Graphs and Network Economics

Shu-Heng ChenAI-ECON Research Center

National Chengchi UniversityTaipei, Taiwan 11623, China

E-mail: chchen@nccu.edu.tw

Abstract-Following the standard probabilistic approach, weshall show explicitly the relation between a microscopic and amacroscopic view of an economy in the context of a discretechoice model. Two crucial issues of the graphical applicationsto the network economy are addressed. The first one concernsthe representation richness of the graph, whereas the second oneconcerns the formation and evolution of the graph when it isapplied to social networks. This study can be a starting pointto see the relevance of agent-based computational economics tothe network economy , in particular, after bringing into theinteraction mechanism associated with a network topology.

I. INTRODUCTIONWhile network economics seems to get its popularity over

the last decade, it remains unclear on some of its fundamentalissues.1 The purpose of this paper is to tackle with these issueswhich we find from a few recent survey articles, and attemptsto clarify some ideas relating to network economics. The issuesconcern us are roughly two-fold.

1) What is the role of the graph, as defined in the graphtheory in mathematics, in the network economy? Whatis the relation between a network and a graph? Can thegraph be a universal representation any kinds of networkmentioned in the network economy? This is an issueperaining to the representation.

2) Where does the network come from, and how will itchange? How does the network at the period t shapethe network coming in period t + 1? This is questionconcerning network formation and network evolution.

In this paper, we will summarize what we may have learnedfrom a series of survey articles, such as Kinnan (1997a, b,1998) and Blume and Durlauf (2001). These papers give usa nice warm-up of what is currently known as agent-basedcomputational economics. One distinguishing feature sharedby these papers is that macroeconomic phenomenon cannot bestudied in terms of the behavior of an individual in isolation.The series of papers contains three essential ingredients,

namely, discrete-choice dynamics, networks and interactions.These three ingredients constitute two important subjects.First, let us focus first on discrete choices. This literaturepoints out the significance of the interaction on decision-making. But, interaction, at least local interaction, is carried

1 The term is used in a very general way so that it may encompassor intersect with some other more focusing areas named differently, e.g.,the interaction-based economics as defined by Lawrence Blume and StevenDurlauf in their series of papers.

Chien-Liang ChenDepartment of EconomicsNational Chi Nan University

Nantou, Taiwan 545, ChinaE-mail: clchen@ncnu.edu.tw

out essentially through an underlying network. Therefore,these three ingredients are combined in a coherent framework.Second, but, how the network on question is formed in the firstplace? One answer is that this is formed by agents' decisions.Agents intentionally choose their trading partners, friends,neighbors,..., etc. Many of these choices can be regarded asdiscrete choices; therefore, the discrete-choice model may helpus to see the formation of the networks. Actually, this is a wayto see what we mean by evolving networks, as discussed inKirman (1997b).

II. DISCRETE-CHOICE DYNAMICSA. Aggregate DynamicsWe start with a general description of the discrete-choice

model. The discrete-choice model has been extensively appliedto models of technology choices and models of evolutionarygames. All discrete-choice models share some common ele-ments which can be briefly summarized as follows.

1) There are a number of choices (technology, restaurants,strategy,...) available for a number of agents (firms,consumers, players).2

2) The number of choice is generally held fixed, say K.33) The number of agents may be fixed, such as in the fixed

population urn model or in the Fbllmer-Kirnan model(Kirnan, 1993), or increase over time, such as in theArthur model (Arthur, 1988) or the scale-free network,say Nt.

4) The choice made by the agent n (n = 1, 2, ..., N) canbe fixed or can change over time.4

With this fundamental setting, discrete-choice models com-monly share some basic questions. It starts with what we shallcall demographic statistics. A major demographic statistic at

2It would be particular useful to start with the binary-choice model, as itwas worked out in the interaction-based economics. See footnote (1).30ne may be curious about this seemingly very restrictive device. Kirman

(1998) has made a remark on this. "...It should of course be said that theset of K technologies available should expand over time to take account ofthe arrival of new candidates. This, however, would add considerably to theanalytical complications." (Italic added). In should be noticed that one featureof genetic algorithms or genetic programming is to remove such a restriction.In the case of increasing number of strategies, see Lindgren (1991). In thecase of increasing number of technologies, see Chen and Chie (2004).

41t should be noticed that tremendous effort has been made as to the choicerevision or renew process (Blume, 1991.). In a standard model, agents tendto synchronize their revision. It is less usual to consider the asynchronousadjustment. One exception is given by Ellison and Fudenberg (1993).

