Friend of a friend models of network growth

One of the best-known models in network science is preferential attachment. In this model, the probability of attaching to a node depends on the degree of all nodes in the population, and thus depends on global information. In many biological, physical and social systems, however, interactions between individuals depend only on local information. Here, we investigate a truly local model of network formation—based on the idea of a friend of a friend—with the following rule: individuals choose one node at random and link to it with probability p, then they choose a neighbour of that node and link with probability q. Our model produces power-laws with empirical exponents ranging from 1.5 upwards and clustering coefficients ranging from 0 up to 0.5 (consistent with many real networks). For small p and q = 1, the model produces super-hub networks, and we prove that for p = 0 and q = 1, the proportion of non-hubs tends to 1 as the network grows. We show that power-law degree distributions, small world clustering and super-hub networks are all outcomes of this, more general, yet conceptually simple model.


Introduction
One of the important breakthroughs of network science at the start of the millennium was to formulate several models of network growth based on simple assumptions on how individuals connect. For example, Barabási and Albert used the assumption of preferential attachment-that individuals attach to or follow other individuals with a probability that increases with the number of links (or followers) they already have-to show that such a behaviour results in a power-law distribution of the number of attachments [1]. This distribution is long-tailed, which means that a few individuals become extremely popular, but not because they are intrinsically more attractive to attach to. Powerlaw distributions describe substantial inequalities between individuals, as a consequence of the rich getting richer [2,3]. There are several empirical examples of networks following power-law distributions [4][5][6][7][8][9][10][11][12][13][14]. discuss our results in the context of fitting power-laws to data, showing that our model produces powerlaws with empirical exponents ranging from 1.5 upwards and clustering coefficients ranging from 0 up to 0.5. This is consistent with many real networks and we discuss how data from networks might be interpreted in light of our results.

Related models
Models similar to, what we call, the friend of a friend model have been studied before in various contexts. The first very closely related model is the one proposed by Dorogovtsev and his colleagues, in which an edge of the graph is selected at random [20]. They found that as time goes to infinity the probability that a randomly chosen node has degree k is equal to 12 kðk þ 1Þðk þ 2Þ , i.e. the degree distribution has a power-law of degree 3. The Dorogovtsev et al. model is similar to the case where p = 1 and q = 1 in the friend of a friend model, but the distribution is different, since choosing one node at random gives a higher probability of choosing a node with lower degree than choosing an edge at random (as Dorogovtsev and his colleagues do). Another related model is triadic closure, studied by for example Bianconi et al. [22]. In this model, the first link is formed by connecting a new node to a randomly chosen existing target node, and a second link is formed by either connecting the new node to a randomly chosen neighbour of the target with a probability r or connecting to a completely different node with a probability 1 − r. When r = 1, this model is equivalent to our model with p = q = 1, but all other parameter combinations differ from our model.
Jackson & Rogers [32] were the first researchers to study the friend of a friend model. They applied continuous time mean-field techniques to derive a master equation for the model of directed networks. They showed that when p = 1 and q = 1, the process gives a power-law of degree 3. Here, we focus on the undirected case, but irrespective of this difference, as we shall soon see, Jackson and Rogers' analysis is incomplete, and in some places incorrect, for the case where p ≪ 1. For example, they incorrectly predict that for p = 0 and q = 1 a power-law with exponent 2 will be obtained (see eqn 5 and fig. 1 in [32]).
A similar mistake is made by Barabási [33], who claims in §5.9 of his book Network Science, that for p = 0 and q = 1, this model is equivalent to the preferential attachment model and should thus give a power-law with exponent 3. The veracity of this last claim is important, because it is used as part of a discussion of what Barabási calls 'the origins of preferential attachment'. The friend of a friend model is used by both Jackson & Rogers and Barabási as an argument to get round the limitation, stressed in the first paragraph of the introduction of this paper, that preferential attachment is a global mechanism, while most social and biological processes are local. In short, there is thus a reasonably widespread misunderstanding of the potential outcomes of the friend of a friend model, which require clearing up.
