复杂网络中的可持续增长
Sustainable growth in complex networks
摘要：
基于对18个Java项目中的依赖网络的实证分析，我们开发了一个新的网络增长模型，它考虑了两个方面：一个附件机制和添加新的节点与它们的初始度的异构分布，$k_0$。经验上，我们发现初始度和最终网络的累积度分布分别遵循幂律行为：$P(k_{0}) \propto k_{0}^{1-α}$和$P(k)\propto k^{1-γ}$。对于作为网络大小的函数的链路总数，我们根据经验发现$K(N)\propto N^β$，其中$β$（在网络演进的开始）在1.25和2之间，而对于大的$N$收敛到$\sim 1$。这表明，从增长制度与网络密度不断增加的过渡到一个可持续的制度，防止崩溃，因为不断增加的依赖。我们的理论框架是能够预测的指数$α$，$β$，$γ$，这也链接软件工程和开发活动的问题之间的关系。这些关系通过计算机模拟和实证研究得到了验证。他们指出，真正的开源软件网络的增长发生在两个政权之间的边缘，这是由添加节点的初始度分布，或由优先连接机制。因此，新增加的节点的异质度分布，经验发现，是必不可少的描述网络的可持续增长的规律。
英文摘要：
Based on the empirical analysis of the dependency network in 18 Java projects, we develop a novel model of network growth which considers both: an attachment mechanism and the addition of new nodes with a heterogeneous distribution of their initial degree, $k_0$. Empirically we find that the cumulative degree distributions of initial degrees and of the final network, follow power-law behaviors: $P(k_{0}) \propto k_{0}^{1-α}$, and $P(k)\propto k^{1-γ}$, respectively. For the total number of links as a function of the network size, we find empirically $K(N)\propto N^β$, where $β$ is (at the beginning of the network evolution) between 1.25 and 2, while converging to $\sim 1$ for large $N$. This indicates a transition from a growth regime with increasing network density towards a sustainable regime, which revents a collapse because of ever increasing dependencies. Our theoretical framework is able to predict relations between the exponents $α$, $β$, $γ$, which also link issues of software engineering and developer activity. These relations are verified by means of computer simulations and empirical investigations. They indicate that the growth of real Open Source Software networks occurs on the edge between two regimes, which are either dominated by the initial degree distribution of added nodes, or by the preferential attachment mechanism. Hence, the heterogeneous degree distribution of newly added nodes, found empirically, is essential to describe the laws of sustainable growth in networks.
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