnear functon of dstance: C + ( N ) C ( N0 0 ) = γd (14) 7
9 When we plot the margnal benefts as a functon of dstance we see that for 0 < δ < 1 the functon s postve, but approaches zero as dstance ncreases. The maxmum of the functon s where 1 1 D = ln( ). (15) αδ 1 δ For δ = 1,.e. the case where dstance frcton s elmnated completely, the margnal beneft ncreases monotoncally and approaches 1. Snce δ = 0 represents the case when dstance frcton s not changed at all by network lnks, the margnal beneft s always zero n ths case. Snce the margnal beneft s postve for all meanngful values of δ, the answer to the queston we have posed depends upon the slope of the cost functon. When the slope of the cost functon s lower than that of the margnal beneft at dstance zero, there wll be a range of dstances where the creaton of a network lnk s benefcal. Snce the slope of the margnal beneft functon at dstance zero s αδ, we fnd that only when αδ > γ (16) a network lnk may be created by ratonal economc agents. When ths condton holds, the range of dstances at whch establshng a network lnk s economcal always begns at dstance zero. For δ < 1 and γ > 0 there must be a maxmum dstance beyond whch the margnal proft s negatve. The optmum dstance, the dstance for whch the margnal proft s hghest, must be somewhere between zero and ths upper bound. Because of the dfference of exponental functons n (5) we cannot calculate an exact soluton for the upper bound and the optmum dstance. When we approxmate the beneft functon by a Taylor seres expanson, we fnd that the upper bound and the optmum dstance are approxmately 2( γ + αδ ) 2 α δ ( δ 2) and ( γ + αδ ). (17) 2 α δ ( δ 2) When the parameters are such that for a certan range of dstances n D postve margnal profts exst, wll there always be a network lnk establshed? Under our set of assumptons the answer s 8
10 yes. The reason s that the network lnk that yelds the hghest margnal proft of all possble network lnks must also yeld the hghest margnal proft for the actors at both ts ends. Therefore, whenever at least one dstance exsts that provdes a postve margnal proft, a network lnk wll be establshed for economc reasons. Because of (16) and the condton that δ s at most equal one, both nomnators and the denomnator n (17) are less than zero. Therefore, when γ ncreases, the optmum dstance and the upper bound decrease. Ths mples, for example, that n a country where telecommuncaton costs are hgh, computer networks wll start off wth shorter connectons than n a country where telecommuncaton costs are low. The argument that we have made so far also holds when network lnks can be establshed for free ( γ = 0 ). Although the margnal proft s postve for all dstances, the maxmum margnal proft s at the dstance gven by (15) above. So, even when network lnks are free, the network wll begn wth that connecton that s closest n dstance to the optmum dstance gven n (15). The reason s that nteracton partners that are further away than ths optmum dstance contrbute less to the beneft of the actor. Only when n addton to free network lnks (γ = 0) also dstance frcton s perfectly elmnated (δ = 1), then actors wll want to start the network by establshng the longest possble network lnk Indvdual vs. Socal Proft The next queston we can dscuss s that of the ndvdual evaluaton n relaton to a socal one. We can pose ths queston n the context of the above dscussed problem and ask ourselves, whether there mght be stuatons when based on the ndvdual evaluaton no frst network lnk wll be created although t would be desrable from a socal pont of vew. A dfference between ndvdual and socal evaluaton may result from two sources: 1. There mght be benefts to other actors that are not taken nto account by the decson maker, and 9
11 2. the costs to socety may dffer from those that the decson maker takes nto account. As mentoned above, both sources exst n our problem. Establshng a network lnk from to not only mproves the nteracton potental of actor, but also that of and all the other actors that are already connected to or. As far as costs are concerned, we have argued above that apror t s unclear who wll have to bear the costs of a network lnk and that therefore each actor should base her decson upon the assumpton that she wll have to bear all the costs by herself. As stated n (7), the ndvdual costs of an addtonal lnk are equal to the socal costs. Therefore, on each sde the margnal beneft of the network lnk must be hgher than the total margnal costs. From a socal pont of vew, however, the margnal benefts to all actors together have to exceed the margnal costs of the lnk. We can derve the socal beneft of a certan network N drectly from (5) above by summng the benefts for all actors n the system. So, the socal beneft can be wrtten as: B ( N) = B ( N) (18) Ths allows us to derve a drect measure of the network externalty at the beneft sde, whch s the sum of all the benefts that are not been taken nto account by the decson maker. When s the decson maker, the network externalty s B ( N) B ( N) = B ( N) (19) Based on these arguments, we can derve the socal margnal proft of the frst network lnk between and as: + + Π( N 0 ) Π( N 0 ) = [ B ( N 0 ) B ( N 0 )] [ C ( N 0 ) C ( N 0 )] (20) k k k k + k = 2 {exp[ α(1 δ ) D ] exp( αd )} γd (21) 10
