Introduction

Massive websites – Facebook, Twitter, MySpace, LinkedIn, Flickr, Orkut, Google+, Weaklink, just to name a few – are booming in the past few years, where millions of users and their interactions naturally form the so called online social networks (OSNs)1,2,3. For OSNs, one important characteristic is the strong interplay between the user behaviour and the network topology4. On the one hand, the user behaviour is affected by the topology-dependent information flowing in the networks5,6,7,8; on the other hand, the network topology continually evolves as a natural consequence of network dynamics8,9,10. Due to this feature, OSNs exhibit certain correlation patterns during evolution, such as the highly skewed degree distributions11,12,13, the generalized Gibrat's Law14, assortativity/disassortativity11,12, etc, which are of great importance for us to understand the possible generic laws governing the organization and evolution in networked systems15.

Recently, two interesting phenomena in OSNs have attracted much attention. The first one is related to the assortativity/disassortativity property of the network, which is an important structural measure characterizing the degree correlation between pairwise nodes. Mathematically, the assortative coefficient can be defined as the Pearson correlation coefficient averaged for all pairs of adjacent nodes in the network. As shown in Table I, it is reported that some OSNs (e.g., Twitter and Cyworld) show negative or neutral assortative coefficients11,12,13,16,17,18,19 and some OSNs, such as Weaklink11 and Google+ (G+)12, even convert from being assortative to being disassortative during evolution. These findings challenge our traditional knowledge20,21 that biological and technical networks (e.g., financial networks22) are disassortative, while social networks (e.g., acquaintance networks23) are assortative. Secondly, the scale-free property is of great importance for a network, which can be characterized by a power law exponent γ as in p(k) ~ k−γ, where k and p(k) are node degree and the distribution of degree, respectively. Under the thermodynamical limit, i.e., the network size N → ∞, the mean degree of a scale-free network will diverge when γ ≤ 2. Therefore, γ = 2 is an important boundary and scale-free networks can be classified into dense (γ ≤ 2) and sparse (γ > 2) accordingly. Previously, many scale-free networks are found to be sparse24. However, as shown in Table I, some large OSNs, e.g., YouTube (YT), Digg and LiveJournal (LJ), turn out to be dense scale-free networks with γ < 213,19,25.

Table 1 Properties of typical OSNs, including the number of nodes N, the average degree 〈k〉, the average shortest path 〈d〉, the exponent of power law for out-degree (in-degree) γout (γin), the average clustering coefficient 〈c〉 and the assortative coefficient r, which is defined as the correlation between out-degree and in-degree as the links in OSNs are directional. The empirical data sets analyzed in this paper are also listed here, i.e., Flickr, FriendFeed (FF), aNobii and Epinions
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In Table I, the basic statistical properties for 14 popular OSNs are listed. It is found that these OSNs basically share common properties observed in real world networks, such as power-law distribution of degrees, large clustering coefficient and small average shortest path. However, two features, i.e., negative or neutral assortative coefficients and γ < 2, also turn out to be typical. In order to obtain insights into the evolution patterns of real OSNs, it is desirable to set up a dynamical model which could reproduce the properties and dynamics observed in real OSNs. Previously, the power law distribution of degrees1,2,3,26,27,28 and the disassortative correlation11,29 have been separately studied in theoretical models and in most models the exponents γ of degree distributions are larger than 2 (see Table III of Ref. 1 and the corresponding references). Recently, some theoretical works discussed the relevant properties of networks with specific functions to determine the degree distribution of the nodes30,31,32. However, attentions have not been paid to the dynamical origin of dense and/or disassortative OSNs, especially the transition from assortative to disassortative during the evolution of real networks.

Recently, we proposed a dynamical model based on empirical analysis of real OSNs such as Flickr and Epinions. It is shown that this simple reaction-diffusion-like model can reproduce statistical properties consistent with real data33. In this present paper we investigate, through modeling and simulations, the two remarkable observations that some OSNs are dense and/or disassortative. Specifically, based on extensive empirical analysis of real network data, including Flickr, FriendFeed (FF), ANobii and Epinions, we set up an evolution model, aiming at reproducing the above properties observed in OSNs. In the model, we characterize the user behaviour (local dynamics) in the OSNs by a state function. By considering a mechanism of local interplay generating new links, i.e., the formation of triadic closure, we are able to describe the network evolution as a reaction-diffusion-like process, in which the network dynamics and topology evolve simultaneously and interdependently. As a natural consequence of the coevolution, the resulting networks exhibit the typical properties observed in real OSNs. Specifically, we show that the network is capable of making the transition from being sparse to dense and from being assortative to disassortative during the evolution. We also offer some heuristic explanation for the above behaviour of the OSNs in our model.

