Graph decompositions for demographic loop analysis

A new approach to loop analysis is presented in which decompositions of the total elasticity of a population projection matrix over a set of life history pathways are obtained as solutions of a constrained system of linear equations. In loop analysis, life history pathways are represented by loops in the life cycle graph, and the elasticity of the loop is interpreted as a measure of the contribution of the life history pathway to the population growth rate. Associated with the life cycle graph is a vector space -- the cycle space of the graph -- which is spanned by the loops. The elasticities of the transitions in the life cycle graph can be represented by a vector in the cycle space, and a loop decomposition of the life cycle graph is then defined to be any nonnegative linear combination of the loops which sum to the vector of elasticities. In contrast to previously published algorithms for carrying out loop analysis, we show that a given life cycle graph admits of either a unique loop decomposition or an infinite set of loop decompositions which can be characterized as a bounded convex set of nonnegative vectors. Using this approach, loop decompositions which minimize or maximize a linear objective function can be obtained as solutions of a linear programming problem, allowing us to place lower and upper bounds on the contributions of life history pathways to the population growth rate. Another consequence of our approach to loop analysis is that it allows us to identify the exact tradeoffs in contributions to the population growth rate that must exist between life history pathways.