EQUALITARIAN SOCIETIES ARE ECONOMICALLY IMPOSSIBLE

BOJIN ZHENG1,2,3, WENHUA DU1, WANNENG SHU1, JIANMIN WANG3, DEYI LI3

1College of Computer Science,South-Central University for Nationalities,Wuhan 430074, ChinaE-mail: zhengbojin@gmail.com

2State Key Laboratory of Networking and Switching Technology, Beijing University of Posts andTelecommunications, Beijing 100876, China

3School of Software, Tsinghua University, Beijing 100084, China

Received December 13, 2012

The inequality of wealth distribution is a universal phenomenon in civilized na-tions and it is often imputed to the Matthew effect, that is, the rich get richer and thepoor get poorer. Some philosophers unjustified this phenomenon and tried to put thehuman civilization upon the evenness of wealth. Noticing the facts that 1) the emer-gence of centralism is the starting point of human civilization i.e., people in a societywere organized hierarchically and 2) the inequality of wealth emerges simultaneously,this paper proposes a wealth distribution model based on the hidden tree structure fromthe viewpoint of complex network. This model considers the organized structure ofpeople in a society as a hidden tree, and the cooperation among human beings as thetransactions on the hidden tree, thereby explaining the distribution of wealth. Thismodel shows that the scale-free phenomenon of wealth distribution can be produced bythe cascade controlling of human society, that is, the inequality of wealth can parasitisein the social organizations, such that any actions in eliminating the unequal wealth dis-tribution would lead to the destruction of social or economic structures, resulting in thecollapse of the economic system, therefore, would fail in vain.

Key words: Wealth distribution, inequality, hidden tree structure, power law.

PACS: 89.75.Fb.

1. INTRODUCTION

The inequality of the wealth distribution is a universal phenomenon in the ci-vilized nations, that is, most of people would have little wealth, and a few peoplewould hold most. All the countries during most of the historical periods would obeythis law [1–3], including the ancient Egyptian dynasty [4], the modern Europe, theup-to-date USA, Japan and India [5].

The inequality is often regarded to following the Matthew effect, which saysthat - the rich get richer and the poor get poorer. This effect implies that most ofpeople, the poor, will face a more and more miserable circumstance; and the rich,will thrive forever. The Matthew effect claims an irresistible process and is a popularmechanism explanation to the inequality of wealth.

Numerous philosophers unjustified this inequality phenomenon and believedthat all the creatures should be equally treated and be assigned the equal wealth. Un-der this idea, some nations were set up, for examples, the Soviet Union, the Taiping

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Heavenly Kingdom (1851-1864 in southern China) and other governments. Thesegovernments legislated the equal wealth distribution, but they failed finally due toserious economic depressions. Seemly the inequality phenomenon is natural andinevitable.

But why such an inequitable phenomenon can dominate human society? Doesthere exist a solution that can put the civilization upon the fair wealth distribution?Scientists have been addressing these problems.

Italian economist Vilfredo Pareto was a pioneer scientist. He proposed that thepersonal wealth would obey the power-law distribution [6]. Then Yule proposed amechanism named “Yule Process” to explain many scaling phenomena in variousfields based on “growth” [7, 8]. Obviously, the growth implies that the power-lawdistribution would depend on the lapse of time. Lately, the physicists applied theprinciples in physics to study the wealth distribution and developed a new subject,econo-physics. For example, statistical mechanics has been proved effective in ex-plaining wealth distribution [9,10]. A series of works named Kinetic Exchange Mod-els by researchers such as B. Chakrabarti and A. Chatterjee suggested that the trade isthe origin of wealth distribution [11–14]. Sorin Solomon et al. also suggested that themarket is the source of the extreme wealth inequality and used the Generalised LotkaVolterra model [15] to explain this phenomenon. Furthermore, Solomon et al. ex-plored the relationship between market size and market instability [16] and drew theconclusion that world-size global markets lead to economic instability [17]. Besides,some factors to impact wealth distribution have also been explored, for instance, theincreased longevity [18]. Other researchers also proposed quite interesting explana-tions.

In general, a lot of researches impute the origin of inequality to trade, market orinvestment. With the lapse of time, the market or the investment would form a pos-itive feedback, which make the rich more successful in the future. Mathematically,this explanation can also fit the curves of wealth distribution quite well.

