The Blind Watchmaker Network: Scale-Freeness and Evolution

It is suggested that the degree distribution for networks of the cell-metabolism for simple organisms reflects a ubiquitous randomness. This implies that natural selection has exerted no or very little pressure on the network degree distribution during evolution. The corresponding random network, here termed the blind watchmaker network has a power-law degree distribution with an exponent γ≥2. It is random with respect to a complete set of network states characterized by a description of which links are attached to a node as well as a time-ordering of these links. No a priory assumption of any growth mechanism or evolution process is made. It is found that the degree distribution of the blind watchmaker network agrees very precisely with that of the metabolic networks. This implies that the evolutionary pathway of the cell-metabolism, when projected onto a metabolic network representation, has remained statistically random with respect to a complete set of network states. This suggests that even a biological system, which due to natural selection has developed an enormous specificity like the cellular metabolism, nevertheless can, at the same time, display well defined characteristics emanating from the ubiquitous inherent random element of Darwinian evolution. The fact that also completely random networks may have scale-free node distributions gives a new perspective on the origin of scale-free networks in general.

Using the Stirling approximation the entropy S = ln Ω takes the form where n(k)=N(k)/N. Variational calculus provides a method for finding the distribution n(k) which maximizes S (and hence also Ω). Since N and M are constant, we want to the maximize s[n(k)] = − k=1 n(k) ln kn(k) with respect to all possible distributions n(k) subject to the two constraints k=1 n(k) = 1 and k=1 kn(k) = k = M/N . The constraints are implemented by two Lagrangian multipliers a and b which means that the functional to be maximized is given by The condition for the maximum is δg[n(k)] δn(k) = 0 and leads to the equation ln kn(k) + 1 + a + bk = 0 which has the solution n(k) = A exp(−bk)/k. The value of the constants A and b follows from the two constraints k=1 n(k) = 1 and k=1 kn(k) = k . In Fig. 2(a) of the main text, this solution is shown to be identical to the corresponding algorithm solution.
Random with respect to the relevant states instead corresponds to the maxi- and has the solution n(k) = A exp(−bk)/k 2 . In Fig. 2(b) of the main text, this variational solution is shown to be identical to the corresponding algorithm solution.
The algorithm which also includes the network constraints goes as follows: 1) pick two boxes (nodes) A and B randomly with probability p ∼ k 2 .
2) pick a random ball in A and move to B.

3) If the attempted move is forbidden by a constraint choose another ball in A. Repeat until one ball is moved. Then choose two new boxes (nodes). 4) If no ball can be moved from A, choose two new boxes (nodes).
The network constraints are introduced through step 3 in the algorithm in such away as to ensure the least possible constraining effect.
A notable difference between the real metabolic networks and the corresponding blind watchmaker network is the number of nodes with just a single link: the number of single link nodes for the metabolic networks is only about 20% of the number for the corresponding blind watchmaker network. In order to investigate how important the number of single-link nodes are for the global statistical properties of the network structure, we introduce an additional constraint into the blind watchmaker network: the average number of single-link nodes are constrained to be the same as for the metabolic networks. Again we choose a constraint which achieves this in an unbiased way. The constraint is again introduced into step 3 of the above algorithm and takes the form of an upper limit: The number of single-link nodes can never exceed a maximum number N max (k = 1). Any move which violates this condition is forbidden. N max (k = 1) is adjusted so as to give the same average n(k = 1) as the metabolic networks. This additional constraint also increases the power-law exponent slightly, from γ ≈ 2.1 to γ ≈ 2.2. The result is presented in Figs 3e and f in the main text. As seen the agreement between the blind watchmaker network and the metabolic networks is now extraordinary. This shows that the number of single-link nodes is not an insignificant detail but a decisive factor reflected in the global statistical properties of the network.