Modified Barabási-Albert networks.

Barabási-Albert networks are relatively sparse networks. A modification of the algorithm can be introduced to grow denser networks. For example, one may replace step 3(b) by

1. 3(b). With default probability 1 − p: Attach x_n by adding bonds 〈x_j ∼ x_n〉 with probability (where λ and A are non-negative parameters of the algorithm);

Since k_j ≪ ∑_j k_j in Barabási-Albert networks, one may assume that λk_j + A ≤ ∑_j k_j for values of λ and A which are not too large (and so q_j ≤ 1).

In Fig 2 two examples of Modified Barabási-Albert networks are shown, one a sparse network with λ = 0.5, A = 0 and p = 0, and the second a denser network with λ = 2.0, A = 0 and p = 0. In both cases the algorithm was iterated 200 times; the sparse network has order 203 and two hubs of degrees {15, 17}, and the dense network has order 172 with seven hubs of degrees {15, 15, 16, 17, 19, 27, 33}.

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Fig 2. Modified Barabási-Albert networks.

The network on the left was grown with λ = 0.1 to order n = 203. It has two hubs of degrees {15, 17} which exceed . The network on the right was grown with λ = 1.5 to order n = 172. This network contains hubs of degrees {15, 15, 16, 17, 19, 27, 33} exceeding . In both cases the algorithm was implemented with p = 0. The arrangement of nodes and bonds in these networks was created using the prefuse force directed lay-out in Cytoscape 3.4.0 [17].

https://doi.org/10.1371/journal.pone.0189866.g002

Variant Barbasi-Albert networks.

A variant Barbasi-Albert algorithm can be introduced by changing step 3(b) in the Barbasi-Albert algorithm to

1. 3(b). With default probability 1 − p: Attach x_n by adding bonds 〈x_j ∼ x_n〉 with probability , (where α and A are non-negative parameters of the algorithm);

The effect of the parameter α is to increase the probability of adding bonds to the hubs of the network if α > 1, and to decrease this probability if α < 1. In the case that α > 1 networks dominated by a single very large hub are obtained (see Fig 3 (right network)), while networks with α < 1 are more sparse and not dominated by a few hubs (see Fig 3 (left network)). The left network in Fig 3 was grown by putting α = 0.15 and A = 0 and has order 327. None of the nodes in this network has degree which exceeds , and so none qualify as hubs. A denser network is obtained if α = 1.15 and A = 0, as shown in Fig 3 on the right. This network is dominated by hubs of degrees {22, 24, 26, 42, 43, 116} and has order 351.

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Fig 3. Variant Barabási-Albert networks.

The network on the left was grown using α = 0.15 and A = 0 to a total to n = 327 nodes. This graph is very sparse, and none of its nodes qualify as hubs. The network on the right was grown to order n = 351 with α = 1.15 and A = 0. This is a dense network with several nodes qualifying as hubs of degrees {22, 24, 26, 42, 43, 116}. The arrangement of nodes and bonds in these networks was created using the prefuse force directed lay-out in Cytoscape 3.4.0 [17].

https://doi.org/10.1371/journal.pone.0189866.g003

Mean field theory for Modified Barabási-Albert networks.

Let k_j(n) be the degree of node j after n iterations. A mean field calculation of k_j(n) is done by assuming that k_j(n) is equal to its expected value for each n; that is, k_j(n) = 〈k_j(n)〉 for each j and n.

The modified Barabási-Albert algorithm appends bonds to a network of order n as follows: Step 3(a) is executed with probability p, and a bond (and the (n + 1)-th node) is appended with uniform probability on one of the n existing nodes. The probability that node j gets a bond in this way is and on average one bond is attached with probability p.

If step 3(b) is done instead, then the expected number of bonds added in the mean field is approximately . The total number of bonds in the network is (8) by handshaking. Thus, the increment in the number of bonds when the next node is appended is (9) Approximate this by a differential equation (10) This can be solved to obtain (11) where C is a function of (p, λ, A) defined by this expression. Notice that E_n grows linearly in n, so that Barabási-Albert graphs will be necessarily sparse as n → ∞ (and by Eq (7) the scaling exponent is γ > 2).

With each iteration the mean field value of k_j(n) (the degree of the j-th node after n iterations) increments by (12) since 2E_n = ∑_j k_j(n) = 2Cn, and since the probabilty of adding a bond to node j is . This can again be approximated by a differential equation: Take n → t, a continuous time variable, and let k_j(n) → k_j(t), the continuous mean field degree of node j. Then (13) The initial condition is to assume that node j is added at time t_j. Putting A = 0 and λ = 1 gives C = 1 and the equation (14) which was also derived in reference [9]. In this event the solution is (assuming the initial condition k_j(t_j) = 1).

