TY - JOUR

T1 - A novel analytical method for evolutionary graph theory problems

AU - Shakarian, Paulo

AU - Roos, Patrick

AU - Moores, Geoffrey

N1 - Funding Information:
P.S. is supported by ARO projects 611102B74F and 2GDATXR042 as well as OSD project F1AF262025G001 . P.R. is supported by ONR grant W911NF0810144 . P.S. would like to thank Stephen Turney (Harvard University) for several discussions concerning his work. The opinions in this paper are those of the authors and do not necessarily reflect the opinions of the funders, the U.S. Military Academy , the U.S. Army , or the U.S. Navy .

PY - 2013/2

Y1 - 2013/2

N2 - Evolutionary graph theory studies the evolutionary dynamics of populations structured on graphs. A central problem is determining the probability that a small number of mutants overtake a population. Currently, Monte Carlo simulations are used for estimating such fixation probabilities on general directed graphs, since no good analytical methods exist. In this paper, we introduce a novel deterministic framework for computing fixation probabilities for strongly connected, directed, weighted evolutionary graphs under neutral drift. We show how this framework can also be used to calculate the expected number of mutants at a given time step (even if we relax the assumption that the graph is strongly connected), how it can extend to other related models (e.g. voter model), how our framework can provide non-trivial bounds for fixation probability in the case of an advantageous mutant, and how it can be used to find a non-trivial lower bound on the mean time to fixation. We provide various experimental results determining fixation probabilities and expected number of mutants on different graphs. Among these, we show that our method consistently outperforms Monte Carlo simulations in speed by several orders of magnitude. Finally we show how our approach can provide insight into synaptic competition in neurology.

AB - Evolutionary graph theory studies the evolutionary dynamics of populations structured on graphs. A central problem is determining the probability that a small number of mutants overtake a population. Currently, Monte Carlo simulations are used for estimating such fixation probabilities on general directed graphs, since no good analytical methods exist. In this paper, we introduce a novel deterministic framework for computing fixation probabilities for strongly connected, directed, weighted evolutionary graphs under neutral drift. We show how this framework can also be used to calculate the expected number of mutants at a given time step (even if we relax the assumption that the graph is strongly connected), how it can extend to other related models (e.g. voter model), how our framework can provide non-trivial bounds for fixation probability in the case of an advantageous mutant, and how it can be used to find a non-trivial lower bound on the mean time to fixation. We provide various experimental results determining fixation probabilities and expected number of mutants on different graphs. Among these, we show that our method consistently outperforms Monte Carlo simulations in speed by several orders of magnitude. Finally we show how our approach can provide insight into synaptic competition in neurology.

KW - Complex networks

KW - Evolutionary dynamics

KW - Moran process

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U2 - 10.1016/j.biosystems.2013.01.006

DO - 10.1016/j.biosystems.2013.01.006

M3 - Article

C2 - 23353025

AN - SCOPUS:84873682650

VL - 111

SP - 136

EP - 144

JO - Currents in modern biology

JF - Currents in modern biology

SN - 0303-2647

IS - 2

ER -