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A central question of evolutionary dynamics on graphs is whether or not amutation introduced in a population of residents survives and eventually evenspreads to the whole population, or gets extinct. The outcome naturally dependson the fitness of the mutant and the rules by which mutants and residents maypropagate on the network, but arguably the most determining factor is thenetwork structure. Some structured networks are transient amplifiers. Theyincrease for a certain fitness range the fixation probability of beneficialmutations as compared to a well-mixed population. We study a perturbationmethods for identifying transient amplifiers for death-Birth updating. Themethod includes calculating the coalescence times of random walks on graphs andfinding the vertex with the largest remeeting time. If the graph is perturbedby removing an edge from this vertex, there is a certain likelihood that theresulting perturbed graph is a transient amplifier. We test all pairwisenonisomorphic cubic and quartic regular graphs up to a certain size and thuscover the whole structural range expressible by these graphs. We carry out aspectral analysis and show that the graphs from which the transient amplifierscan be constructed share certain structural properties. The graphs arepath-like, have low conductance and are rather easy to divide into subgraphs byremoving edges and/or vertices. This is connected with the subgraphs beingidentical (or almost identical) building blocks and the frequent occurrence ofcut and/or hinge vertices. Identifying spectral and structural properties maypromote finding and designing such networks.

Constructing transient amplifiers for death-Birth updating: A case study of cubic and quartic regular graphs