Evolutionary dynamics on any population structure

Evolution occurs in populations of reproducing individuals. The structure of a biological population affects which traits evolve. Understanding evolutionary game dynamics in structured populations is difficult. Precise results have been absent for a long time, but have recently emerged for special structures where all individuals have the same number of neighbors. But the problem of determining which trait is favored by selection in the natural case where the number of neighbors can vary, has remained open. For arbitrary selection intensity, the problem is in a computational complexity class which suggests there is no efficient algorithm. Whether there exists a simple solution for weak selection was unanswered. Here we provide, surprisingly, a general formula for weak selection that applies to any graph or social network. Our method uses coalescent theory and relies on calculating the meeting times of random walks. We can now evaluate large numbers of diverse and heterogeneous population structures for their propensity to favor cooperation. We can also study how small changes in population structure---graph surgery---affect evolutionary outcomes. We find that cooperation flourishes most in societies that are based on strong pairwise ties.Comment: 68 pages, 10 figure

Modeling the propagation of optical beams in three-dimensional photonic crystals

© 2008 Optical Society of AmericaThe definitive version of this paper is available at: http://dx.doi.org/10.1364/JOSAB.25.000785DOI: 10.1364/JOSAB.25.000785We show that the propagation effects of optical beams in three-dimensional photonic crystal structures can be modeled using a direction-dependent effective diffractive index model. The parameters of the model (i.e., the effective diffractive indices) can be calculated using the curvatures of the band structure of the photonic crystal at the operation point. After finding these indices, the wave propagation inside the photonic crystal can be analyzed using simple geometrical optics formulas. We show that the model has good accuracy for most practical applications of photonic crystals. As an example, the application of the model for diffraction compensation in a tetragonal woodpile photonic crystal is demonstrate