Lexical growth, entropy and the benefits of networking (0803.1443v4)

Abstract: If each node of an idealized network has an equal capacity to efficiently exchange benefits, then the network's capacity to use energy is scaled by the average amount of energy required to connect any two of its nodes. The scaling factor equals \textit{e}, and the network's entropy is $\ln(n)$. Networking emerges in consequence of nodes minimizing the ratio of their energy use to the benefits obtained for such use, and their connectability. Networking leads to nested hierarchical clustering, which multiplies a network's capacity to use its energy to benefit its nodes. Network entropy multiplies a node's capacity. For a real network in which the nodes have the capacity to exchange benefits, network entropy may be estimated as $C \log_L(n)$, where the base of the log is the path length $L$, and $C$ is the clustering coefficient. Since $n$, $L$ and $C$ can be calculated for real networks, network entropy for real networks can be calculated and can reveal aspects of emergence and also of economic, biological, conceptual and other networks, such as the relationship between rates of lexical growth and divergence, and the economic benefit of adding customers to a commercial communications network. \textit{Entropy dating} can help estimate the age of network processes, such as the growth of hierarchical society and of language.