- Email: [email protected]
Zipf's law in income distribution of companies
Physica A 269 (1999) 125–131
www.elsevier.com/locate/physa
Zipf’s law in income distribution of companies K. Okuyamaa , M. Takayasub , H. Takayasuc; ∗
a Graduate
School of Information Sciences, Tohoku University, Sendai 980-8579, Japan of Science and Technology, Keio University, Kawasaki 211-0985, Japan c Sony CSL, 3-14-13 Higashigotanda, Shinagawa-ku, Tokyo 141-0022, Japan
b Faculty
Received 28 October 1998; received in revised form 20 January 1999
Abstract Distribution functions of annual income of companies are analyzed based on two company databases. A clear power law distribution consistent with the Zipf’s law can be con rmed for Japanese companies over more than three decades in income scale. Similar distributions can be con rmed in some other countries. It is con rmed that such power laws hold in most of job categories with slightly modi ed exponents. An annual income of a company is about two orders of magnitude smaller than its total assets, and the growth rate distribution of income is c 1999 nearly independent of the income size in contrast to the case of growth rate of assets. Published by Elsevier Science B.V. All rights reserved. PACS: 05.90.+m; 89.90.+n; 05.45.DF Keywords: Zipf’s law; Income distribution; Growth rate of assets
1. Introduction Interactions of companies and complicated money ows may be viewed as the most typical real-world examples of complex systems. It is an important basic step of science to check whether there are any universal laws underlying such economical activities. Here we focus our attention on the distribution of company’s income and show evidences that the statistics of income is following a universal law. The history of study on income distribution is very long. More than one hundred years ago Pareto reported that personal income distribution follows a power law with a possibly universal exponent about 1.5 [1]. Gini studied income distributions of several countries in 1922 and found that the distributions can actually be approximated by ∗
Corresponding author. E-mail address: [email protected] (H. Takayasu)
c 1999 Published by Elsevier Science B.V. All rights reserved. 0378-4371/99/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 0 8 6 - 2
126
K. Okuyama et al. / Physica A 269 (1999) 125–131
power laws but the exponents are not universal [2]. In 1931 Gibrat proposed log-normal distributions of income based on a theoretical assumptions of multiplicative random processes [3]. Mandelbrot rediscovered the power law distributions of income in 1960 [5], and until now there are many open problems related to income distribution in economics [6]. With the increasing interest on power law behaviors in connection with critical behaviors in physical sciences two physicists, Montroll and Shlesinger [4], pointed out in their physics paper that income of rich people follow a power law statistics while that of not rich people follows a log-normal distribution. The power law tails in income distribution have been introduced as an example of fractal behaviors in economics [7,8]. A relating topic is attracting physicists’ interest recently, that is, the company size distributions. Stanley et al. analyzed an American company’s database and found that the size distribution is closer to the lognormal law [9,10]. They also found a plausible scaling relation that the standard deviation of growth rate of company size is inversely proportional to a fractional power of its size, which is consistent with the intuitive empirical impression that uctuation of a large company is smaller than that of a small company. Okuyama and Takayasu analyzed an international company database and reported that the scaling law holds also in other countries but the company size distributions depend on country [11]. Company size can be measured by several quantities such as assets, number of employee, or net sales. It is known that these quantities give consistent results in analysis of company size statistics [9,10]. On the other hand roughly speaking company’s income is de ned by the di erence of the net sales minus net expenses, so it can take a negative value if the net expenses exceed the net sales. Obviously, the statistics of income should be di erent from assets or other quantities proportional to the company size. In the following section we describe our databases. Main results are listed in the Section 3 and we discuss the meaning of such income distributions in the nal section.
2. The databases In this study we analyze two databases. The rst one is a data CD, “Japanese companies the best 85375” which is published in 1998 by a publisher Diamond Inc. in Tokyo, Japan. This CD covers all companies having annual income (to be more precise, the income before taxes reported to the tax oce) more than 40,000,000 yen are listed. Although the number of companies is very large the information for each company is limited, only company’s name, its job category, and incomes for 4 years. We name this database as Data 1 in the following analysis. The other database is “Company data” distributed by Moody’s Investors Service Inc. A pair of CDs cover more than 10,000 companies in the USA and also more than 11,000 companies for other countries. The information for each company is rich
K. Okuyama et al. / Physica A 269 (1999) 125–131
127
and we can nd any nancial information needed for investment, such as assets, sale, income and many other quantities for about 7 years. We call the data as Data 2. Although the information for each company is complete there are demerits in Data 2 for our purpose. First the standard of choice of companies is not speci ed so there is a possibility that we miss companies having big incomes. Second the number of companies are not sucient, for example, there are only 727 Japanese companies in this database.