More generally, Eq (13) can be cast in the general form (15) where and , with solution (16) using again the intial condition k_j(t_j) = 1.

The mean field degree distribution can be determined from this solution. The probability that node j has degree k_j(t) smaller than κ at time t is denoted by P[k_j(t) < κ]. Since k_j(t) < κ if this is also the probability . If the node t_j is chosen uniformly from the n available, then (17) The mean field degree distribution is the derivative of this to κ: (18)

For large κ this shows that the modified Barbasi-Albert network is scale-free with exponent (19) Putting A = 0 gives the exponent (20) This is the mean field exponent of a modified Barabási-Albert network. For small λ < 1 the exponent is large, indicating a network with few nodes (if any) of high degree. For large λ > 1, . This is a lower bound on γ for modified Barabási-Albert networks.

If λ = 1, then the exponent γ is given by (21) In this model one similarly finds that γ ≥ 3, and in fact, if p = 0, then . The parameter A may be used to tune the exponent γ for any given p.

If both λ = 1 and A = 0, then the known expression for γ for Barabási-Albert networks is recovered, namely (22) Notice that γ ≥ 3 and that γ = 3 if p = 0 [9].

The connectivity of modified Barabási-Albert networks is given by (23) where . Since , Eq (5) gives . Inserting the value of P gives the result above as well.

In the Fig 4 the probability P(k), that the degree of a Barabási-Albert network is equal to k, is examined by plotting log P(k)/log(k + 1) against 1/log(k + 1) where P(k) was estimated for values n ∈ {6250, 12500, 25000, 50000, 100000, 200000} and for p = 0. The curves should intersect the vertical axis at −γ. Least squares fit of the data to quadratic curves gives 6 estimates for γ, which average to γ = 3.026 ± 0.076, very close to the theoretical value γ = 3 from Eq (20) (for p = 0 and λ = 1).

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Fig 4. Scaling of Barabási-Albert networks with p = 0.

Data on networks generated by the Barabási-Albert algorithm with p = 0. In each case 100 networks were grown and the average degree sequence P_n(k) computed. The curves above are plots of log P_n(k)/log(k + 1) against 1/log(k + 1) for n ∈ {6250, 12500, 25000, ⋯, 200000}. Least squares fit to the data using a quadratic model gives the y-intercepts which averages to 3.026. This is very close to the value γ = 3 predicted for the scaling exponent in this model by the mean field approach.

https://doi.org/10.1371/journal.pone.0189866.g004

Data collected for the same values of n and for p = 0.5 cannot be successfully analysed by regressions with quadratic curves, but cubic curves give the average value γ = 5.161 ± 0.068, which are not equal to but still fairly well approximated by γ = 5 predicted by Eq (20) for p = 0.5 and γ = 1.

When p = 0.8 the plots are strongly curved and extrapolation to estimate γ is more difficult. In this case a different approach is needed. Putting in Eq (4) gives (24) so that a plot of as k → ∞. That is, plotting ζ(k) against gives a curve with y-intercept equal to −γ. Better results are obtained when plotting against . In this case a linear extrapolation gives γ = 11.67 ± 0.41 and a quadratic extrapolation gives γ = 11.6 ± 2.6. These results are close to the mean field prediction γ = 11 for p = 0.8. Incidently, if p = 0.5 then this kind of analysis show that γ = 5.47 ± 0.14 (linear extrapolation) or γ = 4.4 ± 1.0 (quadratic extrapolation), and if p = 0, then the results are γ = 3.088 ± 0.022 (linear extrapolation) and γ = 2.86 ± 0.18 (quadratic extrapolation).

If λ = 2 and p = A = 0 then the algorithm grows modified Barabási-Albert networks with γ = 3 (the mean field estimate given by Eq (19)). Estimating γ by plotting ζ(k) against gives the estimate γ = 3.019 ± 0.098 (linear extrapolation) and γ = 2.62 ± 0.33 (quadratic extrapolation).

The connectivity of Modified Barabási-Albert networks should converge quickly to a constant with increasing n (by Eq (5)) since γ > 2. Computing it for Barabási-Albert networks (with λ = 1 and A = 0) gives 〈k〉_n ≈ 3.16 for p = 0, 〈k〉_n ≈ 2.28 for p = 0.5 and 〈k〉_n ≈ 2.08 for p = 0.8, and for n = 12500. Increasing n does not change these results.

