Formalism for information transfer in dynamical network

In this paper, we discover a novel approach for defining information transfer in a linear network dynamical system. We provide entropy based characterization of the information transfer where the information transfer from state x to state y is measured by the amount of entropy/uncertainty that is transferred from state x to y over one time step. Our proposed definition of information transfer is based on three axioms. The first axiom has to do with zero information transfer, which says that if state x is not connected (or appears) in the dynamics of y then information transfer from x → y is zero. The second axiom captures the asymmetric nature of information transfer i.e., x is not connected to the dynamics of y but y is connected to the dynamics of x then information transfer from x → y is zero but the transfer from y → x is not zero. The third axiom is on information conservation. Information conservation axiom says that if y space can be split into two subspace, y1 and y2, then the information transfer from x → y will be equal to the sum of the information transfers from x → y1 and x → y2 provided y1 and y2 are “dynamical independent”. Similar conservation property also applies for the case where x is split into two parts x1 and x2 with y intact. We provide an analytical expression for information transfer satisfying these three axioms. Preliminary results are provided for identifying information-based most influential nodes and clusters in network system with small world network topology.