Fig 1.
Illustration of two evolutionary processes on sequential temporal networks.
We consider a sequential temporal network with three snapshots as an example. First, a random individual is selected to be a cooperator (C, blue) in the population full of defectors (D, red). a, In each snapshot, the evolutionary dynamics is sufficient so that the population reaches one of two homogeneous (absorbing) states: all-cooperator and all-defector state. Then, new defectors with links enter the population and continue the evolution. b, In each intermediate snapshot, the evolution proceeds finite rounds g before switching to the next. In this case, the state of the population may not be absorbed when the population structure changes. When g rises to infinity, this process is identical to the first process. In both processes, the evolution ends when the growth stops and the state is homogeneous.
Fig 2.
Fixation probabilities of sequential temporal networks and static networks.
We present three examples for illustrating the analytical result in Eq (6). The number of nodes in the first and second snapshots is denoted as m1 and m2, and the increment of nodes (edges) in the second and third snapshots is denoted as Δm1 (ΔK1) and Δm2 (ΔK2). a, The number of nodes grows exponentially, fulfilling Eq (6a) (i.e. Δm1 = 3 > m1 = 2 and Δm2 = 6 > m2 = 5). Then the fixation probability of sequential temporal network, , is greater than that of its static counterpart,
. b, The increase in the number of nodes and edges fulfills Eq (6b) (i.e. Δm1 = 1 < m1 = 6, ΔK1 = 2 < 2.4 and Δm2 = 2 < m2 = 7, ΔK2 = 6 < 6.4). As a result, the fixation probability is still higher in the sequential temporal network (
). c, When each pair of adjacent snapshots does not satisfy Eqs (6a) and (6b) (i.e. Δm1 = 1 < m1 = 6, ΔK1 = 3 > 2.4 and Δm2 = 2 < m2 = 7, ΔK2 = 8 > 7.2), the fixation probability becomes greater in the static network (
).
Fig 3.
Fixation probabilities by the mean-field approximation.
a, We consider a sequential temporal network with two snapshots. The fixation probabilities and
are identical under neutral drift, i.e.,
(the black dashed line in b). b, We compare these two probabilities
and
under weak selection. We provide numerical simulations for the sequential temporal network (red solid line) and the static network (red dashed line) in the left panel. The probability
is higher than the probability
when the benefit-to-cost ratio b/c is larger than 1. We also calculate the difference between the two probabilities by our mean-field approximation (green dashed line), and the results can well predict the simulations (red diamonds). Furthermore, we calculate the critical benefit-to-cost ratio (b/c)* of the sequential temporal network (solid arrows) and the static network (dashed arrows) by the theoretical formula (red arrows) and approximation (green arrows). The sequential temporal network has a lower positive critical ratio, which shows a better potential to promote cooperation. Parameter values are c = 1, δ = 0.025.
Fig 4.
Evolution of cooperation on synthetic networks.
We analyze four classes of underlying topologies of size N = 100: square lattices with periodic boundaries (abbreviated as SL), random regular graphs with average connectivity k = 6 (abbreviated as RR), Barabási-Albert scale-free networks with the linking number m = 3 (abbreviated as BA), and scale-free networks with initial attractiveness a = 50 and linking number m = 3 (abbreviated as AT). For each network, we obtain the numerical simulation of fixation probabilities by averaging over 106 independent Monte Carlo simulations. a, For the first evolutionary process, the fixation probabilities of these sequential temporal networks (horizontal solid lines) are greater than those of their corresponding static counterparts (horizontal dashed lines) under neutral drift, and, therefore, are also larger under weak selection for a wide range of benefit-to-cost ratios b/c. b, We investigate the relationship between fixation probabilities and the number of rounds g under the second evolutionary process. For these sequential temporal networks, the fixation probabilities are monotonically increasing with respect to g under neutral drift. When g goes to infinite, the fixation probabilities converge to the fixation probabilities under the first evolutionary process (black solid lines). Parameter values are c = 1 and δ = 0.025.
Fig 5.
Evolution of cooperation in four empirical datasets.
We analyze four empirical datasets from different social contexts: a scientific conference in Nice, France [45], the Science Gallery in Dublin, Ireland [46], a workplace in two different years in France [47]. a, the sequential temporal networks promote the evolution of cooperation for any number of rounds g in these datasets. The corresponding fixation probabilities are monotonically increasing with respect to the parameter g. Parameter values are δ = 0.025 for the first dataset, δ = 0.01 for the remaining datasets, and c = 1 for all datasets. b-c, We present the structure of the sequential temporal networks for the first (b) and third datasets (c). The number of nodes and the average degree at time t are denoted as N(t) and k(t), respectively.