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time t is the number of agents who make a specific choice k,say a technology or a strategy, (k = 1, 2, ..., K). This can alsobe summarized by the empirical distribution as

K

Xt = (Xl,t,X2,t,...,XK,t) EZXk,t = 1

k=l(1)

where Xk denotes the fraction of the agents who choose k.For convenience, let us call the vector Xt the social porfolioover a finite number of choices. In a general setting, x k,t isa random variable between [0, 1]. Therefore, it is useful toconsider their associated joint distribution

K

ft(Xl,---,XK), EXk = 11 (2)k=1

and its limit distribution (the equilibrium of this system) if itexists

K

f(Xl,...XK) = liM ft(Xl,...XK), ZXk,t = 1 (3)k=1

When there are only two choices, Equation (3) can be reducedto a uni-variate distrbiution. An example is the Beta distribu-tion well discussed in the famous ant problem as shown inEquation (4) (Kirman, 1993).

f(x) x(a+fl)Xa(l )3-1, x e [0, 1], (4)

and with appropriate assumptions on the parameters of theoriginal model, say a = / = 0.5, the distribution willconcentrate in the tails.5

Particular interesting information of the joint distributionconcerns the life expectancy, the population, the diffusion (lo-gistic curve), the concentration ratio, the cyclic phenomenon,..., etc. One type of limiting distribution particular interestingresearchers is the degenerating case. A limit distribution issaid to degenerate if 3 a fixed vector

K

X*= (xi,...,4K), E k=1k=1

such that

f(X15...- k) ={oif X=X*,otherwise, (5)

This special case known as the existence of a stationarydistribution is just a macroscopic viewpoint of the system.It does not necessary imply that all agents also come to theirrespective stationary states.6 On the contrary, the agents maycontinuously switch in and out among different choices7.Among all degenerating cases, the most noticeable one is

the case where X* is one of the apexes of the simplex. Whenthis happens, the economy converges to a single technology,a single strategy, etc.

51n an extreme case, one can have Kolmogorov's zero-one law.6So far, we have not said anything about how each agent revises their

choices.7See Vriend (1995), for a example.

In addition to the joint density, sometimes one also workswith the marginal density function: fk,t (x), x E [0, 1], or thecondition density fk,t(x Xt_l,Xt2, ...), x E [0,1]. Thecondition density can be reduced to the first order,

fk,t(X Xt-1) = fk,t(X Xt-.1Xt-2,...),x e [0,1], (6)

when the Markovian structure is imposed. Attention may bedrawn on the change of fk,t(x xt-1)9

Afk,t(X) = fk,t(X Xt-1) -fk,t-1(X IXt-2)IxE [0,1]. (

In some cases, when fk,t(x Xt-1) is degenerating to asingle point say Xk,t, then it is equivalent to work with theproportion dynamics itself,

AXk,t = Xk,t -Xk,t-1 (8)Actually, when the discrete-time dynamics is replaced withthe continuous-time dynamics, the above form can be exem-plified by the celebrated Chapman-Kolmogorov equation, orthe master equation:

dXk,t E (W(k l)Xl,t - W(l k)xk,t).dt l#k

(9)

W(k 1) is the Markov transition probability from the state Ito the the state k. In the economic context, the Markov processmay not be time-homogeneous. Very likely, the transitionprobability may depend upon many other variables, such as Xtitself. In principle, it should be connected to some performanceindex, such as profit or utility. Therefore, an alternative, butad-hoc, way is to work with profit directly, which can beexemplified by the familiar deterministic replicator dynamics.The deterministic replicator dynamics basically assume thatthe mapping from the space of Xt-, to the space ofXt followsthe a simple law of motion:

dxk,tdt

=

k,t- 7rt, (10)

where 7rk,t is the expected payoff of the choice k at timet, and 7rt is the expected payoff of the whole society. Sincethe calculation of 7rk,t may generally require the knowledgeof the whole structure of Xt; therefore, it is mathematicallyconnected to its more general form (7).

B. Individual ChoicesThe rest of the whole study concerns how ft(Xl, -, XK)

is determined by the way agents make their choices with orwithout referring to an interaction mechanism which may ormay not depend on an underlying network whose networktopology may or or may be explicitly refereed to, depending onwhether it is a global network or a local network.8 This threadof thought can be further connected to a large pile of learningliterature when we consider learning as a part of interaction.The learning literature in economics typically distinguishesindividual learning from social learning. The social learing

8More on this will be said on Section IV.

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(7)

can be further divided into global learning and local learning.It is the latter that network topologies can matter.

