Modes of Information Flow

Information flow between components of a system takes many forms and is key to understanding the organization and functioning of large-scale, complex systems. We demonstrate three modalities of information flow from time series X to time series Y. Intrinsic information flow exists when the past of X is individually predictive of the present of Y, independent of Y’s past; this is most commonly considered information flow. Shared information flow exists when X’s past is predictive of Y’s present in the same manner as Y’s past; this occurs due to synchronization or common driving, for example. Finally, synergistic information flow occurs when neither X’s nor Y’s pasts are predictive of Y’s present on their own, but taken together they are. The two most broadly-employed information-theoretic methods of quantifying information flow—time-delayed mutual information and transfer entropy—are both sensitive to a pair of these modalities: time-delayed mutual information to both intrinsic and shared flow, and transfer entropy to both intrinsic and synergistic flow. To quantify each mode individually we introduce our cryptographic flow ansatz, positing that intrinsic flow is synonymous with secret key agreement between X and Y. Since the secret key agreement rate is not directly computable, we estimate intrinsic flow by an easily-computed upper bound on it—the intrinsic mutual information—so that the resulting three-way split is relative to this bound. We use it to quantify the three flow modalities in a variety of systems including asymmetric flows, coupled chaotic dynamical systems, leader-follower collective dynamics, and financial markets.