Mean field theory for Variant Barabási-Albert networks.

In this model the increment in the number of bonds when the (n + 1)-th node is appended is given by (25) Approximating this with a differential equation gives (26) The right hand side can be approximated as follows: For α > 1 the algorithm should grow dense networks with nodes of high degree. Assuming that k_j(n) ≈ k_ℓ(n) for all ℓ shows that . Using this approximation gives (27) If A = p = 0, then the differential equation can be solved directly to obtain E_n ≃ 2^{(α−1)/(2−α)} n, provided that α > 1. This shows that E_n is linear in n, which may be expected if α is not too much larger than 1.

Numerical experimentation shows that E_n grows linearly in n for values of α not too much larger than 1. For example, if p = 0.5, A = 1 and α = 1 then , if α = 1.5 then , but if α = 2 then increases slowly with n. Similarly, if p = 0, and A = 1, then, if α = 1, , and if α = 1.5, , but if α = 2 then increases slowly with n and for even larger values of n this growth accelerates.

The recurrence for the degree of the j-th node may be approximated by a differential equation similar to Eq (13): Assuming that E_n = Dn^β, replacing n → t (a continuous time variable), gives the recurrence (28) This can be approximated by the differential equation (29) If α = 1 and β = 1 then the solution of this equation gives the Barabási-Albert case with γ = 3. Proceed by considering the case A = p = 0 and the initial condition k_j(t_j) = 1. Assume that α = 1 + ϵ. Then the equation becomes (30) A perturbative approach for small ϵ can be done by expanding . Truncating this at O(ϵ²) and putting g(t) = log k_j(t) gives the differential equation (31) Using the initial condition g(t_j) = log k_j(t_j) = 0 the solution of this equation is (32) In the case β > 1 suppose that δ = β − 1 and that δ is small. Then approximate With this approximation the solution for g(t) above can be expanded in ϵ and δ to give the first order approximations

Proceed by solving the above quadratics for in terms of g(t). Expand the solution in ϵ and δ and keep only the first few terms. In the case that β = 1 this gives (33) Since g(t) = log k_j(t), the probability that k_j(t) < κ is given by (34) Taking the derivative to κ gives the distribution function in the case that β = 1: (35) These networks are thus not scale-free. For small values of k the log k terms are slowly varying, and the networks will appear to be scale-free with γ = 1 + 2D. However, with increasing k the exponent reduces in value and the connectivity of the network will become dependent on k in the way seen in Eq (5) for small values of γ.

Notice that if D = 1 and ϵ = 0 (or α = 1), then the above reduces to P(k) ∼ k⁻³, as expected for Barabási-Albert networks.

If β > 1, then a similar approach to the above may be considered. Solving the expression for g(t) above for and keeping only terms to O(ϵ) and O(δ) gives (36) This shows that This shows that (37) This gives an effective exponent γ_k = 1 + 2D + δ log t_j + D(2Dδ − ϵ)log k which decreases in size if 2Dδ − ϵ < 0 and increases in size if 2Dδ − ϵ > 0. Since δ = β − 1 and ϵ = α − 1, and for small α numerical simulations show that β ≈ 1, it is normally the case that 2Dδ − ϵ < 0. This means that the networks will first appear scale-free with constant connectivity until k becomes large enough in which case the connectivity will increase with k, as seen above.

Numerical results on Variant Barabási-Albert networks.

In Fig 5 data for networks with p = 0 and α = 1.1 and α = 0.5 is shown. Since α = 1.1 is still very close to 1, the results above show that these networks should still appear scale-free, and with connectivity a constant. This is indeed the case. For n = 6250 the data gives 〈k〉_n = 3.149, and increasing n to n = 200000 gives 〈k〉_n = 3.176. That is, the connectivity of the networks are insensitive to n over this range. Least squares fits to the curves with quadratic polynomials in order to determinate the value of γ give the average γ = 2.857 ± 0.068. This result is consistent with a constant value of the connectivity of networks of these size ranges. With increasing n, it is expected that γ will decrease in value (that is, the value given here is an effective value), and eventually, the connectivity will start to increase.

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Fig 5. Variant Barabási-Albert networks with p = 0.

