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ARTICLE

A Doubly Structural Network Model: BifurcationAnalysis on the Emergence of Money1)

Masaaki KUNIGAMI*,**, Masato KOBAYASHI**, Satoru YAMADERA**,

Takashi YAMADA** and Takao TERANO**

* Joint Staff College, Ministry of Defense JAPAN, 2–2–1 Nakameguro, Meguro-ku, Tokyo153–0061, Japan. E-mail: [email protected]** Department of Computational Intelligence and Systems Science, Tokyo Institute ofTechnology, 4259 Nagatsuda-cho, Midori-ku, Yokohama 226–8502, Japan.

AbstractThis paper presents a new model of micro-macro social learning model for a classicalEconomic problem “the emergence of money”. We propose Doubly Structural Network(DSN) Model, which consists of one global social network of agents and internal networksthat represent agents’ recognition. DSN model enables us to describe the emergence ofproto-money as a self-organization process of the common recognition of exchangeability.We conduct an analytical method and a numerical approach into bifurcation phenomena of anew mean-field dynamics derived from this DSN Model. The main contribution of the paperis summarized as follows. (1) The proto-money can emerge from commodities withoutdistinctive properties. (2) The social network degree is a definitive factor for non-/single-/multiple-emergence of proto-money. (3) The variance of the social network degree(existence of hub-agents) also affects emergence of proto-money.Keywords: emergence of money, network model, self-organization, mean-field dynamics.

1. Introduction

The objectives of this paper are to present Doubly Structural Network (DSN) Model of

social learning on transaction media, and to derive new outcomes by applying this DNS

Model to a classical economic problem called as “the emergence of money from a barter

economy”.

For these objectives, at first, we introduce Doubly Structural Network (DSN) Model.

This new model consists of agents’ social network and inner recognition networks of the

agents, in Section 2.

Next, we apply this DSN Model to “the emergence of money”, in Section 3.

Evol. Inst. Econ. Rev. 7(1): 65–85 (2010)

JEL: D85, Z13.1) Any views expressed herein are solely those of the authors’, and do not represent those of the

Japanese Government or any representative agencies.

We derive approximate dynamics from this DNS Model. This dynamics is improved

from our previous work (Kunigami et al., 2009) in the following two ways. 1) The

dynamics shows that even a commodity with no distinctive properties can become a

medium of exchange. 2) The dynamics enables bifurcation analysis, which shows that

the degree of connection of the social network strongly affects the number of exchange

media, in Section 4.

2. Doubly Structural Network Model

This section introduces our Doubly Structural Network (DSN) Model. This section

introduces our Doubly Structural Network (DSN) Model. This model is unique from

other related work in that it has double structure of inter-agent social networks and inner-

agent recognition networks. This double structure of networks enables us to describe and

to analyze the emergence of common knowledge or organized/collective recognitions in

the society.

Several models for agents’ states and behaviors and their propagation in a society are

known as spatial evolutionary game, infection in network (Masuda and Konno, 2006;

Klemm et al., 2003; Pastor-Satorras and Vespignani, 2002), dynamics of segregation

(Schelling, 1971), dissemination of culture (Axelrod, 1997), the Sugar-scape (Epstein

and Axtell, 1996), and TPM (Tensor Product Model (Kashima et al., 2000).

These models have the following features in common. (i) These models consist of

agents simplified strongly. (ii) Agents’ interaction causes macroscopic change of a state.

(iii) Large changes of these macroscopic states are caused by slight changes of the initial

value or a small number of parameters. (iv) The models are aimed at revealing the

essential mechanism of our societies than quantitative realism on the world. For

example, Schelling’s segregation model has only four rules of the preference to the

neighbors, but it demonstrates that segregation emerged from the initial mixed state and

its infinitesimal fluctuation. Axelrod, Epstein, Kurgman,2) many scientists and

economists has supported this type of approach.

Our study regards these common stances on modeling by them. Axelrod has described

the feature of their researches against prediction by the high fidelity simulations of the

target phenomena as follows. “But if the goal is to deepen our understanding of some

fundamental process, then simplicity of the assumption is important, and realistic

representation of all the details of a particular setting is not.” (Axelrod, 1997, p. 5)

M. KUNIGAMI et al.

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2) Krugman (1996, ch.1) regrets that Schelling’s simple model may be misunderstood as its important

results are achievable by intuition.

