Augmented Graph Models for Small-World Analysis with Geographic Factors

Small-world properties, such as small-diameter and clustering, and the power-law property are widely recognized as common features of large-scale real-world networks. Recent studies also notice two important geographical factors which play a significant role, particularly in Internet related setting. These two are the distance-bias tendency (links tend to connect to closer nodes) and the property of bounded growth in localities. However, existing formal models for real-world complex networks usually don't fully consider these geographical factors. We describe a flexible approach using a standard augmented graph model (e.g. Watt and Strogatz's [33], and Kleinberg's [20] models) and present important initial results on a refined model where we focus on the small-diameter characteristic and the above two geographical factors. We start with a general model where an arbitrary initial node-weighted graph H is augmented with additional random links specified by a generic 'distribution rule' τ and the weights of nodes in H. We consider a refined setting where the initial graph H is associated with a growth-bounded metric, and τ has a distance-bias characteristic, specified by parameters as follows. The base graph H has neighborhood growth bounded from both below and above, specified by parameters β1, β2 > 0. (These parameters can be thought of as the dimension of the graph, e.g. β1 = 2 and β1 = 3 for a graph modeling a setting with nodes in both 2D and 3D settings.) That is 2β1 ≤ Nu(2r)/Nu(r) ≤ 2β2 where Nu(r) is the number of nodes v within metric distance r from u: d(u, v) ≤ r. When we add random links using distribution τ, this distribution is specified by parameter α > 0 such that the probability that a link from u goes to v ≠ u is ∝ 1/dα(u,v). We show which parameters produce a small-diameter graph and how the diameter changes depending on the relationship between the distance-bias parameter α and the two bounded growth parameters β1, β2 > 0. In particular, for most connected base graphs, the diameter of our augmented graph is logarithmic if α ≤ β1, and poly-log if β2 ≤ α 2β2. These results also suggest promising implications for applications in designing routing algorithms.