Data on networks generated by the Variant Barabási-Albert algorithm with p = 0 and α = 1.1 (red curves) and α = 0.5 (blue curves). In each case 100 networks were grown and the average degree sequence P_n(k) computed. The curves above are plots of log P_n(k)/log(k + 1) against 1/log(k + 1) for n ∈ {6250, 12500, 25000, ⋯, 200000}.

https://doi.org/10.1371/journal.pone.0189866.g005

Networks generated with p = 0 and α = 0.5 turned out to be sparse with low connectivity. For example, for n = 100000, the connectivity is 〈k〉_n = 1.036 and this decreases even further for n = 200000, where 〈k〉_n = 1.020. Attemps to extract an exponent γ from the data for these networks were not succesful, the regressions did not settle on a value, but are strongly dependent on n. Notice that the mean field analysis above does not apply to networks with α < 1.

Putting α = 2 gives networks with average connectivity which increases with n. For example, if n = 100, then 〈k〉_n = 43, for n = 500, 〈k〉_n = 260 and for n = 1000, 〈k〉_n = 527. On the other hand, if , then 〈k〉_n = 3.08 if n = 100, 〈k〉_n = 3.27 if n = 500, and 〈k〉_n = 3.31 if n = 1000, and it appears that for small values of n the connectivity does not change quickly with increasing n.

Mean field theory for Duplication-Divergence networks.

Let k_j(n) be the degree of node j after n iterations. The algorithm appends nodes by duplicating them (the probability that a node υ is duplicated in a network of order n is ), adds bonds by inserting a bond between a node and its duplicate with probability p, and remove bonds by selecting one bond between node-duplicate pairs and other nodes independently and deleting it with probability q. Let 2E_n = ∑_j k(n) be twice the total number of bonds after n iterations. Then, if k_j(n) is the degree of node j at time n, and node j is duplicated, the number of bonds in the network E_n increases in the mean field by (38) This follows since k_j(n) bonds are created in the duplication move in the mean field, and another bond is created between the j-th node and its duplicate with probability p. The number of deleted bonds in the mean field is qk_j(n).

Notice that 2E_n = ∑_j k_j(n) = na_n where a_n = 〈k_j(n)〉 is the average degree. In the mean field approximation one substitutes k_j(n) in the recurrence (38) by its network average a_n. Then Eq (38) can be casted as a recurrance for a_n: (39) Let n → t, where t is a continuous time variable, and approximate this recurrence by the differential equation (40) The initial condition is a₁ = 1, and this has solution (41) Since , it follows that (42) Comparison to Eq (7) shows that, if , (43) In this case E_n = O(n^2(1−q)) + O(n) and that while 2(1 − q) > 1, the term O(n) is a strong correction to the growth in E_n for even large values of n. In other words, the degree distribution P(k) of the network will be strongly corrected from the powerlaw distribution in Eq (1).

If , then by solving Eq (40), a_t = 1 + 2p log t (so that a₁ = 1). Since , this shows that (44) In this case γ = 2 by Eq (7), but notice the subtle domination of the n log n term. In numerical work this will be very hard to see.

The case is considered by noting that as t → ∞. This shows that (45) This shows that γ ≥ 2 by Eq (7).

Putting the above together gives (46) with a logarithmic correction if .

Comparing the coefficient in Eq (7) with Eq (45) gives a refined estimate , provided that 2q < 1 + 2p. For example, if q = 0.75 then p > 0.25. However, numerical work shows this estimate to be too small, and estimating γ in this regime for this model remains an open question.

The power law decrease in P(k) in Eq (1) is only asymptotic for this algorithm; and there should be corrections in particular for . From the results above the average connectivity can be computed: Since , (47) From these results P(k) can be calculated. Since , it follows that . Thus, using this approach gives (48) where the case is unknown since the dependence of the exponent γ on the parameters (p, q) is not known. Notice the change in behaviour at the critical value ; this was already observed numerically in reference [11].

The modified Duplication-Divergence algorithm has the same recurrence (41), and so the values for γ and relations for 〈k〉_n and P(k) remain unchanged for this algorithm. Notice that this implementation preserves the degree of the selected node, and tends to give a duplicated node with lower degree (while the (unmodified) implementation tends to lower the degrees of both the selected and duplicated nodes). As a result, networks generated with the modified algorithm have, on average, more nodes of degree equal to one (and so appear more tree-like).

Numerical results on Duplication-Divergence networks.

In Fig 7 two networks grown with the Duplication-Divergence algorithm are shown. Both networks were grown with p = 1 and have order 300. The network on the left was grown with divergence parameter q = 0.4, and that on the right, with the higher mutation rate q = 0.6.

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Fig 7. Duplication-Divergence network.