We proposed the Doubly Structural Network Model (Kunigami et al., 2009) that

handles social propagation of agents’ knowledge and recognition such as exchangeability

or acceptability of commodities. The structure of this model is illustrated in Fig. 1, and is

defined by formula (1).

Similar to the Tag model (e.g., Dissemination of Culture (Axelrod, 1997) on a

network, our model is different from TPM (Kashima et al., 2000) that describes the

social relationship between agents using not inter-agents connection but “groups”. In

contrast to the Tag model, in our model the propagation of inner representation is not

driven by whole similarity of tags but local structural similarity of inner networks. Our

model is advantageous in that it can describe not only autonomous structural change but

also emergence of structure through simple representation. Here “autonomous structural

change” means that each agent’s inner network changes depend on not only neighbors’

inner networks but also on its own topology of connection. Also “emergence of

structure” means a self-organization of inner networks each of which represents

individual recognition on exchangeability.

(1)

In formula (1a), The “social (inter-agent) network” GS represents the social

relationship between agents. The node (vertex) vSi represents the i-th agent. The edge

set ES represents connection or disconnection between these agents.

In formula (1b), Each “internal (recognition) network” GIi represents the internal

landscape or recognition of the i-th agent on certain objects (a , b , · · · ). The node

(vertex) vIa represents the object a . The edge set EI

i represents connection or

disconnection between those objects in the i-th agent’s recognition.

Formula (1c) shows that “doubly structural network” GD is created by attaching (or

mounting) each internal (recognition) network GIi (i�1,2, · · · , N) onto the

G V E V v i N E V VS S S SiS S S S≡ ≡ ⊆( , ), { , } ··········� �1∼ ·········( )

( , ) , { }

a

G V E i N V v MiI I

iI I I≡ ≡� �1 1∼ ∼α α ,, ·····( )

{{( , ) }, }

E V V

G v G i N EiI I I

DiS

iI S

⊆≡

�

�

b

1∼ ···················································( )

( , )·················

c

G F t Gt dtD

tD

� ≡ ·························································( )d

⎛

⎝

⎜⎜⎜⎜

A Doubly Structural Network Model: Bifurcation Analysis on the Emergence of Money

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Fig. 1. A doubly structural network of society.

corresponding node vSi (i-th agent) of the social network GS. In other words, “doubly

structural network” bundles up agents’ internal networks through a network that shows

the external structure of society.3)

Formula (1d) shows that a propagation/learning model of the doubly structural

network is defined by the ways of changing states (connection/disconnection) of

agents’ internal networks via interaction of agents in the social network In this paper,

we use the doubly structural network model of static society if the social network does

not change autonomously. On the other hand, we refer to it as the model of dynamic

society if its social network changes autonomously.

These structure gives us following advantages.

1. To describe directly states of the recognition of the internal network by shape.

2. To define autonomous evolution into the internal networks.

3. To describe the micro/macro interaction4) among agents with these inner

evolutions.

However, the above formula of the DSN model is a conceptual model without a

particular social learning/propagation phenomenon, so we need to implement inter-

agents and inner-agents’ interaction. This enables us to employ an analytical method to

derive interesting results on this important classical problem in economics.

3. Application to the Problem of the Emergence of Money

In this section, we apply this DSN model to the classical problem of “the emergence

(origins) of money” in economics.

In the DSN model of the emergence of money, the social network represents the

exchange/transaction relationship between agents, and the inner network within each

agent represents the recognition on exchangeability of commodities. The phenomenon of

“the emergence of money” is expressed as a self-organization process in which a certain

commodity attains “general acceptability” by a special.

3.1 Problem of the origin/emergence of money

In economics, “money” is usually defined by the following functions (Hayek, 1976,

Hicks, 1967, Mankiw, 1999); i) a medium of exchange, ii) a unit of value, iii) storage of

value. Amongst these, many economists maintain that money is essentially a “medium of

M. KUNIGAMI et al.

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3) Whichever graph (undirected/directed) is available for “external network” or “internal network”.

