Shape of complexity

Abstract

Failure in complex engineered systems does not occur, as a simple or linear accumulation of independent component faults. Their failure modes are often relational: degradation propagates through feedback loops, coupling pathways, and many-body interactions among sensors, controllers, and ac tuators. This creates a gap for prognostics and health man agement, where many established approaches still interpret system health primarily through single-channel indicators or pairwise summaries. This paper argues for a broader PHM perspective in which system health is read from the structure of interactions rather than from isolated signals alone. Simplicial complexes pro vide a natural representation for this purpose because they encode both pairwise and higher-order relations, while topo logical descriptors such as Betti numbers compress that re lational structure into interpretable, threshold-robust signa tures. Within this perspective, we use an interconnectivity pipeline based on mutual information, temporal lag, coupling modal ity, and O-information as one concrete example of how mul tichannel data can be converted into a simplicial complex and analysed topologically. We validate and tune our method using an analytic toy model in which connectivity between components can be controlled. As part of this, we discuss what different interaction measures can and can not recover. A double-loop controller motor experiment then illustrates the PHM value of the approach: edge density, mean edge strength, and persistent loop structure vary systematically across fault conditions even when no single signal provides an equally clear separation. Together these results provide evidence that relational and topological descriptions can extend PHM be yond the single-signal view of system health.