The Size Variance Relationship of Business Firm Growth Rates

The relationship between the size and the variance of firm growth rates is known to follow an approximate power-law behavior $sigma(S) sim S^{-beta(S)}$ where $S$ is the firm size and $beta(S)approx 0.2$ is an exponent weakly dependent on $S$. Here we show how a model of proportional growth which treats firms as classes composed of various number of units of variable size, can explain this size-variance dependence. In general, the model predicts that $beta(S)$ must exhibit a crossover from $beta(0)=0$ to $beta(infty)=1/2$. For a realistic set of parameters, $beta(S)$ is approximately constant and can vary in the range from 0.14 to 0.2 depending on the average number of units in the firm. We test the model with a unique industry specific database in which firm sales are given in terms of the sum of the sales of all their products. We find that the model is consistent with the empirically observed size-variance relationship.

Structure of Business Firm Networks and Scale-Free Models

We study the structure of business firm networks and scale-free models with degree distribution $P(q) propto (q+c)^{-lambda}$ using the method of $k$-shell decomposition.We find that the Life Sciences industry network consist of three components: a ``nucleus, which is a small well connected subgraph, ``tendrils, which are small subgraphs consisting of small degree nodes connected exclusively to the nucleus, and a ``bulk body which consists of the majority of nodes. At the same time we do not observe the above structure in the Information and Communication Technology sector of industry. We also conduct a systematic study of these three components in random scale-free networks. Our results suggest that the sizes of the nucleus and the tendrils decrease as $lambda$ increases and disappear for $lambda geq 3$. We compare the $k$-shell structure of random scale-free model networks with two real world business firm networks in the Life Sciences and in the Information and Communication Technology sectors. Our results suggest that the observed behavior of the $k$-shell structure in the two industries is consistent with a recently proposed growth model that assumes the coexistence of both preferential and random agreements in the evolution of industrial networks.