(e.g.: “Social network” is directed, but “internal network” is undirected. etc.)4) The shape of the social network affects the changes of the internal networks (macro→micro). The

interactions between the internal networks formed t he social attitudes (micro→macro).

exchange” (Hayek, 1976; Hicks, 1967; Iwai, 1996; Mankiw, 1999; Menger, 1871). For

the emergence of money as a medium of exchange, almost all the agents within the

society must recognize that “this commodity is exchangeable with almost all others”.

This nature is called “general acceptability”. In this paper, to focus on the most primitive

form of money we define “proto-money” as a commodity that has general acceptability.

Therefore, the emergence of money means the emergence of proto-money from a barter

economy.

Several theories coexist regarding the origins of money, (Mankiw, 1999). The first is

the “legal theory of money” that states money has its origins in ‘fiat’ by Kings or

governments. The other is the “theory of commodity-money” that states money is

spontaneously specialized from exchangeable commodities. In recent times the later has

become more popular, but the debate is not necessarily settled. In addition, the “metal

theory” maintains that a commodity becomes money due to its appropriate attributes for

exchange, and the “non-metal theory” maintains something can be money despite its

attributes. In this paper, terms such as “metallic theory of commodity-money” or “non-

metallic theory of commodity-money” are used if necessary.

From the following reasons (a)–(c), this paper analyzes the possibility that the proto-

money can emerge under the non-metallic theory of commodity-money.

(a) Money sometimes emerged starting from societies without government

A survey (Quiggin, 1949) shows primitive money was often used in societies without a

government or ruler. It is also known that in modern times primitive money emerged

from a barter economy in prison camps (Radford, 1945) and in the former Soviet Union

(Myerson, 1990, Reynolds, 1993).

(b) Extreme diversity of primitive money

In survey (Quiggin, 1949), the outstanding nature of primitive money is its large

diversity of forms.5) It is futile to search for “common applicable properties” in the

diversity, and difficult to explain that “most of these were not money in other areas/eras”.

(c) Money sometimes did not emerge or collapsed

The Inca Empire did not use money, but people exchanged commodities by “anyi”

(mutual aid) and by “yana” (work by a servant) (Rostworowski, 1988). Bartering arose in


– 69 –

5) Materials: metals (gold, silver, copper, tine, steel, bronze, brass), animals (shell, teeth, dried fish,

fur, scalp, jawbone, crab eyes), crops (rice, barley, beans, corn, cacao, tea leaves, olive/banana seeds),

salt, rock, cloth, paper, etc, Uses: accessories (beads, collar, plastron, armlet, nose ring, hat, shoes,

belt, button), tools (hoe, spade, scale, axe, arrowhead, spear, bell, sword, mace, needle, fishing hook,),

livestock (cows, sheep, goat, pig, chicken) etc.

the former Soviet Union (Myerson, 1990; Reynolds, 1993) during economic breakdown,

and in 18th century Pennsylvania. (Weber, 1920) These facts support (a) also, and that the

model should also support cases where money does not emerge.

This is not a preconception to reject the metallic theory or legal theory. If the

emergence of proto-money is difficult under non-metallic theory, the metallic theory is

indirectly supported. If the metallic theory is also difficult, the legal theory is indirectly

supported.

3.2 Other mathematical models for the emergence of money

The emergence of money is studied in not only economics but also mathematical models.

Some research (Duffy, 2001; Kiyotaki and Wright, 1989; Marimon et al., 1990) shows

that specialization in the exchange media to a certain commodity (e.g. cheapest

preserving) is a rational equilibrium strategy in bartering three commodities. An

evolutionary model (Luo, 1999) shows that using the cheapest preserving commodity is

sometimes unique rational equilibrium also. A matchmaking model of commodities

(Starr, 2003)) shows that commodity-money may spontaneously emerge as the one with

the lowest transaction cost. Such research has a different approach to ours in 3.1, since

they assume the “metallic theory of commodity-money” depending on particular natures

of commodities.

