Ozman, Interactions in Economic Models, Statistical Mechanics and ...

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Mind & Society (2005) 4: 223–238 DOI 10.1007/s11299-005-0014-7

O R I GI N A L P A P E R Section on: COGNITIVE ECONOMICS Mu¨ge Ozman

Interactions in economic models: Statistical mechanics and networks

Received: 14 December 2001 / Accepted: 19 November 2003 Fondazione Rosselli 2005

Abstract During the last decade, the interaction based models have received increased attention in economics, mainly with the recognition that modeling aggregate patterns of behavior requires viewing individuals in their social environments continuously in interaction with each other. The existing literature suggests that statistical mechanics tools can be useful to model interactions among economic agents. In addition to statistical mechanics, the network approach has also gained popularity, as is evident in the rising attention attributed to small world models and scale free network topologies. These developments point to the fact that interdisciplinary research in economics, mainly using the tools of physics has accelerated to a great extent. In this paper, we review the analogies made between molecules and economic agents in statistical mechanics models, as they are utilized in economics literature. We perform a simulation study by using the small world model and statistical mechanics, to demonstrate the inﬂuence of network structure in the spreading of organizational forms, in particular the research collaboration decisions of ﬁrms. Keywords Networks Æ Statistical mechanics Æ Inter-ﬁrm collaborations Æ Small world networks

M. Ozman Maastricht Economic Research Institute on Innovation and Technology (MERIT), Maastricht University, P.O. Box 616, 6200 MD, Maastricht, The Netherlands Present address: M. Ozman Universite´ Louis pasteur, BETA, 61 avenue de La For^et Noire, 67085 Strasbourg, France E-mail: [email protected]

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1 Introduction The key to explain the emergence of aggregate patterns of behavior as a result of the individual decisions of many economic agents lies in viewing individuals in their social contexts as members of social groups, communities or members of social institutions. This is because individuals are not isolated, they are part of social environments in which they engage in informal and formal social relationships. These interactions inﬂuence individual decisionmaking mechanisms, and thereby they have an important role in shaping the aggregate dynamics of population behavior. Moreover, not only individuals but also the behaviour of ﬁrms is inﬂuenced by the external interactions they make within a certain institutional environment. As the growing literature on interaction based models suggest clearly, the classical general equilibrium approach with its emphasis on the behavior of a representative agent is therefore quite inappropriate in modeling aggregate economic and social behavior. Any attempt to explain the aggregate patterns has to take into account the particular interaction structure among heterogeneous agents, and how it evolves over time.1 According to Brock and Durlauf (2000) social interactions refer to the idea that the utility or payoﬀ an individual receives from a given action depends directly on the choices of others in that individual’s reference group as opposed to the sort of dependence which occurs through the intermediation of the markets. In other words, in most cases the individual choices reﬂect the desire to conform to the behavior of a reference group. Viewed in this way, aggregate behavior is not merely the sum of isolated individual agent behaviors. Interaction among individuals does have important consequences for the behavior of the population as a whole. Underlying this dependence of the individual utility on the choices of others is for sure the existence of increasing returns, or network externalities phenomena (Arthur 1989, 1994; David 1985). This is to say that the beneﬁts associated with conforming to the behavior of a group are increasing as more individuals conform. The last decade has witnessed many eﬀorts to model the interaction among economic agents. Blume and Durlauf (2000) indicate three major reasons for the usefulness of interaction-based models. First of all, these models have the ability to formally characterize the feedbacks between individuals within a population as well as the aggregate implications of these feedbacks. Secondly, these models have the ability to model imperfections, and diﬀerent than the general equilibrium framework, they have the ability to model the spillovers and asymmetric information among agents. Thirdly these models can take into account the heterogeneity among agents. In the economics literature the attempts to view the system from such a perspective is quite recent. The network approach and statistical mechanics have been two interrelated ﬁelds that seem to be a platform on which growing importance of interactions shows itself. These models have been

1

See Kirman (1997) for a thorough exploration of interaction based models, in relation to the classical general equilibrium approach.

