# Math Insight

### Applet: Small world model network

A network of $N=200$ nodes spread around a ring. Originally, each node was symmetrically connected to its 8 nearest neighbors along the ring. But then, each edge was rewired with probability $p$. If an edge was selected for rewiring, one end of the edge was disconnected from a node and reconnected with a randomly chosen node. The rewiring creates shortcuts across the network and rapidly decreases the mean path length $l$ across the network. As you increase $p$ (by moving the point on the slider with your mouse), the mean path length decreases rapidly, as shown at the plot on the right. The rewiring also decrease the clustering coefficient $C$, but there is a large range of $p$ where the clustering coefficient is still large and the path length is small, which is the small world network regime (shaded). The path length and clustering coefficient calculated are averages over many realizations of the network for a given $p$ (and $C$ is an approximation for large network size $N$), so they don't exactly correspond to the actual network shown at the left. In the original network with $p=0$, both measures achieve their maximums of $C=0.64$ and $\ell=12.5$.

Applet file: small_world_network.ggb