Implementación del procedimiento para la planificación de la

For the trade-off between object and hierarchical complexity to be properly consid- ered, the analytic solution will not provide sufficient resolution, since what matters is the order in which the mutations happen. This can be shown in an example case

where there are precisely 3 nodes. There are 23 = 8 different mutation orderings,

producing the graphs shown in Figure 6.7. The relative frequencies are such that

graph A is produced by two “n” mutations (new node), which can be represented

by a probabilityn2_{, whilst graphB is produced by two “c” mutations (copy node),}

with probabilityc2. If the process is random, by which is meant that c=n, graphs

A and B should be seen 1₄ of the time, whilst graph C should be seen 1₈ of the

time. Variations of D(i.e. 2 nodes on the base level, followed by 1 node on a level

higher) make up the final 3₈. This disparity arises from the many to one mapping

from combinations to graph outcomes in the case of graphD.

Varying the probabilities of cand nwill skew the relative frequencies of the

graphs produced. For example, in the case where c = p and n = 1−p, then if

increasing object complexity is more common than increasing hierarchical object complexity, the relative frequencies will change. However, this assumes that there are no connections between nodes. If connections exist, then the ordering of the mutations matters. This also affects the average outgoing degree produced. For example, graphDcan produce an average outgoing degree of 2₃ or 1₃.

If connections are restricted to only one level (i.e. nodes on level 3 can only regulate those on level 2), then it is clear thatp= 1 no longer produces the highest

average outgoing degree. Ifp= 1 then the maximum average outgoing degree that

can be reached isn−_n1. Although this is better than the average outgoing degree when

p= 0, it is obviously not as good as a mixed strategy, since a copying step followed by a new node creation, then another copying step at the new level, produces an average outgoing degree of 1. However, this only maximises outgoing degree when

0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 n β

Figure 6.8: Figure showing the inequality derived in Equation 6.13. Shaded area shows combinations of β and n which satisfy the inequality β > 1− _n12 discussed

below.

the subgraph is copied along with the node. If a new subgraph is created, then this has the potential to produce a lower average outgoing degree, whenβ 6= 1.

If β = 1, then the average outgoing degree growth is proportional to the

number of copying events. For example, following a string of (1−p)ncopying steps, the average outgoing degree once the first node on a new level is created will be

βn

n+1. This produces an average outgoing degree higher than p= 1 when

βn n+ 1 > n−1 n βn2 > n2−1 β >1− 1 n2.

This inequality is shown in Figure 6.8. As n increases, β → 1. Because of the

restriction previously stated, this inequality applies for all levels of the hierarchy. However, once a new node has been created on a level, copying can also take place on

that level. Now each node copied on a level connects toβn(l−1) nodes. Following (1−p)n−1 copying steps, the average outgoing degree change from copying will be

either 0 or β(n−2). This is weighted towards 0 as there are more nodes to copy,

therefore the average change in outgoing degree is β(_n−n−₁2), for this initial copying.

It can be noted that this is a similar change as the change produced by p = 1.

However, as more nodes are copied on the upper level, the average change in outgoing degree increases. For example, in the case where the number of nodes at level 1

equals the number of nodes at level 0, n(1) = n(0), then copying a node on level

1 produces a change in average outgoing degree equal to βn₂ . This is clearly a

maximum, since additional copying at level 1 will produce a lower change in average outgoing degree than this. Indeed, once the number of nodes at level 1 exceeds the number of nodes at level 0, the maximum way to increase average outgoing degree is then to create a new node on a new level. This supports a step-wise evolution of meta-regulation with periods of increasing object complexity followed by increasing hierarchical complexity.

6.5

Discussion

This chapter focussed on the evolution of meta-regulation and object vs. hierarchical complexity. These are species level traits which have the potential to direct evolu- tionary trends. Further, meta-regulation is seen as a key component of the major

transitions in evolution [Maynard Smith and Szathm´ary, 1997; Vinicius, 2010].

Using the duplication-divergence framework to analyse the model described in Chapter 4 we found that, for sparse or small graphs, the average outgoing degree

is maximised when p =β = 1. However, the growth rate is also maximised when

β 6= 1. Under these conditions a mixed strategy is produced - where small dense

sub-graphs are sparsely connected to each other. These are analogous to the genetic kernels and modular sub-structures described in Erwin and Davidson [2009], and suggests a system with three simple mutation options and moderate selection for the average outgoing degree is sufficient to develop such structures.

The second question asked how this trade-off between increasing object and hierarchical complexity was bounded. The model does not have a steady state for average outgoing degree, suggesting that either hierarchical or object complexity (or both) is unbounded. A steady state is reached in individual runs of the model, however, suggesting that selection pressure can limit the average outgoing degree. Therefore, although theoretically there is no limit, selection pressure is sufficient to impose limits.

It is important also to note that for individual species the order of the mu- tation steps is important. In particular, for a given selection threshold there will be an optimum point at which meta-regulation develops. However, once this initial meta-regulation is developed, increasing object complexity maximises the average outgoing degree more than increasing hierarchical complexity. This supports our claim that meta-regulation is strongly selected for and agrees with observations re-

garding the nature of meta-regulation [Maynard Smith and Szathm´ary, 1997] - that