Here are some researches consistent with the “non-metallic theory of commodity-

money”. A searching trading partner model (Iwai, 1996) shows that bartering and

monetary economics are different equilibria, and that monetary equilibrium requires the

common recognition that a particular one is accepted as money. The evolution of money

requires a large fluctuation to break the bartering equilibrium. A simulation analysis of

exchanging commodities (Yasutomi, 1995) shows that by adding the Maxim “Accept

what others accept!” (Menger’s ‘salability’), a commodity-money emerges when the

“threshold of exchange” in the “view vector” of the Maxim exceeds certain level.

Another agent simulation (Yamadera and Terano, 2007) in lattice space shows a

commodity becomes money based on the “trust” from agents.

In these points of view, it seems that emergence of money needs some structural

change (change of the “threshold of exchange” in the “view vector”, establishing

common recognition and a “large disturbance”, establishing “trust”) in the society. The

following sections illustrate that our model is useful to describe an emergence

mechanism based on a social structure.

3.3 Doubly structural network model of the emergence of money

Here, we implement a specific mechanism to describe the emergence of money in our

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model. (Kunigami et al., 2009) Upon the implementation, the social (inter-agents)

network reflects the topology of economical/social relationship between agents

(indicated as i,j�1�N). The agents’ inner networks show their own recognition on the

exchangeability between commodities (indicated by a ,b ,g�1�M). Each element of

adjacent matrix is defined by e(i)ab�e(i)

ba�1 if “a and b are exchangeable”, and e(i)ab�e(i)

ba�0

if not.

Among possible stage for emergence of money, this paper focuses on the emergence

of proto-money in which a certain commodity achieves “general acceptability” in the

society. In our doubly structural network model, the emergence of proto-money a is

represented as a self-organizing process in which almost inner-networks become similar

star-shaped networks with a common hub a (Fig. 2).

In Starr (2003) and Yasutomi (1995), they pointed out that star-shaped network of

exchangeability can represent general acceptability of commodity. Our DNS model is

consistent with their work, and gives explicit expression in which the star-shaped inner

(micro level) networks are formed and become coherent in the inter-agent (macro level)

network.

In our model, the agents interact each other in the following manners during each time

step.

1. Exchange: In the social (inter-agent) network, neighboring agents i and j exchange

commodities a and b with probability PE, if both of them recognize that a and b are

exchangeable (i.e. e(i)ab�e(j)

ab�1). All exchanges are assumed to be reciprocal.6)

2. Learning: The learning process of agents consists of the following four methods.

Imitation: If an agent i’s (let e(i)ab=0) neighbor j and j’s neighbor j� succeeded in

exchanging a-b , then i imitate j (i.e. e(i)ab→1) by the probability PI.


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Fig. 2. Emergence of a proto-money: common hub represents general acceptability.

6) In this paper, we intend the simplest explanation of “the emergence of money”, so we focus on the

imitation (or infection) process of the recognition of exchangeability rather than a utility-maximizing

behavior like a matrix game. In our other literature (Kobayashi et al. (2009)), we discuss the case that

the social network evolves dynamically by the agent’s transaction. Both of our static and dynamic

social network models are contributable for the comprehension to the emergence of money.

Trimming: If an agent ‘i’ has cycle recognition of exchangeability (e.g. e(i)ab=

e(i)bg�e(i)

ga�1), then the agent i will trim its inner-network by cutting randomly one of

these cyclic edges by the probability PT. Suppose that an agent can use either of two

equivalent ways (alpha↔beta and alpha↔gamma↔beta) to exchange a commodity

alpha and a beta, will he/she continue to use the two ways equally? To trim down a

cycle of exchangeability is a simple way to implement such partiality of agents’

behavior. Such avoiding cyclic exchanges (Zirkulartausch: Menger, 1923) is

consistent with Kiyotaki and Wright (1989) also.

We consider that these two processes are essential for the emergence of money. In

addition, we introduce two more subsidiary processes as natural fluctuations.

Conceiving: Even if an agent i has no recognition of a-b exchangeability (e(i)ab�0),

it will happen to conceive that (e(i)ab →1) by the probability PC.

Forgetting: Vice versa, even if an agent i has recognition of a-b exchangeability

(e(i)ab�1), it will happen to forget this (e(i)

ab→0) by the probability PF.

