Self-loops in evolutionary graph theory: Friends or foes?
Fig 4
Nodewise analysis of the star graph with self-loops and the directed line with self-loops.
Here, the average fitness trajectories for each node of the self-looped star graph (shown in panel A) and the self-looped directed line (shown in panel B) are shown. Thick lines represent average fitness trajectories at the population level, whereas, thin lines represent average fitness trajectories for the nodes. The effect of self-loops on a node’s fitness depends on the incoming and outgoing weight flowing out of that node. In panel A, self-loops have the least effect on the central node because of relatively higher incoming and outgoing weight. As a result, the central node attains higher average steady-state fitness than the leaf nodes. In panel B, the root node of the directed line has the lowest steady-state average fitness because of the absence of an incoming link to the root node. (Parameters: N = 10, μ = 1, fmin = 0.1, fmax = 10, number of independent realisations is equal to 2000, mutant fitness distribution, . For the directed line with self-loops, every outgoing link from a node (including the self-loop) has the same weight. For the self-looped star graphs, the weights of the links follows Eq (23), such that λ = 1/(N − 1) and δ = 1/(N − 1)2).