Extremal Dynamics Model on Evolving Networks

Abstract

We investigate an extremal dynamics model of evolution with variable number of units. Due to addition and removal of the units, the topology of the network evolves and the network splits into several clusters. The activity is mostly concentrated in the largest cluster. The time dependence of the number of units exhibits intermittent structure. The self-organized criticality is manifested by a power-law distribution of forward avalanches, but two regimes with distinct exponents τ = 1.98±0.04 and τ ′ = 1.65 ± 0.05 are found. The distribution of extinction sizes obeys a power law with exponent 2.32 ± 0.05. [1,[6][7][8][9][10][11]). The dynamical system in question is composed of a large number of simple units, connected in a network. Each site of the network hosts one unit. The state of each unit is described by a dynamical variable b, called barrier. In each step, the unit with minimum b is mutated by updating the barrier. The effect of the mutation on the environment is taken into account by changing b also at all neighbors of the minimum site.General feature of ED models is the avalanche dynamics. The forward λ-avalanches are defined as follows [1]. For fixed λ we define active sites as those having barrier b < λ. Appearance of one active site can lead to avalanche-like proliferation of active sites in successive time steps. The avalanche stops, when all active sites disappear again. There is a value of λ, for which the probability distribution of avalanche sizes obeys a power law without any parameter tuning, so that the ED models are classified as a subgroup of self-organized critical models [12].The BS model was originally devised in order to explain the intermittent structure of the extinction events seen in the fossil record [13]. In various versions of the BS model it was found, that the avalanche exponent is 1 < τ ≤ 3/2, where the maximum value 3/2 holds in the mean-field universality class. On the other hand, in experimental data for the distribution of extinction sizes higher values of the exponent, typically around τ ≃ 2 are found [14]. The avalanche exponent close to 2 was also measured in ricepile experiments [15]. While there are several models of different kind, which give generic value τ = 2 [16,17], we are not aware of any ED model with such a big value of the exponent.The universality class a particular model belongs to, depends on the topology of the network on which the units are located. Usually, regular hypercubic networks [1] or Cayley trees [11] are investigated. For random neighbor networks, mean-field solution was found to be exact [18,7]. Also the tree models [11] were found to belong to the mean-field universality class. Recently, BS model on random networks, produced by bond percolation on fully connected lattice, was studied [19]. Two universality classes were found. Above the percolation threshold, the system belongs to the mean-field universality class, while exactly at the percolation threshold, the avalanche exponent is different. A dynamics changing the topology in order to dri...

Coevolutionary dynamics on scale-free networks

“…The occurrence of two power-law regimes for P (s) was also found in BS dynamics on small-world networks [11] and in an extremal dynamics with evolving networks [12]. However the origins of the two power-law regimes were completely different from ours.…”

“…The origin in the smallworld networks was argued to be the long range connectivity of the networks [11]. The extremal dynamics with evolving random networks [12] changes the network structure and is not exactly the same as BS dynamics. Furthermore the evolving network develop many disconnected clusters.…”

“…Kulkani et al [11] studied BS model on small-world networks. Slania and Kotrla [12] studied the forward avalanches of a sort of extremal dynamics with evolving networks. Moreno and Vazquez [13] studied BS model only on a SFN with γ = 3.…”

Evolution of networks

“…In Refs. [187,188], the idea of preferential linking [55] was combined with partial inheritance (partial copying) of degree of individual vertices by new ones [194,195]. (The papers [194,195] as well as Refs.…”

“…[187,188], the idea of preferential linking [55] was combined with partial inheritance (partial copying) of degree of individual vertices by new ones [194,195]. (The papers [194,195] as well as Refs. [196][197][198][199][200] are devoted to more complex models generically related to the biological evolution processes.…”