Preferential attachment is a stochastic process that has been proposed to explain certain topological features characteristic of complex networks from diverse domains. The systematic investigation of preferential attachment is an important area of research in network science, not only for the theoretical matter of verifying whether this hypothesized process is operative in real-world networks, but also for the practical insights that follow from knowledge of its functional form. Here we describe a maximum likelihood based estimation method for the measurement of preferential attachment in temporal complex networks. We call the method PAFit, and implement it in an R package of the same name. PAFit constitutes an advance over previous methods primarily because we based it on a nonparametric statistical framework that enables attachment kernel estimation free of any assumptions about its functional form. We show this results in PAFit outperforming the popular methods of Jeong and Newman in Monte Carlo simulations. What is more, we found that the application of PAFit to a publically available Flickr social network dataset yielded clear evidence for a deviation of the attachment kernel from the popularly assumed log-linear form. Independent of our main work, we provide a correction to a consequential error in Newman’s original method which had evidently gone unnoticed since its publication over a decade ago.
Citation: Pham T, Sheridan P, Shimodaira H (2015) PAFit: A Statistical Method for Measuring Preferential Attachment in Temporal Complex Networks. PLoS ONE 10(9): e0137796. https://doi.org/10.1371/journal.pone.0137796
Editor: Zhong-Ke Gao, Tianjin University, CHINA
Received: April 1, 2015; Accepted: August 21, 2015; Published: September 17, 2015
Copyright: © 2015 Pham et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited
Data Availability: The Flickr dataset is publicly available at the Koblenz Network Collection: http://konect.uni-koblenz.de/networks/flickr-growth.
Funding: This work was supported by grants from Japan Society for the Promotion of Science KAKENHI (26120523 and 24300106 to HS, URL: http://www.jsps.go.jp/english/index.html). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
The de facto maxim of network science is ‘all systems are networks.’ For those who are well-acquainted with this network view of reality, recognizing the network nature of virtually any real-world system borders on reflexive. But network scientists are not interested in studying just any old networks. Network scientists habitually confine themselves to the study of complex networks, that is, large-scale networks with emergent topological features that are not found to occur in simple networks . Complex network research turns more or less on the study of two related problems: first, the modelling of dynamical processes taking place on static complex networks in a manner that makes judicious use of known topological features , and second, the study of how topological features emerge in temporal complex networks and subsequently evolve over time . It is the latter problem that concerns us in this paper. To be plain: we focus on understanding the extent to which a process know as preferential attachment (PA) explains the emergence of those heavy-tailed degree distributions, best exemplified by power-laws, that are commonly observed in temporal complex networks across nature, society, and technology .
It is reasonable to describe a wide variety of real-world systems as temporal (complex) networks. In formal terms, a temporal network represents an evolving system as a sequence of static (complex) networks. This sequence, which we denote by G0, G1, …, GT, is best envisioned as a progression of snapshots of a given system taken at discrete time-steps, t = 0, 1, …, T. The model at this level of generality is purely descriptive and serves only to frame our thinking about complex systems. Raising it up to the status of an explanatory model with predictive power requires the setting forth of laws governing node generation and the formation of node-to-node connections in the transition from Gt−1 to Gt. This, however, is a subject to which we must return after a brief digression on some important empirical findings.
By the late 1990s, temporal networks in diverse domains of learning were observed to enjoy some universal topological features . The most publicized of these features at the time was the so-called scale-free property . A network, it will be noted, is scale-free when its degree distribution follows a particular kind of heavy-tailed function called a power-law . The purported universality of scale-free networks was soon, however, called into question when it was pointed out that distinguishing a power-law degree distribution from alternative heavy-tailed functional forms, such as the log-normal distribution, is a matter of considerable subtlety [5–8]. In light of this consideration, we will proceed on the supposition that the temporal networks in the world around us are commonly found to have heavy-tailed degree distributions with those following power-laws constituting an interesting special case.
What was interpreted as mounting empirical evidence for scale-free networks had everybody in network science scurrying to propose explanatory temporal network models that account for them in terms of domain independent processes. The first and most influential of these is known as the Barabási-Albert (BA) model . It is essentially a toy model that generates scale-free networks by appealing to a simple growth law for node generation and a special case of PA to govern the formation of node-to-node connections.