Let us review a few often-cited examples here. Maybe themost influential one is the logit model used by Blume (1993).The logit model, basically, treats the K possible choicesexposure to an individual as K strategies, each with a payoff7rk,,t (k = 1 2, ..., K). The 7rk,t is a weighted average of thepayoff of the last period Wk,t-1 and the gain earned mostrecent, Tk,t,

rk,t = (1 - Y)lrk,t-1 + -yTk,t, (k = 1, 2, ..., K), (11)

where -y is also called the forgetting effect.9 Obviously, thelarger the -y, the more myopia the agent is. Given wk,t(k =1,2, ..., K), the logit distribution can be written as follows.

exp 6 7r .,tPk,t = _K 07rk,t (12)

Zk=1 exp/kt(2where Pk,t is the probability assigned to the tth choice atperiod t, and d is a reinforcement parameter which describeshow sensitive the individual to past profits. This is somewhatsimilar to what we know about the mixed strategies used ingame theory, and is also known as the roulette-wheel selectionin genetic algorithms. What, however, is unclear is whether thebehavior model (12) can be derived from the standard utility-maximization framework.

It is useful to reflect upon Blume's logit model to enrichour understanding of agents' discrete-choice behavior. First,Equation (12) can be understood as a normalized fitnessfunction frequently used in evolutionary computation. In thiscase, the original fitness function is simply 7rk,t, where 7rk,t iSsimply the performance (accumulate profits) of the kth choiceat period t. There are, however, different ways to make anormalization. Another familiar transformation is the powerfunction,

Pk,t = (,Kw ,) (13)

When /3 = 1, we have the linear transformation,

Pk,t =7 *, (14)EK '7rl,t'

However, when lrk,t is negative, Equation (14) can cause aproblem, and Pk,t may no longer have probabilistic meaning. 10

Second, during the transformation, the original fitness func-tion can be dilated or diluted to a different degree, and thatis done by the so-called adjusted fitness function in geneticalgorithms. The importance of this step is to characterizedifferent degree of selection pressure (choice intensity). Forexample, in Equation (12), when /3 = 0, then there will besimply no selection pressure.

Third, Equation (12) also can employ different fitnessfunctions. What particular important here is that the fitness

9Using the profit function (11), one actually introduce reinforcementlearning into the discrete choice model.

t0Nevertieless, adjusted linear transformation to protect the probabilitymeasure Pk,t do exist, and they are used quite frequently in genetic algorithms.

function used in Equation (12) is an accumulated one, ratherthan a spontaneous one. In genetic algorithms, it has been oftencriticized that the fitness function used generally does not keepa memory effect, which corresponds to the case -y = 1 in (11).However, there is certainly no particular reason not to use theaccumulated version if one wants.1'

Fourth, the model used in Equation (12) does not referexplicitly to the interaction effect. In a sense, it correspondsto the individual learning model, say, reinforcement learningas demonstrated by Equation (11). One can certainly make in-teraction more explicit if there is such a need. In fact, in somesituations, such as in the models of norms or conventions, thefitness function will be directly related to how far distinguishedone is from the community. Blume and Durlauf (2000) reviewa series of such examples. A particular interesting class ofmodels is the model of social conformation. In this model,agent's fitness function can be decomposed into two parts,namely, the private one and the social one. For a social one,consider the following cross-product of two metric vectors,

rkn= _<Xn,Xm ||- 11 Wn7Wm 11

- S J(Xn,Xm)(Wn-Wm)2m.n

given Wn = k (15)

J(Xnr Xm), as a metric, measures the sensitivity of agent n'sutility upon agent m, and Wn, Wm II measures how distinctbetween agent n's and m's choice. By a proper normalization,one can rewrite Equation (15) into as a weighted average form,so agent n's choice shall be a weighted average of the choicesof his (her) interacting agents. In a special case, when theinteraction does not depend on the similarity metric, Equation(15) is reduced to the uniform weight.

rk = - 1 5 (Wn - Wm)2, given wn k, (16)m.n

where M is the number of agents interacting with agent n.Equation (16) can be rewritten in a less personal way. In

this case, instead of comparing the choice similarity to eachinteracting agents, agent n will only treat the behavior of thewhole society as a summary statistic, namely, mean,

Tkn =- II Wn, In given wn = k (17)

which is the well-known Keynes' beauty contest.There is a practical limitation regarding the synchronization

involved in Equation (15), namely, the unavailability of wmwhile agent n makes his (her) decision. To solve this problem,an estimation of wm must be made, which, in a case, can beconnected to the familiar fictitious play in game theory.

Plus the non-interaction part (private part), H n, the utilityof choosing strategy k can be written as

Tk,t= Hkt+ || Xn,Xm Wn*IIWnxWm ||

= Hknt + EJ(Xn, Xm)(Wn-m)2. (18)mon

"But, whether the accumulated fitness function is desirable is certainly anempirical issue. It depends on the external environment perceived by agents.

