Spatial Heterogeneity of Multivariate Dependence

How do global patterns emerge in complex systems from the underlying interactions or correlation among their composite elements? To what extent do higher order correlations play a significant role in the development of global patterns? In this article, we employ and generalize the recently developed information-theoretic quantity, connected information (Schneidman et al., Phys. Rev. Lett., 91, 238701 (2003)), to quantify the hierarchical multivariate dependence hidden in complex systems. By using as an illustrative example the nonlinear voter model on a two dimensional lattice which shows distinctive spatial patterns in different parameter regions of the system, we demonstrate how different features of the underlying correlations among the elements, such as their orders and strengths, depend on the types of the global patterns, and also how the "microscopic" origin of spatial dynamical heterogeneities can be unveiled in terms of the local decomposition of the connected information. The correspondence of the connected information with various geometrical concepts in the space of probability distribution is also discussed.