One model, which Krapivsky and Redner call, 'growing networks with redirection' or GNR [34], the friend to connect to is chosen uniformly at random with probability 1 − r and the friend of that friend is chosen with probability r [34]. m edges are added for every new node. This model is similar to our friend of a friend model with p = 1 − r and q = r. There is, however, not a one-to-one mapping between our model and this type of model, since we do not have the constraint that p + q = 1 and the average number of links added in our case is m = p + q. These authors appear to claim (in results section III) that this specific model has an exponent of 1 + (1/r), and thus, when p = 0 and q = 1, the exponent is predicted to be 2. A similar result is also argued for a closely related random walk model [35]. This particular prediction is not, though, particularly central to the Krapivsky & Redner [34] article, which focused mainly on what happens when r = A k is a linear function of k.
Krapivsky & Redner [36] went on to formulate a model where a new incoming node connects to a randomly selected target node, as well as to all outgoing neighbours of the target node. Their algorithm is similar to a directed version of the one we will study, but instead of attaching to the uniformly random chosen friend of a friend, it attaches to all nodes which that individual follows. In this case, the network generated by their model does give a power-law in-degree distribution with exponent 2.
In all of the above examples, the models either have a power-law stationary distribution, with an exponent which can be calculated using a master equation or mean-field approach [37] or, in cases where the underlying model does produce a non-stationary distribution, the authors do not analyse royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 9: 221200 this property of the model. Non-stationary networks often take the form of super-hubs, where one or a small number of nodes take all connections [33,37]. This process is also referred to as condensation [38][39][40]. As we outline in the Introduction, such super-hub distributions are one property we will demonstrate in friend-of-a-friend networks.
The first authors to explicitly state that models similar to friend of a friend can produce non-stationary distributions, in the sense that there does not exist a limiting power-law (or any other stationary) distribution, were Gabel et al. [41,42]. In what they call an enhanced redirection model, a new node u is either connected to a target node v with the redirection probability 1 − r or with r, node u is connected to node w that is followed by v. In this model, the parameter r is a function of the degree of node w and has the property that as r → 1, the degree of node w approaches infinity. The choice of r → 1 produces networks with several super-hubs and non-stationary degree distributions with numerically estimated exponent values of 1 < α < 2.
Similarly, Lambiotte et al. [43] and Bhat et al. [44], independently of each other, implemented a copying process by introducing a probability p L of (undirected) link formation between each of the neighbours of a randomly selected target node and the new added node. They observed sparse networks with power-law degree distributions for p L , 1 2 . But, most interestingly, they found that for p L ! 1 2 the resulting degree distribution was non-stationary. After this, Krapivsky & Redner [45] studied the p = 0 and q = 1 case of the friend of a friend model we look at here. The key features of this model is that it generates non-stationary networks with super-hub nodes. They found that the degree distribution has an exponent α that is strictly less than 2, i.e. a powerlaw with no mean. Specifically, their analysis show that the number of nodes with degree k, N k scales as where N is the network size. The value of α was approximated numerically and found to be 1.566.
Taking the literature mentioned above, together with further variations on this theme, we find a rich variety of useful results about models closely related to the 'friend of a friend' model we will now investigate. Most importantly, Krapivsky & Redner [45] have already provided useful numerical results about the case where p = 0 and q = 1.
It is thus important that we are clear about what we see as the contribution of the current work. We see the 'friend of a friend' idea as having a special role, because the mechanism is both local (it requires no information about the degree of any of the nodes attached to) and very simple to state (simpler even than the original network models [1,15,19]). It has also been used to motivate the origins of these more complicated/non-local models. Our contribution then is thus to provide a self-contained description of the important properties of the 'friend of a friend' model-which are surprisingly varied-and a mathematical proof of the non-stationary nature of the distribution when p = 0 and q = 1.

