On Information Links

TL;DR

This paper generalizes the relationship between multivariate mutual information minima and Borromean links to k-variable cases, linking negative information measures to complex system emergence and topological structures.

Contribution

It extends the correspondence between multivariate interaction information minima and Brunnian links to the k-variable case, connecting information theory with topology.

Findings

Negative k-links define collective emergence in complex systems.

Negativity of K-L divergence indicates local decomposition obstruction.

Generalization of link-information correspondence to k variables.

Abstract

In a joint work with D. Bennequin, we suggested that the (negative) minima of the 3-way multivariate mutual information correspond to Borromean links, paving the way for providing probabilistic analogs of linking numbers. This short note generalizes the correspondence of the minima of k-multivariate interaction information with k Brunnian links in the binary variable case. Following Jakulin and Bratko, the negativity of the associated K-L divergence of the joint probability law with its Kirkwood approximation implies an obstruction to local decomposition into lower order interactions than k, defining a local decomposition inconsistency that reverses Abramsky's contextuality local-global relation. Those negative k-links provide a straightforward definition of collective emergence in complex k-body interacting systems or dataset.