Rewiring Without Removal

There are several factors contributing to friendship inflation such as friendship collectors, fakesters and fraudsters, spammers and phishers, as we analysed in the previous chapter.

To be fair to most SNSs, we assume that SNSs are well policed and most members only add friends whom they have actually met offline. Gradually, their offline social activities will bring more friends to their online networks. People have limited time and energy to maintain stable social relationships. In fact, there is a supposed cognitive limit to the individuals with whom people can maintain stable social relationships[38].

As a result, some of the old connections will gradually decay when we acquire new ones. In terms of complex networks, this may be modelled as edge rewiring. Online social networks are capable of preserving old connections, leading to rewiring without removal, a feature that does not exist in real-world social networks but is commonplace in social network sites. The effect of rewiring without removal of decaying connections is illustrated in Figure4.1. Dashed lines represent decaying real-world connections that have been maintained as online social connections. Every time people make new contacts and leave some old contacts obsolete, the old contacts can always be preserved in the social network. As an SNS grows, its social graph will become denser and denser.

On SNSs, people are highly unlikely to explicitly declare the ending of any connections that have actually decayed. The technique of static link employed by most SNSs requires users to articulate their friends publicly by demarcating the borders between friends and non-friends. Therefore, users prefer not to remove any fading connections to avoid offending people. Users also worry that the removal of unused connections will have implications and ramifications that may not be predicted at the time of removal. On the other hand, the popularity of top friend applications and services on social networks like Facebook and MySpace suggests that SNS owners seek to mitigate the problem of friendship inflation by “upgrading relationships” rather than “downgrading them”.

When many users have more connections than they actually do, the topology of the network will increasingly diverge from that of the real-world social network. We propose a model to simulate the growth and evolution of the cumulative network.

The model is based on the Barabasi-Albert network[11] as discussed in the second chap-ter. It has been observed that both conditions in the original model, growth and prefer-ential attachment, apply to social network sites. In addition, two modifications and one condition are added to the model:

(a) In the BA model, the exponent α=3, but in real networks, the number is between 2 and 3. We use 2.3, which is the measure for film actor collaboration based on Internet Movie Database (IMDb). A approximate value to this number has also been found in other real-world social networks[99].

(b) The BA model does not specify the value of m, the average degree of the network.

Figure 4.1: Illustration of the effect of rewiring without removal of decaying con-nections. Dashed lines represent the decaying real-world connections that have been

preserved as online social connections.

Dunbar’s number suggests that people are capable of maintaining regular contact with about 150 friends. The number can be interpreted as the lower bound number of links one can have, for SNSs are usually considered tools for efficient friendship management[42].

Therefore the value of m, which is the number of friends that people claim to have, should be no less than Dunbar’s number. For our convenience, m is set to be 150.

(c) Individuals will make new acquaintances and forget old links after joining the net-work. This is called edge rewiring. The BA model does not take into account the effect of internal edge rewiring. We assume in our model that every node will rewire its m edges to other nodes with probability p_r proportional to d^−r, where d is the social distance (described in chapter 2) between them and r is an adjustable constant. This condition will only be used qualitatively in our model.

With only (a) and (b), we have a new function for probability p_k:

p_k= 2m(m + 1)k^−2.3 = 45300k^−2.3 (4.1)

Figure 4.2shows the graph of Eq 4.1.

0 500 1000 1500 2000 2500 3000

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Degree k

Probability P(k)

Figure 4.2: Degree Distribution in BA model with m=150, α=-2.3

The graph suggests that in a social network with m=150, about 44.78% of the people have about 150 friends. The remaining part of the population are able to maintain stable contact with more than 150 people. This is true regardless of the size of the network as it is scale-free. At the time of writing, empirical data shows none of the social network sites gain a percentage of 44.78% or above, indicating that people have not yet fully moved their real-world relationships online. However, as the social network sites have grown rapidly in the recent years, we would expect the percentage will approach that of the real-world network in a short period. Condition (c) suggests that people will “rewire”

the friend links if they could not afford to keep regular contact with them, thus leaving a long trail of socialising footprints. In cumulative networks, the obsolete connections will not disappear automatically, which is in contrast to real social networks where old relationships will decay gradually when people do not maintain a certain degree of

social interaction with each other. We discuss two scenarios of the consequences for the development of social network sites:

Scenario 1: as the number of friends goes beyond 150 and continues to grow, it is not uncommon to find people who have hundreds of thousands of friends. In the real world, nevertheless, people with many contacts are usually the rich, politicians, celebrities and leaders. Ordinary people may like to make friends with these high-profile figures, but usually find it very difficult to do so. However, on social network sites, the notion of high degree simply does not imply a high social status of the individual, as in the case of the offline world. This will destroy the factor of preferential attachment as described in the BA model: people now do not make friends by looking at their number of contacts.

Model A of the BA network shows that without preferential attachment, the network will lose its scale-free character.

Scenario 2: if at some point, the network stops growing, then the size of the network will remain unchanged or even shrink. This is quite common as social network sites stop growing and start losing members because of a lack of attractiveness. Then members of the network can only make friends with other existing members. This simply increases the clustering coefficient of the network, making it a denser place. In the end, it will become a random graph with an extremely high probability for an edge to be placed between any nodes. In particular, if people still keep making friends in the pattern of preferential attachment, the graph will exhibit a Gaussian distribution. In other words, the number of new friends are proportional to the number of friends already acquired, and this will keep doubling. In both cases, the network will lose the power law degree distribution of a scale-free network.