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By combining the utility function (18) or the profit function(11) with the probabilistic choice model (12), one can thenbase an individual choice on a model integrating the sociallearning model into the individual learning model, and gener-ate the aggregate dynamics, as we discussed in Section (II-A)in a bottom-up manner. 12 In this way, network topologies canmatter for economic theory or practice, since they determinesthe way that human interacts and hence the choices they make,which then further determine the equilibrium or the law ofmotion of the economy. 13

III. ECONOMIC INTERPRETATIONS OF THE GRAPHWhen we talk about the network economy, it is important to

inquire whether there is essentially no other alternatives, if, byusing an appropriate representation, we can represent all theeconomy studied by a network. But, a technical question hereis what a network is. We start with a conventional approachto the network, i.e., treating a network as a graph studiedby mathematicians in graph theory. Within this framework,a network is composed of two entities, namely, vertices andedges. The question then is what the economic interpretationsof these vertices and edges are.

A. VertexA vertex can be interpreted as an economic agent. By

connecting two vertices with an edge, the edge can be inter-preted as the interaction between two agents. So, a networkis basically composed of a set of economic agents and theinteractions among these agents. However, this interpretationmay be too narrow to make networks to represent all possibleeconomy, since economic agents not only interact with eco-nomic agents but can also interact with something else. Asa matter of fact, the simple Walrasian economy shows thateconomic agents may have no direct interaction with eachother, and all their interaction are indirectly made through aprice signal, or in general, a statistic.

In fact, it is very common that decisions made by theagents involve only statistics rather than direct interaction withother agents.14 In herding models (Banerjee, 1992; Kirman,1993), or in technology choice models (Arthur, 1983; Arthur,Ermoliev, and Kaniovski, 1984), what agents take into accountin their decision-making are only some aggregate variables,such as the number of customers already visiting a specific

12I1 fact the joint distribution (2) is a reduced form of the following moregeneral distribution

gt(Wt),where

(19)

Wt=(Wl,...,WNt), and wnCEK=f{1,2,...,k}.

13As an example to show the relevance of this framework, see Chen, Sunand Wang (2005).

141t is interesting to notice that the different taxonomy of literature exists.For example, Blume and Durlauf actually differentiate the direct-interactionmodels with those indirect-interaction models when they define the literatureof the interaction-based economy. "The interactions-based approach focuseson direct interdependence between economic actors rather than those indirectinterdependence which arise through the joint participation of economic actorsin a set of markets." Here, we combine the two.

restaurant, the number of firms already adopting a specifictechnology. This makes us able to see that the target withwhich an economic agent interact can be non-human, such asa market, a company, a statistic or a web page. As a result, avertex is not limited to human agents (decision makers), andit can represent other types of agents. 15

But, since the price signal is basically an aggregate resultof the behavior or decision of a number of economic agents,some of this types of vertices may better be distinguished fromothers. So, we start with primitive vertices, and add on morevertices, called the implied vertices, which are derived fromthese primitive ones. With this distinction, we can say that anetwork is a collection of economic agents who are interactingwith each other and some "agents" implied from them. 16 Assaid, with this extended interpretation, we can include theeconomy which actually involves no human direct interaction,e.g,, the computer matching double-auction (Chen and Tai,2003).

B. EdgesIn addition to vertices, one also need a rich interpretation

of edges. Broadly speaking, edges means interaction. To fiteconomic models, interaction can be carried out in severalways. For example, in the game-theoretic context, it can implycommunication (signaling, cheap talking, tagging, bargaining,negotiation, contracting, cooperating or competing). In theusual market-theoretic context, it can means trading (buying,selling, recruiting, team production). Finally, in an extensivesocial context, it can generally imply influence (externality,strategic interaction, social learning, imitation, conferences,exchange ideas, crossover, norms, control, leadership, club,political party, community, contagious disease). It is the lastcase of interaction, influencing, giving rise to an interdepen-dent relation among agents, and distinguishing what we studyhere from the conventional neo-classical economics.

Furthermore, it is important to maintain the flexibility ofedges. First of all, it can be directed or undirected, uni-directional or bi-directional. Secondly, it can have differentsizes.17 Thirdly, if the edge is bi-direction, the size may beasymmetric, which to a extreme, allows for the possibilityof only one-way direction. 18 Fourth, a more powerful devicecomes from randomizing the size. In this case, the size ofthe edge is a random variable. Of course, that includes thespecial case that the size vanishing (closing) with a positiveprobability.

15Doing in this way, we bridge the gap between economics and computersciences when talking about agents.

16Different colors may be used to represent these implied agents when weconstruct the graph. The fictitious agent, as described by Kirman (1997a), isone example of the implied agents.