3. Distributions of income We rst plot the distribution of company’s income for Japanese companies using Data 1 by a bold line in Fig. 1.The ordinate, P(¿x), shows the cumulative probability in log scale, that is, the probability of nding a company of income larger than x. In the range of income less than 105 million yen the plots can be approximated by a straight line with slope −1 meaning that the distribution follows a power law with exponent very close to −1, the so called Zipf’s law. The distribution tail for income larger than 105 million yen decay more quickly than the power law. We con rmed that these behaviors also hold in other years. In Fig. 1 the income distribution for Italy is also shown using the Data 2 by a dashed line. The plotted points are roughly on a straight line with the same slope −1. Compared with Japanese data there is deviation from the straight line in smaller income range, but this is obviously due to the lack of data of smaller companies in the Data 2. Although the results are not so clear we believe that the Zipf’s law also holds in the case of Italy. We also con rm that the Zipf’s law can be the rst approximation of income distribution for many other countries.
Fig. 1. Cumulative distribution of income of Japanese companies in Data 1 (bold line, x : million yen) and Italian companies in Data 2 (dashed line, x : 105 lira). The two straight lines show the power law with exponent −1, the Zipf’s law.
128
K. Okuyama et al. / Physica A 269 (1999) 125–131
Fig. 2. Income distribution of USA in Data 2 (x : thousand dollars). The dotted line shows the power law with exponent −1:4.
Fig. 3. Income distribution of three categories in Japan; construction (bold line), electrical products (dashed line) and power companies (dotted line). The number of data points is 11805, 2374 and 175, respectively. The least-squares t gives a power law with exponent −1:13 for construction companies (dotted straight line) and −0:72 for Electrical products companies (dash-dot line).
There are exceptions, for example, in the case of USA it is impossible to approximate the plots by the Zipf’s law as shown in Fig. 2. If we dare to approximate the tail part of this distribution by a power law the estimated slope is about −1:4, which coincides with that of Montroll and Shlesinger [4]. This result may mean that the income distribution of USA is signi cantly di erent from that of Japan, or there may be many companies having rather large incomes but they are missed in Data 2 by some reason. With the data now available for us we cannot conclude which is correct. Next, we observe the income distribution in each job category based on Data 1. Fig. 3 shows the income distributions of typical three job categories; construction, electrical products, and power companies, respectively. As typically represented by the curves for construction and electrical products we have power laws in many job
K. Okuyama et al. / Physica A 269 (1999) 125–131
129
Fig. 4. (a) Probability density of income growth rates. The points are distinguished according to their income size x into ve categories to clarify the size dependence. The dotted lines are tting by exponential functions. (b) The standard deviation of income growth rate distribution as a function of income in log–log scale. The dotted line shows the best- t line by the least-squares method. The estimated slope 0.024 can be regarded as 0.
categories. The exponents of the power law scatter around −1 but typically in the range of (−1:2; −0:7). The job categories showing clear power laws other than these two are wholesale, land transport, machine, real estate, service, foods, steel, etc. There are exceptions, for example, the dotted curve for power companies in Fig. 3 clearly deviates from a power law. Such deviations from the power law can be con rmed in the job categories of banks, insurance companies, and medicine companies. For other countries income distributions divided into job categories look similar to that of entire jobs, but the number of data points are not so large and we can not elucidate any clear result from Data 2. In the case of assets Stanley et al. discovered an interesting scaling relation on the statistics of assets growth rate that the distribution of growth rates is approximated by a tent-shaped probability density of which standard deviation is proportional to an inverse fractional power of assets [9,10]. We check whether such relation can also be found in the statistics of income. Let x(t) be a company’s income at year t. We observe the ratio of incomes of successive years in log-scale r = log
x(t + 1) : x(t)
(1)
Here, we neglect the data with negative incomes. In Fig. 4(a) the distribution of r is plotted in semi-log scale. As seen from this gure the probability densities are approximated nicely by a tent-shaped function like the case of assets. However, in contrast to the case of assets there is no obvious size dependence in the plot. The size dependence of probability density is characterized by the plot of the standard deviation as a function of income size. From Fig. 4(b) we cannot nd any coherent relation and we conclude that the standard deviation is approximately independent of income. Namely, the statistics of growth rate of assets and that of income is signi cantly di erent.
