Emergence of Global Preferential Attachment From Local Interaction

Global degree/strength based preferential attachment is widely used as an evolution mechanism of networks. But it is hard to believe that any individual can get global information and shape the network architecture based on it. In this paper, it is found that the global preferential attachment emerges from the local interaction models, including distance-dependent preferential attachment (DDPA) evolving model of weighted networks(M. Li et al, New Journal of Physics 8 (2006) 72), acquaintance network model(J. Davidsen et al, Phys. Rev. Lett. 88 (2002) 128701) and connecting nearest-neighbor(CNN) model(A. Vazquez, Phys. Rev. E 67 (2003) 056104). For DDPA model and CNN model, the attachment rate depends linearly on the degree or strength, while for acquaintance network model, the dependence follows a sublinear power law. It implies that for the evolution of social networks, local contact could be more fundamental than the presumed global preferential attachment. This is onsistent with the result observed in the evolution of empirical email networks.


Introduction
Power law distribution commonly exists in complex networks, including planned network such as Internet [1] and unplanned social networks, such as scientific collaboration networks [2,3,4,5], actor collaboration networks [6], peer-to-peer networks [7], mobile networks [8] and much more. Their degree distributions obey power law, and if applicable, vertex strength(summing the weights of links that connect to a vertex) [4,5] and link weight [9] also follow power law. Many models have been proposed to capture the topological evolution of complex networks (see reviews [10,11]). Especially a class of models based on the idea of global preferential attachment, firstly proposed in BA model [12], is quite successful to reproduce the power law distributions of degree/strength [9,12,13,14]. From the degree-based preferential attachment in BA model, later Barrat[13] developed a model of weighted networks with strength-based preferential attachment. Some networks have hubs or data centers, where global information is collected for future use. For example, WWW has search engines and Internet has routers and DNS servers. For networks with such centers, global preferential attachment mechanism may seem reasonable. However, in social network there is no data centers collecting and providing global information. It is impossible for every individual to know global information of system, such as degree or strength of every individual. For social networks, evolution mechanism based on local quantities would be more natural.
Actually, there are already several evolving models for scale-free networks based on local rules, such as distance-dependent preferential attachment(DDPA) model [15], acquaintance network model [16], connecting nearest-neighbor(CNN) model [17], random walk model [18,19], redirection model [20,21], optimization model [22,23] and so on. For instance, in DDPA model a measure of closeness relation is defined locally within second neighbors and then links are built up preferentially according to the relation. Acquaintance network model evolves via people introduced to know each other by a common acquaintance [16]. In CNN model [17], at every time step, a new vertex is added or a potential edge within second neighbors is converted into an edge. All above models based on local rules can also reproduce the common topological characters of complex networks.
We notice that both the two potentially conflicting types of models successfully capture the main characters of scale-free networks, but one needs global information while the other only requires local information. Here we suggest a wild guess to resolve the conflict and make the whole picture more consistent. We conjecture that global preferential attachment emerges from local contact based models. In this paper, we are going to investigate whether or not the above conjecture holds in the following local models: DDPA model [15], acquaintance networks model [16] and CNN model [17].
This paper is organized as follows. In section 2, we first briefly describe the method used to check preferential attachment. In section 3, we apply this method to DDPA model, acquaintance networks model and CNN model, and present the results. At last, in Section 4 we give some concluding remarks. And it turns out our conjecture does hold in the above models.

Methods for Measuring Preferential Attachment
The preferential attachment hypothesis states that the rate Π(k) with which a vertex with k links acquires new links is a monotonically increasing function of k [12], namely For BA model α = 1 [12].
H. Jeong et al [24] proposed a method to check global preferential attachment from data on evolution process. Here we will briefly describe the method. In evolving process, we record the order of each node and link joining the system within a relatively short time frame after the network evolves steadily in a long time. At a large enough time T 0 , consider all vertices existing in the system, called T 0 vertices. Next select a time Firstly, count the number N(k) of such vertices with exactly k degree in T 0 vertex. Secondly, record all the vertices in T 0 vertex to whom the new links are attached as Ω. Lastly, count the number of vertices with exactly k degree in Ω as A(k). At this condition, Π(k, T 0 , T 1 ) will be independent of T 0 and T 1 but depend on k only [24].
A convenient definition of Π(k, T 0 , T 1 ) function could be defined as, According to mean field theory and Eq.(1), where M is the number of new links attaching to vertices Ω. So Eq.(2) can be written as To avoid the effect of noise, we also study the cumulative function instead of Π(k) If there is a global preferential attachment, then In order to prove the validity of formula (2), we apply this method on BA model, starting from a fully connected n 0 initial network, where k = 2m, α = 1 and γ = 3. The exponent in Eq.(3) should be 1 and the measurement shows that they are approximately 1 (as shown in Fig. 1). This indicates that the formula (2) is fit to measure the form of Π(k). 10 100 1000 Figure 1. The κ(k) function determined numerically for BA model based on Equation (2). The slope of solid line is 2(corresponding to α = 1 as α + 1 = 2), and the slope of dash line is 1 (α = 0). This indicates that the exponent α is almost 1. If not mentioned in the following figures, we run the model for 100 times with the same parameters to count A(k, T 0 ) and N (k, T 0 ), and then we determine κ(k)(κ(s)) according to Eq. In measurement, we only focus on the vertices new links are attached onto, so we only consider ending vertices. Furthermore, measurements on external links from the new vertex and internal links among existing vertices, can be done separately when necessary.

