Transfer entropy for network reconstruction in a simple dynamical - PowerPoint PPT Presentation

Transfer entropy for network reconstruction in a simple dynamical model Roy Goodman NJIT Dept. of Mathematical Sciences

How I Spent My Sabbatical My location My commute My host: Mau Por fi ri

The Por fi ri Lab A big group working in a lot of areas, both theoretical and through laboratory experiments: • Fluid mechanics: fl uid structure interactions during water impact • Arti fi cial Muscles and Soft Robotics • Telerehabilitation • Network-based modeling of infectious diseases • Fish schooling • Using robotics and zebra fi sh to study substance-abuse disorders • Information-theoretic analysis of social science datasets • more… A unifying theme in this work is using methods from information theory for modeling and analysis

What is this talk about? A lot of words to de fi ne here: Transfer entropy for network reconstruction in a simple dynamical model • Network: a graph composed of vertices and edges, the subject at the heart of graph theory • Transfer entropy: a quantity describing the transfer of information from one evolving variable to another, from information theory • The dynamical model: to be described, a simple probabilistic dynamics. • Reconstruction: fi gure out properties of the graph based on the dynamics 18 months ago I knew 𝞋 about any of this

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<latexit sha1_base64="h0CN0sprwUAmbI4BC5SLR0I3FA=">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</latexit> <latexit sha1_base64="+LxPrbxaQrwLNIb/AbHZefKoL54=">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</latexit> Information Theory • Initiated by Claude Shannon’s 1948 “A Mathematical Theory of Communication” • Originally used to study the transmission of signals down noisy channels and to develop optimal strategies for encoding information • Recently become popular tool for analyzing dynamical systems Fundamental quantity: Consider a discrete random variable X drawn from a sample space X The information associated with the event X=x measures how “surprising” it is that X=x I ( x ) = − log ( Pr ( X = x )) The Shannon entropy of the random variable is the expectation value fo the information H ( X ) = E [ I ( X )] = − P x ∈ X Pr ( X = x ) log Pr ( X = x )

Basic example: biased coin toss Consider a biased coin that gives heads with probability p and tails with probability 1 - p • When p≈0 or p≈1 , entropy small since surprising outcomes rarely occur • When p≈0.5 , entropy large since both outcomes equally likely

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Contrived transfer entropy example Source: A Tutorial for Information Theory in Neuroscience Nicholas M. Timme and Christopher Lapish 2018 •Series 1: X & Y uncorrelated: TE ≈ 0 •Series 2 & 3: X tends to fi re before Y: TE large •Series 4: X(t+1) determined entirely from X(t) , no improvement from knowing Y(t): TE=0