130
K. Okuyama et al. / Physica A 269 (1999) 125–131
Fig. 5. (a) Net sales vs. assets of Japanese companies in log–log scale (million yen). The dotted line shows the best- t line by the least-squares method. The estimated relation is given by Eq. (2). (b) Income vs. assets of Japanese companies in log–log scale (million yen). The dotted line shows the best- t line by the least-squares method. The estimated relation is given by Eq. (4).
In order to clarify the di erence of income and assets we next clarify the relation of these quantities using Japanese company data in Data 2. Fig. 5(a) represents the relation of net sales vs. assets in log–log scale. The plotted points indicate that net sales is roughly equal to assets, but the least-squares tting suggests a non-trivial averaged behavior S = 2:9A0:90 ;
(2)
where S is the averaged net sales and A is the corresponding assets (both measured in million yen). As for net expenses we have roughly a similar relation with slightly di erent parameters E = 1:5A0:92 ;
(3)
where E is the averaged net expense (measured in million yen). In Fig. 5(b) we show the relation of income vs. assets. As an income of a company is roughly given by the net sales minus net expenses, the plots of income are much more scattered than the case of Fig. 5(a). By a very rough estimation the income is about two orders smaller than the quantities representing company sizes such as assets, net sales or net expenses. The least-squares method gives the following relation as an average: I = 0:78A0:85 ;
(4)
where I is the averaged income (measured in million yen). Similar relations can be con rmed also for other countries such as USA and Italy. As incomes are generally about two orders of magnitude smaller than assets we believe that our result is not contradicting with Stanley et al.’s empirical relation that growth rates of assets depend on the magnitude of assets [9,10].
K. Okuyama et al. / Physica A 269 (1999) 125–131
131
4. Discussion The main result of this paper is symbolized by Fig. 1, the clear evidence that the Zipf’s law holds for the distribution of income of all companies. Although there has been no theoretical explanation for this empirical law, it is very likely that competitive interactions among companies may be playing an important role. This estimation is derived by considering the job categories in which the power laws do not hold. In Japan, banks, insurance companies and power companies are considered to have special social position. For example, assume that two electric power companies compete in a city and as a result one company collapsed, then it is likely that many people in the city will not have electric power for a while. To avoid such tragedy there have been many restrictions by laws not to make much competition among companies having special social positions. On the other hand, in job categories in which competition among companies of various sizes are allowed such as wholesales or construction companies we have rather clear power law behaviors. Thus we believe that the power law distribution is very closely related with underlying competition mechanisms. In order to understand the meaning of this power law we need to develop a numerical model which is consistent with all empirical laws. References V. Pareto, Le Cours d’Economie Politique, Macmillan, London, pp. 1896 –1897. C. Gini, Indici di concentrazione e di dipendenza, Biblioteca delli’economista 20 (1922) 77. R. Gibrat, Les Inegalites Economiques, Sirey, Paris, 1931. E.W. Montroll, M.F. Shlesinger, J. Stat. Phys. 32 (1983) 209. B.B. Mandelbrot, Int. Econom. Rev. 1 (1960) 79. A.H.Q.M. Merkies, I.J. Steyn, Econom. Lett. 43 (1993) 177. B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman, San Francisco, 1982. H. Takayasu, Fractals in the Physical Sciences, Wiley, Sussex, 1992. M.H.R. Stanley, L.A.N. Amaral, S.V. Buldyrev, S.V. Havlin, H. Leschhron, P. Maass, M.A. Salinger, H.E. Stanley, Nature 397 (1996) 804. [10] L.A.N. Amral, S.V. Buldyrev, S. Havlin, H. Leschhorn, P. Maass, M.A. Salinger, H.E. Stanley, M.H.R. Stanley, J. Phys. I France 7 (1997) 621. [11] H. Takayasu, K. Okuyama, Fractals 6 (1998) 67. [1] [2] [3] [4] [5] [6] [7] [8] [9]
www.elsevier.com/locate/physa
Zipf’s law in income distribution of companies K. Okuyamaa , M. Takayasub , H. Takayasuc; ∗
a Graduate
School of Information Sciences, Tohoku University, Sendai 980-8579, Japan of Science and Technology, Keio University, Kawasaki 211-0985, Japan c Sony CSL, 3-14-13 Higashigotanda, Shinagawa-ku, Tokyo 141-0022, Japan
b Faculty
Received 28 October 1998; received in revised form 20 January 1999