P7Here, it is pretty much like the synapse in the neural network. Somesynapse are much thicker, and some are thinner. Also like the road connectingtwo cities, it can be a high-way, paved road or just dirt road, each with differentroad conditions which can severely restrict the flow between the two cities.Furthermore, the learning dynamics of neural networks actually provides usa way to reflect upon the network evolution.

181t is quite often seen from a social-hierarchy viewpoint, agents with a ahigher social status influence agents with a lower one, but not vice versa.

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Another remark which we would like to make here isthe interacting relation can be transparent, but can also beanonymous. This relaxization enables us to use the graphto represent a number of game experiments conducted withhuman subjects, and to acknowledge the significance of theanonymous design on the experimental results. One would,therefore, not be surprised by that anonymity implies a differ-ent interaction as opposed to the transparent ones.

C. Metrics

The last thing to say about a graph is how the verticesshould be placed into a physical or an abstract space. First ofall, we have to notice that this space must be a metric space.Since, without it, we cannot conceptualize neighbors and theconsequent local interaction, which will make us unable tocapture a large number of economic models working with localinteraction. However, to enhance the economic representationof graphs, we do admit different metrics. In some simplestcases, the metric is the physical distance, such as the latticemodels (Ising models, cellular automata) or transportationnetwork.

But, in a social and economic context, it would be toonarrow to consider only the physical distance. This is particularthe case when who is to interact with whom shall dependon the correlation of their individual attributes (idiosyncraticelements). Therefore, a more general interpretation of distanceis needed, which calls for different metrics. In literature, amore general metric known as the attribute (characteristic)-based approach, or the correlation approach, is developed(Kalai, Postlewaite, and Roberts, 1978; Gilles abd Ruys, 1989;Haller, 1990).19 In this case, agents' attributes are known, andbased on that they are situated at a attribute space associatedwith a defined metric.More precisely, consider each agent (more specifically,

economic agent) as a multi-dimensional object, possibly, aninfinite-dimensional. Each dimension can be an attribute of anagent. So, a typical agent can be represented by a vector, say,Xt, where i refers to the ith agent, and

How two agents, say i and j are related or close, is measuredby their distance

1IXi - Xj 11 (21)

Once the metric is defined, a physical distance between anytwo agents is given. Ideally, we can transform this informationinto a physical map so that we can visualize their spatialposition. Nevertheless, in the graph theory, the distance isusually defined in terms of edges. As a result, agent A and Bmay be close in a physical or attribute space, but if there doesnot exist an edge between them, then, by the graph-theoretic

19Many attributes have been addresses in other research areas within anetwork-like framework, such as language in the sociolinguistic theory (Jupp,Roberts, Cook-Gumperz, 1982; Akinasou and Ajirotutu, 1982; Labov, 1996;Akerlof and Kranton, 2000),

definition, they are not neighbors to each other.20 This is amajor problem when the mathematical graph theory is appliedto economics: the appearance of edges between two agentsare uncorrelated with their social distance. This may causeno problem from the mathematical viewpoint, in particular,for those early literature on random graphs. But, one needsto be careful on dealing with the idea of random graph inthe context of economics, because it is a little counterintuitivefrom the economic viewpoint so long as we think hardly onthe incentives to construct these edges and the associated costsspent.21Whether or not the edge determination should be disen-

tangled from the metric depends on whether we would liketo take a top-down or bottom-up approach to the formationof the graph structure. In a top-down approach, the globalnetwork (a full-connected graph) is already formed at thebeginning so the distance is no longer a variable: everyonein the system can be potentially connected to every other onein the system with a minimum length 1. The issue then is tosee what configuration being realized, out of so many possibleconfigurations.22 Vriend (1995) is an example. In this case, theissue left for the agent is the choice of making connections.The choice can be completely random as the random graphdid, or can be rational as the scale-free network did. For thelatter case, the idea of distance may still matter as we shallsee more in the next section.

IV. IS THE GRAPH NECESSARY OR SUFFICIENT?

After enriching our possible economic and social interpre-tations of the three fundamental elements of a graph, namely,vertices, edges and metrics, we can almost visualize a groupof agents scattering over a physical or an attribute (feature,characteristic) space. The next question is whether theseelements are necessary for the study of a network economy.On the other hand, whether this extended interpretation ofvertices and edges can make the graph sufficient powerful torepresent any economy. Our answer is somewhat conservative:the graph, despite the enhancement, is neither necessary norsufficient to represent an economy model, as we shall discussin this section.

