In (Williams and Beer, 2010), they define the partial information decomposition (PID) as a generalization of Shannon's Mutual Information for multiple information sources. Their key insight is that information sources can jointly provide unique information, redundant information, and synergistic information about a result. To systematize this insight they define a lattice of information sources, from which the partial information decomposition is derived. Roughly, the partial information of a set of sources for a given result is the Möbius inversion of the minimum information that any source provides about each outcome. This framework has been useful for example in showing why some informations can take on negative values and challenges in interpreting transfer entropies (James et al. 2016)
In a separate stream of research, Carmi and Moskovich have developed Tangle Machines as a topological framework for manipulation of information and computation. In this framework they have considered information fusion networks (IFN), which are abstractly represented as knot-like layouts. Information is stored as colors on each thread and when one thread goes over another it updates the information of the lower thread. They define axioms for the updates that allow for an equivalence of diagrams connected via a sequence of Reidemeister moves to convert one diagram to another without changing in the information content.
How do these two programs relate?
It seems that the axioms of the IFN is more restrictive than in the PID setting--e.g. requiring interactions to be dyadic and having a certain group like structure what they call a quandloid. However, perhaps it is possible to relax axioms slightly to coincide with requirements for PID?