Consider a generalization of the BA model that allows for networks with a wider range of heavy-tailed degree distributions . Let Gt denote a static network at time-step t = 0, 1, …, T. Starting from a seed network G0, for each time-step 1 ≤ t < T, the network Gt−1 is ‘grown’ by a new node v′ that is subsequently connected to an already existing node to form Gt. Note that, in order to review the precise results about the degree distribution, here we only consider the case in which one node is added at each time-step. Later in the Materials and Methods section, we will describe a model that more closely reflects real-world networks, in which multiple new nodes can appeared at the same time. The PA rule then states that v′ connects to a node vk of degree k in Gt−1 with probability proportional to Ak: (1) The function Ak is called the attachment kernel. Strictly speaking, PA is only said to occur when Ak is an increasing function on average. It has been shown that the functional form of the attachment kernel can have dramatic effects on temporal network topology. Let us consider the log-linear model Ak = kα with attachment exponent α > 0 as an important example. When Ak = k (α = 1) the resulting network is scale-free . This is called linear PA and it is this form that it is assumed in the BA model. On the other hand, sub-linear PA (0 < α < 1) gives rise to a class of stretch-exponential degree distributions, while super-linear PA (α > 1) leads to a winner-take-all phenomena whereby a handful of nodes acquire all the connections . The limiting case Ak = 1 for all k (α = 0) corresponds to a version of the classical Erdös-Rényi random network model . In this case, the network is also not scale-free.
The point is that the Barabási-Albert family of models explain the emergence of a robust class of heavy-tailed degree distributions in temporal networks in terms of two generic processes: growth and PA. It is, moreover, plausible that these processes operate in many real-world temporal networks. In expanding social networks, for example, it is reasonable to presume that a person with many acquaintances already will tend to make new ones at a faster rate, than a person with few. Or, in citation networks of research papers, to take another example, a paper with many citations may acquire new citations more readily, than a comparatively lesser known paper. The same argument is easily specialized to other sorts of temporal networks by reasoning along similar lines.
Let us suppose for the sake of argument that heavy-tailed degree distributions of this kind prevail in growing temporal networks in the real world. This would not prove that PA was responsible for their generation, because it is always possible that a different process was at work. Rather, the presence of PA in the generative process must be confirmed by observation. In practice, this amounts to estimating the attachment kernel, Ak, from observed data.
The problem of estimating Ak from observed data is one of great importance to the working network scientist. For one thing, there are practical applications in link prediction algorithms , and more generally, it gives us valuable insights into the global characteristics of networks . But the core questions surrounding the PA process are these: does PA actually exist in real-world temporal networks; or, in other words, is Ak really increasing in k? If so, then what elementary functional form, if any, does it take? Is it the widely accepted log-linear model Ak = kα, or something else? In order to investigate these questions scientifically, a rigorous statistical method to estimate the functional form of Ak with confidence bounds is necessary. However, to the best of our knowledge, no such method has yet been advanced in the literature.
Nevertheless, detecting the hand of PA in temporal networks has attracted the attention of many researchers, and a number of estimation methods for Ak have been proposed [12–17]. Table 1 shows a summary of the existing methods. The primary drawback of most of these methods is that they explicitly assume the log-linear form Ak = kα and focus only on estimating the attachment exponent α [14–17]. Massen et al.  used a fixed point iterative algorithm for the purpose, Sheridan et al.  a Markov chain Monte Carlo method, Gomez et al.  a maximum likelihood (ML) approach to estimate the value of α. Lastly, Kunegis et al.  estimated α by fitting a function that relates the number of new connections a node obtains in the future with its current degree.
The remaining methods [12, 13] do not assume any functional form for Ak. As invaluable as these methods are, they are ad-hoc methods that are not without their problems. Newman’s method  is among the first methods proposed to estimate Ak. In real-world examples, Newman’s method appears to be able to estimate Ak well for small k, but systematically underestimates Ak when k is large [18, 19]. This is thought to be an artifact of the method . We will explain the reason for this artifact in detail below. On the other hand, while Jeong’s method  is the most widely-used method in practice [19–31], as we will explain in the Materials and Methods section, it suffers from a kind of bias-variance trade-off.