20The discrepancies between the physical distance and the relation distanceis also found in the initial period of self-organizing maps. Self-organizingmaps is a map consisting of agents whose attributes are close (similar) be-coming neighbors. However, at initial periods, agents with different attributes,stochastically determined, are randomly distributed over a physical space. Ittakes time for them to learn and adjust, known as a competitive learningprocess, before an order emerges. During the competitive learning process,each agent will change their attributes so that in the end only agents withsimilar attributes will have small edge-based distance.

210ne can take an ad hoc approach to be combined with the random graph,i.e., the edge between any vertices is associated with a probability which is adecreasing function of the distance between the vertices. So, the far the twovertices, the less likely that they will be connected. See Gabszewicz, Thisse,Fujita, and Schweizer (1986) as an example.

22Suppose there are n vertices, which means that there is a total of (n- 1)!potential undirected edges, and hence a total of 2(n-1)! network topologiesthere. Kirman (1998) actually used the mapping from this space of 2(n-1)!network topologies to same space when he tried to formulate a general ideaof evolving networks.

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A. Global vs. Local NetworksIn his series of papers, Kirman carefully distinguishes the

economies based on a global network from those based ona local network. Graph is indispensable for the latter, butnot necessary for the former. The global-network economybasically assumes away the significance of distance, be itphysical or social, in interaction. In other words, who is goingto interact with whom is independent of the distance betweenthe two, as already discussed in Section HI-C. The interactingprobability is equally for those agents who are sitting closerand for those agents who are sitting far away.

This discussion reminds us what is the real function ofa graph. The graph basically helps us to think about thenetwork formation when it is crucially dependent upon theconcepts of neighbors. When the network formation can bedisentangled with the concepts of neighbors, then the graphbecomes irrelevant. The global network models are exactlythe situation that network formation does not depends on theconcepts of neighbors, no matter how they are defined (throughdifferent metrics). Maybe the question which one should askhere is under what circumstance can we assume that neighborsor distance is insignificant in network formation. Or, underwhat circumstance, a global random matching model can be arealistic description of the real economy. We shall talk aboutmore on this on Section IV-C.

Apart from the global network models, the graph is alsonot so important when agents interact with some non-humanagents, such as some summary statistics or an organization. Inthis case, there is no needfor networkformation. Examples ofthis kind are abound, such as a computerized double auctionmarket, Diamond's search model (Diamond, 1982), Kirman'sant problem (Kirman, 1993), Banerjee's restaurant problem(Banerjee's, 1992).

In sum, when there is no need for the network formation orthe network formation is independent of the distance, the useof graph is then not necessary or is at most implicit. Needlessto say, a great number of economic models satisfy one of thesetwo criteria; as a result, it is not surprising to see that the graphis not frequently seen in the standard economic models.

B. Network FormationThe discussion above leads us to another intriguing issue:

what is the use ofnetworkformation? To answer this question,one have to make a distinction between the networks whichare formed naturally from those networks which are formedintentionally. This distinction is important as far as the evolu-tion of the~network is concerned.

First, there are a lot of economic models whose networksare formed naturally, usually subject to the Euclidean distance.So, obviously, distance or metric plays an important role if theformation of the network, at least, its initial formation. Maybe,the lattice model, the Ising model, or the cellular automatamodel are the best examples. In this case, neighbors are formednaturally by their minimum distance to the agent on question.Interactions are only allowed with the neighbors. A series ofstudies on the spatial game is a case in point (Blume, 1991;

Ellison, 1993). In this-like models, the graph definitely playsan important role in understanding the system dynamics. Theother class of graphs that networks are formed "naturally' isthe random graphs in which agents' interaction are determinedin a stochastic way and involves no human decision.

But, the real interesting case should be the networks whichare formed by agents' intentional choices: the choices of aneighbor, a partner,..., etc. It is at this point the literature ofdiscrete-choice model becomes relevant to the graph theory.Vriend (1995) and Kirman and Vriend (2001) are some exam-ples on the formation of marketing networks. What, however,should be noticed here is that while networks was formedeventually in their models, the initial metric was not involvedinto their analysis. In other words, who interacts with whom isindependent of the initial distance between the two, and hencethe initial graph, maybe randomly generated, plays no role inthe formation of the final graph. As the observation made byKirman (1998) on Virend (1995), "...although the interactionis global at the outset it becomes local as certain consumersbecome tied to certain firms. (p.33)" This corresponds to thetop-down approach mentioned in Section Il-C.Maybe the best example to show economic motives of

network formation is the concept of core (coalition forma-tion) well taught in general equilibrium analysis. Coalitionformation provides another example of network formation withintentional choices which are directly welfare-oriented. In thiscase, a group of agents can agree to form a coalition which canfacilitate their resource reallocation plans, such as an exchangeand production plan. Core also illustrates the dynamics ofthe network formation. Once a coalition is formed, it willencounter a series of threats (broken down by the some ofthe members) or opportunities (expanding with new membersjoining) so that we experience the life and the death of anetwork as well as the evolution of networks.