Abstract Distribution functions of annual income of companies are analyzed based on two company databases. A clear power law distribution consistent with the Zipf’s law can be con rmed for Japanese companies over more than three decades in income scale. Similar distributions can be con rmed in some other countries. It is con rmed that such power laws hold in most of job categories with slightly modi ed exponents. An annual income of a company is about two orders of magnitude smaller than its total assets, and the growth rate distribution of income is c 1999 nearly independent of the income size in contrast to the case of growth rate of assets. Published by Elsevier Science B.V. All rights reserved. PACS: 05.90.+m; 89.90.+n; 05.45.DF Keywords: Zipf’s law; Income distribution; Growth rate of assets
1. Introduction Interactions of companies and complicated money ows may be viewed as the most typical real-world examples of complex systems. It is an important basic step of science to check whether there are any universal laws underlying such economical activities. Here we focus our attention on the distribution of company’s income and show evidences that the statistics of income is following a universal law. The history of study on income distribution is very long. More than one hundred years ago Pareto reported that personal income distribution follows a power law with a possibly universal exponent about 1.5 [1]. Gini studied income distributions of several countries in 1922 and found that the distributions can actually be approximated by ∗
Corresponding author. E-mail address: [email protected] (H. Takayasu)
c 1999 Published by Elsevier Science B.V. All rights reserved. 0378-4371/99/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 0 8 6 - 2
126
K. Okuyama et al. / Physica A 269 (1999) 125–131
power laws but the exponents are not universal [2]. In 1931 Gibrat proposed log-normal distributions of income based on a theoretical assumptions of multiplicative random processes [3]. Mandelbrot rediscovered the power law distributions of income in 1960 [5], and until now there are many open problems related to income distribution in economics [6]. With the increasing interest on power law behaviors in connection with critical behaviors in physical sciences two physicists, Montroll and Shlesinger [4], pointed out in their physics paper that income of rich people follow a power law statistics while that of not rich people follows a log-normal distribution. The power law tails in income distribution have been introduced as an example of fractal behaviors in economics [7,8]. A relating topic is attracting physicists’ interest recently, that is, the company size distributions. Stanley et al. analyzed an American company’s database and found that the size distribution is closer to the lognormal law [9,10]. They also found a plausible scaling relation that the standard deviation of growth rate of company size is inversely proportional to a fractional power of its size, which is consistent with the intuitive empirical impression that uctuation of a large company is smaller than that of a small company. Okuyama and Takayasu analyzed an international company database and reported that the scaling law holds also in other countries but the company size distributions depend on country [11]. Company size can be measured by several quantities such as assets, number of employee, or net sales. It is known that these quantities give consistent results in analysis of company size statistics [9,10]. On the other hand roughly speaking company’s income is de ned by the di erence of the net sales minus net expenses, so it can take a negative value if the net expenses exceed the net sales. Obviously, the statistics of income should be di erent from assets or other quantities proportional to the company size. In the following section we describe our databases. Main results are listed in the Section 3 and we discuss the meaning of such income distributions in the nal section.
2. The databases In this study we analyze two databases. The rst one is a data CD, “Japanese companies the best 85375” which is published in 1998 by a publisher Diamond Inc. in Tokyo, Japan. This CD covers all companies having annual income (to be more precise, the income before taxes reported to the tax oce) more than 40,000,000 yen are listed. Although the number of companies is very large the information for each company is limited, only company’s name, its job category, and incomes for 4 years. We name this database as Data 1 in the following analysis. The other database is “Company data” distributed by Moody’s Investors Service Inc. A pair of CDs cover more than 10,000 companies in the USA and also more than 11,000 companies for other countries. The information for each company is rich
K. Okuyama et al. / Physica A 269 (1999) 125–131
127
and we can nd any nancial information needed for investment, such as assets, sale, income and many other quantities for about 7 years. We call the data as Data 2. Although the information for each company is complete there are demerits in Data 2 for our purpose. First the standard of choice of companies is not speci ed so there is a possibility that we miss companies having big incomes. Second the number of companies are not sucient, for example, there are only 727 Japanese companies in this database.