Although the current model shares the same framework as in Ref. 33, i.e, based on the reaction-diffusion-like local interaction pattern, we emphasize that there are differences between them. Firstly, the modeling in Ref. 33 deals with typical dual-component networks consisting of users and items, while in this work, we only consider the social network, i.e, the user connections in the OSNs. Secondly, Ref. 33 mainly investigated how the user connections, i.e., the social network, influence the formation of cross links (connections between users and items) and the dynamical correlations and patterns among different types of degrees. In this paper, we focus on the dynamical origin of the transition from assortative to disassortative and from sparse to dense in the OSNs characteristics. In addition, in the current model, we introduce the general Fermi function to simulate the diversity of user dynamics, which should be more reasonable than the random connection in Ref. 33.

Results

Empirical analysis

The mechanism of link formation is the central dynamical process during network evolution. In the seminal work, Barabási and Albert proposed a general rule governing the growth of networks, the preferential attachment (PA), which can explain the scale-free properties observed in many real world networks1,2,3. Since then, much attention has been paid to the investigation of possible microscopic mechanisms underlying the PA phenomenon1,2,3. So far, this important question is still open and challenging. In this paper, we first carry out empirical study on four typical OSNs, including Flickr34,35, FF36,37, aNobii38 and Epinions39 (see Methods for data description). Our particular interest is on the patterns of link creation during network evolution.

To facilitate the analysis, we divide the new links into two mutually-exclusive types: the balance links and the distant links based on the topological distance40. If a new link is formed between a user and one of his second neighbours, i.e., the user who is two hops apart from him in the network, it is regarded as a balance link40. Otherwise, it belongs to the distant links. Obviously, generating a balance link always contributes a triangle in the network. By distinguishing between these two types of new links, we can investigate the dependence of new links on the topological distance.

The main method we use to analyze the pattern of link growth is to measure the conditional probability that nodes acquire (create) new links with respect to their existing in-degree (out-degree)41,42 (see Methods for details). The main empirical results for the four OSNs are summarized in Table II and illustrated in Fig. 1. Interestingly, the relative probabilities of acquiring or creating new links satisfy a power law with respect to the existing degrees, indicating that the users with larger out-degree (in-degree) are more likely to create (acquire) new links. Moreover, it is found that the exponents α for the balance links are significantly larger than that for the distant links, as shown in Figs. 1(a) and 1(b). This suggests that the balance links depend on the local topological structure more than the distant links. We attribute this preferential formation of balance links in the OSNs to the locality of information in such networks, i.e., usually users within a neighbourhood tend to influence each other.

Table 2 Exponents α for empirical networks, characterizing the dependence of balance links and distant links (in the parentheses) on the degree and the number of common neighbours, i.e., κ(x) ~ xα+1. Here αA for PA, αC for preferential creation and αN for common neighbours. For comparison, exponents α for balance links in the model networks are also listed in the brackets
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Figure 1
figure 1

The influence of the current topological status on the formation of balance links and distant links in the aNobii and FF (in the insets) networks.

(a) The cumulative functions of the relative probability κin(kin) for PA versus the in-degree of the destination nodes; (b) κout(kout) for preferential creation versus the out-degree of the source nodes; (c) The cumulative functions of the relative probability κ(u) for a pair of users to build a social link given that they have already shared u common neighbours for all balance links. The exponents are obtained by fitting the curves of κ(k) averaged over different initial snapshots. The straight lines are guide to the eye throughout this paper.

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To further examine the micro-dynamics in the process of link formation, we measure the probability of forming balance links with respect to the number of common neighbours between the source node and the destination node. As shown in Table II and Fig. 1(c), the probability is (approximately) linearly proportional to the number of common neighbours. Thus the preferential formation of balance links can be understood as a two-step random walk in the network. Here, by carefully examining the four OSNs mentioned above, we obtain empirical evidence that the preferential formation of triadic closure, i.e., the formation of balance links, can be one possible micro-dynamical process leading to the PA phenomenon in OSNs.

