Power laws in cities population, financial markets and internet sites (scaling in systems with a variable number of components)

Blank, Aharon; Solomon, Sorin

doi:10.1016/s0378-4371(00)00464-7

Cited by 135 publications

(105 citation statements)

References 9 publications

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“…Obviously, this normalization does not change the exponent of the power law distribution of sizes, if it exists. Furthermore, elaborating on Krugman (1996)'s argument about the non-convergence of the distribution of firm sizes toward Zipf's law in Simon (1955)'s model, Blank and Solomon (2000) have shown that Gabaix (1999)'s argument suffers from a more technical problem. Based on the demonstration that the two limits, the number of firms N → ∞ and s min (t)/Ω(t) → 0 4 (or equivalently the limit of large times t → ∞) are noncommutative, Blank and Solomon (2000) showed that Zipf's exponent m = 1 as obtained by Gabaix (1999)'s argument requires (i) taking the long time limit s min (t)/Ω(t) → 0 over which the economy made of a large but finite number N firms grows without bounds, while simul-…”

Section: Comparison With Gabaix's Modelmentioning

confidence: 99%

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Zipf's law and maximum sustainable growth

Malevergne

Саичев

Sornette

2013

Journal of Economic Dynamics and Control

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Zipf's law states that the number of firms with size greater than S is inversely proportional to S. Most explanations start with Gibrat's rule of proportional growth but require additional constraints. We show that Gibrat's rule, at all firm levels, yields Zipf's law under a balance condition between the effective growth rate of incumbent firms (which includes their possible demise) and the growth rate of investments in entrant firms. Remarkably, Zipf's law is the signature of the long-term optimal allocation of resources that ensures the maximum sustainable growth rate of an economy. JEL classification: G11, G12Keywords: Firm growth, Gibrat's law, Zipf's law. * Y. Malevergne acknowledges financial support from the French National Research Agency (ANR) through the "Entreprises" Program (Project HYPERCROIS n o ANR-07-ENTR-008). 1Zipf's law and maximum sustainable growth AbstractZipf's law states that the number of firms with size greater than S is inversely proportional to S. Most explanations start with Gibrat's rule of proportional growth but require additional constraints. We show that Gibrat's rule, at all firm levels, yields Zipf's law under a balance condition between the effective growth rate of incumbent firms (which includes their possible demise) and the growth rate of investments in entrant firms. Remarkably, Zipf's law is the signature of the long-term optimal allocation of resources that ensures the maximum sustainable growth rate of an economy. JEL classification: G11, G12

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Section: Comparison With Gabaix's Modelmentioning

confidence: 99%

“…Blank and Solomon (2000) showed that this inconsistency can be resolved by allowing the number of firms to grow proportionally to the total size of the economy.…”

Section: Comparison With Gabaix's Modelmentioning

confidence: 99%

Zipf's law and maximum sustainable growth

Malevergne

Саичев

Sornette

2013

Journal of Economic Dynamics and Control

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“…In spite of the huge literature dealing with firms' exit available in economics, models of firm dynamics adding an exit process to multiplicative growth use rather unplausible exit mechanisms, that is they assume that firms exit as they reach a minimum size [12] or that the number of exits is time-invariantly proportional to the total population of firms [13]. Note that none of the reasons for exit listed above are by themselves fully compatible with neither the minimum size nor with the proportionality assumptions.…”

Section: Evidence On the Bad Debt Distributionmentioning

confidence: 99%

“…In particular, recent research [12,13] has shown that power laws naturally emerge from systems where a random multiplicative process is combined with the entry end exit of systems' components. While simulations show that results are robust to different specifications for entry and exit mechanisms, scarce attention has been paid so far to understand how realistic such processes are.…”

Section: Introductionmentioning

confidence: 99%

Bankruptcy as an exit mechanism for systems with a variable number of components

Gatti

Guilmi

Gaffeo

et al. 2004

Physica A: Statistical Mechanics and its Applications

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Dynamical systems with components whose sizes evolve according to multiplicative stochastic rules have been recently combined with entry and exit processes. We show that the assumptions usually made in modeling exits are at odds with the available evidence. We discuss a recently proposed macroeconomic model with random multiplicative shocks and a mechanism for exit based on bankruptcy, which displays several observed stylized facts for firms' dynamics, like power law distributions for firms' sizes and Laplace distributions for firms' growth rates.

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“…This one may be vulnerable to the frequency analysis, but can also resist to, [1]. The client may help, e.g., through decoy values, obfuscating power laws possibly serving disclosures, [2]. To get best of both worlds an approach perhaps practical is to use Paillier's scheme and a deterministic encryption like AES on every column subject to both: value expressions and SPJ-queries, [17].…”

Section: Introductionmentioning

confidence: 99%

Numerical SQL Value Expressions Over Encrypted Cloud Databases

Jajodia

Litwin

Schwarz

2015

Lecture Notes in Computer Science

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Cloud databases often need client-side encryption. Encryption however impairs queries, especially with numerical SQL value expressions. Fully homomorphic encryption scheme could suffice, but known schemes remain impractical. Partially homomorphic encryption suffices for specific expressions only. The additively homomorphic Paillier scheme appears the most practical. We propose the homomorphic encryption for standard SQL expressions over a practical domain of positive values. The scheme uses a version of Paillier's formulae and auxiliary tables at the cloud that are conceptually the traditional mathematical tables. They tabulate encrypted log and antilog functions and some others over the domain. The choice of functions is extensible. We rewrite the expressions with any number of SQL operators '*', '/' '^" and of standard aggregate functions so they compute over encrypted data using the tables and Paillier's formulae only. All calculations occur at the cloud. We present our scheme, show its security, variants and practicality.

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Power laws in cities population, financial markets and internet sites (scaling in systems with a variable number of components)

Cited by 135 publications

References 9 publications

Zipf's law and maximum sustainable growth

Zipf's law and maximum sustainable growth

Bankruptcy as an exit mechanism for systems with a variable number of components

Numerical SQL Value Expressions Over Encrypted Cloud Databases

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