Intersections of complex networks in real world and infinite particle systems Ⅱ

Reference No.	20210005
Type/Category	Grant for Young Researchers-Short-term Joint Research
Title of Research Project	Intersections of complex networks in real world and infinite particle systems Ⅱ
Principal Investigator	Shota Osada（Institute of Mathematics for Industry, Kyushu university・Postdoctoral Researcher）
Research Period	August 19,2021. ～ August 19,2021. November 22,2021. ～ November 26,2021. February 15,2022. ～ February 20,2022.
Keyword(s) of Research Fields	Complex networks, scale-freeness, small world, large scale interacting systems
Abstract for Research Report	In this research, we consider large-scale interacting particle systems on complex networks and multi-layered graphs to mathematically understand various transport phenomena in the real world. In particular, optimization problems for a transportation cost on networks and graphs. Since it is expected that singular phenomena occurs in particle systems on spaces with special structures, we believe that modelings in this research is useful to elucidate nontrivial phenomena in the real world. The connections between people in the real world are modeled by networks (mathematical graphs), and the logistics of goods and the transmission of information can be regarded as motion of particles on a network. In general, it is believed that communication networks are (1) small-world (the average distance in the graph between two vertices is short, and the clustering coefficient is large) and (2) scale-free (the degree distribution is heavy-tailed). Graphs with these features have a different aspect from the square lattice or Erd?s?Renyi random graphs, which are dealt with for a long time in mathematics. It is called complex networks. To consider diffusion processes on complex networks, we will generalize the mathematical method due to square-integrable energies so far and choose some proper functional space characterizing the complex networks. We aim to establish new mathematical methods and results by the generalization. The analysis of multi-layered graphs are currently being studied intensively for applications to deep learning models and artificial generation of data. It is known that such a graph is naturally viewed as a statistical mechanics model called the Ising model, and we expect to obtain rigorous results for multi-layered graphs through the analysis of the Ising models. Specifically, we consider the learning process in multi-layered graphs as the learning process of Gibbs distribution corresponding to the Ising model, and then analyze the latter by formulating it as a gradient flow on a metric space consisting of probability measures on the set of configurations. The objective of our research is to prove the convergence to the global optimum solution after a long time through the space-time scaling limit, and to derive the order of the convergence time. In addition, as a statistical mechanics interest, we aim to derive the exact equation of evolution corresponding to the learning process to understand how to converge to the global optimum solution.
Organizing Committee Members (Workshop) Participants (Short-term Joint Usage)	Hayate Suda（Faculty of Science and Technology, Keio University・Researcher） Yuta Arai（Graduate school of Science and Engineering, Chiba university・PhD student） Shota Osada（Institute of Mathematics for Industry, Kyushu university・Postdoctoral Researcher） Kohei Hayashi（Graduate school of mathematical sciences, The university of Tokyo・PhD student） Takahiro Mori（Research Institute for Mathematical Sciences, Kyoto University・PhD student） Yoshinori Kamijima（Department of mathematics, Graduate school of science, Hokkaido university・PhD student） Satoshi Hayakawa（Mathematical Institute, University of Oxford・DPhil student）
Adviser	Takayuki Osogami (IBM Japan Ltd)