In any physically constructible network there is a structure-function relationship between the geometry of the network and its dynamics. The network's geometry, i.e. its physical structure, constrains and bounds signaling and the flow of information through the network. In this paper we explore how the physical geometry of a network constrains and ultimately determines its dynamics. We construct a formal theoretical framework of the relationship between network structure and function and how information flows through a network. We show how a strictly local process at the scale of individual node pairs directly affects the behavior of the system at the whole network scale. Individual nodes responding to information from the upstream nodes it is connected to produce the observable emergent dynamics of the entire networkat a global scale, independent of and without any knowledge of what all the other nodes in the network may be doing. We then provide empirical evidence that at least some important examples of both naturally occurring and engineered networks are capable of approaching a state of optimal dynamical efficiency, one of the key results of the theory. While we progressively build up to what this formally means, informally it means that these networks have evolved or have been designed to optimize how they are able to handle the processing of information by matching the dynamical requirements of individual nodes to the flow of information (i.e. signals) between nodes in the network . In different ways, we use the theory to investigate properties of pyramidal neurons in biological neural networks in the visual cortex, the prevalence of the small world network topology, and the internet router network  .
The functional and computational power of a network emerges as a property of the system operating as a coherent whole at a global scale, not at the scale of individual nodes. A natural question to ask is what are the physical principles that allow a collection of nodes to interact in such a way that they produce emergent global network dynamics? There is in general a lack of sufficient theory capable of providing insights into universal physical principles that bridge individual node dynamics to the behavior of the network as a whole in a deterministic way. Here, we show that in any physically constructible geometric network, the geometry of the network constrains and bounds signaling and the flow of information through the network, and develop a theory derived from foundational principles of neural signaling that can both describe and predict the dynamics of geometric networks from considerations of node dynamics. We provide empirical evidence that at least some important examples of widely different naturally occurring and engineered networks , specifically, the prevalence of the small world network topology, axonal branching of pyramidal neurons in the visual cortex, and the internet router network , are capable of approaching a state of optimal dynamical efficiency, a concept that is derived from the theory. The framework we develop has a wide range of (non-biological) engineering and neurophysiological applications .
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In any physically constructible network there is a structure-function relationship between the geometry of the network and its dynamics. The network's geometry, i.e. its physical structure, constrains and bounds signaling and the flow of information through the network. In this paper we explore how the physical geometry of a network constrains and ultimately determines its dynamics. We construct a formal theoretical framework of the relationship between network structure and function and how information flows through a network. We show how a strictly local process at the scale of individual node pairs directly affects
The functional and computational power of a network emerges as a property of the system operating as a coherent whole at a global scale, not at the scale of individual nodes. A natural question to ask is what are the physical principles that allow a collection of nodes to interact in such a way that they produce emergent global network dynamics? There is in general a lack of sufficient theory capable of providing insights into universal physical principles that bridge individual node dynamics to
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system at the whole network scale. Individual nodes responding to information from the upstream nodes it is connected to produce the observable emergent dynamics of the entire networkat a global scale, independent of and without any knowledge of what all the other nodes in the network may be doing. We then
network as a whole in a deterministic way. Here, we show that in any physically constructible geometric network, the geometry of the network constrains and bounds signaling and the flow of information through the network, and develop a theory derived from foundational principles of neural signaling that can both describe and predict the dynamics of geometric networks from considerations of node dynamics. We
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both
widely different
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are capable of approaching a state of optimal dynamical efficiency, one of the key results of the theory. While we progressively build up to what this formally means, informally it means that these networks have evolved or have been designed to optimize how they are able to handle the processing of information by matching the dynamical requirements of individual nodes to the flow of information (i.e. signals) between nodes in the network . In different ways, we use the theory to investigate properties
, specifically, the prevalence of the small world network topology, axonal branching
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biological neural networks in
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the prevalence of the small world network topology,
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, are capable of approaching a state of optimal dynamical efficiency, a concept that is derived from the theory. The framework we develop has a wide range of (non-biological) engineering and neurophysiological applications