The network on the left is a network generated with p = 1 and q = 0.40. It has order 300 and it has 114 nodes with degrees exceeding and so qualify as hubs. The largest few of these hubs have degrees {43, 45, 47, 47, 50}. The network on the right is similarly a network generated with p = 1 and q = 0.60. It is more extended but has only one node of degree equal to one. Its order is 300, and it has 5 nodes of degrees {18, 18, 19, 20, 23} which qualify as hubs. Networks generated with the Modified Duplication-Divergence algorithm have a similar appearance, with the exception that more nodes of degree 1 are seen. The arrangement of nodes and bonds in these networks was created using the prefuse force directed lay-out in Cytoscape 3.4.0 [17].

https://doi.org/10.1371/journal.pone.0189866.g007

In Fig 8 data for networks grown with p = 0.75 and q = 0.4 are shown. The curves on the right were obtained by plotting (log P(k))/log(k + 1) averaged over 100 networks of sizes {3125, 6250, 12500, 25000, 50000, 100000, 200000} against 1/log(k + 1). The mean field value of γ is denoted by the bullet on the left-hand axis. These data show that convergence to this value is very slow—this indicates strong corrections to scaling arising in Eq (42).

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Fig 8. The distribution of degrees in Duplication-Divergence networks with p = 0.75 and q = 0.40.

Data on networks generated by the Duplication-Divergence algorithm. In each case 100 networks were grown and the average degree sequence P_n(k) computed. The curves on the right are plots of log P_n(k)/log(k + 1) against 1/log(k + 1) for n ∈ {3125, 6250, 12500, ⋯, 200000}, while those on the left are plots of (log P(2k) − log P(k))/log2 as a function of log(k + 1)/k. The mean field estimate for the exponent γ is marked at −γ = −1.8 on the left hand axis. The strong correction to scaling evident in these curves makes it difficult to extrapolate to the mean field value for γ.

https://doi.org/10.1371/journal.pone.0189866.g008

An alternative approach is to estimate γ by plotting ζ(k) = (log P(2k) − log P(k))/log2 as a function of log(k + 1)/k (see Eq (4) with α = 2). The results are also strongly curved data (left in Fig 8), and while the results are not inconsistent with the mean field value γ ≈ 1.9 in this model, however, it seems difficult to extrapolate these curves to a limiting value of γ.

If then the results in Fig 9 are seen. The curves of ζ(k) = (log P(2k) − log P(k))/log2 as a function of log(k + 1)/k have straightened considerably, and each can be extrapolated by a quadratic least squares to obtain an estimate γ_n for each value of n = 3125 × 2^ℓ (for ℓ = 0, 1, 2, …, 6). This gives estimates {9.68, 8.52, 7.99, 7.95, 7.82, 7.58, 7.05} which can be extrapolated by a least squares fit of γ_n = γ + A/log n, giving the estimate γ ≈ 2.87, which is slightly larger than the value predicted by the mean field formula (see the paragraph following Eq (46)). This suggests that the approach to limiting behaviour in this model is quite slow, consistent with the remarks after Eq (46) in the previous section.

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Fig 9. The distribution of degrees in Duplication-Divergence networks with p = 0.75 and q = 0.60.

Data on networks generated by the Duplication-Divergence algorithm. In each case 100 networks were grown and the average degree sequence P_n(k) computed. The curves on the right are plots of log P_n(k)/log(k + 1) against 1/log(k + 1) for n ∈ {3125, 6250, 12500, ⋯, 200000}, while those on the left are plots of (log P(2k) − log P(k))/log2 as a function of log(k + 1)/k. Each of these curves can be extrapolated by a quadratic least squares fit to obtained estimates of γ. This gives the estimates γ_n for n = 3125 × 2^ℓ for ℓ = 0, 1, 2, …, 6. Extrapolating the γ_n to n = ∞ by a least squares fit γ_n = γ + A/n gives γ ≈ 7.4.

https://doi.org/10.1371/journal.pone.0189866.g009

The average connectivity 〈k〉_n is expected to behave according to Eq (47). In Table 1 〈k〉_n is listed for p = 0.75 and q = 0.40, q = 0.50 and q = 0.60. If q = 0.4, then Eq (47) suggests that 〈k〉_n ≃ 8.5n^0.2. Computing 〈k〉n × n^−0.2 from the data in Table 1 gives {5.18, 5.45, 5.65, 5.91, 5.96, 6.01, 6.12}. Plotting these results against 1/log n and then linearly extrapolating as n → ∞ gives 7.98, close to the value of 8.5 predicted in Eq (47).