Modelling

The above empirical analysis has shown that in the OSNs studied, user behaviour is essentially influenced by each other within the neighbourhood and such an interplay in turn regulates the global evolution of the network. This suggests that local dynamics plays a leading role in the formation of new links during evolution. Based on this finding, in the following we set up a coevolving model, which is only driven by local interactions at the microscopic level, i.e., preferential formation of triadic closures43,44,45 and influence within neighbourhood. For simplicity, we neglect the link directions in the modelling, i.e., we only consider an undirected network.

In order to describe the dynamics of the users, we introduce a state function ϕ(i, t) for each user in the network. Here i and t denote the nodes and time, respectively. The values of the state functions describe the willingness of the users to create links. For each user in the network, we assume that his state function satisfies the following reaction-diffusion-like equation:

where two parameters μ and ϕ0 are constants; kj(t) is the degree of user j at time t. The LHS of the equation is the change of state function with time, which is driven by two “forces”: reaction and diffusion. The first term on the RHS, i.e., ϕ0, is a source term denoting the reaction, which means that a user can change his state on his own. The second term describes the diffusion process, i.e., how the interplay in the neighbourhood of the user i changes his state function. Basically, if the neighbours of user i build new links, his state function will be increased as a result of this influence. We set a threshold Θ for the state function of each user. If the state function exceeds the threshold, the user will be activated and has a probability F(ki) to actively create a new link. Once a user has built a new link or his state function has exceeded the threshold, his state ϕ(i, t) is reset to zero at the next time step. Essentially, the model simulates the user logins and activities in the OSNs in terms of the state functions.

As shown in Fig. 1(b), users with more friends, i.e., with larger degrees, turn out to be more active in generating new links. To characterize the diversity of users' activities, we adopt a general Fermi function, which has been extensively used in evolutionary games models as the adaptive acceptance probability for each activated user45,46:

Here ki is the degree of user i, 〈k〉 = 2(m + 1) is determined by the parameter m in the model, representing the average degree of the whole network and 0.001 denotes the intensity of selection. F(ki) monotonically saturates to 1 with the increase of ki, modulating the acceptance probability of nodes with different degrees. The parameter values (20 and 0.001) do not affect the qualitative behaviour of the model. In this paper, we choose the parameter values to allow the assortative coefficient vary in a relatively wide range. We emphasize that the acceptance probability F(k) may take other forms as long as it has similar behavior as the Fermi function.

Specifically, the algorithm to realize the model works as follows: (1) At the very beginning, the initial network consists of a few users (N0), forming a small connected random network. The state functions of users in the network evolve according to Eq. (1). (2) Adding users: at every time step, one new user is added to the network and randomly connects to an existing user. (3) Adding links: at each time step, m users are randomly selected from the activated users with the acceptance probability F(ki) (Eq. (2)) and each connects to one of his second neighbours if they are not connected. If the number of activated users is less than m, the remaining users are randomly chosen from the network. The above procedure is schematically illustrated in Fig. 2, where the states and the topology coevolve for one step driven by the local dynamics. As shown in Figs. 2(d)–2(f), with the increase of k, the period of the state ϕ(i, t) for a user could become smaller, indicating that the users with larger degrees are more frequently activated.

Figure 2
figure 2

Schematic illustration of the coevolution of both topology (a–c) and dynamical states (d–f) in the model.

The numeric tags are the values of the state functions. (a) The network at time t, when some users (solid) are activated according to their states. (b) One step updating of the network topology: New user U16 joins and randomly connects to user U11; Some activated users connect to their second neighbours according to the acceptance probability F(k), e.g., U2 to U13 and U12 to U3. The arrows represent the diffusion process. (c) At time t + 1, the states of users are updated according to equation (1). The states of activated users (U9 and U10) and those users building new links (U2, U13, U3, U14, U11, U16) at time t are reset to 0, but some nodes are activated again according to their states at time t + 1. (d)–(f) Illustrating the evolution of the degree and the state function for a specific user during certain time period in the model. (d) Evolution of the degree k(t). (e) The activities of the user. The solid lines indicate the moments when the user initially increases his social degree (e.g., U2 to U13 in (b)) and the dotted lines represent the moments when the user passively increases his social degree (e.g., U3 was connected by U12 in (b)), respectively. (f) Evolution of the state function ϕ(t). Parameters for the model: m = 10, μ = 1, ϕ0 = 0.1, Θ = 100.

