## 6 Experimental Study
### Convergence and Reconvergence  
The mean value and standard deviation of convergence speed for a variety of network sizes can be found in Figure 3. It is easy to see that the convergence speed depends on the type of the network. For example, BA networks of a size of 1,000 take almost twice the amount of time to converge than networks of a size of 100, while ER networks of a size of 1,000 need less than twice the amount of time needed by networks of a size of 100 and only a slightly higher amount of time than the networks of a size of 500 and 750. Likewise, the time needed for reconvergence can be seen in Figures 4 and 5, for networks of a size of 1,000 of both types and for the four types of operations under consideration. By focusing on low percentages of affected nodes (_i.e._, up to 20% of the network size, which is a high volume of change), it  
Figure 3: Convergence speed per network size. is easy to see that reconvergence is obtained in most cases for a _fraction_ of the rounds needed for initial convergence. For join and leave operations, reconvergence is expressed not in rounds, but in relation to the convergence time, since network sizes change significantly. Note here that this extreme change in network sizes leads in some cases to percentages greater than 100%, _i.e._, more rounds are needed for the reconvergence than for the initial convergence. For the preference change and churn operations, this is not the case: the network size remains the same, either because no node joins or leaves (preference change case) or the amount of nodes joining and leaving is the same (churn case), and the reconvergence time is expressed in rounds. In the case of join operations, nodes arrive at the network and want to join the already established equilibrium of connections by being more attractive choices for some of the nodes with whom they are neighbors. This creates cascade effects of nodes rejecting old connections in favor of the newcomers, the rejected nodes trying to repair their lost connections, and so on. Naturally, the more nodes wanting to join the network, the bigger upheaval is created. A similar effect is generated during leave operations, where previously rejected nodes suddenly become attractive choices for nodes that were left behind by departing nodes. Note here that the BA networks reconverge much faster than the corresponding ER networks in the case of join operations. This happens because new nodes (being of a low degree) connect preferentially to relatively few high degree nodes, limiting the extension of the upheaval in the network. In the case of leave operations, the same behavior poses a challenge, since departing nodes may happen to be of a high degree themselves, leaving behind a lot of low degree nodes to repair their connections. Notice though that in both cases (ER or BA networks), when a substantial percentage of the network departs (_i.e._, above 35%), the remaining nodes repair their connections much more easily, since they have more unformed connections than established ones. Preference change affects both network types in the same way: a node that changes preferences destroys some connections, creating waves of changes in its neighborhood. For the case of BA networks, it may happen that a node changing preferences is a high degree one, causing a lot of nodes to repair their connections. However, this effect dies off quickly, since most of its neighbors are of a low degree, leading to an overall performance similar to the ER case. Figure 4: Reconvergence speed per operation, joins/leaves, \(n=1,000\). The two network types show their differences more prominently under churn (Figure 5). For the ER networks, a joining node under churn can be seen as a \"reincarnation\" of a leaving node with all its previous connections dropped and its preferences changed, since both of them have comparable node degrees. However, the churn operation is detrimental for the BA network, since the joining nodes are of a low degree and the departing ones of a potentially much higher degree.