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Table 1. Connectivity data for Duplication-Divergence networks.

https://doi.org/10.1371/journal.pone.0189866.t001

If q = 0.5, then Eq (47) suggests that 〈k〉_n ≃ 1.5log n since p = 0.75. Dividing the results in Table 1 by log n for each value of n gives the results {1.42, 1.44, 1.44, 1.42, 1.43, 1.46, 1, 45}. The average of this is close to the predicted value of 1.5.

Finally, if q = 0.6 then the data appear to approach a constant. Extrapolating these results using the model A + B/log(n) gives the estimated limiting value 8.72. By Eq (5) this indicates that γ = 2.13, a value which is quite close to 2.15, the value predicted by the formula in the paragraph following Eq (46).

Mean field theory for Solé networks.

Let E_n be the total number of bonds in a Solé network after n iterations of the algorithm, and let 〈k〉_n be the connectivity of the network (that is, the average degree of nodes) after n iterations (so that 2E_n = n〈k〉_n). In the mean field approximation the node in step 2 of the algorithm has degree 〈k〉_n and this number of bonds is added in step 3, while, in a similar way, δ〈k〉_n bonds are removed in step 4. In step 5 there are n − 〈k〉_n choices in the mean field for the node u not adjacent to υ′ and each bond 〈u ∼ υ′〉 is added with probability α. This shows that the number of bonds after n + 1 iterations is given by the recurrance relation (49)

Since 2E_n = n〈k〉_n this becomes (50) which is a mean field recurrence relation for E_n.

Taking n → t, a continuous time variable, and approximating E_n by E_t, and approximating the finite difference as a derivative, gives the following differential equation for E_n: (51) Solving this equation and letting t → n again gives the approximate mean field solution for E_n: (52) Eq (52) shows that the number of bonds is proportional to n², so that networks created by this algorithm are dense, except when α = 0. Comparison to Eq (7) suggests that γ ≤ 1 in this model. Notice that there is no logarithmic factor in the denominator, and that E_n = Θ(n²). This is consistent with a mean field value γ < 1 (and this requires that P_n(k) be modified so that it is a normalisable probability distribution). With these results, it is reasonable to expect that, in the mean field, (53) If α = 0 then Eq (52) gives E_n ∼ n^2−2δ and comparison to Eq (7) gives (54)

Numerical results for Solé networks.

Similar to Barabási-Albert and Duplication-Divergence networks, Solé networks can be grown numerically by implementing the algorithm as given above, using sparse matrix routines to efficiently store the adjacency matrix of the network. The larger size of networks makes these more difficult to grow, and our algorithms sampled efficiently to networks of size 51200 bonds.

Solé networks are rich in bonds. This is seen, for example, in Eq (52), which shows that E_n ∝ n² if α > 0. In Fig 11 two examples of networks generated by the Solé algorithm are shown. If δ < 0.5, then the networks have a dense appearance dominated by a few hubs. If δ > 0.5, then the networks appear more extended, often with no nodes qualifying as hubs under the definition that the degree of a hub in a network of order n is at least . The networks in Fig 11 were generated with α = 0.005, and increasing the value of α quickly increases the number of bonds.

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Fig 11. Solé evolutionary networks.

The network on the left was generated with δ = 0.25 and α = 0.005. Its has order 279 and has 47 nodes with degrees exceeding and so qualify as hubs. The largest few of these hubs have degrees {40, 41, 62, 80}. This algorithm creates dense networks as seen here, even for small values of α. Increasing the value of δ gives more extended networks. The network on the right was generated with δ = 0.75 and α = 0.005 and grown to order 230. None of its nodes qualify as hubs. The arrangement of nodes and bonds in these networks was created using the prefuse force directed lay-out in Cytoscape 3.4.0 [17].

https://doi.org/10.1371/journal.pone.0189866.g011

The mean field result that γ ≤ 1 has implications for the scaling of Solé networks. In particular, P_N(k) in Eq (1) is not normalisable for infinite networks if γ ≤ 1 and so is not a valid candidate degree distribution in this model. The degree distribution can be modified to (55) where D(x) is a function of the combined (or scaled) variable x = n^−ϕ k. That is, as n → ∞, k is rescaled by n^−ϕ and k^γ P(k) approaches a limiting distribution proportional to D(x).

This can be tested numerically by plotting n^γ P(k) as a function of x = n^−ϕ k. For the proper choices of γ and ϕ it is expected that n^γP(k) ≃ C_oD(x) for a wide range of values of n (that is, the data should approach a limiting curve as n → ∞). The result is shown in Fig 12 for (δ = 0.25, α = 0.005) and (δ = 0.75, α = 0.005). These are plots on the same graph for n = 100 × 2ⁿ for n ∈ {6, 7, 8, 9} (other curves at smaller values of N are left away to give a clearer picture).