On the other hand, the researches on complex network can contribute someuseful ideas to our understanding of the wealth distribution [19–21]. All of us knewthat complex systems can often be modelled as Complex Networks [22]. Since thereexist many scaling phenomena in complex systems, hence, some complex networks,such as the Scientific Collaboration Networks [23], the World Wide Web [19] etc.,are scale-free. That is, their degree distributions obey the power-law distribution. Toillustrate the mechanism of the Scientific Collaboration Networks, Price proposedthe concept of accumulative advantage [23]. Later, Merton coined the accumulativeadvantage processes as the Matthew effect [24]. To explain the mechanism of theWorld Wide Web, the pioneer scientist Barabasi also proposed a model named thegrowth with Preferential attachment. Both explanations are similar to the positivefeedback. For the scale-free networks, notice that we map the degree to the fortune,

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so that the mechanisms of the scale-free networks can be regarded as the mechanismof wealth distribution and explain the origin of wealth inequality. A. Chatterjee et al.also explored the trade on the networks and obtained interesting results [25, 26].

From above, the positive feedback, or the Matthew effect is a popular and rea-sonable explanation.

However, the Matthew effect is more likely a phenomenon, not the fundamentalfact.

Seemly, when we focus on the phenomenon of wealth condensation, this phe-nomenon is easy to understand, since the rich can invest in new sources of creatingwealth, so that the richer the rich are, the more they can earn. This process wouldrepeat and repeat, the richer would get richer.

But the reality is not like this. John D. Rockefeller was the first billionaire onEarth, but his heirs have to envy the eminence of their ancient now, even if they aresmart enough and the economy of USA increases continually. Not only the Rocke-feller family, but also the other richest families in that age encounter this problem. Ofcourse, the richest men/women in current world can not yet escape from this princi-ple. Probably, at least, the Matthew effect does not work on the richest families. Thewinners can not take all, the poor can become the richest, for example, the foundersof Google, Facebook and Twitter, etc..

On the other hand, suppose that we leave a small amount of money on depositand then sleeping for several hundred years, theoretically our off-springs will havea great amount of wealth through the action of compound interest. Unfortunately,historical real rates of return show that the effects of taxation and inflation wouldlikely cut the money off, less than the amount that had been deposited. This resultimplies that the Matthew effect does not hold in long run and does not work on thecommon population.

From the cases above, seemingly the Matthew effect is often a theoretical pro-cess from the viewpoint of investment. Considering that most of current top rich inUSA didn’t accumulate their fortune from the capital market, this viewpoint may notbe fully reasonable for all circumstances.

Actually, the inequality of wealth has an another source. The economists havedeeply explored a universal phenomenon, the hierarchy in the firms. With the hierar-chy, the company managers are paid with much more salaries and bonus than theirless paid employees, however, the owners of a company would have much more profitthan the salaries and bonuses of the managers. Here, the social position is positivelyrelated to the amount of wealth. Furthermore, if we regarded the society as a set ofcompanies, then the wealth distribution could rely on the hierarchy of company. Thatis, the disparity of wealth may come from the structure of the companies.

If the disparity of wealth could come from the structure of the companies,strictly, the tree-like cascade controlling structure, that means we cannot eliminate

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this kind of disparity, unless we destroy the structure. That is, if there is no structurei.e., no company, so that no labour division, the society would return to the pre-historical state, hence, economical products would fall into an ignorable level. Thatis, the hypothesis on cascade controlling will not only explain the wealth inequality,but also tell the reason that the inequality is natural and not frangible.

In this paper, we design a model based on the hidden tree structure [27, 28],and proved that the trading actions on this structure can produce a power law distri-bution of wealth, meaning that the social cascading controlling structure is one originof wealth inequality. In contrast to the explanation to the Matthew effect based oninvestment, the hidden structure model can explain more phenomena on wealth dis-tribution, and also produces phenomenon similar to the Matthew effect.

2. MODELLING

When we consider the emergence of the wealth, we can find that the wealthobviously relates to the structure in ancient society. For example, the cacique in atribe would probably be the richest man in the eolithic age, and the biggest slaveowner, the king would probably be the richest man in the slavery society. From theexperiences of early human societies, the wealth may be closely positively related tothe social position in the society. In modern societies, the richest men actually arethe kings in their fields.