Simulation model
We now formalize the model described in the introduction. The mechanism of formation of the network is illustrated in figure 1. We start with a network of three mutually connected friends, as shown in figure 1a. We add one friend (or node) to the network per time step, using t to denote both the time step and the number of individuals. The new node (D) is added as follows. First, it picks one of the existing nodes (B) uniformly at random from all the t nodes as shown in figure 1b. The node D then forms an attachment to B with probability p. After that it picks randomly a neighbour (C) of the original node (B) and attaches to it with probability q. In the simplest version of this model, p = q = 1, i.e. D attaches to both B and C. Figure 1c shows a typical network example after t = 16 time steps for the same extreme situation p = q = 1.
In order to get an overall impression of the degree distributions generated by this model, we first run the simulation for representative parameter values and look at the resulting degree distribution. We use logarithmic binning to plot the histograms on logarithmic scales using the method of Newman [6]. The outcome of the simulations is shown as the blue lines in figure 2.
royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 9: 221200 We see that for p = 1 and q = 1 (figure 2a), where we attach to both the randomly chosen person and the randomly chosen friend, for degree k > 10, we get a relatively straight line in the log-log plot of the frequency distribution. As we decrease q, while keeping p = 1 constant, the frequency of nodes with higher degree decreases (figure 2b,c). Conversely, if we decrease p, while keeping q = 1 constant, higher degree nodes become more frequent (figure 2d,e). For p = q = 0.5, some nodes are not connected, but the ones that are have a frequency distribution reasonably similar to p = q = 1.

Master equation
In order to better understand how the degree distribution depends on the parameters p and q, we create a master equation estimate of the proportion of individuals with k friendships at integer time t as P k (t) [6]. We treat t = 3 as the first time step at which P 2 (3) = 3 and P k (3) = 0 for all other values of k (as in figure 1a). After t > 3 time steps, there will be t nodes and on average t( p + q) − 3 edges. In the limit of t → ∞, the expected ratio of edges to nodes tends to ( p + q).
We now consider the probability distribution for the number of neighbours of the chosen node B and C in turn. Since B is chosen uniformly at random, the probability that it has k neighbours is simply P k (t), i.e. the proportion of nodes with k neighbours. Now, let K C denote the number of neighbours of node C, the neighbours of B which are chosen. We want to know P(K C = k|B friends with C), i.e. the probability that C has k neighbours given that it is friends with B. This can be obtained using Bayes' theorem, If we assume that B is equally likely to be friends with any person in the network (we will come back to this assumption later, which does not always hold), then the probability B friends with C is Nodes are labelled alphabetically and linked with edges labelled in numbers. As shown in figure 1a, network formation starts at t = 3. Figure 1b shows that when the fourth node (labelled D) is added it first picks node B as a friend (blue dashed line of link number 4) then it picks node C as a friend of node B (red dotted line of link number 5). Figure 1c illustrates how this formation continues in which new incoming node join two earlier nodes with p = 1 and q = 1. So, node E links to two old nodes by first picking node B as a friend and then A as a friend of B, node F make link 8 with D then link 9 with node C etc. The formation continues until the last node P is connected to node A as a friend and H as a friend of A.
royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 9: 221200 5 since C friends with k out of all the t individuals in the network. It thus follows that where P k (t) is the prior probability of a node having k neighbours. We can calculate E[K C ] by noting that the mean or expected number of degrees added each time step is 2( p + q) (since edges are bidirectional). royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 9: 221200 Thus, finally, gives the probability of attaching to a node with degree k. The average number of attachments to a node with degree k is thus where the first term is attachment to the randomly chosen node, B, and the second term is attachment to the neighbour of B, the node C. Following Newman [6], we can now write a master equation for the evolution of the proportion of individual with k ≥ 3 friendships as The first term on the right-hand side (tP k (t)) gives the number of nodes with degree k. The second term describes the outcome when a node with degree k − 1 is linked to, and the third term is the outcome when a node with degree k is linked to.