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Fig 12. Scaling of Solé evolutionary networks.

Plotting k^γ P_N(k) against N^−ϕ k for networks generated by the Solé Evolutionary algorithm gives the distributions above. On the left the results are shown for networks grown with δ = 0.75 and α = 0.005. The choices γ = 1/2 and ϕ = 1 uncovers a distribution as shown where the order of the networks are N = 100 × 2ⁿ for n = 6, 7, 8, 9. A similar distribution, but with γ = 2/3 and ϕ = 1, is seen when networks are grown with δ = 0.25 and α = 0.005. It is not known that the value of γ changes discontinuously as δ increases from 0.25 to 0.75.

https://doi.org/10.1371/journal.pone.0189866.g012

The data for δ = 0.75 are the cluster of peaks to the left, rescaled by choosing ϕ = 1 and , while the cluster of peaks to the right is for δ = 0.25 with ϕ = 1 and . With increasing n the data appear to approach a single underlying curve if in the one instance, and in the other instance. Both these values are consistent with the mean field expectation that γ ≤ 1 in this model. Further refinements in this scaling assumption may be necessary, since the curves are still becoming narrower with increasing n. It is not clear that these approach a limiting curve as n → ∞, although the data for δ = 0.75 suggest this to be the case. In these cases the curves are sharply peaked with a mean of about 0.02 if δ = 0.25 and about 0.007 if δ = 0.75.

Since the curve D(x) is sharply peaked at a constant value c_o of the rescaled variable x, the connectivity of Solé networks is estimated by treating D(x) as concentrated at c_o and then (assuming that ϕ = 1 and approximating the connectivity) (56) In other words, the connectivity of Solé networks should increase linearly with n^ϕ (and since ϕ = 1, linearly with n). In Table 2 the connectivities of Solé networks for δ = 0.25 and δ = 0.75 (with α = 0.005) are listed. Non-linear least squares fits to the data show that ϕ = 1.01 when δ = 0.25 and ϕ = 0.99 when δ = 0.75. That is, these results are consistent with the value ϕ = 1 seen above.

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Table 2. Connectivity data for Solé networks.

https://doi.org/10.1371/journal.pone.0189866.t002

Mean field theory for the iSite model.

Let nodes in the network correspond to proteins, and let i_j(n) be the number of active iSites on node j after n iterations of the algorithm. Denote the degree of node j by k_j(n) (that is the total number of bonds with one end-point in node j), and let E_n be the number of bonds of the network (this is the size of the network). Then 2E_n = ∑_j k_j(n).

The average number of active iSites per node is . With each iteration i(n) iSites are created, of which qi(n) are silenced, in the mean field. This gives the following recurrance relation for i(n): (57) The exact solution of this recurrance is (58) where Γ is the gamma function with the property that Γ(x + 1) = xΓ(x) and Γ(1) = 1. Notice that i(0) = I, where I is the number of iSites on the source node x₀.

For large n the Γ-function and the factorial have well known asymptotics (namely the Stirling approximation [27]), so that (59) This shows that with increasing n the total number of iSites grows proportionally to n^1−q. If q = 0, then this is linear in n since no iSites become silenced, and if q = 1, then the number approaches a constant.

The total number of bonds in the network increases after n iterations by the recurrance (60) since there are on average bonds incident to each node, and the probability that each one of them is duplicated is 1 − r, and there are on average i(n) iSites per node, and the probability that each of these is self-interacting is p.

Using the asymptotic solution for i(n) and approximating this recurrence by a differential equation gives (61) This equation can be solved, and using the initial condition E₁ = 0, the result is (62) Thus, the average degree of a node is equal to , so that the connectivity of iSite evolutionary networks is given by (63) in the mean field. This shows that the large n value of 〈k〉_n is dominated by the larger of −q and 1 − 2r. In particular, if , then 〈k〉_n ∼ n^1−2r.

By Eq (7) one may determine γ for this model: (64) If 2r = 1 + q, then a different solution is obtained, namely (65) This shows that γ = 2 + q in this case as well, but there is also a logarithmic correction to the growth of E(t), and so there is a logarithmic factor in the expression for 〈k〉_n.

Modified iSite evolutionary algorithm.

The subfunctionalization of proteins can be refined by introducing in the iSite algorithm the probability of creating new iSites on the progeny node with a probability s. This changes the algorithm as follows.

