Two stage approach to functional network reconstruction for binary time-series

Abstract

The deduction of network connectivity from the observed node dynamics is costly in large networks. The theoretical number of possible networks containing N nodes connected by binary links grows exponentially with N square. This problem is often termed “the curse of dimensionality”. In practice, unfeasible long time-series and a high computational cost are required to detect the connectivity of a network from its observations. Given the large number of time-series currently assembled in all domains of science, a solution to this inverse problem in large networks is required. We here propose a solution to the inverse problem in large networks of binary variables through a redefinition of the problem. Instead of attempting to deduce the links of a network, we redefine the problem into the prediction of future dynamics. Specifically, we show that links between nodes can be divided into links affecting the future dynamics and links that do not. We further show that hard-to-predict links belong to the second group, and as such can be ignored when predicting future dynamics. This division is applied through a two stage algorithm. In the first stage, the vast majority of potential links (pairs of nodes) is removed, since even if they exist they do not affect the dynamics. At the second stage, a rapid high-precision estimate of the predictable links is performed using a modified partial correlation algorithm. A good predictor for the classification of potential links is the mutual information between a node-pair. Similarly, some nodes have practically no variability and as such have practically no effect on the dynamics of other nodes. The links to and from such nodes are hard to predict. We show that a two stage algorithm can be applied to these nodes with similar results. This methodology does not reproduce the network that originally induced the dynamics, but its prediction of future dynamics is similar to the one of the real network. The current analysis is limited to reconstruction using partial correlation methods. However, the same principle can be applied to other reconstruction methods.