Fundamental interactions in self-organised critical dynamics on higher order networks

In functionally complex systems, higher order connectivity is often revealed in the underlying geometry of networked units. Furthermore, such systems often show signatures of self-organised criticality, a specific type of non-equilibrium collective behaviour associated with an attractor of internal dynamics with long-range correlations and scale invariance, which ensures the robust functioning of complex systems, such as the brain. Here, we highlight the intertwining of features of higher order geometry and self-organised critical dynamics as a plausible mechanism for the emergence of new properties on a larger scale, representing the central paradigm of the physical notion of complexity. Considering the time-scale of the structural evolution with the known separation of the time-scale in self-organised criticality, i.e., internal dynamics and external driving, we distinguish three classes of geometries that can shape the self-organised dynamics on them differently. We provide an overview of current trends in the study of collective dynamics phenomena, such as the synchronisation of phase oscillators and discrete spin dynamics with higher order couplings embedded in the faces of simplicial complexes. For a representative example of self-organised critical behaviour induced by higher order structures, we present a more detailed analysis of the dynamics of field-driven spin reversal on the hysteresis loops in simplicial complexes composed of triangles. These numerical results suggest that two fundamental interactions representing the edge-embedded and triangle-embedded couplings must be taken into account in theoretical models to describe the influence of higher order geometry on critical dynamics. Graphical abstract