The study of motifs in networks can help researchers uncover links between structure and function of networks in biology, the sociology, economics, and many other areas. Empirical studies of networks have identified feedback loops, feedforward loops, and several other small structures as "motifs" that occur frequently in real-world networks and may contribute by various mechanisms to important functions these systems. However, the mechanisms are unknown for many of these motifs. We propose to distinguish between "structure motifs" (i.e., graphlets) in networks and "process motifs" (which we define as structured sets of walks) on networks and consider process motifs as building blocks of processes on networks. Using the covariances and correlations in a multivariate Ornstein-Uhlenbeck process on a network as examples, we demonstrate that the distinction between structure motifs and process motifs makes it possible to gain quantitative insights into mechanisms that contribute to important functions of dynamical systems on networks.
A major challenge for causal inference from time-series data is the trade-off between computational feasibility and accuracy. Motivated by process motifs for lagged covariance in an autoregressive model with slow mean-reversion, we propose to infer networks of causal relations via pairwise edge measure (PEMs) that one can easily compute from lagged correlation matrices. Motivated by contributions of process motifs to covariance and lagged variance, we formulate two PEMs that correct for confounding factors and for reverse causation. To demonstrate the performance of our PEMs, we consider network interference from simulations of linear stochastic processes, and we show that our proposed PEMs can infer networks accurately and efficiently. Specifically, for autocorrelated time-series data, our approach achieves accuracies higher than or similar to Granger causality, transfer entropy, and convergent crossmapping-but with much shorter computation time than possible with any of these methods. Our fast and accurate PEMs are easy-to-implement methods for network inference with a clear theoretical underpinning. They provide promising alternatives to current paradigms for the inference of linear models from time-series data, including Granger causality, vector-autoregression, and sparse inverse covariance estimation.
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