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employed in many ﬁelds, including ﬁnancial economics, competing technologies, as well as models of trade among heterogeneous buyers and sellers. Recently, the network models have gained popularity to a large extent, mainly with the understanding that mostly, underlying the complexity that we observe in many real world networks, are some basic rules that are straightforward to model (see for example, Barabasi and Albert 2000). Apart from the scale free networks, it has been shown that some real world networks exhibit the properties of small world networks. In this structure, there is high degree of clustering in the network, meaning that neighbours of an agent are also neighbours themselves, but via a few shortcuts among distant agents, the average distance between agents reduces to a large extent (Watts and Strogatz 1998). This is also conceptually similar to the strong ties and weak ties argument by Granovetter (1973). In the most general sense, statistical mechanics refers to the statistical basis of thermodynamics. The emerging literature on interactions based models suggests that statistical mechanics tools can be appropriate for modeling interactions in socioeconomic sphere. The mathematical tools, which have been initially derived to investigate the relationship between macroscopic and microscopic phenomena in the physical sciences, have been generalized to explore the aggregate patterns of activity resulting from individual decisions in social sciences. A detailed survey of the use of statistical mechanics approach in economics and sociology can be found in Durlauf (1997) and Brock and Durlauf (2000) and for interactions based models, in Blume and Durlauf (2000). One of the objectives of this paper is to depict the analogies made between statistical mechanics and economics, mainly by focusing on the main equations like the Boltzmann-Gibbs distribution and the Ising model. An investigation of the particular analogies made between physics and economics is interesting in the sense that from a methodological point of view one has the chance to see the explicit analogies made between molecules and economic agents. The discussion of whether the use of physics tools in economics is appropriate from a methodological point of view is beyond the scope of this paper. However, in the economics literature, the last decade has witnessed an increasing use of physical tools mainly in the area of networks and interactions. Do we know in-depth about how the tools that we use were originally developed? This paper takes an initial step in this direction and, with a simple approach, aims to give a summary of these analogies. The approach that is followed is based on the statistical basis of neural networks2 which is expected to be useful since the artiﬁcial neural network, although in its infancy in economics literature, can have useful applications in eﬀorts to model interaction among individuals. In Sect. 2 of the paper, the scope of statistical mechanics is explained brieﬂy, with the Ising model. The equations that are covered are those that are most commonly used in socioeconomic interaction based models. Also in this part a brief survey of the use of statistical mechanics approaches in industrial economics, mainly in the context of technology adoption processes 2

See Hertz et al. (1991) and Weisbuch (1991).

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and trading relationships is provided. Sect. 3 gives a brief review of small world models as they are used in economics. Section 4 is devoted to a simulation study. In this model, we focus on the networking activities among ﬁrms. It is known that, ﬁrms imitate the behaviour of other ﬁrms who occupy similar strategic positions in the industry, which is a factor eﬀecting the spread of certain organizational structures.3 With a simple model, we model ﬁrm behaviour by the Ising model, and we utilize the small world approach in positioning ﬁrms in the network. In particular, under diﬀerent network topologies and diﬀerent levels of heterogeneity among ﬁrms, how does the aggregate behaviour of the system changes, when ﬁrms are inﬂuenced by the behaviour of similar ﬁrms in the industry? Some concluding remarks follow.

2 Statistical mechanics 2.1 Some introductory concepts in thermodynamics Statistical mechanics refers to the statistical basis of thermodynamics. A physical system is composed of many interacting particles. In general these systems are characterized by the volume in which the particles are conﬁned to, the energy of the system and the number of particles. Basically, this characterization describes a particular macrostate of the system. Each characterization of a macrosystem is associated with many possible microstates. More precisely, if the total energy of the system is E, then there are a ﬁnite number of diﬀerent ways in which E can be allocated among particles. Each of these states characterizes a particular microstate of the system. The number of diﬀerent microstates consistent with a particular macrostate is usually quite high. In the most general sense, statistical mechanics deals with the relationship between the macrostate and the microstate. One of the basic equations in statistical mechanics is the BoltzmannGibbs distribution. It gives the probability of a particular microstate of the system, given by the following equation:4 eEa b PrðaÞ ¼ P E b e a

