Preferential attachment and growth dynamics in complex systems

Complex systems can be characterized by classes of equivalency of their elements defined according to system specific rules. We propose a generalized preferential attachment model to describe the class size distribution. The model postulates preferential growth of the existing classes and the steady influx of new classes. According to the model, the distribution changes from a pure exponential form for zero influx of new classes to a power law with an exponential cut-off form when the influx of new classes is substantial. Predictions of the model are tested through the analysis of a unique industrial database, which covers both elementary units (products) and classes (markets, firms) in a given industry (pharmaceuticals), covering the entire size distribution. The model’s predictions are in good agreement with the data. The paper sheds light on the emergence of the exponent $\tau \approx 2$ observed as a universal feature of many biological, social and economic problems.

Figure 1

(Color online) Empirical results for the cumulative probability distribution, $p\left(k\right)$, of class size $k$ at different levels. (a)–(d) correspond to levels A–D, respectively. Symbols represent data points in each level (a)–(d), while dashed lines are predictions of the model. The cumulative probability distributions for all levels are reasonably well fit by pure exponentials, as predicted by the model.

Figure 2

(Color online) Comparison of empirical results for firms of the PHID and the simulation results. (a) and (c) Log-log plots of the cumulative probability distribution of the class sizes show a power-law decay $k^{-(\tau -1)}$ with $\tau \approx 2$ for $k<200$. The fit (dashed line) in (a) and (b) is obtained by numerical calculation of Eq. (4) with $b=0.1$, $t∕N=200$, $\lambda =0.77$, and $\mu =0.33$. (b) and (d) Log-linear plots of the cumulative probability distribution, showing exponential decay for $k>200$. In (c) and (d), $엯$, $◻$, and $▵$ show the distribution for $t=200000$, $20000$, and $2000$, respectively. Note that the exponential function gradually changes into a power-law function.