Our contributions in this paper are twofold.
First and foremost, we propose a nonparametric estimation method called PAFit that employs the ML approach of  to fit the functional form of the attachment kernel. PAFit is nonparametric in the sense that it does not make any assumptions on the functional form of Ak. The method works instead by estimating the value of Ak separately for each k. The algorithm underlying PAFit is the Minorize-Maximization [33, 34] (MM) algorithm. We prove that our algorithm converges to a global maximizer of the log-likelihood function and provide a fast approximation to the confidence intervals of the estimated attachment kernel . This last step is essential for the analysis of large datasets. What is more, we present the results of extensive Monte Carlo simulations that demonstrates how PAFit outperforms the state-of-the-art methods, namely Jeong’s method and Newman’s method, at the task of estimating Ak for a variety of different functional forms. Lastly, we used PAFit to analyze a publicly available Flickr social network dataset , and found that the estimated attachment kernel differed considerably from the functional form Ak = kα. This result immediately suggests that it is important to look beyond the classical log-linear form when modelling the attachment kernel.
A second contribution of ours that should not be overlooked is an explanation of the artifact observed in the Newman’s method. In particular, we show that this artifact can be completely eliminated by a simple mathematical correction. This is important because researchers have been using Newman’s method to estimate Ak in real-world networks for the better part of 15 years. It will be noted that while PAFit outperformed the corrected version of Newman’s method in the simulated examples, the corrected version was found to work better than uncorrected version.
In summary, PAFit is a statistically sound method for estimation of the attachment kernel in temporal complex networks. We have demonstrated its potential in this paper to uncover interesting findings about real-world complex networks. This method is implemented in the R package PAFit .
Before entering into the details of PAFit in the next sections, we note some recent developments in the field of complex networks. In this paper, we estimate the growth mechanism, in particular the PA mechanism, of a network given its growth process, that is, the observed data are the final network and all previous networks in the growth process. There is another line of research that considers a somewhat opposite problem: the underlying network is unknown and the task is to estimate it. This line of research includes, for example, estimating the unknown gene regulatory network from observed omics data [37, 38]. Complex networks have also found applications in many other areas, including economics , image analysis  and epidemiology [2, 3, 41, 42]. Beside the scale-free property, which only concerns the degree distribution of a network, other topological features which concern higher order of information have also been studied extensively in the literature [43–46].
Materials and Methods
In this section, we begin by introducing the temporal network model underlying the PAFit method. We follow that with an illustrative example to demonstrate the workings of PAFit in comparison with previous methods. From there we explain the mathematical details of PAFit and call attention to its important properties. The similarities between PAFit and previous methods are discussed. It is there that the correction to Newman’s method is outlined. Next, fast calculations for the confidence intervals of the estimated values are explained. We then proceed to discuss binning and regularization, which are two important techniques for stabilizing the PAFit estimation. Finally, we mention how to estimate the attachment exponent α of the log-linear model Ak = kα from .
The general temporal model
The statistical estimation method we present in this paper is tailored for the following temporal model for directed networks. Note, however, that our method can be easily adapted to work for undirected networks. The model is a simplified version of a model that we introduced in a previous publication . Starting from a seed network at time-step t0 = 0, we grow the network by adding n(t) nodes and m(t) edges at every time-step t, for t = 1, …, T. Note that our method allows m(t) to be consisted of both new edges that emanate from the n(t) new nodes and emergent edges between existing nodes. This is important since a large number of real-world networks have emergent edges between existing nodes. At time-step t, the probability that an existing node v with in-degree k acquires a new edge is (2) where Ak is the value of the attachment kernel at degree k. We call this model the general temporal (GT) model. The GT model includes a number of important network models as special cases. When Ak = kα, the GT model corresponds to Price’s model [47, 48] or Barabási-Albert (BA) model  in the undirected case. Furthermore, when Ak = 1 for all k, then it reduces to the classical Erdös-Rényi random network model .
