arXiv subjects

S. Havlin

Publications and source records attributed to S. Havlin.

41 records · Page 3Linked to original sources

arXiv1997

Power Law Scaling for a System of Interacting Units with Complex Internal Structure

We study the dynamics of a system composed of interacting units each with a complex internal structure comprising many subunits. We consider the case in which each subunit grows in a multiplicative manner. We propose a model for such systems in which the interaction among the units is treated in a mean field approximation and the interaction among subunits is nonlinear. To test the model, we identify a large data base spanning 20 years, and find that the model correctly predicts a variety of empirical results.

cond-mat.stat-mech↗

arXiv1997

Scaling behavior in economics: I. Empirical results for company growth

We address the question of the growth of firm size. To this end, we analyze the Compustat data base comprising all publicly-traded United States manufacturing firms within the years 1974-1993. We find that the distribution of firm sizes remains stable for the 20 years we study, i.e., the mean value and standard deviation remain approximately constant. We study the distribution of sizes of the ``new'' companies in each year and find it to be well approximated by a log-normal. We find (i) the distribution of the logarithm of the growth rates, for a fixed growth period of one year, and for companies with approximately the same size $S$ displays an exponential form, and (ii) the fluctuations in the growth rates -- measured by the width of this distribution $σ_1$ -- scale as a power law with $S$, $σ_1\sim S^{-β}$. We find that the exponent $β$ takes the same value, within the error bars, for several measures of the size of a company. In particular, we obtain: $β=0.20\pm0.03$ for sales, $β=0.18\pm0.03$ for number of employees, $β=0.18\pm0.03$ for assets, $β=0.18\pm0.03$ for cost of goods sold, and $β=0.20\pm0.03$ for property, plant, & equipment.

cond-mat.stat-mech↗

arXiv1997

Scaling behavior in economics: II. Modeling of company growth

In the preceding paper we presented empirical results describing the growth of publicly-traded United States manufacturing firms within the years 1974--1993. Our results suggest that the data can be described by a scaling approach. Here, we propose models that may lead to some insight into these phenomena. First, we study a model in which the growth rate of a company is affected by a tendency to retain an ``optimal'' size. That model leads to an exponential distribution of the logarithm of the growth rate in agreement with the empirical results. Then, we study a hierarchical tree-like model of a company that enables us to relate the two parameters of the model to the exponent $β$, which describes the dependence of the standard deviation of the distribution of growth rates on size. We find that $β= -\ln Π/ \ln z$, where $z$ defines the mean branching ratio of the hierarchical tree and $Π$ is the probability that the lower levels follow the policy of higher levels in the hierarchy. We also study the distribution of growth rates of this hierarchical model. We find that the distribution is consistent with the exponential form found empirically.

cond-mat.stat-mech↗

arXiv1994

Avalanches and the Directed Percolation Depinning Model: Experiments, Simulations and Theory

We study the recently-introduced directed percolation depinning (DPD) model for interface roughening with quenched disorder for which the interface becomes pinned by a directed percolation (DP) cluster for $d = 1$, or a directed surface (DS) for $d > 1$. The mapping to DP enables us to predict some of the critical exponents of the growth process. For the case of $(1+1)$ dimensions, the theory predicts that the roughness exponent $α$ is given by $α= ν_{\perp} / ν_{\parallel}$, where $ν_{\perp}$ and $ν_{\parallel}$ are the exponents governing the divergence of perpendicular and parallel correlation lengths of the DP incipient infinite cluster. The theory also predicts that the dynamical exponent $z$ equals the exponent $d_{\rm min}$ characterizing the scaling of the shortest path on a isotropic percolation cluster. The exponent $α$ decreases monotonically with $d$ but does not seem to approach zero for any dimension calculated ($d \le 6$), suggesting that the DPD model has no upper critical dimension for the static exponents. On the other hand, $z$ appears to approach $2$ as $d \rightarrow 6$, as expected by the result $z = d_{\rm min}$, suggesting that $d_c = 6$ for the dynamics. We also perform a set of imbibition experiments, in both $(1+1)$ and $(2+1)$ dimensions, that can be used to test the DPD model. We find good agreement between experimental, theoretical and numerical approaches. Further, we study the properties of avalanches in the context of the DPD model. We relate the scaling properties of the avalanches in the DPD model to the scaling properties for the self-organized depinning (SOD) model, a variant of the DPD model. We calculate the exponent characterizing the avalanches distribution $τ_{\rm aval}$ for $d = 1$ to $d = 6$, and compare our results with recent theoretical

cond-mat↗

arXiv1994

Dynamics of Surface Roughening with Quenched Disorder

We study the dynamical exponent $z$ for the directed percolation depinning (DPD) class of models for surface roughening in the presence of quenched disorder. We argue that $z$ for $(d+1)$ dimensions is equal to the exponent $d_{\rm min}$ characterizing the shortest path between two sites in an isotropic percolation cluster in $d$ dimensions. To test the argument, we perform simulations and calculate $z$ for DPD, and $d_{\rm min}$ for percolation, from $d = 1$ to $d = 6$.

cond-mat↗