  Modern theories of phase transitions and scale-invariance are rooted in path
integral formulation and renormalization group (RG). Despite the applicability
of these approaches on simple systems with only pairwise interactions, they are
less effective on complex systems with un-decomposable high-order interactions
(i.e., interactions among arbitrary sets of units). To precisely characterize
the universality of high-order interacting systems, we propose simplex path
integral and simplex renormalization group (SRG) as the generalizations of
classic approaches to arbitrary high-order and heterogeneous interactions. We
first formalize the trajectories of units governed by high-order interactions
to define path integrals on corresponding simplices based on a high-order
propagator. Then we develop a method to integrate out short-range high-order
interactions in the momentum space, accompanied by a coarse graining procedure
functioning on the simplex structure generated by high-order interactions. The
proposed SRG, equipped with a divide-and-conquer framework, can deal with the
absence of ergodicity arised from the sparse distribution of high-order
interactions and renormalize a system with intertwined high-order interactions
on the $p$-order according to its properties on the $q$-order ($p\leq q$). The
associated scaling relation and its corollaries support to differentiate among
scale-invariant, weakly scale-invariant, and scale-dependent systems across
different orders. We have validated our theory in multi-order scale-invariance
verification, topological invariance discovery, organizational structure
identification, and information bottleneck analysis. These experiments
demonstrate the capacity of our theory for identifying intrinsic statistical
and topological properties of high-order interacting systems during system
reduction.