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Verifications

In our model, although we consider only simple local rules as the force driving network evolution, numerical experiments have shown that the model can exhibit the main properties observed in empirical OSNs, such as the large clustering coefficient, small average shortest path and the power-law distributions of degrees, etc. In order to verify our model, we first compare the degree distributions of the model network with that of the empirical networks in Fig. 3. It is found that the distributions are qualitatively consistent with each other under appropriate parameters. In empirical networks, the probability to build a new link depends on the existing degrees, as shown in Fig. 1 and Table II. To compare the dynamics of our model with that of the empirical networks, we also applied the same analysis to the model under the same parameters of Fig. 3 and summarized the results in Table II (in the brackets). It is seen that the characteristic exponents α are qualitatively consistent with the empirical ones.

Figure 3
figure 3

Comparing the degree distributions of the empirical networks with that of the model network.

Since the model network is undirected, we ignored the direction of links in the empirical networks for comparison. (a) Flickr, where the parameters of the model are m = 25, N = 500, 000, μ = 1, ϕ0 = 0.01, Θ = 100. (b) FriendFeed, where the parameters of the model are m = 8, N = 200, 000, μ = 1, ϕ0 = 0.02, Θ = 200. (c) aNobii, where the parameters of the model are m = 7, N = 100, 000, μ = 1, ϕ0 = 0.02, Θ = 180. (d) Epinions, where the parameters of the model are m = 25, N = 100, 000, μ = 1, ϕ0 = 0.01, Θ = 100.

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We now focus on the two major properties of the model network: the power-law exponent γ and the assortative coefficient r. First, we investigate how the exponent γ varies with respect to the model parameter m. In this work, the best power-law exponents γ are calculated using the maximum likelihood method47. As shown in Fig. 4 (a), for small parameter m, the distribution of degrees follows a stretched power law with the exponents γ larger than 2; while for large m, the exponent γ turns out to be smaller than 2. As we know, many real world OSNs are characterized by γ < 2. The present model can produce this important feature in flexible parameter regimes. In Fig. 4 (a), we show the degree distributions for different network sizes. It is found that they are almost the same, indicating that the statistical properties of the model network are stable after long time evolution. We further find that, as parameter m increases, the exponents γ go down across 2, as shown in Figs. 4(b) and 4(c), indicating that the generated network makes a transition from a sparse scale-free network to a dense network24. To justify the power law fitting, we compute the p-value for the power law model, which measures how good the power law fitting is suitable for the data47. As shown in the insets of Figs. 4(b) and 4(c), the p-values are generally larger than 0.25 and the averages are 0.60 and 0.63, respectively, indicating the power law model is a plausible fit to the data.

Figure 4
figure 4

Transition from sparse to dense in the model network.

(a) Degree distribution for m = 3 and m = 25 with different size N. (b)–(c) Power law exponents γ with respect to parameter m for different values of ϕ0 (b) and for different values of Θ (c). The insets are the p-values from the maximum likelihood method. If not specified, the parameters in our simulations are N = 500, 000, μ = 1, ϕ0 = 0.1, Θ = 100 throughout the paper. Results are averaged over 10 realizations.

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We then investigate the assortative coefficient r in the model21. Since the links in the model are undirected, the assortative coefficient r is defined as the correlation between degrees of pairwise nodes. As shown in Figs. 5(a) and 5(b), with the increase of parameter m, r changes from positive to negative, indicating that the model networks convert from being assortative to being disassortative. There are two important points to emphasize. First, as shown in Fig. 5(a), the change of the sign of r occurs at larger m as parameter ϕ0 increases. Second, as shown in Fig. 5 (b), the value of Θ has significant influence on r.

Figure 5
figure 5

Transition from assortativity to disassortativity in the model.

(a)–(b) The assortative coefficient r with respect to the parameters m for different values of ϕ0 (a) and for different values of Θ (b). The coefficients r are calculated at the final stage in the model when N = 500, 000. (c) The temporal evolution of the assortative coefficient r(N) for different m at Θ = 100 and for different values of Θ at m = 9. The error bar is the standard deviation.

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In the above, we have shown that r in the model could convert from positive to negative when parameter varies. As reported in Refs. 11, 12, some OSNs convert from being assortative to being disassortative during evolution. How does r in the model behave with the increase of time in our model? First we note that the final network size N is proportional to the total evolution time. As shown in Fig. 5(c), the coefficients r become almost stationary when the model evolves for sufficiently long time. In particular, in certain parameter regimes, the generated networks evolve from the initial assortativity to the subsequent disassortativity with the increase of time. Therefore, the current model can characterize the distinct dynamical stages observed in the OSNs such as Weaklink and Google+11,12.

