Graph neural networks for integrated information and major complex estimation

This study investigates the potential of graph neural networks (GNNs) for estimating system-level integrated information and major complex in integrated information theory (IIT) 3.0. Owing to the hierarchical complexity of IIT 3.0, calculating the integrated information and identifying the major complex are computationally prohibitive for large systems. To overcome this difficulty, we propose a GNN model with transformer convolutions characterized by multi-head attention mechanisms for estimating the major complex and its integrated information. For evaluation, we begin by obtaining exact solutions for integrated information and major complexes in systems with 5, 6, and 7 nodes, and conduct two experiments: (1) a non-extrapolative setting in which the model is trained and tested on a mixture of systems with 5, 6, and 7 nodes, and (2) an extrapolative setting in which systems with 5 and 6 nodes are used for training and systems with 7 nodes are used for testing. We then examine the scaling behavior for tree-like, fully connected, and loop-containing graph topologies in larger systems. Although accurate estimation is difficult, our approximate estimates for larger systems generally preserve the qualitative patterns of integrated information and major complex size that are observed in small systems. Finally, based on this observation, we qualitatively analyze a split-brain-like system of 100 nodes. The system consists of two weakly coupled subsystems of 50 nodes each, representing a structurally meaningful, brain-inspired configuration. When the connectivity between the subsystems is low, "local integration" emerges, and a single subsystem forms a major complex. As the connectivity increases, local integration rapidly disappears, and the integrated information gradually rises toward "global integration," in which a large portion of the entire system forms a major complex. Our analysis suggests that the proposed GNN-based framework provides a practical approach to qualitative analysis of integrated information and major complexes in large systems.