Here we note a quick remark on an assumption about the joint probability distribution of n(t) and m(t). We will return to this assumption later when we discuss the MLE for Ak. Let θ(t) denote the parameter vector that governs the joint distribution of n(t) and m(t). We assume that θ(t) does not involve Ak. This very mild assumption allows broad and realistic models for m(t) and n(t). For example, m(t) and n(t) can be random variables whose means and variances depend on Gt−1.
A final remark is that, although we do not incorporate deletions of nodes and edges into the formal specification of the GT model, this is purely for simplicity. The PAFit method is able to estimate Ak even when there are deletions of nodes and edges, as long as the deletion mechanism is independent of the addition mechanism and the parameters govern the deletion mechanism do not involve Ak, which are reasonable assumptions. The current R implementation of PAFit  can be easily adapted to work in these cases.
An illustrative example
In this section we present a simulated example of a GT model network. The purpose being to demonstrate the workings of our proposed method, PAFit. The network generative process follows the GT model with Ak = 3(log(max(k, 1)))2 + 1. Starting from a seed network with 20 nodes, at each time-step, m(t) = 5 new edges and n(t) = 1 new node were added, until a total of T = 2000 nodes is reached. We measure how well a method performs by using the average relative error between the true Ak and the estimated , defined as where K is the maximum degree that appears in the growth process.
We first apply Jeong’s method. Here we choose the time window between when the 1500-th and 2000-th nodes are added. After a suitable normalization, the histogram of the degrees of nodes to which new edges appeared in this window connect will give us . The average relative error eA for Jeong’s method is 0.337, which is the highest among four methods considered in this example. From Fig 1A, one can see that Jeong’s method did capture the shape of Ak, but the estimated function was sparse and fluctuated considerably. These are inherent drawbacks of the method that follow from using only a small time window to estimate Ak.
A: Jeong’s method. B: Newman’s method. C: Corrected Newman’s method. D: PAFit. The solid line depicts the true model. The plots are on a log-log scale. The gray vertical lines are the estimated confidence intervals of the estimated values by PAFit. Confidence intervals are not available in the remaining methods.
Second, we apply Newman’s method. His method can be interpreted as estimating the attachment kernel by a weighted sum of multiple histograms created at multiple time-steps. In Fig 1B, follows the true function very closely, but the value of Ak starts to fall off when the degree k becomes large. This phenomenon can also be seen in other papers [18, 19], and is thought to be an artifact of the method . The value of eA for Newman’s method is 0.168, which is the second highest error among four methods used in this example. With a simple correction, we can eliminate this artifact. As can be seen from Fig 1C, the corrected Newman’s method estimated the attachment kernel well the entire range of k. The value of eA for this corrected version is 0.016, which is much smaller than those of Jeong’s method and the original method of Newman.
Unlike all the other methods, PAFit estimates the attachment kernel by maximum likelihood estimation (MLE), and can give quantifications of the uncertainties in the value of in the form of confidence intervals. In Fig 1D, the estimated attachment kernel follows the true function comparatively well, even in the high degree region. The average relative error eA of PAFit is 0.015, which is the smallest error among four method considered here.
Maximum likelihood estimation
Here we derive the MLE for the GT model as described in the previous section. Recall that K is the maximum degree that appears in the growth process. Let A = [A0, A1, …, AK] be the parameter vector we want to estimate. The likelihood of the data at time-step t ≥ 1 is the product of P(m(t), n(t)∣Gt−1, θ(t)) and P(Gt∣Gt−1, m(t), n(t), A). The likelihood of the data at the starting time t = 0 is the product of P(m(0), n(0)∣θ(0)) and P(G0∣m(0), n(0), θ*), with θ* the parameter vector governing the distribution of G0. Thus the log-likelihood function of the dataset is (3) As previously mentioned, we assume that θ(t), the parameter vector governing the joint distribution of m(t) and n(t), does not involve A. We also assume that θ* does not involve A. When deriving the MLE of A, these two assumptions allow us to ignore all but the first term in the right hand side of Eq (3). At the onset of time-step t, denote mk(t) and nk(t) as the number of new edges that connect to a node with degree k and the number of existing nodes with degree k, respectively. The term P(Gt∣Gt−1, m(t), n(t), A) corresponds to a multinomial distribution. This follows from the observation that given m(t), the quantities m0(t), m1(t), …, mK(t) follow a multinomial distribution with parameters p0(t), p1(t), …, pK(t), where pk(t), the probability that a newly added edge at time t connects to a node with degree k, is (4) After dropping constant terms, we can write down the log-likelihood function in detail: (5) Note that the Ak are only identifiable up to a multiplicative constant, as can be seen from Eq (4). We can enforce uniqueness by fixing Ak = 1 for some k that satisfies . In practice, we can often set, for example, A1 = 1.