Modified iSite evolutionary algorithm:

Implement the algorithm as above but introduce the parameter s and create new active iSites by replacing step 2 in the iSite evolutionary algorithm by

1. 2. Choose a progenitor node υ uniformly in the network and duplicate it, and its associated iSites A, to a progeny node υ′ with duplicated iSites A′;
   1. (a). A duplicated iSite A′ ∈ υ′ is active with probability q if it is duplicated from an active iSite on A ∈ υ, and silenced otherwise;
   2. (b). An active duplicated iSite A′ ∈ υ′ is self-interacting with probability p if it is duplicated from a self-interacting iSite on A ∈ υ, and not self-interacting otherwise;
   3. (c). If a silenced iSite A is duplicated to iSite A′, then A′ is also silenced;
   4. (d). With probability s create an active iSite C on the progeny node υ′, where C is self-interacting with probability p.

The recurrence for the average number of active iSites per node i(n) (see Eq (58)) is modified to (66) in the Modified iSite evolutionary algorithm. The exact solution is obtained by replacing q by q − s in Eq (58), and the asymptotic approximation of the solution is given by (67) as seen in Eq (59).

The total number of bonds in the network, E_n, still satisfies Eq (60), and so it follows from Eqs (62)–(64), that for the modified iSite evolutionary algorithm (notice the condition that q < r + s): (68) This shows that the connectivity of Modified iSite networks is given by (69) The value of the scaling exponent is seen from above to be given by (70) with a correction factor in the expression for 〈k〉_n if 2r = (1 + q − s).

Numerical results for iSite networks.

The iSite algorithm was coded and networks were grown to compute averaged statistics. Examples of iSite networks generated by the algorithm are shown in Fig 14. The algorithm was then used to sample networks of size up to 200,000.

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Fig 14. iSite evolutionary networks.

The network on the left was generated with 4 iSites per node, p = 0.5, q = 0.1 and r = 0.8, and the network on the right was generated with 2 iSites per node, and with p = 0.5, q = 0.1 and r = 0.8. The order of the network on the left is 501 and on the right, 491. The network on the left has two nodes qualifying as hubs, of degrees {23, 25}, while the network on the right has none. The arrangement of nodes and bonds in these networks was created using the prefuse force directed lay-out in Cytoscape 3.4.0 [17].

https://doi.org/10.1371/journal.pone.0189866.g014

The connectivity 〈k〉_n of iSite networks for I = 3 iSites per node, and with p = 0.5, q = 0.4 and r = 0.3, is shown in Table 3. By Eq (5), . Least squares fit to the data in Column 2 gives , and (2 − γ) = 0.258. Solving for γ gives in the first instance γ = 1.735 and in the second γ = 1.742. Since 2r < 1 + q in this case, the mean field value of γ is γ = 1 + 2r = 1.6, close to these estimated values.

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Table 3. Connectivity data for iSite networks.

https://doi.org/10.1371/journal.pone.0189866.t003

Data for I = 5 and with the same values of (p, q, r) = (0.5, 0.4, 0.3) are shown in Table 3 as well (see Fig 15). Changing the value of I (the number of iSites per node) should not change the value of γ, and this appears to be the case here. A least squares fit to the data in Column 3 and determining γ as above gives γ = 1.737 and γ = 0.7498, very close to the values above.

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Fig 15. iSite evolutionary networks with I = 3, p = 0.5, q = 0.4 and r = 0.3.

Data on networks generated by the iSite evolutionary algorithm. In each case 500 networks were grown and the average degree sequence P_n(k) computed. The curves are plots of log P_n(k)/log(k + 1) against 1/log(k + 1) for n ∈ {3125, 6250, 12500, ⋯, 200000}. As k → ∞, then the curves are expected to pass through −γ on the y-axis, and its mean field value is γ = 1 + 2r = 1.6—this value is marked on the y-axis.

https://doi.org/10.1371/journal.pone.0189866.g015

If p = 0.5, q = 0.05 and r = 0.8, then 2r > 1 + q, and in this case γ = 2 + q. If the number of iSites per node is I = 3, then the data in Table 3 gives a constant value for 〈k〉, and for I = 5 a slightly decreasing numerical estimate. The mean field value of γ in these cases is 2.05, and a least squares fit gives γ ≈ 2.009 if I = 3 and γ ≈ 2.022 if I = 5 (where the coefficient of log n in the least squares fit is 2 − γ). These results are consistent with the mean field results obtained above, since it shows that the value of γ is close to 2 + q.