ð1Þ

Here, P is the probability that the system will be in any state a at a particular instant, E is the total energy of the system at state a. The denominator term is the summation over all states. Equation 1 imposes an equilibrium condition of the system such that the probability that the system will be in a particular state is just equal to the proportion of that state to the sum of all states. Equation 1 is useful since it enables calculation of some expected values characterizing the physical system. 3

See, for example, DiMaggio and Powell (1983) for the concept of mimetic isomorphism Here b=1 /kT where k is the Boltzmann constant and T is the temperature. There are several derivations for the Boltzmann-Gibbs distribution. See for example, Pathria (1972) and Landau and Lifshitz (1969). Also see Cowan and Cowan (1998a, 1998b) and Weisbuch et al. (2000) within the context of industrial economics. 4

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Denoting the number of possible microstates accessible to a particular macrostate by n, the absolute entropy of the system is given by: S¼

n X

PrðaÞ log PrðaÞ

ð2Þ

a¼1

As it stands, than, each of the possible macrostates has a speciﬁc entropy and a number of possible microstates, denoted by n. Equation 2 is the absolute entropy of the system.5 When the number of possible microstates of the system is one (Pr(a)=1, n=1), the entropy vanishes, that is to say order prevails. On the other hand maximum entropy is achieved when n ﬁ ¥. Entropy is a measure of disorder in the system. The more is the number of microstates accessible to the system, the more is disorder. Particular relevance of statistical mechanics in binary discrete choice problems is evident in statistical mechanics of ferromagnetic particles, or the Ising model. In the Ising model the system is composed of a ﬁnite number of magnetic particles, called spins, arranged on a lattice, where each of them either point up or down. The magnetic ﬁeld acting on a spin and the physical temperature are the two variables that determine the direction of the spins. This magnetic ﬁeld is a function of the interaction energy between the spins. At low temperatures, a spin moves consistent with the magnetic ﬁeld acting on it. Therefore all spins that are in interaction are likely to point to the same direction, and form a magnetic ﬁeld. Whereas in high temperatures, the tendency of the spins to align with the direction of their magnetic ﬁelds are disturbed, meaning that they start to reverse their directions. As it is shown below, these models are subject to phase transitions, determined by the critical temperature separating order and disorder. The function of the magnetic ﬁeld is given by the following formula; X wij zj ð3Þ mi ¼ j

where m is the magnetic ﬁeld around spin i, wij is the strength of the interaction between spins i and j, and zj is the value of the spin, taking the value of +1 if the spin is pointing up, and 1 if the spin is moving down. Basically, we are interested in ﬁnding out the probability of a particular state of the system. If we consider a single spin, and the magnetic ﬁeld mi acting on it, the probability that it is up or down is given by the following logistic function:6

5

Note that, the case of maximum entropy applies to closed systems, since in these systems entropy does not decrease. 6 Since b=1 /kT, it is easy to see that as T ﬁ ¥, Pr (z=±1) ﬁ 1/2. This means that, irrespective of the magnetic ﬁeld sorrounding the spin, it has equal chance of pointing up or down. On the other hand, as T ﬁ 0, and sgn(mi)=+ (magnetic ﬁeld sorrounding the spin is positive), Pr (z=+1) ﬁ 1 and Pr (zi= 1) ﬁ 0. On the other hand when sgn(mI)=, Pr (z= 1) ﬁ 1 and Pr (zi=+1) ﬁ 0. Therefore at low temperatures, the spins align with the magnetic ﬁeld acting on them.

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Prðzi ¼ 1Þ ¼

1 1 þ expð2bmi Þ

It is then easy to compute the expected spinh zi ¼ Prð1Þð1Þ þ Prð1Þð1Þ that is equal to hzi i ¼

