We develop a theoretical framework for defining and identifying flows of information in computational systems. Here, a computational system is assumed to be a directed graph, with "clocked" nodes that send transmissions to each other along the edges of the graph at discrete points in time. A few measures of information flow have been proposed previously in the literature, and measures of directed causal influence are currently being used as a heuristic proxy for information flow. However, there is as yet no rigorous treatment of the problem with formal definitions and clearly stated assumptions, and the process of defining information flow is often conflated with the problem of estimating it. In this work, we provide a new information-theoretic definition for information flow in a computational system, which we motivate using a series of examples. We then show that this definition satisfies intuitively desirable properties, including the existence of "information paths", along which information flows from the input of the computational system to its output . Finally, we describe how information flow might be estimated in a noiseless setting, and provide an algorithm to identify information paths on the time-unrolled graph of a computational system.
We develop a theoretical framework for defining and identifying flows of information in computational systems. Here, a computational system is assumed to be a directed graph, with "clocked" nodes that send transmissions to each other along the edges of the graph at discrete points in time. We are interested in a definition that captures the dynamic flow of information about a specific message, and which guarantees an unbroken "information path" between appropriately defined inputs and outputs in the directed graph. Prior measures, including those based on Granger Causality and Directed Information, fail to provide clear assumptions and guarantees about when they correctly reflect information flow about a message. We take a systematic approach---iterating through candidate definitions and counterexamples---to arrive at a definition for information flow that is based on conditional mutual information, and which satisfies desirable properties, including the existence of information paths . Finally, we describe how information flow might be detected in a noiseless setting, and provide an algorithm to identify information paths on the time-unrolled graph of a computational system.
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A few measures of information flow have been proposed previously in the literature, and measures of directed causal influence are currently being used as a heuristic proxy for information flow. However, there is as yet no rigorous treatment of the problem with formal definitions and clearly stated assumptions, and the process of defining information flow is often conflated with the problem of estimating it. In this work, we provide a new information-theoretic
We are interested in a definition that captures the dynamic flow of information about a specific message, and which guarantees an unbroken "information path" between appropriately defined inputs and outputs in the directed graph. Prior measures, including those based on Granger Causality and Directed Information, fail to provide clear assumptions and guarantees about when they correctly reflect information flow about a message. We take a systematic approach---iterating through candidate definitions and counterexamples---to arrive at a
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in a computational system, which we motivate using a series of examples. We then show that this definition satisfies intuitively
that is based on conditional mutual information, and which satisfies
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"information paths", along which information flows from the input of the computational system to its output
information paths
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estimated
detected