We investigated the possibility of estimating network link weights from the multivariate time series of phase oscillators on a complex network. The inverse phase synchronization index of the coupled oscillator network is found to grow in proportion to the corresponding link weight, as network synchronization occurs for a strong coupling strength. This implies that the network link weights can be estimated from the measurement of the inverse phase synchronization indices. By adopting this estimation method, we successfully reconstructed the minimal spanning tree of the original network from the inverse phase synchronization indices. Even for the weak coupling case, the estimation of the network link weights could be improved significantly by taking the average of a sufficiently large number of configurations.

-A Physics textbook is composed of numerous physics terminologies which are organized to form new concepts of physics. For the purpose of understanding the global organization of the science-knowledge system in science textbooks, we analyzed the binary network of science textbooks, where the pair appearance of two science terminologies is taken as the link between two nodes. By analyzing the characteristics of binary network of the textbook, we can be understood the structure characteristics of physics knowledge in the textbook. We found that a physics knowledge network possesses the characteristics of complex network: 1) short mean distance between terminologies; 2) power law degree distribution; 3) hierarchical modular structure.

We propose a method of estimating inter-modular connectivity in a hierarchical modular network. The method is based on an analysis of inverse phase synchronization applied to the local field potentials on a hierarchical modular network of phase oscillators. For a strongcoupling strength, the inverse phase synchronization index of the local field potentials for two modules depends linearly on the corresponding inter-modular connectivity defined as the number of links connecting the modules. The method might enable us to estimate the inter-modular connectivity in various complex systems from the inverse phase synchronization index of the mesoscopic modular activities.

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