Proposition 1 Any A′ that satisfies ∂l(A)/∂A = 0 is a global maximizer of l(A).
The implication of this proposition is that even though the log-likelihood function l(A) might be not concave, any local maximizer is guaranteed to be a global maximizer.
Proof. We consider the one-to-one re-parametrization β = β(A) = [logA0, …, logAK]T. From Eq (5) and the inverse relation Ak = expβk, the log-likelihood function under this new parametrization is The second derivative of the log-sum-exponential term can be shown to be non-negative semi-definite (a similar calculation can be found on page 74 of ). Therefore the second derivative Δ2 lβ(β) of lβ(β) is non-positive semi-definite. Thus lβ(β) is concave. Now the likelihood equation of the re-parametrized log-likelihood function lβ is (7) One can see that any A′ satisfying Eq (6) will give a reparametrization β′ that satisfies Eq (7). This β′ is a global maximizer since lβ(β) is concave. Thus the A′ is also a global maximizer.
Turning now to how to solve for Ak, since Ak appears in both sides of Eq (6), an explicit solution for A is difficult to obtain. Define as the estimated parameter vector at iteration i. Starting from some initial value A(0), at iteration i ≥ 1 we can iteratively update using Eq (6): (8) We repeat Eq (8) until the following convergence condition is met. (9) This iterative algorithm turns out to be an instance of Minorize-Maximization (MM) [33, 34] algorithms, which are used in maximization problems. In MM algorithms, instead of maximizing a complicated objective function, at each iteration we find and maximize a minorize function of the objective function. A minorize function is a function that is equal to the objective function at the current point, and is the lower bound of the objective function at all other points. The minorize function is often chosen so that we can maximize it easier than the original objective function. For our problem, we have the following proposition:
Proposition 2 Using the aforementioned algorithm, the log-likelihood function is guaranteed to increase with number of iterations. In particular, Furthermore, the stopping point A(M) which satisfies A(M) = A(M−1) is a global maximizer of l(A).
Proof. We find a minorize function Qi(A), which is a function that satisfies l(A) ≥ Qi(A) for all A, and l(A(i)) = Qi(A(i)). Define (10) Applying the inequality −logx ≥ −logy − x/y + 1, ∀x, y > 0 with and , one can verify that l(A) ≥ Qi(A). By substitution, one can check that l(A(i)) = Qi(A(i)). It is also easy to verify that the A(i+1) computed by Eq (8) is the solution of ∂Qi/∂A = 0, and thus satisfies A(i+1) = argmax Qi(A). Therefore, Suppose that the algorithm stops at iteration M such that A(M) = A(M−1). From Eqs (8) and (6), we can see that A(M) satisfies Eq (6). Combining this with Proposition 1, this means that A(M) is a global maximizer l(A).
It is instructive to look at Eq (8). If we only have one time-step t, then the MLE solution is Ak = mk(t)/nk(t), which is an intuitive formula. When there are multiple time-steps, things become less obvious. In this case, the MLE solution for Ak sums up all the mk(t) then divides this number by a weighted sum of the nk(t). Looking at the weights, we notice that a weight is larger when the number of new edges m(t) is large and the normalizing factor is small. This can be interpreted as the MLE solution naturally emphasizing nk(t) of the time-step t that has a lot of information.
In this section, we discuss in detail the two most important methods for estimating the attachment kernel: Jeong’s method and Newman’s method. We then provide a simple mathematical correction to the Newman’s method. Finally, we discuss how PAFit fits in the framework of these previous methods.
