We introduce a statistical system on random networks of trivalent vertices for the purpose of studying the canonical tensor model, which is a rank-three tensor model in the canonical formalism. The partition function of the statistical system has a concise expression in terms of integrals, and has the same symmetries as the kinematical ones of the canonical tensor model. We consider the simplest non-trivial case of the statistical system corresponding to the Ising model on random networks, and find that its phase diagram agrees with what is implied by regrading the Hamiltonian vector field of the canonical tensor model with N=2 as a renormalization group flow. Along the way, we obtain an explicit exact expression of the free energy of the Ising model on random networks in the thermodynamic limit by the Laplace method. This paper provides a new example connecting a model of quantum gravity and a random statistical system.