The assortative to disassortative change in our model can be heuristically understood based on Eq. (1). Basically, it is the result of the competition between two factors in our model: the reaction factor denoted by parameter ϕ0 and the diffusion factor denoted by parameter μ. Parameter m is important because it controls the diffusion and thus can change the ratio of these two factors. When m is small, i.e., the number of new links formed at each time step is small, the local influence is weak due to the small average degree, i.e., 〈k〉 = 2(m + 1). In this case, the factor of reaction is relatively more important and the user's own motive plays a dominant role in the evolution of the state function. Consequently, the activation probability of a user is almost independent of the degree. Users thus have almost equal chance to be activated and connect to others, leading to the assortative mixing pattern. This may correspond to the situations in some OSNs where users tend to establish links with people they know in real life, resulting in assortativity in the acquaintance network during the initial stage. On the other hand, when m is large, according to Eq. (1), the local influence, i.e., the diffusion, then plays a dominant role in the evolution of state function. In this case, users with larger degrees have more chance to be activated and connect to others, leading to the disassortative mixing pattern. In real situations, this may correspond to certain OSNs where the celebrities attract their fans to connect to them.

To further illustrate how parameter m regulates the assortative mixing pattern in the model network, we calculate the average nearest neighbours' degree knn(k) in the generated networks23. As shown in Fig. 6, it is seen that knn(k) increases with respect to degree k for small m, corresponding to positive assortativity in model networks. This is consistent with the situation in acquaintance networks23. However, for larger m, knn(k) increases first and then decreases when the degree is large enough, corresponding to neutral and negative assortativity, as in some real OSNs12,13,19. Similarly, the above analysis can also explain the results shown in Fig. 5(a), where a larger m is required for the transition of r when ϕ0 increases. Since ϕ0 represents the reaction factor, to overcome the outcome of increasing ϕ0 in the model, the diffusion factor needs to increase too. As a result of this competition, we observe that the transition occurs at a larger value of m.

Figure 6
figure 6

Characterizing the average nearest neighbours' degree knn(k) for different values of the parameter m.

The corresponding assortative coefficients are 0.21, 0.15, 0.12, 0.09, 0.07, 0.007, −0.10, −0.23 and −0.41 for increasing m, respectively.

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In the evolution of real OSNs, generally the average degree increases with time12,48. This roughly corresponds to the increase of m in the present model due to 〈k〉 = 2(m + 1). As shown by our model in Fig. 5(c), this will cause the diffusion factor gradually to be dominant and the network may convert from being assortative to being disassortative with the increase of time. Similarly, the decrease of parameter Θ is equivalent to the increase of parameter m and the behaviour of the model in Fig. 5(c) can also be explained from the viewpoint of competition between reaction and diffusion factors.

To support our argument above, we apply empirical analysis to the aNobii network. Specifically, we regard it as a hybrid of a real world social network and a virtual online network. The former subnetwork consists of the acquaintance links connecting users knowing each other in real life, e.g., their family members and friends, such as “Acquaintances” in Google+ and “Friendship” in aNobii; and the latter comprises the stranger links connecting their online virtual friends, such as “Following” in Google+ and “Neighbourhood” in aNobii. In terms of the reaction-diffusion process, the generation of these two types of links is mainly due to the reaction factor (i.e., user's personal desire) and the diffusion factor (i.e., the local influence) respectively. Interestingly, we find that the subnetwork consisting of the acquaintance links is assortative with r = 0.06, like real world social networks. On the contrary, the subnetwork consisting of the stranger links is disassortative with r = −0.09. As shown in Fig. 7, the relative probabilities forming stranger links are significantly larger than that forming acquaintance links, implying that the diffusion factor is dominant in aNobii. As a result, the aNobii network as a whole turns out to be disassortative with r = −0.05. The above results provide empirical evidence that the competition between diffusion and reaction might determine the mixing pattern of degrees in an OSN. Reasonably, during the evolution of the OSNs, if the diffusion factor dominates over the reaction factor, a transition from assortativity to disassortativity could be expected as in Weaklink11 and Google+12.

Figure 7
figure 7

Characterizing the difference between the acquaintance and stranger links in the aNobii network.