Jeong’s method  estimates Ak by building a histogram of the degrees of nodes to which new edges connect. They choose a time window T0 and call all nodes in the network at that time T0-nodes. Next they choose a time window T1 > T0 and a window length ΔT ≪ T1, then call all nodes added in the interval [T1, T1 + ΔT] the T1-nodes. For a degree k, they record the number of times a degree k T0-node is linked to by a T1-node, and then normalize this number by nk(T0) to give . No guidelines are offered for choosing T0, T1 and ΔT.
This method very clearly suffers a type of bias-variance trade-off. In order to negate the effect of fixing the degrees of T0-nodes at time T0, one should use a very small window time ΔT. But if the number of new edges in ΔT is small, then the estimation will be unstable. We can increase the number of new edges by increasing the length of ΔT, but this will inevitably introduce a bias into the estimation.
One natural way to overcome this drawback is to use multiple histograms calculated at multiple time-steps, but then the question arises as to how to combine those histograms into a single estimation of Ak while the normalizing factor at each time-step depends on A (and thus unknown). Newman’s method provides an answer to this question.
Instead of using only one histogram at a single time window, Newman’s method can be interpreted as estimating Ak by a weighted sum of multiple histograms created at multiple time-steps. Let N(t) denote the number of nodes in the network at the onset of time-step t. Consider creating a histogram at time-step t. Denote Ak(t) as the value of Ak at time t. Newman’s argument, written in the notation used here, is that Ak(t) can be estimated as (11) if nk(t) ≠ 0 or Ak(t) = 0 if nk(t) = 0, using the following approximations . (12) The final result of the procedure proposed by Newman  is equivalent to the formula (13) with the weight wk(t) = m(t)1nk(t) ≠ 0, using the convention 0/0 = 0. Note that, in Newman’s argument, the normalized constant C in Eq (13) is a constant that does not depend on k. In real-world examples, this procedure seems to work well for small k, but the value of Ak starts to fall off when k is large [18, 19] (cf. Fig 1B). Although this phenomenon is often thought to be an artifact of the method , an explanation of its cause has yet to be provided.
The corrected Newman’s method.
Surprisingly, we can completely eliminate the artifact in Newman’s original method by a applying a simple mathematical correction (cf. Fig 1C). Immediately from Eq (13), one can recognize that the summation is incorrectly normalized. The constant C should depend on k. The correct normalization is as follows: (14) Newman’s original method (see Eq (13)) implicitly assumes that all the degrees already appear in the network from time t = 0. This is of course not generally true. In a typical network that grows from a small initial network, the small degrees will appear far sooner than the large degrees. In such cases, the sum ∑t m(t)1nk(t) ≠ 0, which is a weighted sum of the number of time-steps that contain nodes with degree k, is larger for small k and smaller for large k. Forgetting to normalize each Ak in Eq (13) by ∑t m(t)1nk(t) ≠ 0 will cause the waterfall artifact as observed in Fig 1B. In Eq (14), correctly normalizing each Ak by this number completely eliminates the artifact. We will prefer to Eq (14) as the corrected Newman’s method.
Relation of PAFit to previous methods.
Using almost the same route as in deriving Newman’s method, one can deduce a similar equation to PAFit’s MLE Eq (6). Starting from Eq (12), mk(t) is equal to Ak(t)m(t)nk(t)/N(t). Summing over all time-steps t, we see that is equal to . Assuming Ak(t) = Ak for all t, we have (15) The only difference between Eqs (15) and (6) is the normalizing factor at time t: versus N(t). This difference is the result of the approximation in Eq (12).
If the ratio is independent of t, then Eqs (15) and (6) are equivalent. Notice that is equal to where nj(t)/N(t) is the proportion of nodes with degree j at time t. Interestingly, this proportion has been shown to be independent of t when t is large in the case of asymptotically linear or sub-linear attachment kernels, that is, Ak ∼ kα with α ≤ 1 . In such cases, Eqs (15) and (6) are asymptotically equivalent. The two equations are not asymptotically equivalent when Ak is super-linear, that is, Ak ∼ kα with α > 1, since in this case the proportion nj(t)/N(t) depends on t .