Although these probabilities are constant data in the model, their values can be

dependent on the kind of commodities (i.e. PE(a , b)i is not always equal to PE

(a , g)i ). To

simplify the notation, we sometimes omit superscripts(a , b) or subscriptsa , b.

Thus, we introduced the specific mechanism for the micro-macro interaction

announced in Section 2. In this model, unless the agents interact each other through the

social network, we cannot observe macroscopically the coherence of the agents’ internal

states. This implies that our model of the emergence of proto-money is a proper

framework of the “first order emergence” in the Hyperstructure (Bass, 1992).

4. Mean-field Dynamics and Bifurcation Analysis on the Emergence of

Money

We will derive the dynamics that uses mean-field approximation, in order to find the

important nature on the emergence of proto-money. Under some idealized assumptions

and approximation, a more simplified dynamics of the mean-field dynamics enables us to

analytical approach on the system bifurcation. The numerical results by the original

mean-field dynamics validate our finding from the analytical approach of bifurcation. In

addition, the simulation model from our other literature without the mean-field

approximation supports our results from analytical and numerical approach.

4.1 Mean-field dynamics

Here, we derive some mean-field dynamics and analyze the behavior of the doubly

structural network model of the emergence of money by the mean-field approximation.

M. KUNIGAMI et al.

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Mean-field approximation substitutes the overall agent average state for state around

each agent. Instead of ignoring the specific local structure of the network, it makes some

analytical approaches possible.

At first, we denote parameter “k” as the degree of nodes (agents) on the social

network. This k is assumed to have certain distribution function p(k). Next, we introduce

the state variable xa ;k which represents the average acceptability of commodity a with

respect to agents with degree k. Each xa ;k means the probability that an agent with degree

k recognizes exchangeability between a and another arbitrary commodity.

The following dynamics describes the time-evolution of these mean-field states.

(2)

The right hand side of (2) consists of 4-interaction processes (at 3.2) as shown below.

* 1st term�Imitate: A transaction occurs between neighbors and neighbors of neighbors.

Inside the large brackets, the imitation (by the agent with degree k who doesn’t use a)

occurs depending on the expected number of the edges (k�) with the neighbors (who

use a) and the expected number of the edges (k�) with the neighbor of the neighbor

(Pastor-Satorras and Vespignani, 2002). (Fig. 3)

* 2nd term�Trim: A cyclic recognition from a to a via b and g of M commodity types

(Fig. 3).

* 3rd terms�Conceive and 4th terms�Forget: Obviously, these represent the mutational

dx

dtP P x k

k k p k x

kk

E I kk kα

αα,

,

,( )

( ) ( )� �

� ��

11∑

kk

k k

kp k

k p k x

k p k�

� �

��

� �

� �∑∑∑

⎛

⎝⎜⎜

⎞

⎠⎟⎟( )

( )

( )

,α⎛⎛

⎝⎜⎜

⎞

⎠⎟⎟

≠∑� � � �P Mx x P x P xT k k c k Fα ββ α

α α, , , ,( )2 1 kk

kx M k N0 1 1 2 1 2 1 � � �α α, ( , , , , , , )� �


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Fig. 3. (i) The imitation occurs depending on the edges (k�) with the neighbors and the

number of the edges (k�) of the neighbor of the neighbor (above). The trimming occurs in

proportion to the expected number of the cyclic exchangeabilities (below).

obtainment and loss of exchangeability.

4.2 Bifurcation analysis on the emergence scenario

This section discusses the effect of social network structure on the emergence of proto-

money by contriving ideal settings that simplified mean-field dynamics (2). We consider

that distribution of the social network has only k-degree node (k has a point distribution

(p(k)�d k,k0), e.g. Regular Network (Watts and Strogatz, 1998). Although generality is

lost, it seems to be appropriately the ideal type for relationship between the emergence of

proto-money and the density of social connections.

Figure 4 is one of the numerical outcomes. This result shows that even though all

properties of commodities are homogeneous (P*s do not depend on commodities), proto-

money may emerge spontaneously from infinitesimal fluctuations in the initial condition. This

supports the “non-metallic theory of money” that we discussed in the previous section (3.1).

