Resistance and relatedness on an evolutionary graph

Abstract

When investigating evolution in structured populations, it is often convenient to consider the population as an evolutionary graph-individuals as nodes, and whom they may act with as edges. There has, in recent years, been a surge of interest in evolutionary graphs, especially in the study of the evolution of social behaviours. An inclusive fitness framework is best suited for this type of study. A central requirement for an inclusive fitness analysis is an expression for the genetic similarity between individuals residing on the graph. This has been a major hindrance for work in this area as highly technical mathematics are often required. Here, I derive a result that links genetic relatedness between haploid individuals on an evolutionary graph to the resistance between vertices on a corresponding electrical network. An example that demonstrates the potential computational advantage of this result over contemporary approaches is provided. This result offers more, however, to the study of population genetics than strictly computationally efficient methods. By establishing a link between gene transfer and electric circuit theory, conceptualizations of the latter can enhance understanding of the former.

Hamilton's inclusive fitness in finite-structured populations

“…A significant property of G is that it can be readily calculated recursively and examples are found in tables 1, 3 and 4. (But see Maciejewski [43] for an interesting alternative approach that uses established results from the theory of random walks.) The equations are obtained by asking for the G-coefficient just before the most recent replacement affecting the pair.…”

“…(But see Maciejewski [43] for an interesting alternative approach that uses established results from the theory of random walks.) The equations are obtained by asking for the G-coefficient just before the most recent replacement affecting the pair.…”

Flow-loss

“…Electrical flows have been used for graph algorithms in various fields: modeling random walks[7], developing more efficient algorithms for approximating the maximum flow problem[6,22,28], modeling landscape connectivity in ecology[33], and inferring relatedness in evolutionary graphs in biology[27].…”