Since edge formation between the new incoming node and the existing ones depend on p and q, it is also possible for the growing network N(k, t) to have nodes with k < 3. We can deduce the equations for the proportion of nodes with k = 0, k = 1 and k = 2 as follows. For k = 0, isolated nodes appear in the network when neither the target nor the neighbour node is chosen, i.e. with probability (1 − p)(1 − q), the network may be disconnected and may consist of several isolated components. At each time step, the proportion of isolated nodes P 0 (t) will increase on average by (1 − q)(1 − p) and decrease with pP 0 (t), giving ðt þ 1ÞP 0 ðt þ 1Þ ¼ tP 0 ðtÞ þ ð1 À qÞð1 À pÞ À pP 0 ðtÞ, for k ¼ 0: ð3:6Þ There are three possible cases for which a new incoming node chooses the target node and creates only one edge. The first case occurs with probability p(1 − q), i.e. when a target node is chosen but the neighbour is not. The second case, occurs with probability (1 − p)q, i.e. when a target node is not chosen but the neighbour is. As a result, the proportion of nodes with k = 1 is increased on average by p(1 − q) + q(1 − p). The third case is when the incoming node chooses an existing isolated (k = 0) target node and this occurs with probability pP 0 (t). The master equation of such nodes with degree k = 1 is given by ðt þ 1ÞP 1 ðt þ 1Þ ¼ tP 1 ðtÞ þ pð1 À qÞ þ qð1 À pÞ þ pP 0 ðtÞ À p þ q 2ðp þ qÞ P 1 ðtÞ, for k ¼ 1: ð3:7Þ Nodes with degree k = 2 appear when the new incoming node makes connection with both the target and the neighbour, i.e. with probability p · q. Otherwise, the master equation is similar to that for k ≥ 3, i.e.
ðt þ 1ÞP 2 ðt þ 1Þ ¼ tP 2 ðtÞ þ pq þ p þ q 2ðp þ qÞ P 1 ðtÞ À p þ q ðp þ qÞ P 2 ðtÞ, for k ¼ 2: ð3:8Þ Taking the limit t → ∞ so that P k (∞) → P k , equations (3.5), (3.6), (3.7) and (3.8) simplify to Expanding equation (3.9), we obtain a general asymptotic degree distribution, where P 2 is a constant value which depends on the values of p and q. It is computed from ð3:11Þ Using the gamma function, we write equation (3.10) as follows: Multiplying Gðð2ðp þ qÞ=qÞ þ 1Þ on both the numerator and denominator of equation (3.12) to obtain Using the beta function, we write equation (3.13) as follows: The beta function has the property that when x → ∞ and x ≫ y then B(x, y) ≈ x Ày GðyÞ. Thus, when k is very large, P k can be approximated as When the values of (2( p + q)/q) and (2p( p + q)/q) are integers we have where α = 3 + (2p/q) ≥ 3. In appendix A, we derive this same result in another way, using the mean-field approach. When p = q = 1, the solution of equation (3.9) for the case k ≥ 3 becomes exact given by The red lines in figure 2 show the distribution derived from the above master equation. For p = q = 1, we find a similar degree distribution as in the simulation, with a power-law exponent α = 5 ( figure 2a). However, the numerical estimate of the exponent of the power-law (α = 5) is not obtained for p = q = 1 because the maximum degree is approximately k = 100, which is not sufficiently large to reliably measure a power-law.
Decreasing q to 0.75 and 0.25 while holding p = 1 ( figure 2a and 2b, respectively), we again see similar trend in the analytical results as those observed in numerical calculations, although the slope of the distribution in the simulation is slightly steeper. However, figures 2d and, especially, 2e show that when q is maintained at 1 and p is reduced to 0.75 and 0.25 the slope of the distribution is less steep in the simulation than in the theory (we will return to this point in a later section). In figure 2f where p = q = 0.5, we still have α = 5 as that of figure 2a. We can also see that the slope of the numerical line is reasonably close to the analytical one.