ð4Þ state

ebm ebm þ ¼ tanh bm ebm þ ebm ebm þ ebm

of

a

single

ð5Þ

Expected value of a single spin, subject to a particular magnetic ﬁeld m is given by [5]. However the single spin is only an isolated case and we are interested in the behavior of the system as a whole, including all the spins. Normally a spin is surrounded by other spins and their magnetic ﬁelds. It is therefore quite more complicated to ﬁnd the expected value of the system as a whole. For this purpose, the mean ﬁeld theorem is used, replacing the magnetic ﬁeld of a single spin by the mean value of all the other spins surrounding it. The basic idea behind the mean ﬁeld theorem is that instead of dealing with all the conﬁgurations individually we replace them by their average P value. Simply, the average magnetic ﬁeld is then given as hmi i ¼ wij zj : Finally, we can replace [5] in the following way. j X ð6Þ wij zj Þ hzi i ¼ tanhðb j

so that the stochastic terms are replaced by deterministic ones and the problem remains to solve N equations in N unknowns. A particular case of the Ising model is the ferromagnets, where the interaction weights are assumed to be the same among all spins, so that wij=w=J/N where J is a constant and hzi i ¼ h zi; h zi ¼ tanhðbJ h ziÞ

ð7Þ

It was mentioned above that the behavior of the system depends critically on temperature level, since b=1/kT. These systems are usually characterized by phase transitions at critical values of the temperature, which is found by solving [7]. The solution to [7] depends on whether BJ>1 or not. The critical temperature is therefore Tc=J/k. When BJ £ 1 (T > Tc), there exists one solution whereh zi ¼ 0 so that a spin may move up or down with equal probability. On the other hand when T £ Tc (BJ > 1) there are two stable solutions,h zi ¼ 1; so that either all the spins are directed up, or all of them are directed down. Above this critical temperature, the tendency of the spins to align with their magnetic ﬁelds is disturbed, so the spins become independent of each other, some of them directing up and some of them down. This means that the number of states that may characterize the system increases, and the probability that a spin points up or down approaches 1/2. Entropy increases, and the expected value of a spin, h zi ! 0: Therefore the magnetic ﬁeld surrounding spins also approach zero, the magnet looses the magnet properties. On the other hand, below the critical temperature level, the spins have the tendency to align with their magnetic ﬁelds formed by the other spins.

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Either all the spins point up or all of them point down. The system is characterized by a single possible microstate, so that there is order and the entropy is minimized. The energy function of these physical systems is expressed in the following way:7 E¼

1XX wij zi zj 2

ð8Þ

When all the spins have the same sign, there is only one microstate accessible to the system. This happens, as it was explained above, when temperature is zero, and also energy of the system is minimized. Therefore, as temperature decreases, order increases and entropy and energy of the system also decrease. On the other hand at maximum temperature, each spin has equal probability of pointing up or down, which is 1/2, there is maximum disorder and entropy. Having made a brief account of the statistical mechanics equations which are most commonly employed in economics, below a summary of how statistical mechanics have been employed in interaction based models is provided.

2.2 Statistical mechanics in industrial economics Most of the phenomena that represent the physical systems found its counterparts in the social and economic sphere, so that the mathematical tools used to model physical phenomena have been widely employed in social sciences. These tools have signiﬁcant explanatory power when they are used to describe the aggregate dynamics of socioeconomic systems based on individual interaction. The interaction based models gained popularity in explaining the macrodynamics resulting from micro level interactions among heterogeneous populations. The earliest example of the use of statistical mechanics in economics goes back to Follmer (1974). Among many others, these models have been used to analyze the role of learning and experience in the trading relationships (Weisbuch et al. 2000), the technology adoption process in the existence of local and global externalities (An and Kiefer 1995; Dalle 1997; Cowan and Cowan 1998a) as well as clustering in R&D activity (Cowan and Cowan 1998b). The major question is, how do we observe certain aggregate patterns emerging as a result of individual decisions of heterogeneous agents, acting on their own self interest? The relationship between macrobehaviour and microbehaviour is central here and it is trying to explain this relationship

7 The notation used here is that of Hertz et al. (1991) and Weisbuch (1991), where they focus on the statistical mechanics of cellular automata and neural networks. The minimization of energy in this case occurs only when the weights of interaction between two nodes are symmetric. In other words, when the magnetic force that two spins exert on each other is the same.