Let us come back to our original issue: does distance matterin this process, so that coalition are more likely to formedthrough "neighbors" rather than "strangers"? We believe thatthe answer is largely yes, and we shall say more on this inSection IV-C.23

23Kirman, Oddou and Weber (1986) uses the diameter of the graph asa pre-condition of the coalition formation. In one case, the diameter is at amaximum of one; the other case, it is two. So, economic theory already noticedthe possible interest of using distance as a predictor of coalition formation. Theonly problem here is that diameter is defined in terms of edges, which meansa initial network already exists before the coalition. But, then what causes theformation of the initial network is left unexplained. In their case, the originalgraph is determined by using a random graph with a common probability, p.The interesting thing is that they set the probability p as 1 and L ratherthan N. This allows for the absolute scale of network increases. This settingprovides us some empirical thoughts on the "feeling" of an agent living in abigger and bigger city. They may have more and more "friends", but on theother hand, they become more isolated as the number of strangers increasingin an even faster speed. Psychologically, they live in a world with a sharpcontrast. Some of the time they feel quite warm, but then suddenly lost again.It is the switch between the two extremes define the feeling of city man (citywoman).

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C. Search Engine and Network Formation

The discussion of coalition formation above motivates amore general approach to think about network formation. Inthis new approach, an (active) network can be regarded as apattern extracted by a search engine directly from a databasein which individual characteristics are registered. For example,if a Ph.D student who is interested in doing agent-based com-putational economics, then this search engine automaticallyconnect yourself to a network of ACE researchers so thatyou can interact with them. Search engine can, therefore,be regarded as a metric by which a nearest neighbor canbe identified. But, the distance measure is not objectivelydetermined; instead, it can be subjectively determined. Apowerful search engine enables us to assume that agentsclearly know whom they want to interact with. In other words,the active network cognitively perceived by the agent can behighly deterministic rather than totally stochastic.The assumption of the perfect search engine is equivalent to

the assumption of a completely deterministic network forma-tion. Therefore, with the absence of a perfect search engine,completely deterministic network formation is also infeasible.The fundamental difficulty for the design of a powerful searchengine is not just because of the availability of the data andits size, but more on the computation of the distance of twovertices which can be difficult when the distance is basedon various possible measures. For example, it is hard to seehow far between an economist and a physicist before one canenvisions the emergence of the field Econophysics.

V. CONCLUDING REMARKS

Based on a few recent overarching articles on the networkeconomics, this paper singles out and addresses two crucialissues concerning the further development of the networkeconomics, i.e., its representation and dynamics. The choicesmade by agents as to their interacting partners determine theset of edges and hence the network topology. This networktopology then have further impacts on the adaptation of agents,via the interaction mechanism implied from the given networktopology, which in turn change itself again. It is left forthe future study to see how agent-based modeling can beeffectively integrated within this framework so as to makethe dynamics of both agents and networks, and hence thedynamics of micro- and macro- structure explicitly.

REFERENCES

[11 Akerlof, G. and R. Kranton (2000), "Economics and Identity," TheQuarterly Joumal of Economics, 115 (3), pp. 715-753.

[2] Akinasou, F. N., and C. S. Ajirotutu (1982), "Performance and EtnicStyle in Job Interviews," in J. Gumperz (ed.), Language and Social Identiy,Cambridge University Press, Cambridge.

[31 Arthur, W. B. (1983), "Competing Technologies and Lock-in by HistoricalEvents: The Dyanmics of Allocation under Increasing Returns," HIASAPaper WP-83-90, Laxenburg, Austria. Received as C.E.P.R. Paper 43,Stanford University.

[4] Arthur, W. B. (1988), "Self-reinforcing Mechanism in Economics," in P.Andrew, K. Arrow, D. Pines (eds), The Economy as an Evolving ComplexSystem, SF1 Studies in the Sciences of Complexity. Addison-Wesley, NewYork.

[5] Arthur, W. B., Y. Ermoliev and Y. Kaniovski (1984), "Strong Lawsfor a Class of Path-Dependent Stochastic Processes, with Applications,"in Arkin, Shiryayev and Wets (eds.), Proceedings of the Conference onStochastic Optimization, Kiev, Lecture Notes in Control and InformationSciences, Springer.

[6] Banerjee, A. (1992), "A Simple Model of Herd Behavior," QuaterlyJounral of Ecnomics, 108, pp. 797-817.

[7] Blume, L. (1991), "The Statistical Mechanics of Social Interaction,"mimeo, Cornell University.