Why does the wealth strongly relates to the social position? We can use a sim-ple example to illustrate it. If one find that he can buy some apples from the providersand then sell them to customers to make money, he would repeat his business, if theprofit is enough. When the market is big enough, he would find that he can employanother person to help him. Obviously, the profit which is made by the employerwould be shared between the employee and the employer. The larger the business is,the more the employees can be employed, until the employer can employ managersto help him, so a tree structure is built. In this process, the wealth strongly relates tothe social position, does not strictly relate to the initial wealth distribution, becauseof the cascade controlling. In this case, the poor does not have a poor destiny, andthe rich does not have a fate to be richer. We can find many similar cases in thereal world. In a sense, this kind of tree-like cascade controlling, actually is a hiddenorder [27–29] in the society, therefore, we termed it as “hidden tree structure”.

The hidden tree structure means that: 1) it is a hierarchical structure and reflectsthe social positions of the individuals; 2) it is cascade controlling and by it the randomactions are organized; 3) it underlies in the social phenomena and is the hidden order.

The hidden tree structure can produce the phenomenon of the Matthew effect.When the business of the employer is growing bigger and bigger, he/she gets richerand richer in contrast to his employees even if the employer does not have a lot of

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cashes in the whole process. Of course, the employer would hold much more fortunethan the employees finally.

Though the effect of this mechanism satisfies the Matthew effect roughly, wouldthis mechanism explain the wealth distribution? If it works, the distribution made byit should obey the power law distribution.

Assume that the society is an isolated system and every person in this system isorganized into a tree structure. In the society everyone survives on the services of theothers. This means everyone should trade with the others. When two persons have atransaction, they have to take orders from their employers. To minimize the cost, theparticipants should be minimized. That is, both sides in a transaction would choosethe shortest route. Because of cascade controlling, this shortest path actually is thepath between these two persons in the tree structure.

Therefore, we design a model as follows.Suppose that there are N individuals in the society. Let Fi represents the wealth

of the i-th individual. All individuals are organized as a tree. Assume that everyindividual except the leaf nodes has K son nodes. Moreover, every individual has aranking number, representing its level in the tree. Let the ranking number of root be1, denoted as r(1) = 1, the ranking number of the son nodes of root be 2, and so on.The hidden tree structure is illustrated in Fig. 1.

Level 1

Level 2

Level 3

Level 4

Fig. 1 – The sketch map of the hidden tree structure. The hidden tree structure reflects the hidden orderin the society, and the cascade controlling relationships among the individuals. Every node controls afew son nodes.

We then randomly choose two individuals to simulate the transactions betweenthem. For each transaction, there is only a shortest path in the tree structure. More-over, for every transaction, we set the total value of one transaction as 1, and thevalue would be assigned among the individuals who participate the transaction. Theassignment function would be relative to the ranking numbers of all participatingindividuals. For simplicity, assume that the assignment function follows

h(r) = r−t. (1)

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Here r is the ranking number and −t is the exponent.Let the numbers of all the participating individuals be l1, l2, ..., lp, then the

assigned value of the lj node in one transaction would follow

Flj =h(r(lj))

s. (2)

Here,s=

∑p

h(r(lp)) (3)

We use Fig. 2 to illustrate the rules of transactions.

Level 1

Level 2

Level 3

Level 4

Fa=h(4)/s

Fb=h(3)/s

Fc=h(2)/s

Fd=h(1)/s

Fe=h(2)/s

Ff=h(3)/s

a

b

c

d

e

f

g

Fg=h(4)/s

Fig. 2 – The sketch map of a transaction on the hidden tree. The node a proposes the transaction to nodeg. The red arrow indicates that node a and node f are chosen, the profit should be distributed among allthe participants. The purple arrows indicate the shortest path and the participating nodes in the shortestpath b, c, d, e, f between a and g, each of them will obtain profit h(r)/s. s is the normalized factorand s= 2h(4)+2h(3)+2h(2)+h(1). Every node would accumulate its wealth when the transactionscontinue to occur.