Fast approximation of the confidence intervals
We provide a fast approximation for the confidence intervals of . Our approximation works quite well in practice. In many real-world datasets, our approximation can provide a huge gain in speed. In standard statistical theory, the confidence intervals of can be calculated as the diagonal entries of the inverse of D, where D = −∂2 l(A)/∂A∂A. Define the following matrices B, C, U as: then we have (16) The form of Eq (16) makes it quite natural to calculate the inverse of D by the Woodbury formula : (17) Note that B and C are both diagonal matrices. This makes inverting them trivial. A direct inversion of D will require O(K3) operations, while using Eq (17) will require O(KT2). We propose to approximate D by B. This reduces the required number of operations to O(K).
Estimation of the attachment exponent
Given and their estimated variances vk, if we assume that the attachment kernel is indeed Ak = kα, it is an important task to correctly estimate the attachment exponent α. Jeong’s method proposes to fit kα+1 to the cumulative function Π(k) = ∑j:j ≤ k Aj in order to estimate α.
To take advantage of the estimated variances given by PAFit, we propose to find α by weighted least squares method. The variance of can be estimated by . The uncertainties in can be naturally incorporated into the fitting process by using weights that are inversely proportional to variances of . In particular, we minimize the following objective function with respect to α and d: It is important to note that the idea of using estimated variances as weights in weighted least squares methods can also be applied when estimating parameters of models that are different from the model Ak = kα.
For more stable estimation of the attachment kernel, we use logarithmic binning in order to let the Ak borrow information from the ones that are near them. In logarithmic binning, the length of the i-th bin is c times the length of the (i − 1)-th bin. We freely choose the number of bins B, then c is determined from B and the maximum degree K. We set Ak = ωi for all k in the i-th bin. Choosing B small helps with stabilizing the estimation result at the risk of loosing fine details in the attachment kernel. For the sake of simplicity we have been using Ak in our equations, but the readers should keep in mind that unless stated otherwise, we always use binning, so in fact Ak should be replaced by ωi.
After binning is performed, another important technique to reduce the variances of the estimated result is regularization. We add the following regularization term (18) to the log-likelihood function l(A) (see Eq (5)), and maximize the resulted function. This penalty term penalizes the second order differentiation of logAk, and by doing so, it encourages linearity in logAk.
The weight wk can be any positive number. In this paper, we set wk to be proportional to the number of data points of the degree k: . Intuitively, we want to emphasize the regularization where there is plenty of data (when k is small) and de-emphasize it where the data is scarce (when k is large).
We are still able to derive an MM algorithm for maximizing the penalized log-likelihood function. The details are given in S1 Appendix. As a property of MM algorithms, the penalized log-likelihood function is also guaranteed to increase with number of iterations.
Turning now to how to choose λ, the ratio of the strength of the regularization term (measured by λ) and the number of observed data (measured by ) can be used as a heuristic, reasonable criterion for choosing λ. The ‘right’ amount of regularization depends on many factors: the number of data points for each degree, the number of time-steps, the shape of the true attachment kernel, etc. We found that a reasonable amount of regularization with ratio ranges from 0 to 1 often leads to satisfactory results.
Results and Discussion
Monte Carlo simulation
Here we compare five methods: Jeong’s method, Newman’s method, the corrected Newman’s method, PAFit without regularization (ratio = 0) and PAFit with regularization (ratio = 0.1). We examine two different bin settings: B = 100 and B = 20. And we perform comparisons in three different functional forms of the true attachment kernels Ak. Table 2 shows the true attachment kernels we used in this experiment. For each true attachment kernel, we generate 100 networks according to the GT model. Starting from a seed network with 20 nodes, at each time-step, m(t) = 5 new edges and n(t) = 1 new node were added, until a total of 2000 nodes is reached. We compare the average relative error, defined as , between the true value Ak and .
We note some remarks about the implementation of the methods. First, due to the inherent sparse nature of Jeong’s method, even with binning, sometimes the estimated result of Jeong’s method has more 0 than those of the other methods. In such cases, to ensure fair comparison, we perform linear interpolation on the log scale to interpolate the value of those zero-value . Second, in Jeong’s method we choose T0 as the time when the 1500-th node is added, T1 = T0 + 1 and ΔT = 500. Third, convergence condition for PAFit is ϵ = 10−5.