[8] Blume, L., and S. Durlauf (2001), "The Interaction-Based Approachto Socioeconomic Behavior," in S. Durlauf and P. Young (eds.), SocialDynamics, MIT Press, Cambridge, MA, pp. 15-44.

[9] Chen, S.-H., and B.-T. Chie (2004), "Agent-Based Economic Modelingof the Evolution of Technology: The Relevance of Functional Modularityand Genetic Programming," International Journal of Modern Physics B,18 (17-19), pp. 2376-2386.

[10] Chen, S.-H., L.-C. Sun, and C.-H. Wang (2005), "Network Topologiesand Consumption Externalities," Working Papers, AI-ECON ResearchCenter, National Chengchi University.

[11] Chen, S.-H., and C.-C. Tai (2003), "Trading Restrictions, Price Dy-namics and Allocative Efficiency in Double Auction Markets: AnalysisBased on Agent-Based Modeling and Simulations," Advances in ComplexSystems, Vol. 6, No. 3, 2003. pp. 283-302.

[12] Diamond, P. (1982), "Aggregate Demand in Search Equilibrium," Jour-nal of Political Economy, 15, pp. 881-894.

[13] Ellison, G. (1993), "Learning, Local Interaction and Coordination,"Econometrica, 61(5), pp. 1047-1071.

[14] Ellison, G. and D. Fudenberg (1993), "Rules of Thumb for SocialLearning," Jounral of Politcial Economy, 101 (41), pp. 612-643.

[15] Gabszewicz, J. J., J. F. Thesis, M. Fujita, and U. Schweizer (1986),"Spatial Competition and the Location of Firms," in J. Lesourne and H.Sonnesche (eds), Location Theory: Fundamentals of Pure and AppliedEconomics, Harwood Academic Publishers.

[16] Gilles, R. P., and P. H. M. Ruys (1989), "Relational Constraints inCoalition Formation," Department of Economics Research Memorandum,FEW 371, Tilburg University, 1989.

[17] Haller, H. (1990), "Large Random Graphs in Pseudo-Metric Spaces,"Mathanetical Social Scineces, 20, pp. 147-164.

[18] Hildenbrand, W. (1971), "On Random Preferences and EquilibriumAnalysis," Jounral of Economic Theory, 3, pp. 414429.

[19] Jupp, T., C. Roberts, and J. Cook-Gumperz (1982), "Language andDisadvantage: The Hidden Process," in J. Gumperz (ed.), Language andSocial Identiy, Cambridge University Press, Cambridge.

[20] Kalai, E., A. Postlewaite, and J. Roberts (1978), "Barriers to Trade andDisadvantageous Middlemen: Nonmonotonicity of the Core," Jounral ofEconomic Theory, 19, pp. 200-209.

[21] Kirman, A. P., C.Oddou, and S. Weber (1986), "Stocastic Communica-tion and Coalition Formation," Econometrica, 54, pp. 129-138.

[22] Kirman, A. P. (1993), "Ants, Rationality, and Recruitment," QuaterlyJounral of Ecnomics, 108, pp. 137-156.

[23] Kirman, A. P. (1997a), "The Economy as an Interactive System'" in W.B. Arthur, S. Durlauf, and D. Lane (eds.), The Economy as an EvolvingComplex System II, Addison Wesley, Reading, MA, pp. 491-531.

[24] Kirman, A. P. (1997b), "The Economy as an Evolving Network," Jounralof Evolutionary Economics, 7, pp. 339-353.

[25] Kirman, A. P. (1998), "Economies with Interacting Agents," in P.Cohendet, P. Llerena, H. Stahn, and G. Umbhauer (eds.), Th Economicsof Networks, Springer Verlag, Berlin, pp. 17-51.

[26] Kirman, A. P., and N. Virend (2001), "Evolving Market Structure:An ACE Model of Price Dispersion and Loyalty", Journal of EconomicDynamics and Control, 25, pp. 459 - 502.

[27] Labov, W. (1996), "The Organization of Dialect Diversity in NorthAmerica," mimeo, Department of Linquistics, University of Pennsylvania.

[28] Lindgen, K. (1991), "Evolutionary Phenomenon in Simple Dynamics,"in C. Langton, C. Taylor, J. Farmer, and S. Rasmussen (eds.), ArtificalLife, Vol. 11. Addison-WesleX. Redwood City, CA.

[29] Verhulst, P. F., (1838), "Noace Sur La Loi Que La Population PursuitDans Son Accroissement," Correspodence of Mathematical Physics, 10,pp. 113-121.

[30] Vriend, N. (1995), "Self-Organization of Markets: An Example of aComputational Approach," Computational Economics, 8, pp. 205-231.

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