Next, we derive more simplified dynamics to approve the above outcome more

generally and to illustrate the effect of social network structure on the emergence of

proto-money. The “mean-field 1st–2nd dynamics” (3) is given by focusing on only the 1st

and the 2nd acceptable commodities and fixing the amount of others in a small constant s .

dx

dtP P k k x x P Mx x PE I T C

αα α α β σ� � � � � �( ) ( ) ( ) (1 1 12 2 ��

� � �

x P x f x x

dx

dtP P k k

Fk

E I

α α α α β

β

) ( , )

( ) (

( )≡ 2

1 1 xx x P M x x P x P x f xT Ck

β β β β β β β ασ) ( ) ( ) (( )2 2 21� � � � � ≡ ,, )

,

, ,

x

x x x x

x

k k

K

β

α α β β

γγσ

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

≡ ≡

≡,

,≠� 1

ααβ

α β γ

∑⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟, , , ....

( )

�1 2

3

M

M. KUNIGAMI et al.

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Fig. 4. A numeric example of mean-field dynamics (2) in Regular social network shows

proto-money emerges from a homogeneous set of commodities. (In the legend, xaa (aa�1��4)

indicate xaa (t) (aa�1��4))


– 75 –

Fig. 5. Bifurcations of the 1st–2nd dynamics (3) on the case of “monotonic isoclines”, the

horizontal axis is xaa (largest acceptability of commodity), vertical axis is xbb (2nd largest one).

Fig. 6. Bifurcations of the 1st–2nd dynamics (3) on the case of “isoclines with minima and

maxima” (above) and “unimodal isoclines” (below) show similar scenarios on the emergence

of proto-money with the emergence scenario in Fig. 5.

Assuming enough small s (total of below 3rd), the position and stability of the

equilibria of the “1st–2nd mean-field dynamics” (3) shows the “non-emergence/

independent-emergence/double-emergence of proto-money. Global analysis with the

isocline method is possible for two-dimensional dynamics (3). The isoclines xb (xa) and

xa(xb) are derived by solving f (k)a (xa, xb)�0 and f (k)

b (xa, xb)�0 in (3).

The isoclines have several shapes according to coefficients (simplifying as s�0). At

first we describe the case of “monotonic isoclines”.

In Fig. 5, while social parameter k (number of each agent’s friends) is small, the

equilibrium point Q1 stays around a small level of acceptability. Once k exceeds a certain

value k�*, equilibrium splits up, so Q2 moves towards the area where a takes almost all

acceptability (emergence). Furthermore, if k exceeds k�*, the two equilibria are united, so

both of a & b take large acceptability (double emergence). Here, the critical value k* is

found as the value that leads Q1�(x0(k); x0(k)) (: the intersection of iso-clines on the line

xa�xb) to change from a stable equilibrium to a saddle point or vice versa.

Similarly, in the case of “isoclines with minima & maxima” and “unimodal (concave)

isoclines” (Fig. 6) they have common proto-money emergence scenario as follows:

* Non-emergence: no commodities with general acceptability emerge if the network

degree k is small enough.

* Single-emergence: only one of the commodities emerges as proto-money if the degree

k grows larger than the lower critical value.

* Multiple-emergences: Two (or more) commodities emerge as proto-money if the

degree k grows larger than the higher critical value.

4.3 Numerical approach and simulation

We can observe numerically such the bifurcation in emergence, by the mean-field

dynamics without “the 1st–2nd approximation”. Under the condition that sigma is small

enough, the mean-field dynamics (2) with constant degree k confirms the analytical

prediction given by the isocline method to the “1st–2nd” dynamics (3). The mean-field

dynamics (2) in Figs. 7, 8 and 9 correspond to the typical shapes of isoclines of (3). In

these figures, we can find again the emergence of proto-money depend on the degree of

the social network.

In another literature (Kobayashi et al., 2009), we have investigated on the emergence

of money by conducting agent-based simulation with doubly structural network. This

simulation is built from individual agents’ behavior without mean-field approximation.

The following figure (Fig. 10) shows an outcome of the simulation under the situation

corresponding with the analytical and numerical approaches above. It is similar with our

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analytical/numerical outcomes that the simulation illustrates the degree of social network

is an essential parameter for the emergence of proto-money. It implies our approach

using these approximated dynamics is valid.


