Distributed computing network-systems are modeled as graphs with vertices representing compute elements and adjacency-edges capturing their uni- or bi-directional communication. Distributed function computation has a wide spectrum of major applications in distributed systems. Distributed computation over a network-system proceeds in a sequence of time-steps in which vertices update and/or exchange their values based on the underlying algorithm constrained by the time-(in)variant network-topology. For finite convergence of distributed information dissemination and function computation in the model, we present a lower bound on the number of time-steps for vertices to receive (initial) vertex-values of all vertices regardless of underlying protocol or algorithmics in time-invariant networks via the notion of vertex-eccentricity in a graph-theoretic framework. We also address lower bounds on vertex-eccentricity and its maximum version in terms of common graph-parameters such as maximum degree, and order and size.