Preferential Attachment: Beyond Reciprocity

People with higher social capital can attract more people with less effort than those with lower social capital. This is called preferential attachment and is considered as one of the two important factors in the Barabasi-Albert model which theorises that a complex network exhibits a power-law degree distribution. Since the social capital accruing from a relationship is much greater to the extent that the person who is the object of it is richly endowed with capital, the possessors of an inherited social capital, symbolised by a great name, are able to transform all circumstantial relationships into lasting connections[16]. They are so well known that they do not even have to make the acquaintance of the acquaintances.

The power-law degree distribution of social network, which is due to the effect of pref-erential attachment, reflects our living experience that the rich get richer. In the study of distribution of wealth, 20% of the population in a given society hold 80% of total wealth of the society. The richest people can grow even richer by taking advantage of their existing capital. This topology can also explain the spreading of disease. Research has found that there is no epidemic threshold for viruses to spread all over a network with a power law degree distribution, regardless the rate of infection[104]. This implies that power law structure can facilitate the dissemination of information and knowledge.

As long as this information exists, it will eventually spread all over the network if it has any value.

We therefore propose that individuals with more connections should be able to acquire new connections with less effort. There should be no universal frequency of communica-tion but a decreasing range of frequency over the individuals with increasing conneccommunica-tions.

The use of universal frequency was a sound decision at first thought. It is intuitive to argue that the more one communicates, the more connections one can establish. Some may take for granted that there is a linear relationship between the number of contacts and the effort one has spent on social grooming. However, this is not true. If we take the topology of the social network into account, the hallmarks of any social network of human beings are power-law degree distribution and assortativity. The power-law degree distribution, or scale-free character, is the base for the social network to spread the in-formation and knowledge quickly. It also reflects the self-organising nature of a network that is robust and resistant to random attacks. Unfortunately, the message network of online social network does not follow the power-law degree distribution. As a case in point, the network of testimonials on Cyworld exhibits exponential degree distribution, rather than power-law degree distribution as presented in the real-world social network, as indicated in Figure 5.2.

Research on Facebook indicates that the probability distribution of number of messages sent per user does not show power law distribution either, as illustrated in Figure5.3. It is in-between the heavy-tail Pareto or power-law and thin-tailed exponential distribution

Figure 5.2: Cumulative distribution of in-degree and out-degrees of Cyworld’s testi-monial network. Picture taken from [4]

in terms of its asymptotic behaviour[55]. This is also true for the social network of the University of Southampton. About 10% of people have more than 500 connections and 30% of them have between 200 and 500 connections. However, it is untrue that all of them are actively engaging on Facebook activities such as public wall posting, photo and video commenting, gift exchanging and blog commenting. On the contrary, most of these users are fairly inactive and rarely engage in these social grooming activities.

However, a significant portion of people will actively engage with these activities. For these people, the group with connections above 500 will have exchanged information slightly more than those with connections between 200 and 500. It is unlikely for people with more connections to spend equally exponential time and effort for social grooming, and is not achievable even if they spend all of their total time.

The statistics confirm that the social network purely based on universal frequency of communication will not exhibit power-law degree distribution. That is, f_k is not a constant. In fact, as we argued earlier, highly connected users simply do not have sufficient time for such expensive social grooming. In order to design an algorithm to transform the message network into a network with complex network topology, we would like to find out the mapping between the degree and its frequency of communication.

First, the power law degree distribution goes as follows:

Figure 5.3: Number of messages sent versus number of users sending. Picture taken from [55]

p_k∼ k^−γ 2 < γ < 3 (5.1)

Second, we need to find out the relationship between the p_nand the number of messages.

In our study of the social network of the University of Southampton, we are not able to access either the data of private messaging or those of instant messaging. Therefore, we do not have first-hand data to draw the connection. However, according to the research of Golder et al., who did manage to access the Facebook messages, the relationships between p_n and number of messages is as follows[55]:

p_n∼ n^−αnβ α > 0, 0 < β < 1 (5.2)

Given the formula above, we introduce the frequency function f_k, so that the previous distribution becomes a power law distribution. Replacing n by kf_kin eq.5.2 and making index equivalent to γ, we have:

γ ∼ kf_k^β (5.3)

Solving eq.5.3, we have:

f_k∼ (^γ_α)

1 β

k (5.4)

Thus, the relationship between frequency and degree is:

f_k+1∼ kf_k

k + 1 (5.5)

Eq.5.4 suggests that the frequency of communication is inversely proportional to the degree. In other words, the more friends we have, the less effort we need to spend on communication and social grooming. This appears to be true by our intuition. People who are highly popular or are opinion leaders can maintain tens of hundreds of times more meaningful connections that ordinary people can manage to establish. However, while they are very likely to spend more time on social grooming than most ordinary people, they are very unlikely to spend orders of magnitude more time and effort than ordinary people on social grooming. If we put the value for ordinary users on social grooming to be 20% of their total time, then people who are highly connected or are opinion leaders can only spend five times of that at maximum. If ordinary users can have a number of 150 connections, as described by the rule of 150 or Dunbar’s number, then people who are centres and hubs of the social network can only have a maximum of 750. This figure is much smaller than we have found on most social network sites, where users with more than one thousand connections are not unusual. Although, as we analysed earlier, a significant part of these users are friendship collectors, fakesters and fraudsters and even spammers and phishers, a decent percentage of them are genuinely popular figures who enjoy a high reputation in the social network. These are usually bloggers, journalists and celebrities. They may simply acquire connections passively without much effort. The inverse relationship between the frequency of communication and the degree of individual users, as we concluded earlier, explains the fact that these Facebook whales require less time than ordinary users to acquire more connections, as they can leverage the connections they have already acquired.

Because the method reflects the real activities of genuine social network, the configu-ration can single out the highly connected nodes from the whole pool of nodes. With the information hubs and opinion leaders that have more connections, information and knowledge can spread much faster and further.