Emerging Composition: Being and Becoming envisions a work in continuous transformation, never reaching an equilibrium, a complex dynamic system whose components permanently fluctuate and adjust to global changes. The process never produces a definitive version, but provides at any arbitrary point in time a plausible variant of the work - a transitory being. Directed Graphs are used to represent the structural levels of a composition (vertices) and the relationships between them (edges); parent-children and ancestor-descendant type connections describe well potential hierarchies in a piece of music. By determining adjacencies and degrees of vertices and introducing weights for edges, one can define affinities and dependencies in the complex and flexible structure that is a musical composition. Ways in which the all-incidence matrix of a graph with weighted edges can evolve are discussed including the use for that purpose of elements of Information Theory. The Emerging 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.