The result is shown in Fig 2. Overall, PAFit with regularization outperformed all remaining methods. This suggests that a small amount of regularization is indeed needed to reduce the error in the estimated result. We also notice that binning helps reduced the error in all methods. The fewer number of bins we used, the better the estimated result was found to be.
A: B = 100. B: B = 20. See Table 2 for the details of the true attachment kernels Ak used here.
PAFit without regularization performed better than the corrected Newman’s method. In the functional form Ak = kα, when α is up to about 1.2, PAFit without regularization delivered almost the same error as that of the corrected Newman’s method, but when α > 1.2, PAFit without regularization was better, especially when the number of bins is 20. In the remaining two functional forms, PAFit without regularization also performed better or at least similar to the corrected Newman’s method.
When B = 100, except the functional form Ak = min(100, max(k, 1))β, Jeong’s method performed reasonably well in comparison with PAFit without regularization and the corrected Newman’s method. When B = 20, Jeong’s method became worse than those two methods. Newman’s method (uncorrected) performed the worst in almost all cases, due to its underestimating Ak when k is large.
The Flickr social network
In this section we present the results from our analysis of a publicly available Flickr social network dataset . It consists of a simple directed network of friendship relations between Flickr users. The dataset contains T = 133 days of growth. Table 3 shows some important summary statistics of the dataset. The results are shown in Fig 3. We also implemented a quasi-Newton speed-up scheme  for the MM algorithms used in PAFit. The convergence condition for PAFit is ϵ = 10−7.
A: Jeong’s method. B: Newman’s method. C: Corrected Newman’s method. D: PAFit. The plots are on a log-log scale. The solid line corresponding to Ak = k is plotted as a visual guide.
For Jeong’s method, we use all the available data by choosing T0 = 0, T1 = 1 and ΔT = 132. It found a sub-linear attachment kernel . The value of fluctuated considerably for large value of k, however. Note that the domain of of Jeong’s method is smaller than in other methods, since the degrees of T0-nodes are fixed at T0.
In the estimated result of Newman’s method, we once again spotted the falling off when k is large. This phenomenon is, of course, completely eliminated in the corrected Newman’s method.
For PAFit, we performed regularization with ratio equal to 0.1. Although the estimation result of PAFit looks very similar to that of the corrected Newman’s method, the in the high degree region of PAFit are less fluctuated and more compact than those of the corrected Newman’s method. It is worth noting that we spotted a clear signal of deviation from the log-linear model Ak = kα here.
We proposed a statistically sound method, called PAFit, for estimating the attachment kernel Ak in temporal networks. The method is nonparametric in the sense that it does not assume any particular functional form for Ak. In this way it is able to detect different types of functional forms. We proved that the log-likelihood function is concave under a suitable re-parametrization, and provided a Minorize-Maximization algorithm for its maximization. The proposed algorithm is shown to increase the log-likelihood function monotonically with the number of iterations. It also has the property that if it converges, it will converge to global maximum of the log-likelihood function. We also investigated binning and regularization, and showed that these two simple techniques considerably improve the quality of the estimation. We reported clear evidence for the presence of PA in the Flickr social network. We also found that the functional form of the attachment kernel differs from the classically assumed log-linear form, Ak = kα.
In this paper, we focused on estimating the attachment kernel. Another important ingredient related to temporal network evloution is node fitness [52–55]. In the future, we would like to consider a natural generalization of the method of this paper, in which we can propose a method to estimate jointly the attachment kernel and node fitness. Another potential research direction is theoretical justifications of the superiority of PAFit over Jeong’s method and Newman’s method. We expect that PAFit and its future extensions will become useful tools for analyzing many types of temporal networks in the real-world.
Conceived and designed the experiments: TP PS HS. Performed the experiments: TP PS HS. Analyzed the data: TP PS HS. Contributed reagents/materials/analysis tools: TP PS HS. Wrote the paper: TP PS HS.
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