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Fig. 7. This numerical illustration shows the bifurcation of the mean-field dynamics (2) (the

number of the commodities; M�16, the degree of the social network; k�5��8 & 24 (without

the 1st–2nd approximation)) corresponding to the case of “monotone isoclines” in Fig. 5.

Fig. 8. This numerical illustration shows the bifurcation of the mean-field dynamics (2)

(M�16, k�6��10 (without the 1st–2nd approximation)) corresponding to the case of “isoclines

with minima and maxima” in the upper half of Fig. 6.

5 Discussions and Further Research

5.1 Discuss the emergence and collapse of money

We summarize the results in this section 4 and clarify the relation with section 3.

First, we conducted bifurcation analysis and illustrated the proto-money can emerge

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Fig. 10. The simulation (Kobayashi et al., 2009) without mean-field approximation shows

that increase of the social network k induces increase of commodities that emerge as proto-

money. (PI�0.2, PE�0.8, PT�0.1, PC�0.01, PF�0.01, M�32 Commodities, 250 Agents/run,

20000 Steps/run, 100 runs/k)

Fig. 9. This numerical illustration shows the bifurcation of the mean-field dynamics (2)

(M�16, k�4��7 & 24 (without the 1st–2nd approximation)) corresponding to the case of

“unimodal isoclines” in the lower half of Fig. 6.

spontaneously (depend on infinitesimal initial fluctuation, without any exogenous

governance) even from the commodities without any particular advantages for exchange.

This result consists with the natures of primitive-money (a) and (b) that we required at

section 3.1 and also supports “non-metallic theory of commodity money.”

In (a) of section 3.1, the literature (Radford, 1945) said that alternative primitive-

money (e.g. cigarette) emerged in prison camps instead of the lost currency. We can

observe what will happen in such kind of the situation by small modification

(substituting the depressed component by an appropriate dynamics e.g. logistic equation

after particular time) in the dynamics (2). In Fig. 11, a numerical outcome of the

modified dynamics illustrates that an alternative proto-money spontaneously emerges if

someone compulsorily depresses the exchangeability of the proto-money that was

already recognized.

Next, our bifurcation analysis also explains that the emergence of proto-money (as

non-metallic theory of commodity-money) strongly depends on the degree of social

network k. This result consists with the natures of primitive-money (c) required at

section 3.1 and the social structure affecting the emergence of money we remarked at

section 3.2. The following figure illustrates that exogenous change of the social network

degree k�k(t) affects not only emergence but also collapse of proto-money that already

emerged.

This result on the collapse mechanism of money is a novel contribution, compared to


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Fig. 11. If we depress compulsorily (substitute the corresponding component of (2) by a

logistic type dynamics dx/dt��bb(x�aa)(1�x) after t�150) the proto-money (commodity #1)

that emerged naturally, another commodity (#2) emerges as an alternative proto-money.

the related work (Yasutomi, 1995) mentions that the emergence and collapse of money

can be driven by stochastic fluctuation of agents’ decisions.) It is not difficult to build

models that include the endogenous mechanism changing k�k(t, x(t)).

5.2 Hub-effect on the emergence

In 4.2, we discussed the common emergence scenarios based on the assumption that the

social network is regular. Here, we show another aspect of the degree of social network

affects on the emergence of money.

Literature on epidemiology and complex networks mention that existence of hub

nodes is an important factor for spreading disease. We can also expect our model of the

emergence of money shows such a kind of hub-effect. Applying the 2-points (the hub

agents and the others) distribution of the social network to our mean-field dynamics (2),

we can find a simple example of the hub-effect.

In the upper half of Fig. 13, the regular social networks don’t show the emergence

when these degrees k are less than 6. In the lower half, although all the average degrees

are 5.0, the modified regular networks (If it has enough population of hub agents or if

each hub agent has enough large degree k.) show the emergence of proto-money. This

outcome implies that the emergence of proto-money essentially depend on the variance

of the network degree of the society as well as on the expected value. This hub-effect

shows that some people who have high centrality of exchange of commodities have an

M. KUNIGAMI et al.

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Fig. 12. This figure illustrates how the degree of the social network k affects the proto-

money. In the dynamics (2), if the degree k�k(T) increases more than a certain level, a

commodity emerges as proto-money. If k(T) decreases lower than another level, the proto-

money collapses.

important role in the emergence of money. We often call such hub people of

commodities-exchange “merchants”. Therefore the network analysis of the emergence of

money implies the important existence of merchants.