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that the statistical mechanics of particles is useful. Brock (2000) states that the microcategory is separated from the macrocategory by the degree of survival to aggregation. If the phenomenon at hand is to be called macro, there must be strong enough dependence across individual micro units so that the averaging eﬀect of aggregation does not wash out the phenomenon of interest. The positive returns associated with being in compliance with the decisions of other agents and the resulting dynamics, probable lock-in and path dependence have been investigated in economics in several ways, mainly within the context of technology adoption (Arthur 1989, 1994; David 1985). Brieﬂy, externalities refer to the idea that the net beneﬁts of conforming to the decisions of a reference group increases as the number of individuals conforms. These externalities may carry the system to a ﬁxed point where complete standardization results, which is the case in Arthur’s lock-in phenomena in technology adoption decisions. The system is path dependent, as long as the increasing returns are unbounded, the system locks in to a state of complete adoption of one of the technologies and standardization obtains. The technology that is the ‘‘winner’’ is largely determined by the initial conditions of the system. Arthur’s model of complete standardization has been modiﬁed during the last decade, and statistical mechanics tools have widely been employed for this purpose. More precisely this model has been modiﬁed considering two factors. Firstly, agents are heterogeneous. Secondly, the distinction between local and global externalities does change the dynamics of the system. Distinguishing between local and global externalities shows that two technologies may co-exist at the same time (Cowan and Cowan 1998a). These models show that the technology adoption process can be subject to phase transitions. Below a critical level of heterogeneity among agents, increasing returns dominate as in Arthur’s model, so that there is complete order in the system and standardization is obtained. On the other hand, above the critical level of heterogeneity among agents, rival technologies co-exist and disorder prevails8 and agents are equally likely to select any of the technologies without paying attention to the increasing returns phenomena (Dalle 1997). In this case, the heterogeneity of agents dominate the tendency to conform to the behavior of the reference group. In summary there are two forces working in opposite directions, one of them is agglomeration economies, or the beneﬁts from the externalities associated with conforming to the behavior of others, and the other is the heterogeneity of economic agents. The precise dynamics of the models is of course largely inﬂuenced by the particular functional forms employed. It is clear from the discussion above that statistical mechanics tools are appropriate to explain this transition between order and disorder, by the help of the mathematics of the phase transitions for physical systems. In this sense, the mathematics of increasing temperature in physical systems is used

8 As it was mentioned above, entropy is associated with increasing disorder. For the issue of entropy maximization in the context of ﬁnancial markets, see Brock (1993). Also in Weisbuch et al. (2000) entropy maximization associated with increasing disorder in trading relations is derived.

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to model the eﬀect of increasing heterogeneity among agents in social systems. In these models, the equilibrium condition is imposed such that the expectations of each of the interacting agents about the average state of the system is the same, and consistent with the average behavior of the system. The mean ﬁeld approximation is particularly useful at this point. The state of the agents, other than a particular agent is averaged over and the average state of the system is assumed to be the same for all agents, as it was the case for the Ising spins above. In the context of interacting agents models, mean ﬁeld approximation as a condition of equilibrium is familiar in discrete choice random utility functions.

3 The network approach and small worlds The last decades have witnessed a vast amount of research being carried out about networks in many diverse disciplines. This has largely been triggered by the understanding that essentially many micro and macro systems from a social, economic or technological perspective are networks, and need to be analysed as such. There exist a tremendous literature on networks, including empirical work in many diverse disciplines and theoretical work on their structural properties. More recently, research in the latter area has intensiﬁed especially in the area of evolving networks. Empirical work covers many areas in which diﬀusion of innovations, inter and intra organizational relations, the World Wide Web, the Internet, ecological networks, co-authorship networks, movie actor networks are only a few examples. The central emphasis of economists is on innovation, how networks inﬂuence innovative activity in general and the processes behind emergence of networks. In this context, work is diversiﬁed. One strand of literature focuses on inter-ﬁrm networks and various forms of arrangements between ﬁrms, including strategic alliances, R&D collaborations, and many other types of interaction among ﬁrms in an industry. The coordination among many actors is obviously accompanied by social relations. These social communications are understood to inﬂuence the eﬃciency with which knowledge diﬀuses to a large extent, which has implications for economic outcomes, and thus the networks over which the interactions among parties occur requires a more in-depth analysis. The literature reveals that social network analysis tools are being widely used for this purpose. Watts and Strogatz (1998) have been the ﬁrst to systematically explore in mathematical terms what was already known to many people: that the world is small. Since then, a vast amount of research has been conducted on the socalled small world networks. Small world networks are characterised by two parameters, cliquishness and path length. Watts and Strogatz (1998) demonstrate that the small world region lies somewhere in between a perfectly cliquish regular network with high clustering and high path lengths, and random networks with no clustering and short path lengths. Small worlds, which lie in between these two extremes are characterised by high clustering (local connections, where nodes have ties with nearest nodes), but at the