Every individual would trade many times with the other individuals, so we useA to represent the number of transactions that every node proposed. The wealth ofarbitrary node would be the summary of all the participated transactions.

Based on the model above, we prove that the distribution of wealth would obeythe power law distribution by the simulation and theoretical analysis. That is,

p(F )∼ F−γ (4)

3. EXPERIMENTAL RESULTS AND ANALYSIS

There are four factors to the wealth distribution in this model: first, the pop-ulation of individuals N ; second, the number of son nodes of every individuals K;third, the times of transactions for every individual A; finally, the exponent value ofthe assignment function t.

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For the first experiment, we set the population of the society as N = 1000,2000, 5000, 10000, 20000, and the other parameters K = 2, A= 0.8, t= 0. We havesuch distributions as in Fig. 3.

100

101

102

103

104

105

10−5

10−4

10−3

10−2

10−1

100

wealth

p(w

ealth

)

Wealth Cumulative Distribution

N=1000N=2000N=5000N=10000N=20000

Fig. 3 – Wealth Cumulative Distribution When N Varies. This figure uses log-log coordinates. All thecurves approximate a straight line, indicating that the distribution of wealth is a power-law. When thenumber of nodes N varies, the exponent γ does not vary, meaning that N does not affect the exponent.At the tails of the curves there is an exponential cutoff. When N increases, the location of the cutoffmoves right. That is, the finity of nodes leads to the cutoff.

From Fig. 3, we can see that no matter how the population varies, the distribu-tion still follows the power law.

The next experiment is designed to check the impact of the number of son nodesK. Here we set K = 2, 3, 4, 5, 6, respectively, the other parameters N = 10000,A= 0.8, h(r) = r0 = 1. We obtain the distributions as in Fig. 4.

100

101

102

103

104

10−4

10−3

10−2

10−1

100

wealth

p(w

ealth

)

Wealth Cumulative Distribution

K=2K=3K=4K=5K=6

Fig. 4 – Wealth Cumulative Distribution When K Varies. For this figure, the curves approximate astraight line, indicating the power law distribution for different K. Some curves have waves whichcome from the fact that every node has integral son nodes and the wealth values are discrete.

The third experiment is designed to check the impact of the number of tran-

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sactions A. Here we set A = 0.08 , 0.16, 0.32, 0.64, 1.28, respectively, the otherparameters N = 10000, K = 2, t= 0. We obtain such distributions as in Fig. 5.

100

101

102

103

104

10−4

10−3

10−2

10−1

100

wealth

p(w

ealth

)

Wealth Cumulative Distribution

A=0.08A=0.16A=0.32A=0.64A=1.28

Fig. 5 – Wealth Cumulative Distribution When A Varies. For this figure, all the curves also approximatea straight line. When the number of transactions increases, the wealth of every node would increasewith the same proportion, therefore, the curves can shift right when A increase. Because the layernumber is limited, the curves show a cut-off.

The last experiment is designed to check the impact of the assignment function.We set the parameter t = 1, 0.5, 0, −0.5, −1 respectively, and the other parametersN = 10000, K = 2, A= 0.8.

10−2

10−1

100

101

102

103

104

10−4

10−3

10−2

10−1

100

wealth

p(w

ealth

)

Wealth Cumulative Distribution

t=−1t=−0.5t=0t=0.5t=1

Fig. 6 – Wealth Cumulative Distribution When t Varies. For different t, all the curves approximatestraight lines with different slopes suggesting that the assignment function can change the exponentvalue.

From these experiments above, we can see that the wealth distributions satisfypower law when the parameters varies. These results imply that the wealth distribu-tions are robust when the individuals are organized as a tree.

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4. THEORETICAL ANALYSIS

Assume that the hidden tree is a complete K-tree, the number of nodes N willbe determined by the number of layers M , so we have,

N=KM+1−1

K−1(5)

To simplify the theoretical analysis we consider a special example i.e., assumethat only the leaf nodes can be chosen randomly and A is fixed, although the simu-lations allow all the nodes to be selected randomly. For any node, its ranking valuer determines the number of its offspring nodes. Since the hidden tree is regular andcomplete, the nodes in every layer would have the same probability to be selectedto participate the transactions. Let us consider the root node in a 3-tree, assume thatthe root node has three branches a, b and c, we select two nodes in the leaf layer toperform one transaction. Assume that the first node belongs to the a branch, thuswhen the second node also belongs to the a branch, the root would not participate inthe transaction; when the second node belongs to the other branches, the root wouldparticipate. Therefore, the root node has a probability of 2/3 to participate in thetransaction. Similarly, the probability of nodes in r-th layer is (2/3)r. Because thenumber of nodes in r-th layer is 3r−1, according to the exponential combination me-thod [30–33], the curve of the probability distribution of the participating transactionsis linear in the log-log coordinates.