Clustering
The local clustering coefficient (C v ) of an arbitrary node v with degree k v is measured as the ratio between the number of all edges that exist among all first neighbours of v (E v ) and the maximum number of potential links between the neighbours of v [15], which is k v (k v − 1)/2. Thus, We are interested in how the average local clustering coefficient depends on degree k. Hence we define a mean clustering over all nodes with degree k, The blue points in figure 3 show how the mean local clustering coefficient changes with degree k for different attachment probabilities, p and q. In all cases, the local clustering decreases as degree increases. When p = 1 and q = 1 (figure 3a), the clustering decays more slowly than when p < 1 or q < 1 (figure 3b-f ). When p = 0.25 and q = 1, or when p = q = 0.5 there is much less clustering, especially for nodes with high degree (figure 3e,f ).
We now derive an analytical approximation of the local clustering coefficient based on a method similar to the master equation. We consider the attachment illustrated in figure 1b, where the network has started growing. We first look at the case where p = q = 1 and will then adapt this result to the more general case. Whenever a new node is attached to an existing v, the degree of v increases by 1 and E v also increases by 1, because we also attach to one of v neighbours (assuming p = q = 1). Furthermore, all nodes have an initial value of E v = 1 and k v = 2 (since the two nodes they attach to are attached to each other with one edge, as in figure 1a). Thus, ð3:22Þ Therefore, This equation matches the simulation results, suggesting that the approximation is accurate (red line in figure 3a). When either p < 1 or q < 1, there are two ways in which a node v can increase the total number of links between its neighbours. The first is when v is the first node chosen (at random from all nodes), and then a connection is made to both v and its chosen neighbour. This occurs with probability pP k ðtÞq: The second way is when the first node is a neighbour of v, from which v is then chosen as the second potential attachment. This occurs with probability q k Á P k ðtÞ 2ðp þ qÞ p: Summing these two possible cases gives pP k ðtÞq þ q k Á P k ðtÞ 2ðp þ qÞ p, ð3:24Þ which is the expected average number of triangles attached to a node with degree k. The ratio between this (equation (3.24)) and the overall average number of attachments to the node (equation (3.4)) gives the expected increase in E v each time an edge is added to a node of degree k. As a result, The prediction based on this approximation is shown as the red line in figure 3b-f and partially captures the numerical results (given by blue squares). The network clustering coefficient C T is the average of local clustering coefficients C k given that k ≥ 3. This can be written as  royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 9: 221200 When p = q = 1, we have

Super-hub networks
While the master equation method gives a good approximation to the degree distribution and clustering for a range of values of p and q, it fails when q is close to one and p is close to zero (for example, figure 2e). This is at first surprising, because in, for example, Barabási's book Network Science, there is a claim that this model with p = 0 and q = 1 should produce a power-law with degree 3 (see 'related models' for details) [33]. Similar claims can be found in, for example, [32]. Indeed, Barabási refers to the friend of a friend model, which he calls the copying model, as providing the origins of preferential attachment. Why then does the simulation not agree with the master equation?
The way to answer this question is to look at whether or not the degree distribution is stationary (see 'related models' for background on non-stationary distributions [41][42][43][44][45]). To this end, we started by visualizing six network topologies, each generated with only 300 nodes, with p = 0 and q = 1. Figure 5 shows that diverse network structures arise, from tree-like ( figure 5a,b,e,f ) to star-like networks (figure 5b,d). They are, in general, characterized by one or more central hubs to which the majority of nodes are connected. The emergence of these structures can be explained heuristically, as follows. At the start of the simulation, we have three connected nodes. Initially, and quite quickly, one of these (or another of the nodes added near the start) becomes a hub: attracting more connections than the other. From then on, the added nodes will only have one attachment, i.e. to the hub, while the hubs are attached to a large number of nodes. For example, imagine the situation where we have 20 nodes, 2 of which are hubs and the others, non-hubs, have only one connection, directly to a hub. In this case, the probability that when we make the first random choice (i.e. the random target node) in the model that we choose a non-hub is 9 in 10 (since only 1 in 10 of the nodes are hubs). Then, since all non-hubs are attached to hubs, our new node will also attach to a hub (i.e. the non-hub has only one neighbour, so the random neighbour must be a hub). As a result, hubs increase their degree even further and are thus even more likely to be attached to. Far from being standard preferential attachment, where attachment is proportional to the number of edges, this attachment disproportionally prefers hubs over non-hubs. royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 9: 221200 Returning to our original derivation of the master equation (in particular, equation (3.3)), with an assumption that B (i.e. the random target node) was equally likely to be friends with any person in the network, leading us, in equation (3.3) to assume that the probability B is friends with C is proportional to k. While this assumption appears to hold when p is close to one, because there are lots of random attachments in the network, the simulations where p = 0 and q = 1 show that this assumption is wrong. Indeed, the probability that B is friends with a non-hub is very small.