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same time, via a few short cuts between distant agents, low average path lengths. As empirical work in many areas like the World Wide Web, scientiﬁc collaboration, movie actor networks reveal, most of the real world networks exhibit small world properties.9 The application of small world phenomena in the economic sphere takes place along several dimensions. One of these is their eﬀect on knowledge diﬀusion. Cowan and Jonard (2001), in a simulation study, demonstrate that small worlds outperform regular and random structures in the extent of knowledge diﬀusion. In a diﬀerent framework, one in which the collective invention phenomena is related to the network structure, Cowan and Jonard (2003) ﬁnd that small world produces highest knowledge growth, yet the most uneven distribution of knowledge among the actors. Some other examples include Wilhite (2001) where he models trade negotiations in a simulation model, and his results indicate that small world networks are more eﬃcient in terms of the period elapsed until the emergence of an equilibrium price. Latora and Marchiori (2003) devise the concept of an ‘‘economic small world’’, where they modify Watts and Strogatz framework to take into account diﬀerent cases (including growing networks) where the ties among agents are weighted, and there are costs to establishment of links.

4 A model of research collaborations Network based industries are characterised by intensive interactions taking place across ﬁrm boundaries. The domain of these interactions can take countless forms ranging from legal ownership agreements to informal know how trading. A major process that accompanies the inter-ﬁrm relations is the signiﬁcant knowledge ﬂow that takes place between the actors, which is usually considered to be an important engine for rapid innovation. As ﬁrms rely more on external collaboration, the boundaries of the ﬁrms get blurred, and the ﬁrm becomes a dense collection of communication links within a larger network through which there is continuous information and knowledge ﬂow. The literature on network-based industries reveal that various approaches like transaction cost economics, strategic management, and social network perspective focuses on diﬀerent aspects of these industries. When looked from a dynamic network perspective, the continuous interaction of ﬁrms through time forms an evolving network, and it is generally accepted that the state of this network in one period inﬂuences the state of the system in subsequent periods, as a result of a dynamic process of experience accumulation and learning (Garcia-Pont and Nohria 2002; Gulati 1999, Powell et al. 1996). In this sense, networks provide a feedback mechanism among the nodes through time, which facilitates the spread of certain organizational forms.

9 For small world literature, see Watts and Strogatz (1998); Frommer and Pundoor (2003) for a review of three recent books on networks; Barabasi and Albert (2000); Newman (2000) for a review covering research on small worlds.

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The concept of mimetic isomorphism was ﬁrst put forward by Di Maggio and Powell (1983), and in the context of alliances Kogut (1988) mentions that this might be an important explanation in explaining increased external collaboration of ﬁrms. This phenomena have been studied systematically by various scholars in relation to network structure and/or embeddedness perspectives. For competitive advantage, ﬁrms imitate the behaviour of other ﬁrms, thus the emergence of certain organizational structures happens through a positive feedback mechanism provided by networks (Garcia-Pont and Nohria 2002). At the same time, the positions of ﬁrms within the network give an insight into the structural similarities among ﬁrms. More speciﬁcally, ﬁrms are found to be more likely to be inﬂuenced by other ﬁrms with similar structural locations, as revealed by their position within the network; like membership in the same clique (so that the neighbours of a ﬁrm are also neighbours themselves), or having similar centrality measures in the network. Garcia-Pont and Nohria (2002), in their dynamic analysis of the automotive industry ﬁnd support for the hypothesis of ‘‘local mimetism’’ as an important motive for alliance formation, where, as the number of prior alliances increase, the competitive pressure on similar ﬁrms to do the same also increases and so is their likelihood to be involved in collaboration. Partitioning the ﬁrms in the network to strategic groups (as in Garcia-Pont and Nohria 1991), they measure the eﬀect of similarity by the local network density of strategic groups. Within a social embeddedness perspective, Galaskiewicz and Wasserman (1989) ﬁnd also empirical evidence that managers tend to follow the behaviour of others in their social networks in their decisions regarding donations to charity organizations.