12 When we compare (21) to (13) and (14), we see that the socal margnal proft dffers from the ndvdual one by the ndvdual margnal beneft of the lnk. Ths s the beneft that actor enoys, but whch s not consdered by actor n the decson makng. When we do the same calculatons as above, we fnd that the equvalent to (16), the condton that a network lnk may be socally desrable, s 2αδ > γ. (22) Obvously, the costs for the network lnk could be twce as hgh when socal costs and benefts are taken nto account than n the case of an ndvdual calculaton. So, when 2αδ > γ > αδ (23) a socally desrable network lnk may be possble, but wll not be created based on the actors mcroeconomc calculus. We can also derve approxmate solutons for the upper bound of the dstance range and the optmum dstance under a socal calculus. They turn out to be ( γ + 2αδ ) 2 α δ ( δ 2) and ( γ + 2αδ ). (24) 2 2α δ ( δ 2) When we compare these results to (17), we fnd that both the upper bound and the optmum dstance s hgher under socal evaluaton. So, from a socal pont of vew, the range of dstances for whch a network lnk may be created s too narrow, and when a network lnk s created based on the ndvdual calculus, t wll tend to be too short from a socal pont of vew Network Structure Another queston we may ask s, whether a network that emerges step by step from ths mcroeconomc mechansm wll be effcently servng the system of actors. In order to answer ths queston, we wll have to look at the end pont of the process. Suppose that the spatal constellaton and parameter values are such that based on the decson process n the end all actors are connected 11
13 to the network. Let us denote ths network as are actually lnked, ths network wll yeld a vector of benefts wth The socal beneft s N n. It s easy to see that rrespectve of how the actors B ( N ) = exp[ α (1 δ ) D ]. (25) n B ( N n ) = exp[ α (1 δ ) D ]. (26) The proft of the network therefore depends only upon the costs. The network N n wll be optmal when the actors are connected such that the sum of the costs of these connectons s mnmal. In graph theoretc terms ths means that the vertces V must be connected by a mnmumweght spannng tree (see e.g. Gbbons, 1985). Prm (1957) has shown that a mnmumweght spannng tree can be constructed by always fndng the shortest dstance between a vertex n the tree and one not n the tree and connectng the two. So, when we want to construct a network that serves the system at least costs, we should begn by selectng an arbtrary vertex and connect t to that vertex whch s nearest to t. Ths provdes the bass for answerng the queston we have rased above. As we have seen, nether based on the ndvdual nor based on the socal calculus wll want to establsh a network lnk to her nearest neghbor. Instead, she wll want to establsh a lnk to that actor that s nearest to the optmum dstance, where the latter dffers from zero when there exsts a range of dstances for whch a network lnk s desrable. So, the network that emerges from a mcroeconomc calculus rrespectve of whether t s based on ndvdual or socal costs and benefts wll not connect the actors at least costs. There mght always exst another set of connectons that yelds the same level of benefts at lower costs. As some expermentaton shows, the mutual best mechansm that we have dscussed may not even connect the actors by a tree. The resultng network may even dsplay loops. 4. Conclusons 12
14 Ths paper represents a frst step toward an economc analyss of the queston of the formaton of computer networks. Stmulated by the tremendous growth of the Internet, we try to nvestgate whch economc factors drve ths development toward global connectvty and how these factors nfluence the process of network formaton. In a frst part (secton 2) we develop a formal model of the key economc mechansms that drve network formaton. The model focuses on ndvdual mcroeconomc decsons based upon a comparson of costs of network lnks and benefts n the form of an ncrease n the nteracton potental. The model s stll farly smple, and there s room for mprovement. For example, the model does not take nto account capacty constrants, dfferences between actors, dfferent decson strateges, etc. In secton 3 we analyze the behavor of ths model and the mplcatons t has for the process of network formaton. We show under what condtons the economc ncentve s suffcent to ntate the process of network formaton, that there mght be constellatons where a network would be socally desrable, but the economc ncentves are nsuffcent to generate any network lnks, and that the economc process of network formaton does not lead to an optmal network topology. As has been mentoned above, we vew ths paper as a frst step n the drecton of the topc and ntend to develop more elaborated versons of the model later. Because of the complex relatonshps between the varous actors n the process of network formaton and because of network externaltes and capacty constrants, we wll probably have to use smulaton experments when analyzng these more complex model versons. 13
15 References Capello, Roberta, 1994: Spatal Economc Analyss of Telecommuncatons Network Externaltes. Avebury Castells, Manuel, 1996: The Rse of the Network Socety. Blackwell Gbbons, Alan, Algorthmc Graph Theory, Cambrdge, Cambrdge Unversty Press Gllespe, Andrew: Advanced communcatons networks, terrtoral ntegraton and local development. In: Camagn, Roberto (ed.), 1991: Innovaton networks  spatal perspectves. Belhaven Press; p MacLuhan, Marshall and Powers, Bruce R., 1992: The global vllage  transformatons n world lfe and meda n the 21st century. Oxford Unversty Press Prm, R. C., Shortest connecton networks and some generalzatons, Bell System Tech. Journal, Vol. 36, pp
16 Abtelung für Stadt und Regonalentwcklung Wrtschaftsunverstät Wen Abtelungsleter: o.unv.prof. Edward M. Bergman, PhD Roßauer Lände 23/3 A1090 Wen, Austra Tel.: /4777 Fax: /705 EMal:
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