Structural Information and Dynamical Complexity of Networks

In 1953, Shannon proposed the question of quantification of structural information to analyze communication systems. The question has become one of the longest great challenges in information science and computer science. Here, we propose the first metric for structural information. Given a graph <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, we define the <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>-dimensional structural information of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> (or structure entropy of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>), denoted by <inline-formula> <tex-math notation="LaTeX">$\mathcal {H}^{K}(G)$ </tex-math></inline-formula>, to be the minimum overall number of bits required to determine the <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>-dimensional code of the node that is accessible from random walk in <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>. The <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>-dimensional structural information provides the principle for completely detecting the natural or true structure, which consists of the rules, regulations, and orders of the graphs, for fully distinguishing the order from disorder in structured noisy data, and for analyzing communication systems, solving the Shannon&#x2019;s problem and opening up new directions. The <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>-dimensional structural information is also the first metric of dynamical complexity of networks, measuring the complexity of interactions, communications, operations, and even evolution of networks. The metric satisfies a number of fundamental properties, including additivity, locality, robustness, local and incremental computability, and so on. We establish the fundamental theorems of the one- and two-dimensional structural information of networks, including both lower and upper bounds of the metrics of classic data structures, general graphs, the networks of models, and the networks of natural evolution. We propose algorithms to approximate the <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>-dimensional structural information of graphs by finding the <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>-dimensional structure of the graphs that minimizes the <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>-dimensional structure entropy. We find that the <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula>-dimensional structure entropy minimization is the principle for detecting the natural or true structures in real-world networks. Consequently, our structural information provides the foundation for knowledge discovering from noisy data. We establish a black hole principle by using the two-dimensional structure information of graphs. We propose the natural rank of locally listing algorithms by the structure entropy minimization principle, providing the basis for a next-generation search engine.