The node degree (i.e., number of neighbors) of each node is close to the mean node degree of the network.
The instantaneous node degree is then defined as the number of simultaneous valid single-hop receptions of the same message, and finally the mean node degree is computed analytically in both AWGN and block-fading channels.
Thus, the real-world networks fulfill the prediction of constant mean node degree.
However, for the sample of networks used here, as noted above, the mean node degree 〈k〉i is approximately constant.
We tested the dominance of linear S∝n scaling given an approximately constant mean node degree 〈k〉 and path length scaling β.
For the outgoing links the mean node degree was higher for the LH than for the RH (27 vs. 22.7), F 1,252) = 10.20, p<0.01.
We could start with the empirically observed approximate constancy of mean node degree 〈k〉 and calculated rewiring parameter p for the real world networks, and deduce a linear scaling of SΔ for the WS models.
We conclude here that the linear relationship between small-world-ness and system size does not hold for an arbitrary collection of networks, but is highly likely if all such networks have a similar mean node degree.
Using the definition of S and (10) for random graphs, <img src="http://journals.plos.org/plosone/article/asset?id=info?doi/10.1371/journal.pone.0002051.e032.PNG" class= inline-graphic"/> where Ci, Li, 〈k〉i are the clustering coefficient, mean shortest path length, and mean node degree of system i respectively.
Topological parameters: D, density; MND, mean node degree; MNDSD, standard deviation of MND; MCC, mean clustering coefficient of the network.
