Fig 6. Causal powers analysis of various network architectures.
Each panel shows the network’s causal model and weights on the left. Blue regions indicate complexes with their respective φs values. In all networks, k = 4 and the state is Abcdef. The Φ-structure(s) specified by the network’s complexes are illustrated to the right (with only second- and third-degree relation faces depicted) with a list of their distinctions for smaller systems and their ∑φ values for those systems with many distinctions and relations. All integrated information values are in ibits. (A) A degenerate network in which unit A forms a bottleneck with redundant inputs from and outputs to the remaining units. The first-maximal complex is Ab, which excludes all other subsets with φs > 0 except for the individual units c, d, e, and f. (B) The modular network condenses into three complexes along its fault lines (which exclude all subsets and supersets), each with a maximal φs value, but low Φ, as the modules each specify only two or three distinctions and at most five relations. (C) A directed cycle of six units forms a six-unit complex with φs = 1.74 ibits, as no other subset is integrated. However, the Φ-structure of the directed cycle is composed of only first-order distinctions and few relations. (D) A specialized lattice also forms a complex (which excludes all subsets), but specifies 27 first- and high-order distinctions, with many relations (>1.5 × 106) among them. Its Φ value is 11452 ibits. (E) A slightly modified version of the specialized lattice in which the first-maximal complex is Abef. The full system is not maximally irreducible and is excluded as a complex, despite its positive φs value (indicated in gray).