  Recent studies of networks representing complex systems from the brain to
social graphs have revealed their higher-order architecture, which can be
described by aggregates of simplexes (triangles, tetrahedrons, and higher
cliques). Current research aims at quantifying these hidden geometries by the
algebraic topology methods and deep graph theory and understanding the dynamic
processes on simplicial complexes. Here, we use the recently introduced model
for geometrical self-assembly of cliques to grow nano-networks of triangles and
study the filed-driven spin reversal processes on them. With the
antiferromagnetic interactions between the Ising spins attached to the nodes,
this assembly ideally supports the geometric frustration, which is recognized
as the origin of some new phenomena in condensed matter physics. In the
dynamical model, a gradual switching from the pairwise to triangle-based
interactions is controlled by a parameter. Thus, the spin frustration effects
on each triangle give way to the mesoscopic ordering conditioned by a complex
arrangement of triangles. We show how the balance between these interactions
changes the shape of the hysteresis loop. Meanwhile, the fluctuations in the
accompanying Barkhausen noise exhibit robust indicators of self-organized
criticality, which is induced by the network geometry alone without any
magnetic disorder.