4.1 The simulation model In this part of the paper, we develop a model to depict how the activity of networking may spread among the ﬁrms in an industry, by using the approaches above. We assume that ﬁrm i has two possible strategies to pursue: to take part in external research collaboration, or to pursue its R&D activities in-house. The choice of ﬁrm i is represented by zi which takes a value of 1 for the ﬁrst case, and 1 for the latter. The probability that ﬁrm i will chose strategy i is given by; Prðziq Þ ¼

1 P 1 þ expðb wij zi zj Þ

ð9Þ

j

Here b measures the inverse of heterogeneity among research activities of ﬁrms. We assume that ﬁrms that are similar in terms of their research activities are more likely to be inﬂuenced by each other’s decisions to be involved in research collaboration. In other words, ﬁrm i, which is active in one ﬁeld will ﬁnd it more convenient to collaborate with other ﬁrms, if it’s competitors pursuing research in a similar ﬁeld are following such a strategy.

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As it was mentioned above, b captures the idiosyncratic eﬀects. As b ﬁ 0, the ﬁrms become increasingly heterogeneous in their research activities, i.e what they can contribute to each other decreases.10 Then the probability of choosing a particular strategy approaches 1/2. When this is the case, ﬁrm’s decision is not inﬂuenced by what the other ﬁrm’s are doing, but by exogenous factors. On the other hand as b ﬁ ¥, the eﬀect of idiosyncracy eﬀects are eliminated, and with near certainty the ﬁrm chooses a similar strategy as neighbours. Other cases fall in between. The term inside the summation measures the extent of the interaction eﬀect surrounding ﬁrm i. If this sum is high compared to b, agglomeration eﬀect dominates heterogeneity eﬀect and the probability ofP conforming to the selection of the majority outweighs and Prðsgnðzi Þ ¼ sgnð wij zj ÞÞ ! 1: Here we take wij=1. One of the parameters that we vary in the simulations measures the structure of the network. Here we follow the small world algorithm, by Watts and Strogatz (1998).11 The agents are located around a circle. At one extreme of the space of network structures, there is the regular structure: each agent is connected to his/her s nearest neighbours, s/2 on each side. The number of connections for each agent is the same on the average. At the other extreme each agent is connected to, on average, s agents located at random on the circle. To interpolate, ﬁrst the regular structure is created. With probability p each edge of the graph is rewired. Here, each edge of the graph is examined sequentially; with probability p one of its vertices is disconnected, and connected to a vertex chosen uniformly at random. By this algorithm, the degree of randomness in the graph is tuned with parameter p 2 [0,1]. In this case, when the network structure is completely regular, there is high cliquishness (neighbours of a ﬁrm are neighbours themselves), as well as high average path lengths (distance) between two agents in the network. On the other extreme, when the structure is completely random, cliguishness is minimum, whereas path lengths are signiﬁcantly shorter. A completely regular network structure can be associated with industrial districts with high degree of clustering. On the other hand, a random structure is associated with ﬁrms distributed evenly across space. In between these two extremes lies the small world region, characterized by high cliquishness and low path lengths. In the simulation study we aim to compare the critical level of heterogeneity (below which all ﬁrms pursue the same strategy) as well as the period that elapses until stability is achieved, for diﬀerent network structures. In other words, how does the geographical positioning of the ﬁrms in the network inﬂuence the critical level of heterogeneity and the time elapsed until there is complete standardization?

10

Here we assume that common knowledge between parties and similar knowledge bases facilitates the transfer of knowledge, so that ﬁrms ﬁnd it more eﬃcient to collaborate with other ﬁrms researching in similar ﬁelds (Stuart 1998). 11 See also various papers by Cowan and Jonard (2001, 2003) in the context of innovation.

