I am generating some network samples by using small-world package provided by igraph in R. When I keep the number of nodes and the neighborhood parameter the same, the number of edges do not change when changing the rewiring probability. I was wondering if someone could explain me the exact role of this parameter. For instance, what does exactly happen if I constantly increase the rewiring probability? It is obvious that the density of the graph does not change.
2026-03-27 13:40:56.1774618856
The role of the rewiring probability in small world networks
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Small-world networks start out with nodes that can be visualized as lying on a circle in the plane. Each node is initially connected to some number $k$ of nearest neighbors. Then each edge is considered for re-wiring. With some probability (the re-wiring probability) each edge is disconnected from one of its ends and then connected to a random node in the network instead. These “long-distance” edges give the network its small-world behavior (short paths between arbitrary nodes).
The details can be found in a famous paper by Watts and Strogatz or on Wikipedia.