This model can also illustrate the effect of hub agents (confederate) to induce

artificially emergence of second-currency. In introducing a new additional currency or

starting a virtual money fraud (“Enten” fraud case: Mainichi Daily News, 2009), it seems

effective that the confederates give a demonstration of exchanging them to many kinds of

commodity. The following figure illustrates an experiment of adding hub agents who

have certain level of exchangeability to particular commodity into the society in which

another proto-money has emerged.

Although it might seem that the emergence is a trivial process driven by simple

diffusion dynamics, however, such a oversimplified intuition they cannot predict the

result of collapse of money (Fig. 12) or limited success of induced emergence (Fig. 14).

These results are come from not only agents’ interaction but also inner-agent model that

we mentioned in section 3.3. We inherited this mechanism from the essential insights on

money by Menger and Kiyotaki.


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Fig. 13. A numerical example of the hub-effect via mean-field dynamics (2) with two-points

distribution of the social network; (i) In this case, without the hub-agents, the proto-money

doesn’t emerge when the degree k is less than 6 (above). (ii) Enough population or degree of

the hub-agents promotes the emergence of proto-money (below).

5.3 Further research: ABS

The mean-field dynamics is an applicable approach to find analytically the existence and

sketchy behavior of emergence scenario. However, mean-field approximation requires

strong assumptions that situation around particular agent and natures of the inter-agents

networks can be substituted by the global mean of agents and the mean degree of nodes.

Then this approach has little effectiveness to research a locally heterogeneous system or

complex networks that have long tailed or specific distributions of node degree.

The doubly structured network model is straightforwardly implemented as agent-

based simulation (ABS) on the set of assumptions and situations in previous section. So

we expect that the ABS can show further natures of the emergence of money depending

on specific complex social networks of agents.

We have conducted intensive simulation studies on in the emergence of money on the

regular networks and complex networks such as small-world (Watts and Strogatz, 1998)

and scale-free (Barabasi and Albert, 1999) networks. Furthermore, the other results have

suggested the complexities of those networks promote the emergence of proto-money.

The detailed discussion will be published elsewhere. (Kobayashi et al., 2009)

M. KUNIGAMI et al.

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Fig. 14. The numerical experiment of dynamics (2) in which we add (at t�150, 3% of total

population) the hub agents who recognize #3 and #4 commodity into non-hub population in

which #1 commodity already emerged. The commodity #4 emerges as second proto-money but

#3 failed.

6. Summary

This paper proposes the “doubly structural network model”. This model can easily

describe microscopic propagation/learning along with macroscopic emergence/self-

organization in a society. Modeling and analysis of the “emergence of money” is carried

out as application of this model.

As a result, the approximated dynamics of the model can illustrate the emergence of

money (establishment of general acceptability). The bifurcation analysis of the dynamics

can also illustrate that a nature (degree of social network: k) of social structure plays an

important role in emergence. The numerical approach implies also a possibility of hub-

effect.

Insight gained by this analysis on emergence of money suggests that the doubly

structural network model can be a valid analytical method when applied to other fields

by looking at money as a communication medium in the form of transactions.

Acknowledgements

Valuable suggestions and opinions were put forth through discussions and questions during

seminars and conferences by; Prof. Tamotsu Onozaki of Aomori Public College, Dr. Hideki

Takayasu of SONY Computer Science Laboratory, Prof. Naoki Masuda and Prof. Yukio Osawa

of Tokyo University, Prof. Hiroshi Deguchi of Tokyo Institute of Technology, Prof. Hisatoshi

Suzuki, Prof. Naoki Makimoto, Prof. Hua Xu and Prof. Setsuya Kurahashi of Tsukuba

University. The helpful opinions for improving this paper were also given us by the anonymous

reviewers of EIER. I would like to take this opportunity to express my gratitude.

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