3. Distributions of income We rst plot the distribution of company’s income for Japanese companies using Data 1 by a bold line in Fig. 1.The ordinate, P(¿x), shows the cumulative probability in log scale, that is, the probability of nding a company of income larger than x. In the range of income less than 105 million yen the plots can be approximated by a straight line with slope −1 meaning that the distribution follows a power law with exponent very close to −1, the so called Zipf’s law. The distribution tail for income larger than 105 million yen decay more quickly than the power law. We con rmed that these behaviors also hold in other years. In Fig. 1 the income distribution for Italy is also shown using the Data 2 by a dashed line. The plotted points are roughly on a straight line with the same slope −1. Compared with Japanese data there is deviation from the straight line in smaller income range, but this is obviously due to the lack of data of smaller companies in the Data 2. Although the results are not so clear we believe that the Zipf’s law also holds in the case of Italy. We also con rm that the Zipf’s law can be the rst approximation of income distribution for many other countries.
Fig. 1. Cumulative distribution of income of Japanese companies in Data 1 (bold line, x : million yen) and Italian companies in Data 2 (dashed line, x : 105 lira). The two straight lines show the power law with exponent −1, the Zipf’s law.
128
K. Okuyama et al. / Physica A 269 (1999) 125–131
Fig. 2. Income distribution of USA in Data 2 (x : thousand dollars). The dotted line shows the power law with exponent −1:4.
Fig. 3. Income distribution of three categories in Japan; construction (bold line), electrical products (dashed line) and power companies (dotted line). The number of data points is 11805, 2374 and 175, respectively. The least-squares t gives a power law with exponent −1:13 for construction companies (dotted straight line) and −0:72 for Electrical products companies (dash-dot line).
There are exceptions, for example, in the case of USA it is impossible to approximate the plots by the Zipf’s law as shown in Fig. 2. If we dare to approximate the tail part of this distribution by a power law the estimated slope is about −1:4, which coincides with that of Montroll and Shlesinger [4]. This result may mean that the income distribution of USA is signi cantly di erent from that of Japan, or there may be many companies having rather large incomes but they are missed in Data 2 by some reason. With the data now available for us we cannot conclude which is correct. Next, we observe the income distribution in each job category based on Data 1. Fig. 3 shows the income distributions of typical three job categories; construction, electrical products, and power companies, respectively. As typically represented by the curves for construction and electrical products we have power laws in many job
K. Okuyama et al. / Physica A 269 (1999) 125–131
129
Fig. 4. (a) Probability density of income growth rates. The points are distinguished according to their income size x into ve categories to clarify the size dependence. The dotted lines are tting by exponential functions. (b) The standard deviation of income growth rate distribution as a function of income in log–log scale. The dotted line shows the best- t line by the least-squares method. The estimated slope 0.024 can be regarded as 0.
categories. The exponents of the power law scatter around −1 but typically in the range of (−1:2; −0:7). The job categories showing clear power laws other than these two are wholesale, land transport, machine, real estate, service, foods, steel, etc. There are exceptions, for example, the dotted curve for power companies in Fig. 3 clearly deviates from a power law. Such deviations from the power law can be con rmed in the job categories of banks, insurance companies, and medicine companies. For other countries income distributions divided into job categories look similar to that of entire jobs, but the number of data points are not so large and we can not elucidate any clear result from Data 2. In the case of assets Stanley et al. discovered an interesting scaling relation on the statistics of assets growth rate that the distribution of growth rates is approximated by a tent-shaped probability density of which standard deviation is proportional to an inverse fractional power of assets [9,10]. We check whether such relation can also be found in the statistics of income. Let x(t) be a company’s income at year t. We observe the ratio of incomes of successive years in log-scale r = log
x(t + 1) : x(t)
(1)
Here, we neglect the data with negative incomes. In Fig. 4(a) the distribution of r is plotted in semi-log scale. As seen from this gure the probability densities are approximated nicely by a tent-shaped function like the case of assets. However, in contrast to the case of assets there is no obvious size dependence in the plot. The size dependence of probability density is characterized by the plot of the standard deviation as a function of income size. From Fig. 4(b) we cannot nd any coherent relation and we conclude that the standard deviation is approximately independent of income. Namely, the statistics of growth rate of assets and that of income is signi cantly di erent.