(a) The cumulative functions of the relative probability κin(kin) versus in-degree of destination nodes; (b) κout(kout) versus out-degree of source nodes. (c) The cumulative functions of the relative probability κ(u) for a pair of users to build a social link given that they have already shared u common neighbours.

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Discussion

In this work, based on some empirical analysis of four typical OSNs, we set up a reaction-diffusion-like model, in which the evolution of the network is governed by both the users' personal motives and the influence within neighbourhood. As a natural consequence of the coevolution of dynamics and topology, the model is able to qualitatively reproduce the major properties observed in real world OSNs. In particular, the generated networks can convert from being sparse to dense and from being assortative to dissassortative with appropriate parameters. The model provides explanations of these two important features in real world OSNs in terms of the competition between reaction and diffusion factors in network evolution.

We believe that the current work is enlightening in modeling the evolution of the OSNs as well as of other real world networks. For example, other mechanisms of link formation, such as collective action and the structural hole mechanism, etc40, can be readily formulated and investigated. The idea of the model might be applicable to a wide range of social networks and can be easily generalized to treat multi-layer networks, weighted networks and social-attribute networks, etc. For example, recently, we have carried out a modeling for Flickr, with a typical dual-component and dual-connection OSN and obtained satisfactory results33.

Methods

Data description and notations

Flickr is one of the most famous websites sharing photos. The data set for our study is collected by daily crawling the Flickr network over 2.3 million users from Nov 2, 2006 to Dec 3, 2006 and again daily from Feb 3, 2007 to May 18, 2007. In total, there are 104 days in the time window of data collection34,35 (http://socialnetworks.mpisws.org/). There are more than 2.3 million users and 33 million directed links among them. FriendFeed (FF) is a content aggregation site where users discover and discuss the interesting contents found on the web by their friends. The data set is collected by crawling the FriendFeed network once within every five days between Feb 26 and May 6, 200936,37. There are 14 snapshots or 70 days in the time window. More than 200 thousand users were found and about 4 million directed links among them were identified. ANobii is a website where readers can rate, review and discuss books with others. The data set is collected by crawling the neighbourhood (stranger in real life) and friendship (acquaintance in real life) networks of aNobii. Six snapshots of the network, 15 days apart, are collected starting from Sep 11, 200938. Users connect to each other through two mutually-exclusive types of ties: friendship and neighbourhood links. At last, the aNobii network includes 86,800 users, 429,482 stranger links and 268,655 acquaintance links. Epinions is a consumer review website where users can write reviews of products and also “trust” or “distrust” each other. The data set contains the trusted relationships among users before Aug 12, 200339 (http://www.trustlet.org/wiki/Extended_Epinions_dataset), including 114,467 users and 717,667 trusted relations. Mathematically, we can use the adjacency matrix AN×N to characterize the topology of the online social networks, where aij = 1 if user i declares user j as friend, otherwise 0. Since the links among users are directional in these four networks, we accordingly define two types of degrees: the out-degree , i.e., the number of friends claimed by user i and the in-degree , i.e., the number of users who claim user j as friend. The statistical properties of these four data sets are listed in Table I.

Measuring preferential attachment

In Refs. 41, 42, a numerical method is used to measure the preferential attachment (PA) growth of a network. Given that we know the temporal order in which the nodes join the network, the essential idea of the method is to monitor to which existing node the new nodes connect, as a function of the degree of the old node. We take an example to briefly explain the method as follows: (1) At time t0, we mark the nodes with kout out-degree as “t0 nodes”, denoting their number as C(kout). (2) After the evolution of a period Δt, the out-degrees of the “t0 nodes” have increased due to the evolution of the network (of course, the in-degrees also change). We count the out-degree created by the “t0 nodes” as A(kout). Since we divide the newly generated links into two types, i.e., the balance and the distant, we have A(kout) = AB(kout) + AD(kout), where the subscripts B and D denote the two types, respectively. (3) The histogram providing the number of out-degree acquired by the “t0 nodes” with exact kout out-degree, after normalization, defines a function:

where, i can be either B or D. It has been proven that if PA mechanism exists, the conditional probability with which the out-degree grows with respect to the existing out-degree follows a power law, namely . Numerically, it is convenient to examine the cumulative function of , which will also follow a power law, i.e.,

Similarly, the above numerical method can be applied to the calculation of the probability for acquiring new links with respect to in-degrees kin and the treatment of the acquaintance and stranger links is straightforward.