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Motivated by the prospect of attaining Majorana modes at the ends ofnanowires, we analyze interacting Majorana systems on general networks andlattices in an arbitrary number of dimensions, and derive various universalspin duals. Such general complex Majorana architectures (other than those ofsimple square or other crystalline arrangements) might be of empiricalrelevance. As these systems display low-dimensional symmetries, they arecandidates for realizing topological quantum order. We prove that (a) theseMajorana systems, (b) quantum Ising gauge theories, and (c) transverse-fieldIsing models with annealed bimodal disorder are all dual to one another ongeneral graphs. As any Dirac fermion (including electronic) operator can beexpressed as a linear combination of two Majorana fermion operators, ourresults further lead to dualities between interacting Dirac fermionic systems.The spin duals allow us to predict the feasibility of various standardtransitions as well as spin-glass type behavior in {\it interacting} Majoranafermion or electronic systems. Several new systems that can be simulated byarrays of Majorana wires are further introduced and investigated: (1) the {\itXXZ honeycomb compass} model (intermediate between the classical Ising model onthe honeycomb lattice and Kitaev's honeycomb model), (2) a checkerboard latticerealization of the model of Xu and Moore for superconducting $(p+ip)$ arrays,and a (3) compass type two-flavor Hubbard model with both pairing and hoppingterms. By the use of dualities, we show that all of these systems lie in the 3DIsing universality class. We discuss how the existence of topological ordersand bounds on autocorrelation times can be inferred by the use of symmetriesand also propose to engineer {\it quantum simulators} out of these Majorananetworks.

Arbitrary Dimensional Majorana Dualities and Network Architectures for Topological Matter