Diffusion of Innovation over Social Networks under Limited-trust Equilibrium

The paper’s Theorem 1 states the closed-form absorption probabilities P_n^y and P_0^y and the expected absorption time τ_y for a 1D birth–death chain on {0,…,n} with absorbing endpoints, in terms of the product-sum quantity Δ, and proves them by explicitly inverting I−Q (Appendix B) . The candidate solution derives exactly the same formulas using the standard birth–death recurrence (scale function) for P_n^y and a discrete Green’s function for τ_y. The results align, and the proof strategies differ but are both valid. The model also makes an assumption (a_i+b_i>0 on the interior) that the paper implicitly uses when calling interior states transient; making it explicit would improve clarity.