Connectivity inference with asynchronously updated kinetic Ising models

Abstract

This thesis focuses on the inference of network connections from statistical physics point of view. The reconstruction methods of the asynchronously updated kinetic Ising model with an asymmetric Sherrington-Kirkpatrick (SK) model is studied theoretically. Both approximate and exact learning rules for the couplings from the generated dynamical data are developed. The approximate formulae are based on naive mean field (nMF) and Thouless-Anderson-Palmer (TAP) equations respectively. The exact learning rules are derived for two cases: one in which both the spin history and the update times are known and one in which only the spin history. One can average over all possible choices of update times to obtain an averaged learning rule that depends only on spin correlations. We studied all the learning rules numerically. Good convergence is observed in accordance with the theoretical expectations.

The developed inference learning rules are applied to two data sets. One is spike trains recorded from 20 retinal ganglion cells and the other is generated by transactions of 100 highly traded stocks on the New York Stock Exchange (NYSE).

For the neuron data set, we compared the inferred asynchronous couplings with the equilibrium ones. The results show that the inferred couplings from these two models are very similar. This implies that real dynamical process of the neuron system satisfies the Gibbs equilibrium conditions and that the final distribution of states is the Gibbs stationary distribution.

For the financial data set, three inference methods are applied to reconstruct the coupling matrices between traded stocks. They are equilibrium, synchronous and asynchronous inference formula respectively. All of them are based on mean-field approximation. Synchronous and asynchronous Ising inference methods give results which are coherent with equilibrium case, but more detailed since the obtained interaction networks are directed.