Algebraic bounds for heterogeneous site percolation on directed and undirected graphs

We analyze heterogeneous site percolation and the corresponding uniqueness transition on directed and undirected graphs, by constructing upper bounds on the in-/out-cluster susceptibilities and the vertex connectivity function. We give separate bounds on finite and infinite (di)graphs, and analyze the convergence in the infinite graph limit. We also discuss the transition associated with proliferation of self-avoiding cycles in relation to the uniqueness transition. The bounds are formulated in terms of the spectral radius and norms of the appropriately weighted adjacency and non-backtracking (Hashimoto) matrices associated with the graphs.