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In the following we construct a model for the collapse of OSN-s, which incorporates both exogenous and collective effects on churning. The parameters of the model will be adjusted to the iWiW data.
In order to keep the underlying structure simple, the model is defined on a random network where 40% of the links are created in an uncorrelated random way and the rest is generated by placing random triangles, as high clustering is typical for social networks39,40. Finally we have a random network with N nodes and an average degree of 〈k〉. At the beginning all nodes are part of the OSN. In the Supplementary Information we present results using part of iWiW network as underlying structure.
Social contacts have different intimacy levels building a hierarchy of concentric layers around the persons in their egocentric networks41. It is natural to assume that only the set of closest friends matter in the decision of well embedded individuals about staying with the service or leaving it. Therefore we will consider only a network with reduced average degree, which is considered as a parameter.
For the endogenous, collective effects we implement a threshold mechanism18,19: Whenever the fraction of active friends of a node drops below its threshold λ the node will feel inclined to leave the OSN. This does not happen immediately but with a rate 1/τ, so τ is the timescale of leaving the OSN after the threshold condition is fulfilled. All nodes are given a predefined uncorrelated threshold value λ = 0.5 ± 0.2 with uniform variations.
The spontaneous churning is defined as follows: At each timestep nodes leave the service with a rate linearly increasing with time: γ = μt/τ, reflecting the growing interest in the competing service. We select nodes to leave the service with the probability which was a decreasing function of the node degree (see Supporting Information for details). This was motivated by the fact that more embedded users are more reluctant to leave the site due to exogenous effects.
In summary the model is defined as follows:
The dynamics defined this way has two timescales: τ and 1/μ. The latter determines the exogenous timescale and can be fitted to the initial quadratic increase of the number of inactive agents, which relates the model time to real time, therefore it has no influence on the dynamics. The value of the threshold can be adjusted by measuring the peak in the r
end curves.
We are left with two parameters: τ and 〈k〉. The timescale τ controls the speed of the collapse of the service together with the average degree of the original network. If either τ is small (it was zero in the original threshold model19) of the network has a high average degree the collapse of the site is almost instantaneous. We found that either large waiting time or low degree is needed to recover the empirical results.
Fitting our model to the empirical data gives a prefect match as shown in FigNagpur Stock. 7(a) with values of 〈k〉 = 10 and τ = 14.5 days. Both values are realistic: First, it is natural to assume that people check back for two more weeks after they start getting motivated to leave the service. Second, it was already suggested in41 that people have a circle of intimacy of about 12–15 friends and relatives with whom they have regular communication; these are the relationships, which are particularly important for them. In Fig. 7(b) we show the best fit with 〈k〉 = 200. It is clear visually that this is a worse fit. For more details see Supporting Information.
Here we used an artificial random network for the model studiesBangalore Investment. Since the contact network is available from iWiW we may use part of it (the whole being too large for simulation) to study the model. It was shown that political borders inside Hungary are apparent through the contact network42 therefore we have decided to use the users in one of the counties to simulate the model.
We have introduced a probability with which the links are kept in order to be able to decrease the degree of the network. After decimating the links we remove nodes with degree 0 and 1. We have studied the evolution of the active users and the results is shown in Fig. 7(c). The best fit was obtained with an average degree 9.1 similar to the model network.
In Fig. 7 we have also shown the time evolution of the ratio of the active users without avalanches and with no waiting time. It can be seen that the time evolution of the three curves depart at middle 2010 where the concurrent service becomes popular and collective effects become important. The model generates cascades similar to those observed in the empirical data, see FigJaipur Stock. 4(b).
A further verification of the model besides Fig. 4 can be obtained by calculating the degree dependence of the number of churned neighbours at leaving the network FigAgra Investment. 3(a). In order to compare the results of the model with the empirical ones we scaled up the model in the following way: We introduced further (socially less important) links such that the average degree became the same as in the OSN (220). The links were attached to the nodes with a probability proportional to the degree in the small network and the triadic closures were also applied with the same density. The similarity between FigMumbai Investment. 3(a) and (b) are rather convincing.
Jaipur Investment
