Modes of Information Flow
- 1Complexity Sciences Center and Physics Department, University of California at Davis, One Shields Avenue, Davis, CA 95616, USA
- 2Institute of Science and Technology Austria, Maria Gugging, 3400 Klosterneuburg, Austria
Abstract
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.
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BibTeX
@article{james2026modes,
title = {{Modes of Information Flow}},
author = {Ryan G. James and Blanca Daniella Mansante Ayala and Bahti Zakirov and James P. Crutchfield},
journal = {Entropy (Basel, Switzerland)},
year = {2026},
month = oct,
doi = {10.3390/e28101162},
url = {https://europepmc.org/article/PMC/PMC13703659},
note = {CC BY},
}
RIS
TY - JOUR
TI - Modes of Information Flow
AU - Ryan G. James
AU - Blanca Daniella Mansante Ayala
AU - Bahti Zakirov
AU - James P. Crutchfield
PY - 2026
DA - 2026/10/01
JO - Entropy (Basel, Switzerland)
DO - 10.3390/e28101162
UR - https://europepmc.org/article/PMC/PMC13703659
AB - 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.
SN - 1099-4300
ER -