name of the model: a box always contains at least one ball. Obviously, any association of the balls corresponds to a network. However, it is customary to include additional constraints in the definition of a network. The most common for an undirected network are: 1) a network must be connected, 2) only one link between two nodes, 3) no self-loops (which means that the two link-ends of the same link cannot belong to the same node). We will first discuss the model without the additional network constraints. In order to make connection to standard statistical mechanics we will consider the case with a fixed number of balls M and a fixed number of boxes N. The total number of ways you can distribute M balls into N M! boxes, V, is by elementary combinatorics V~ where Pk~0N(k)! the factors N(k)! in the denominator is the number of identical ways you can place the same balls in the same order into N(k) boxes of size k . However, in the CBB-model one ball is assigned to each box to start with. This means that the distinguishable number of ways you can distribute the remaining k-1 balls into a box which already contains one ball, is k times less the number of ways to put k balls into the same box. Thus the total number of distinguishable ways you can distribute balls into the boxes is for the CBB-model reduced by a combinatorial factor Pk~1kN(k) (more details in Supporting Information Text S1). The most unbiased estimate of V corresponds to the maximum and this maximum corresponds to one particular distribution N(k). This is completely equivalent to the maximum entropy principle in statistical mechanics [18]: Once the statistical states are identified, the most unbiased system corresponds to the maximum of such states or equivalently to the maximum of the entropy S = lnV. This maximum is achieved when all statistical states are equally probable. The last statement is the counterpart of the postulate of a priory equal probabilities is statistical mechanics [18]. From an information perspective the maximum of S gives a measure of the maximum information which can be contained in the system [18]. For the CBB-model two statistical states are equal provided that there exists a one to one mapping between boxes containing the same balls and furthermore that for each such mapped pair of boxes the time order in which the balls arrived to the boxes differ by at most a cyclic permutation. The point is that this degeneracy of the statistical states is enforced by the constraint that the boxes always contain at least one ball: The statistical state of the CBB-model and its network counterpart contain a k-degeneracy. The practical issue is to determine the distribution N(k) which maximizes V[N(k)] subject to the appropriate constraints. One way is to find an update algorithm which picks the statistical states with equal probability, since such an algorithm automatically yields N(k). For the CBB-model the obvious algorithm is as follows: Pick two balls randomly and then move one to the same box as the other. If the attempted move involves emptying a box you try again with two new randomly picked balls. If you start with a random distribution with at least one ball in each box, then the restriction of ‘‘always at least one ball in all boxes’’ is in this way implemented without any additional bias. The resulting distribution, N(k), is shown in Fig. 2(a). An alternative is to directly find N(k) using a mathematical standard method called variational calculus (see supporting information Text S1). Both alternatives give the same as result (see Fig. 2(a)). The advantage with the algorithm method is that it directly gives a complete characterization of the unbiased situation: the average SN(k)T which maximizes SðSN(k)TÞ and at the same time qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ maximizes the noise D~ SN(k)2T{SN(k)T2 [19]. As will be shown below this noise, or equivalently the spread of the data, provides important characteristics of the metabolic networks. The definition of the statistical state is a direct consequence of the maximum number of distinguishable ways you can distribute The Blind Watchmaker Network the balls in the boxes: The statistical states, when picked with equal probability, give the global maximum of the entropy. However, our requirement is that the relevant states are picked with equal probability: in our case the rankings of the balls in the boxes is relevant. As emphasized above, this is a direct consequence of the sequential element implicit in the natural selection process and which imposes a time-order ranking on the balls in the boxes. The crucial assumption is that the blind watchmaker network is random with respect to these relevant states. The key observation is that many such relevant states correspond to the same statistical state. Thus unbiased, or equivalently random, with respect to the relevant states inevitably means biased with respect to the statistical states. Furthermore, biased with respect to the statistical states means smaller entropy S. What bias is imposed on the statistical states? In order to see this we note that the entropy S~ is related to the probability p~ of obtaining a new statistical state when choosing a new relevant state: S~~ ln p~V. When the relevant states and statistical states are identical, this reduces to p~~1 and S = lnV. If several relevant states correspond to the same statistical state then ~pv1 and S~vS. In the present case p~~1=Pk~1kN(k) because Pk~1kN(k) is the number of different time-orders which gives the same statistical state. Thus S~~lnV=Pk~1kN(k)~lnV{ Sa where Sa~ Pk~1 N(k)lnk. Consequently, the most unbiased situation in terms of the relevant states corresponds to the N(k) which maximizes S~. The corresponding least biased algorithm which achieves this maximization goes as follows: 1) 2) 3) 4) pick two boxes (nodes) A and B randomly with probability p,k2 pick a random ball in A and move to B. 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) If no ball can be moved from A, choose two new boxes (nodes). The important point here is that this algorithm incorporates the constraints in the most unbiased way and consequently yields the optimal N(k). Figure 2(b) illustrates that the algorithm solutions which have the functional form N(k),exp(2bk)/k2 (see Supporting Information in Text S1). Note that the distribution has an exponential decay for smaller values of average degree SkT~M=N but becomes very power law like for larger SkT. In order to find the variational solution for the network case, one needs to introduce the network constraints. These constraints can be directly implemented into the corresponding algorithm method: Moves which violates the network constraint are discarded.(see Supporting Information in Text S1) When the network constraints are included the distribution follows a power law SN(k)T*k{c over a large range with c.2, as illustrated in Fig. 2(c). An important point to notice is that average distribution SN(k)T is different from the individual network configurations N(k). The striking difference is that a single network configurations always contains large split-off nodes, as illustrated in Fig. 2(d). These split-off nodes constitute an essential characteristic of the network. Discussion How does the blind watchmakers random network compare to real networks? Fig. 3 illustrates this for the case of metabolic networks: Fig. 3(a) shows the average distribution obtained from 107 such networks and Fig. 3(b) the metabolic network for the E. Coli. bacteria (the data is taken from Ref. [14][15]). In Fig. 3(c) PLoS ONE | www.plosone.org 4 February 2008 | Volume 3 | Issue 2 | e1690