### Abstract

We study the spreading of a single mutant in graph-structured populations with a birth-death update rule. We use a mean-field approach and a Markov chain dynamics to investigate the effect of network topology on the path to fixation. We obtain approximate analytical formulas for average time versus the number of mutants in the fixation process starting with a single mutant for several network structures, namely, cycle, complete graph, two- and three-dimensional lattices, random graph, regular graph, Watts–Strogatz network, and Barabási–Albert network. In the case of the cycle and complete graph, the results are accurate and in line with the results obtained by other methods. In the case of two- and three-dimensional lattice structures, some efforts are made in other studies to provide an analytical justification for simulation results of the evolutionary process, but they can explain just the onset of the fixation process, not the whole process. The results of the analytical approach of the present paper are well fitted to the simulation results throughout the whole fixation process. Moreover, we analyze the dynamics of evolution for a number of complex structures, and in all cases, we obtain analytical results which are in good agreement with simulations. Our results may shed some light on the process of fixation during the whole path to fixation.

### Graphic Abstract

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## Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: All relevant data are reflected in the figures and the body of the paper.]

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## Acknowledgements

The authors would like to thank Arne Traulsen for reading the first draft of this paper and for useful comments and suggestions.

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## Appendix

### Appendix

In this Appendix, we illustrate some details of calculating the average number of interface edges in regular and small-world graphs.

In regular graph by cluster-like growth assumption, as shown in Fig. 12, in the most of the interface, edges are on the sides of the mutant clusters, and since each edge connects just with its neighbors, there is just a single mutant cluster. Since each node connects with its *m* neighbors, then the first node on the side has \(\frac{m}{2}\) interface edges, the second one has \(\frac{m}{2}-1\), the third one has \(\frac{m}{2}-2\), and so on. Then, in each side, the number interface edges are:

since each cluster has two sides, the number of interface edge in the sides is \(\frac{m^{2}}{4}+\frac{m}{2}\). This enumeration is exact for \(m=2\), but for \(m>2\), it may happen that among the cluster of mutants, resident nodes appear, as well.

To approximate the number of these residents, suppose there is no resident in the cluster and the mutant in the side of clusters chosen for reproduction. If it chooses one of its resident neighbors for substituting its offspring, some residents will appear in the mutant cluster. The number of these residents varies from 0 to m/2 which its average is *m*/4. Other nodes also could create residents among the mutant cluster, but the average number of residents that they create is close to *m*/4 and then *m*/4 is a good approximation for the number of residents that appear among the mutant cluster. Since each one is connected to *m* mutants, we should add \(m^2 /4\) to the number of interface edges and the final approximation for the number of interface edges is \(\frac{m^{2}}{2}+\frac{m}{2}\).

In the case of the small-world graph, as shown in Fig. 13, there is more than one mutant cluster in the graph during the Moran process. The reason is random rewiring edges while building the small-world graph from a regular graph. It is possible that reproduction happens through a rewired edge and it makes a new mutant cluster in the graph in the Moran process. Therefore, we approximate the number of interface edges as \(n_{I}\approx \langle c_{i} \rangle (\frac{ m^2}{2}+\frac{m}{2})\) where \(\langle c_{i}\rangle \) is the average number of clusters in a graph with *i* mutants, \(\langle c_{i} \rangle =\sum P_{i}(c)c\). Here, \(P_{i}(c)\) is the probability for a graph with *i* mutants to have *c* clusters. Now, the problem is approximating the \(P_{i}(c)\).

Since each node has *m* edge in average, the maximum number of clusters is \(R=\lfloor \frac{N}{m}\rfloor \) approximately, where \(\lfloor . \rfloor \) is the floor function. Every time that the number of mutants increases by one, there is a chance that newborn mutants belong to the new clusters. When there is *i* mutants in the graph, this changing cluster process happened \(S:=\lfloor iP_{\scriptscriptstyle \mathrm WS}\rfloor +1\) times approximately where \(P_{\scriptscriptstyle \mathrm WS}\) is the probability of rewiring edges in the small-world graph. Here, we are facing a selection with replacement problem. There are R clusters that we want to select S of them and the reparation is allowed. What is the probability to get c different clusters at the end of selection? This is a simple question in combination theory and the answer is:

where:

The approximation used above works well when the number of mutants *i* is small. When the number of mutants increases, the approximation is not valid anymore, so we use the trick that applied before in case of lattice graphs; since the number of average interface edges between *i* mutants and \(N - i\) residents is equal to the number of interface edges between \(N - i\) mutants and *i* residents. Therefore, when \(i>N/2\), \(n_I(i) = n_I(N-i)\).

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Hajihashemi, M., Aghababaei Samani, K. Path to fixation of evolutionary processes in graph-structured populations.
*Eur. Phys. J. B* **94, **51 (2021). https://doi.org/10.1140/epjb/s10051-021-00061-7

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