Emergent activity networks in a model of punctuated equilibrium

We revisit the simple, yet very influential Bak-Sneppen model of biological evolution known to yield a self-organizedstate exhibiting features of punctuated equilibrium. We consider a variant of the model with varying degrees of random links inthe underlying connection network of biological niches, mimicking a scenario that is expected to be more typical than the strictlynearest neighbour interactions. First, we investigate the robustness of self-organized criticality under random links and demonstratethat the randomly rewired system also attains a self-organized critical state, for probability of random rewiring ranging from p similar to 0 (i.e. close to a ring as in the Bak-Sneppen model) to p similar to 1 (where the underlying connection graph is almost completely random).The robustness of the self-organized state under random links is manifested in the emergent power-law scaling of the frequency ofmutation distances as defined by the path length between mutating sites, irrespective of the extent of randomness in the network of niches. The critical fitness in the system is also found to decrease as a power-law with increasing random links. We then explore anew way to understand the activity of the system through the characteristics of the emergent network of active sites, which we denoteas an activity network. We demonstrate how the structure of this activity network is significantly different from the network of the niches, thus lending a different understanding of the system's activity in general. Interestingly, the mean path length of the activity network has a weak dependence on the presence of random links, while in contradiction the network of niches changes sensitively with respect to the probability of random rewiring in the small-world limit. More importantly, the system evolves to an activitynetwork whose mean path length is typically 2 orders of magnitude smaller than the network of niches. This implies that the systemself-organizes to a network of active nodes where there is a very efficient transfer of information. The size of the activity networkis also very weakly dependent on random links and time of evolution. More surprisingly, it has a markedly small characteristicsize, independent of the system size. This indicates that, counter-intuitively, the set of niches where mutation takes place is alwaysvery small, irrespective of system size, with most niches in evolutionary stasis over significantly long times, interspersed by smallsub-sets of nodes that undergo repeated mutations.

Automated summary

The article examines a variant of the Bak-Sneppen model with random links, demonstrating the robustness of self-organized criticality and exploring the characteristics of the emergent activity network. The findings show that the activity network's mean path length remains stable with random links, while the niche network is sensitive to the probability of random rewiring. The size and characteristic size of the activity network are independent of the system size.