Small-world effect in a uniform-radius broadcasting network
We studied the properties of a network formed from nodes that are randomly deployed over an L × L area. All nodes are broadcasting at a uniform transmission radius, βL. Connections are established when nodes are within their respective communication ranges. We found a critical value, βc, equivalent to the percolation threshold of the system, that marks a transition in network characteristics. For β < βc, the network exhibits Poisson properties that are also found in Erdos-Renyi networks, and is consistent with the random geometric network model, G(N, r), for small values of r. For βc < β < 1, the network exhibits the small-world property
of high average clustering coefficient in conjunction with low average shortest path length. For β > 1, we obtained a regular network where each node is connected to all other nodes in the network. We also show a fitting function for the mean degree, 〈k〉, as a function of β which captures the quadratic dependence of 〈k〉 for small β. We offer an explanation why such a transition occurs in the context of percolation and use the results for optimal communication operation in broadcasting systems.