The computational problem of setting up simulations for equations of the form (4.4) has recently been addressed, e.g., by [127, 129], in the context of evolutionary models. We develop a similar computational framework, coded in Python 3.9.7 and freely available on the GitHub folder EE-graph-dyn 22.

Once again, the core idea of the simulations is to mimic the evolutionary process by simultaneously tracking the dynamics of an entire population of individuals. Each individual is associated to a graph, i.e., to a binary strings with bit-like entries = 0/1 (dyads). Our population-based simulations keep track of all the individuals existing within the population, at each time 23 . To speed up the simulations, we group similar individuals into a clone, that is a pair (, ), where is the number of individuals associated with the same graph . At time , the population is thus defined as the set of existing clones P() = ((), ()). The population size (total number of individuals) (cid:80) () = is held fixed while the total number of clones () fluctuates. At each simulation step, the population is updated, new clones are created by dyadic mutations (exploration), their size updated by functional selection (exploitation), fig. 4.5. In particular:

24: In the case of an edge toggle, one has . Later in ch. 5, we will use growth-only dyadic mutations, i.e., | |. Exploration. Each dyad of each individual in the population mutates24 with probability 1 . The exploration rate is uniform across dyads. Exploitation. The values of the graphs associated to each clone are computed. The clone sizes are then updated by extracting independent samples from a multinomial distribution where each graph is selected with probability = ()/(cid:88) () , 1, . . . , () .

25: In practice, the simulation step can always be defined as = 1 by rescaling accordingly: Our simulations have six parameters, summarised in tab. 4.2. The structural parameters , set the geometry of the simulations. The former is the (fixed) number of nodes of each graph; the latter is the size of the time window to be simulated. There are two internal degrees of freedom: the population size and the time step for technical convenience, it is often preferable to set the inverse time step = 1. Finally, two parameters control the dynamics of the system, the exploration rate and the relative strength of exploitation 25 . / . (4.22) Algorithm: EE graph dynamics, forward simulations (pseudocode). P(0) = (0, 0) = 0 while t<T do Exploration: with probability (, ), update P = (, ) compute ( ) Exploitation: draws from a multinomial distribution with ( = set P() = (, ) += ) compute new counts ( )/(cid:80) Parameter Description