We give a general existence and convergence result for interacting particle systems on locally finite graphs with possibly unbounded degrees or jump rates. We allow the local state space to be Polish, and the jumps at a site to affect the states of its neighbours. The two common assumptions on interacting particle systems are uniform bounds on degrees and jump rates. In this paper, we relax these assumptions and allow for vertices with high degrees or rapid jumps. We introduce new assumptions ensuring that such vertices are placed sufficiently apart from each other and hence the process does not blow up. Our assumptions involve finitude of certain weighted connective constants on the square graph of the underlying graph and our proofs proceed by showing that these assumptions imply non-percolation of the Poisson graphical construction. For some random graph models, we give practically verifiable sufficient conditions under which our assumptions hold almost surely. These conditions involve exponential growth of a fractional moment sum of probabilities of self-avoiding walks from each vertex and that of product moments of fixed powers of jump rates. Using these conditions, we show the existence of interacting particle systems with possibly unbounded jump rates like contact processes, consensus formation models, evolutionary models, etc., on random graphs which can lack uniform bounds on degrees almost surely, e.g., long-range percolations on quasi-transitive graphs, and geometric random graphs on Delone sets.