130
K. Okuyama et al. / Physica A 269 (1999) 125–131
Fig. 5. (a) Net sales vs. assets of Japanese companies in log–log scale (million yen). The dotted line shows the best- t line by the least-squares method. The estimated relation is given by Eq. (2). (b) Income vs. assets of Japanese companies in log–log scale (million yen). The dotted line shows the best- t line by the least-squares method. The estimated relation is given by Eq. (4).
In order to clarify the di erence of income and assets we next clarify the relation of these quantities using Japanese company data in Data 2. Fig. 5(a) represents the relation of net sales vs. assets in log–log scale. The plotted points indicate that net sales is roughly equal to assets, but the least-squares tting suggests a non-trivial averaged behavior S = 2:9A0:90 ;
(2)
where S is the averaged net sales and A is the corresponding assets (both measured in million yen). As for net expenses we have roughly a similar relation with slightly di erent parameters E = 1:5A0:92 ;
(3)
where E is the averaged net expense (measured in million yen). In Fig. 5(b) we show the relation of income vs. assets. As an income of a company is roughly given by the net sales minus net expenses, the plots of income are much more scattered than the case of Fig. 5(a). By a very rough estimation the income is about two orders smaller than the quantities representing company sizes such as assets, net sales or net expenses. The least-squares method gives the following relation as an average: I = 0:78A0:85 ;
(4)
where I is the averaged income (measured in million yen). Similar relations can be con rmed also for other countries such as USA and Italy. As incomes are generally about two orders of magnitude smaller than assets we believe that our result is not contradicting with Stanley et al.’s empirical relation that growth rates of assets depend on the magnitude of assets [9,10].
K. Okuyama et al. / Physica A 269 (1999) 125–131
131
4. Discussion The main result of this paper is symbolized by Fig. 1, the clear evidence that the Zipf’s law holds for the distribution of income of all companies. Although there has been no theoretical explanation for this empirical law, it is very likely that competitive interactions among companies may be playing an important role. This estimation is derived by considering the job categories in which the power laws do not hold. In Japan, banks, insurance companies and power companies are considered to have special social position. For example, assume that two electric power companies compete in a city and as a result one company collapsed, then it is likely that many people in the city will not have electric power for a while. To avoid such tragedy there have been many restrictions by laws not to make much competition among companies having special social positions. On the other hand, in job categories in which competition among companies of various sizes are allowed such as wholesales or construction companies we have rather clear power law behaviors. Thus we believe that the power law distribution is very closely related with underlying competition mechanisms. In order to understand the meaning of this power law we need to develop a numerical model which is consistent with all empirical laws. References V. Pareto, Le Cours d’Economie Politique, Macmillan, London, pp. 1896 –1897. C. Gini, Indici di concentrazione e di dipendenza, Biblioteca delli’economista 20 (1922) 77. R. Gibrat, Les Inegalites Economiques, Sirey, Paris, 1931. E.W. Montroll, M.F. Shlesinger, J. Stat. Phys. 32 (1983) 209. B.B. Mandelbrot, Int. Econom. Rev. 1 (1960) 79. A.H.Q.M. Merkies, I.J. Steyn, Econom. Lett. 43 (1993) 177. B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman, San Francisco, 1982. H. Takayasu, Fractals in the Physical Sciences, Wiley, Sussex, 1992. M.H.R. Stanley, L.A.N. Amaral, S.V. Buldyrev, S.V. Havlin, H. Leschhron, P. Maass, M.A. Salinger, H.E. Stanley, Nature 397 (1996) 804. [10] L.A.N. Amral, S.V. Buldyrev, S. Havlin, H. Leschhorn, P. Maass, M.A. Salinger, H.E. Stanley, M.H.R. Stanley, J. Phys. I France 7 (1997) 621. [11] H. Takayasu, K. Okuyama, Fractals 6 (1998) 67. [1] [2] [3] [4] [5] [6] [7] [8] [9]