To investigate what type of distributions arise when p is close to zero and q is close to one, we simulated the model and measured the properties of the resulting distribution. First, we simulated the p = 0 and q = 1 model with t = 1 000 000 nodes, five times. The average frequency of degree over all five simulations is shown as the blue line in figure 6a, while the degree distribution for each simulation that appeared is shown by the scattered green points. Computing the power-law exponent by the maximum-likelihood method proposed by Newman [6], with a lower cut-off value k = 10, gave α = 1.61. This is very different from the value of α = 3 obtained from theoretical predictions (shown as the red line in figure 6a). They are, however, relatively consistent with those of Krapivsky & Redner [45] who, as discussed in §2, found that the p = 0 and q = 1 generates non-stationary networks with super-hub nodes, and estimated an exponent α ≈ 1.566.
Some care is required in how we interpret the power-law fitting of α. Due to the effects of the condensation phenomenon [38][39][40], and, as we already saw in figure 5, the hub-like networks where P 1 (t) is the proportion of nodes with degree k = 1. The undefined quantity in this equation is P C (t), which is the probability that the second node (C) chosen has degree k = 1, given that the first node (B) chosen does not, i.e. P C (t) = P(K C = 1|K B > 1).
In appendix B, we prove that P C (t) ≤ P 1 (t). The key to this proof is to show, using Lagrange multipliers, that the case where the hubs (in this case defined as all nodes which have k > 1) have an equal number of connecting non-hubs (i.e. nodes with k = 1). In this case, if m is the number of hubs and t − m is the number of non-hubs, then Given that P C (t) ≤ P 1 (t), we can define a new master equation that has the property that P Ã 1 ðtÞ P 1 ðtÞ for all t. Now, as t → ∞, which has solution P Ã 1 ¼ 1. Thus the number of non-hubs goes to 1 in the limit of large t. Since P Ã 1 ðtÞ P 1 ðtÞ it thus follows that equation (3.28) also goes to 1. In an infinitely large graph, all but a finite number of nodes, become non-hubs.
Real-world graphs are, of course, finite. So it is still relevant to look at how clustering and the empirical power-law exponent depend on the parameters p and q. To do this, we simulated the model for t = 100 000 nodes for different values of p and q. Figure 6b shows the values of C T and α obtained. The red colour intensity in figure 6b is proportional to the product of p and q, highlighting that a high clustering coefficient and α values close to approximately 3.47 occur when both p and q are close to royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 9: 221200

Discussion
We have investigated the friend of a friend model for network formation and found a surprisingly rich diversity of resulting networks. The mechanism behind our model is both local and simple. We would claim that it is difficult to find a simpler mechanism: connections are made by first picking one person at random and then picking one of their friends at random. Yet, at a global network level, this model produces many properties which are considered very important for complex networks. By just varying p and q, we can simulate networks spanning from random networks through scale-free networks to super-hub networks. These networks can have average clustering coefficients ranging from 0 to 0.47 and exhibit small-world characteristics.
