Composition as an Evolving Entity envisions a work in continuous transformation, never reaching equilibrium, a complex dynamic system whose components permanently fluctuate and adjust to global changes. The process never produces a definitive version, but at any arbitrary point in time provides a plausible variant of the work - a transitory being. Directed Graphs are used to represent the structural levels of any composition (vertices) and the relationships between them (edges). By determining adjacencies and degrees of vertices and introducing weights for edges, one can define affinities and dependencies. Ways in which the all-incidence matrix of a graph with weighted edges can evolve are discussed including the use of Information Theory. The Evolving Composition model is closer to the way composers actually write music and refine their output; it also creates the equivalent of a live organism, growing, developing, and transforming itself over time.