Measuring preferential attachment of Local Contact Model
In this section, we are going to check the possibility of such global attachment emerging from the following three local contact models.
DDPA model : The evolution of networks starts from a fully connected n 0 initial network. At every time step, one new vertex is added into the network by connecting it randomly to one old vertex. And then other l old vertices are randomly activated. Every one (denoted as vertex n) of these 1 + l vertices can attempt to build up m connections. The probability for every link (except the first link from the new vertex) from a vertex n to a vertex i is given by where ∂ 2 n means neighbors of vertex n up to the second separation. L ni is the similarity distance [5,9], which means that the larger is the similarity distance, the closer is the relation between the two end vertices. If two vertices i and j are connected directly, L ij = w ij , where w ij is the similarity weight, meaning the larger the closer, such as the number of collaboration in scientific collaboration networks [2,3,4,5], and the duration of calls in mobile networks [8]. Otherwise, L is connected to vertex µ (j) with similarity link weight w iµ (w µj ) then L The final similarity distance between vertex i and vertex j is L ij = max L (µ) ij , the maximum one via all vertices {µ}. In other words, it is sufficient to know the degree of familiarity between second neighbors. In addition, the reconnection between linked vertices is allowed. When building up connection between vertices i, j represents an event happening between the vertices, the number of occurrences of such an event may be defined as the connection count T ij , which will be transferred into similarity link weight w ij = f (T ij ), e.g., w ij = T ij . This DDPA mechanism does not use vertex degree or vertex strength, i.e., s i = j w ij , as the reference, not even locally. It only makes use of the information about local similarity distance within second neighbors, but scale-free phenomenon does appear in DDPA model for intermediate degrees/strengths(see Fig.6 in Ref. [15]).
For DDPA model, we need to do this measure for three kind of links: the first external links from the new vertex, all other external links from the new vertex and internal links. The same method can also be applied to check preferential attachment according to vertex strength. Since the ending vertex of first external link is selected randomly, the probability of a vertex selected should be independent of its degree (k) or strength(s). In the inset of Fig.2, we see the first external link does follow flat distribution. But for all other external links, we find that κ(k) (measuring according to degree) and κ(s) (measuring according to strength) for intermediate degrees/strengths follow a straight line on a log-log plot, indicating Π(k) ∝ k α k or Π(s) ∝ s αs . The curves of κ(k) for DDPA model are consistent with those for BA model (Fig.1). And they are parallel with different T 0 (as shown in Fig.2), indicating that Π(k) and Π(s) are independent of T 0 and T 1 , and depend on k or s only. Their power law exponents are  respectively α k = 0.85 and α s = 0.91.
In the measurement of internal links, we find that global preferential attachment is also valid. The curves of κ(k) and κ(s) follow a straight line on a log-log plot. Their power law exponents are respectively α k = 1.07 and α s = 1.1 (as shown in Fig.3). Actually, if checking all links together, including the first external link, the curves of κ(k) and κ(s) follow a straight line on a log-log plot, where exponents are respectively α k = 0.98 and α s = 1.04 when l = 1. The parameters almost have no effect on the exponents α k and α s .
Acquaintance model : At first, a random network with N vertices, where every pair vertices are connected with probability p l , is generated. Then acquaintance networks evolve according to the following rules: (i) One randomly chosen person introduces two random acquaintances of his to each another. If they have not met before, a new link between them is formed. If he has less than two acquaintances, then he introduces himself to another random person. (ii) With probability p one randomly chosen person is removed from the network, including all links connected to this vertex, and replaced by a new person with one randomly chosen acquaintance [16]. The model generates degree distributions spanning scale-free and exponential regimes(see Fig.1 in Ref. [16]). Though the size of network is fixed, this model includes birth and death process. New links emerge between individuals frequently. After evolving for T 0 times, again we record and analyze evolution process between T 0 and T 1 . We can see that κ(k) also follows a straight line on a log-log plot, where α ≃ 0.80(as shown in Fig. 4).
CNN model : Next we consider a variant of the acquaintance model, connecting nearest-neighbor model [17] with increasing network size. CNN model starts from a single vertex and evolves according to the following rules: (i) With probability 1 − u, a new vertex(denoted as n) is introduced into the network by connecting it randomly to In the process of measurement, we record the degree of vertex j and the other two selected end vertices. We see that κ(k) follows a straight line on a log-log plot, and they are parallel with different T 0 (as shown in Fig.5). For different parameter u, the value of exponent α are different, for example, α ≈ 1.11 for u = 0.3 and α ≈ 0.95 for u = 0.7. In Fig.5, we show the measurement on u = 0.4. And there we can see that α ≈ 1.00.
Although α is close to 1 for DDPA and CNN model and α = 0.80 for acquaintance model, we find that κ(k) ∼ k α+1 holds in all models. And this indicates that although links are created by local information, it seems as if the network evolves according to global preferential attachment.

Concluding Remarks
In this paper, we measured global degree/strength preferential attachment of DDPA model, acquaintance network model and CNN model. These models make use of local quantities instead of global degree/strength preferential attachment mechanism. However, our measurement shows that they still can be seen as if the networks evolving according to global preferential attachment. From this point of view, preferential attachment is an emergent phenomenon from the more fundamental local rules. Empirical study on e-mail networks in Ref. [25] also suggests that cyclic closure, especially triadic closure, plays an important role in building up new links. In modeling social networks, this observation may suggest that local rules are preferred rather than the global degree/strength preferential attachment. For people who believe in the later, our discovery is especially meaningful. Our results indicate that one does not need to worry about the widely believed and used global preferential attachment because it can emerge from properly designed local rules.