It is impossible to reconcile our results with some of the previous claims in the literature about complex networks. It appears that in the first articles investigating the model as we study here, the authors did not concern themselves closely enough looking into the discrepancy between numerical models and the mean-field approximation they made [32]. Instead, they claimed that the p = 0 and q = 1 model had the same mean-field model, and thus a power-law exponent of 3, as the Barabási and Albert's preferential attachment model [1]. Similar claims are repeated in Barabási's Network Science book [33], in a statement regarding the copying models. Both our numerical simulations and the proof that the proportion of nodes with degree k = 1 tends to 1 contradict these claims.
We believe that our model should prove useful help in navigating data collected on complex networks. There are still ongoing discussions on how scale-free networks should be characterized [26][27][28]. In the context of our model, we would simply encourage researchers to measure the slope of the empirical power-law α in their data together with the average clustering coefficient C T and see if it lies within the red shaded area in figure 6b. Power-laws with empirical exponents as large as approximately 1.5 are possible in our model, as are average clustering coefficients up to approximately 0.5. It is thus possible to identify (reverse engineer) the values of p and q consistent with empirical measurements of clustering and power-law exponent. The friend of a friend model with those values of p and q thus becomes a candidate mechanism for generating the network.
It is also worth pausing to think about our model in the context of the idea that the rich get richer. Often preferential attachment is associated with this concept, under which individuals who are seen by others to have social resources (i.e. friends) are more likely to attract more attachments. Power-laws, or long-tail distributions, are thus associated with the rich getting richer. The friend of a friend model generates both small-world clustering and power-laws without individuals explicitly preferring to attach to 'richer' individuals. It is the recommendation of others that leads to the power-law. In the most extreme variation of the model (p = 0 and q = 1), the distribution is even more skewed towards individuals who happened to attract the most attachments initially.
Data accessibility. The data are provided in electronic supplementary material [46]. Initially at time step t = t i , a new node with an average of k(t i ) = E[K C ] new degrees is added. In one time step, the total rate of increase of degree k is given by The solution of equation (A 1), gives the prediction of the degree distribution of the network, kðtÞ þ 2pðp þ qÞ q ¼ 2ðp þ qÞ þ 2pðp þ qÞ q t t i q=2ðpþqÞ : (A 2Þ Equation (A 2) can also be written as Suppose that new nodes are added to the network at equal time interval. The probability density function for any time t i falling within this particular time interval [t i , t] may be assumed to be Therefore, the probability that a node has degree k(t i ) smaller than k(t) (cumulative degree distribution) can be written as Substituting equation (A 3) into equation (A5), we have P(kðt i Þ , k) ¼ 1 À (2ðp þ qÞ þ ð2pðp þ qÞ=qÞ) 2ðpþqÞ=q t ð2ðp þ qÞ þ tÞ(k þ ð2pðp þ qÞ=qÞ) 2ðpþqÞ=q : (A 6Þ The probability density function (degree distribution) P(k(t)) is given by the partial derivative of equation (A 6) with respect to k. Thus, P(kðtÞ) ¼ @P(kðt i Þ , k) @k ¼ 2ðp þ qÞ q (2ðp þ qÞ þ ð2pðp þ qÞ=qÞ) 2ðpþqÞ=q t ð2ðp þ qÞ þ tÞ(k þ ð2pðp þ qÞ=qÞ) 1þð2ðpþqÞ=qÞ : (A 7Þ As t → ∞, P(k(t)) gets closer and closer to P(k). Therefore, P(k) ¼ 2ðp þ qÞ q 2ðp þ qÞ þ 2pðp þ qÞ q 2ðpþqÞ=q k þ 2pðp þ qÞ q Àð1þð2ðpþqÞ=qÞÞ : (A 8Þ In the limits of very large k, the degree distribution exhibits the extended power-law given by P(k) k þ 2pðp þ qÞ q Àð1þð2ðpþqÞ=qÞÞ k Àð1þð2ðpþqÞ=qÞÞ , for all k: (A 9Þ From equation (A 9), the scaling parameter α can be computed using the exponent of k, The value of α must always fall within the interval [3, + ∞). When p = 0 and q = 1, the model predicts α = 3. The master equation for directed networks is similarly derived, but with E[K C ] = p + q. This gives, α = 2 + ( p/q).