Generally, the probability of root node of a K-tree is (K−1)/K and the pro-bability of nodes in r-th layer would be ((K − 1)/K)r so the number of nodes inthe r-th layer is Kr−1; the curve of the probability distribution of the participatingtransactions still is linear in log-log coordinates.

Mathematically, the probability of the participating transactions of a node inthe r-th layer is,

D(r)∼ (K−1

K)r

(6)

For any node, the probability that it has a ranking number r can be calculatedby equation 7.

p(r)∼Kr−1−Kr−2 ∼Kr (1≤ r ≤M) (7)

So we have,

p(D)∼D−1− ln(K)−ln(K−1)

ln(K) (8)

and the exponent −γ = 2− ln(K−1)ln(K) . When K = 2 we see that γ =−2.

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Notice the equation 1, when t= 0, we have,

p(F )∼ F−2+

ln(K−1)ln(K) (9)

That is, the wealth distribution would obey power law.In general, the proposed model can produce the power-law wealth distribution.

5. DISCUSSIONS AND CONCLUSIONS

From the theoretical analysis and the simulation experiments, we can draw theconclusion that the hidden tree structure does produce the power-law wealth distri-bution. Compared to the positive-feedback-like models, this model has some advan-tages. Firstly this model has total parallelism. The actions of every node are notdependent on the others’, so that the growth is not yet a necessary assumption. Se-condly the investment is not the only source to determine the wealth distribution.That is, the final individual wealth does not certainly come from the advantages ofinitial wealth. Creativity, i.e., creating a new branch of the hidden tree, can alsoproduce immense wealth. Thirdly the wealth distribution does not rely on the un-stable positive feedback. On the contrary this model allows the stable structure, thehidden tree. Finally the model is compatible to the Matthew effect. The process ofgenerating a branch can be regarded as an example of the Matthew effect.

Moreover, this hidden tree model means a lot to the ideas on the wealth in-equality. From this model, we can see that the social structure leads to the wealthinequality. That is, according to the inverse theorem, if we want to achieve the resultof wealth evenness, we have to destroy all the social/economic structures. Actually,the social/economic structure relates to the labour divisions. Since the labour divi-sions can empower the labour efficiency hundreds of thousands of times, once thelabour divisions were destroyed, the products would drop drastically, that means, thesociety would experience a serious economic depression. This conclusion can besupported by the early civilization. Thousands of years ago, human beings acceptedthe civilizations even if it often meant the cruel slavery only because the social orga-nization provided more products i.e., more surviving opportunities for all. Since ourancestors abandoned the wealth evenness, or say, zero wealth, we can not go back.

When the exponent of the assignment function varies, the exponent of powerlaw of wealth distribution also varies. That is, the assignment function can not fun-damentally change the power-law wealth distribution, however, it still can adjust theinequality to some extent by increasing more assigned value to the low-level nodes.

In conclusion, although this paper draws a pessimistic conclusion, it simul-taneously proves that human beings are unnecessary to defy the natural principles,because we can modify the wealth distributing processes, by taxing or other policies,

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so that the products increase and are shared. Although the economic inequality isinevitable, even some regard it vicious, it still contributes much to our world.

Acknowledgments. The authors would like to thank the anonymous reviewers for their valuablecomments and suggestions, and thank Dr. Oskar Burger, Dr. Chunlai Zhou, Dr. Baobin Wang andDr. Zhongxun Zhu for valuable discussions and Dr. Ting Hu for English improvement. Supported bythe State Key Laboratory of Networking and Switching Technology (No. SKLNST-2010-1-04) and theNational Natural Science Foundation of China (No. 61273213) and (No. 60803095).

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