Edge removal balances preferential attachment and triad closing.

Most network formation analysis studies are centered on edge addition. However, edges in real world networks often have a rapid turnover with a large number of edges added and removed between each node addition or removal steps. In such a case, quasiequilibrium is obtained between edge addition and deletion. Edges have been shown to be added to nodes with a high degree and between pairs of nodes with a high number of common neighbors. If not balanced by a degree dependent edge removal, the preference for high degree nodes and node pairs with many common neighbors is expected to increase the average degree of high degree nodes and their clustering coefficient until very large cliques will be formed. Since such large cliques are not formed in real world networks, we conclude that the edge removal probability around high degree nodes and between node pairs with many common neighbors should be higher than around other nodes. We here show the existence of such a balancing mechanism through the relation between the future edge removal probability around nodes and their degree and a similar relation between the edge removal probability and the number of common neighbors of node pairs. In some networks, this preferential detachment process represents an explicit saturation process, and in others, it represents a random deletion process accompanied by a sublinear edge preferential attachment process. A more complex mechanism emerges in directed networks where the preferential detachment can be proportional to the in and out degrees of the nodes involved. In such networks, preferential detachment is stronger for the incoming edges than for the outgoing edges. We hypothesize multiple possible mechanisms that could explain this phenomenon.