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5 Results The population is composed of N=100 ﬁrms, whose state is determined randomly at period t=0.12 First we take the case where ﬁrms are arranged regularly on the circle, having links to the closest s=4 neighbours (rewiring parameter p=0). On this regular structure, each agent is taken sequentially and his state is updated according to (9), and the period elapsed until standardization is recorded. This procedure is repeated for diﬀerent levels of heterogeneity (1/b 2 [0.01,1.91]). Then we apply the rewiring algorithm, and for a range of rewiring probabilities p (where p 2 [0,1]) we apply the same procedure as above, each time recording the periods until standardization for diﬀerent values of heterogeneity. The Figure 1 shows the time elapsed until standardization for a range of b and p values used in the simulations. In the ﬁgure, the light areas denote higher values. The critical value of heterogeneity, below which standardization always obtains is clearly shown in the ﬁgure, separating black and white regions. What is interesting for our purposes is that the network structure over which agents are positioned has a clear inﬂuence on the time elapsed to achieve standardization in the system. When the structure of the network is regular (low values of p) it clearly takes a longer time to achieve standardization on average (as evident by the gray areas in the lower left part of the Fig. 1). As the network structure gets more random, with a decrease in average path lengths, we see that it takes very short period until standardization (all ﬁrms pursuing the same strategy).

1

rewiring probability (p) logarithmic

0.3

0.06

0.01

0.003

0.001 0.01

0.41

0.81 1.21 heterogeneity

1,41

Fig. 1 Periods elapsed until stability 12

Here, state of ﬁrm i refers to its initial tendency to be involved in collaborative activity.

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However, network structure does not inﬂuence the critical level of heterogeneity at which phase transition occurs. The average value of 1.21 holds the same, regardless of the network structure. Why do we see this pattern, and what are the implications? Clearly, short path lengths are associated with more rapid diﬀusion in the network. Therefore, randomness in the network brings faster transition. On the other hand, cliquishness reduces the rate at which transition occurs. Here the basic Ising model corresponds to a regular structure of the network. But our results reveal that the structure of the network can be critical in determining the pace at which all ﬁrms pursue the same strategy.

6 Concluding remarks The importance of interactions in inﬂuencing aggregate economic outcomes have been acknowledged by many economists today. Consequently, a variety of approaches have been undertaken to model this phenomena. Among these, in this paper ﬁrstly we focused on the use of statistical mechanics, and explained how some of the main formulas are used in the thermodynamics, as well as in industrial economics. Secondly, we presented a brief overview of the literature on small world networks, and its use in economics literature. Finally, we combined the two approaches in a simple simulation model where ﬁrms decide to form a research collaboration or not, by looking at what their neighbours do. The results revealed that, the time elapsed until all ﬁrms pursue the same strategy depends critically on the network structure over which ﬁrms are positioned, and takes shorter time when the structure of the network is random. The current paper is an attempt to give an overview of how interactions are perceived in economics literature, mainly by focusing on currently popular concepts, like the small worlds and Boltzmann Gibbs distribution. The evaluation of whether the use of physics tools are appropriate to model economic agents and economic phenomena is out of the scope of this paper. However, it is important to note that a sound methodological evaluation of these analogies requires knowing how the tools are used and developed originally. But there is a trade-oﬀ involved here; interdisciplinary work clearly increases creativity, but on the other hand, specialization in a particular discipline renders a scientist less ﬂexible in absorbing the developments in other, less related ﬁelds (like economics and physics). Therefore we think that it is useful, even if simply, to know how the approaches that we use were developed originally in other disciplines that we borrow our tools from. Acknowledgements An initial version of this paper was presented at the workshop ‘‘Cognitive economics’’ held in Alessandria and Turin from the 15th to the 18th of November 2000, and coordinated by the Fondazione Rosselli as the ﬁfth initiative of the scientiﬁc network of the European Science Foundation called ‘‘Human reasoning and decision making’’.

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Thermodynamics and Statistical Mechanics