No abstract

The problem of finding birth-death fixation probabilities for configurations of normal and mutants on an N-vertex graph is formulated in terms of a Markov process on the 2 N -dimensional state space of possible configurations. Upper and lower bounds on the fixation probability after any given number of iterations of the birth-death process are derived in terms of the transition matrix of this process. Consideration is then specialized to a family of graphs called circular flows, and we present a summation formula for the complete bipartite graph, giving the fixation probability for an arbitrary configuration of mutants in terms of a weighted sum of the singlevertex fixation probabilities. This also yields a closedform solution for the fixation probability of bipartite graphs. Three entropy measures are introduced, providing information about graph structure. Finally, a number of examples are presented, illustrating cases of graphs that enhance or suppress fixation probability for fitness r > 1 as well as graphs that enhance fixation probability for only a limited range of fitness. Results are compared with recent results reported in the literature, where a positive correlation is observed between vertex degree variance and fixation probability for undirected graphs. We show a similar correlation for directed graphs, with correlation not directly to fixation probability but to the difference between fixation probability for a given graph and a complete graph.

This paper presents an adaptation of the Moran birthdeath model of evolutionary processes on graphs. The present model makes use of the full population state space consisting of 2 N binary-valued vectors, and a Markov process on this space with a transition matrix defined by the edge weight matrix for any given graph. While the general case involves solution of 2 N -2 linear equations, symmetry considerations substantially reduce this for graphs with large automorphism groups, and a number of simple examples are considered. A parameter called graph determinacy is introduced, measuring the extent to which the fate of any randomly chosen population state is determined. Some simple graphs that suppress or enhance selection are analysed, and comparison of several examples to the Moran process on a complete graph indicates that in some cases a graph may enhance selection relative to a complete graph for only limited values of the fitness parameter.

No abstract

Relativistic kinetic theory i s used in conjunction with the theory of relativistic surface layers in order to study relativistic disks of matter. After a brief general discussion, attention i s restricted to the case of counter-rotating disks. The general surface stress-energy tensors of such disks a r e exhibited and a distribution function which generates these s t r e s senergy tensors i s deduced. This i s followed by a discussion of stability, and a criteria for the stability of particle orbits i s derived. Finally, the question of central red shift i s considered. It is shown that all counter-rotating disks without singularities a t the r i m will have a finite central red shift, but the question of the existence of a maximum central red shift remains open.