(a) Size of the first (circles) and second (squares) largest components as a function of the obesity prevalence threshold

*s* in 2008. As we lower

*s*, the largest component increases abruptly indicating absorption of whole clusters, as also evidenced by the peaks in the second largest cluster. We observe two main transitions at

and

in the real data (red) and a single second-order transition in the randomized data (blue). The maps show the progression of the obesity clusters with at least 5 counties for a given

*s*. (b) Percolation tree representing the hierarchical formation, growth and merging of obesity clusters. Each dot represents a cluster at a given

*s* with a size proportional to the logarithm of the cluster's area. Cluster colors follow and we indicate their geographic regions. As we lower

*s* from right to left, regions of high obesity prevalence appear first in the tree. The main percolating cluster starts in the lower Mississippi basin (red) at high

*s* and absorbs clusters until percolating through all US. In particular, we note the two main transitions at

, where it absorbs the two Appalachian clusters, and at

, where it absorbs the West US cluster. (c) Detail of the evolution of obesity clusters near percolation as indicated. The map shows the shape of the first (red), second (yellow), and third (violet) clusters around

, and the largest (green) cluster at

, together with the location of the red bonds responsible for the transitions. The epicenter is Greene county, AL with 43.7% obesity prevalence. (d) Box fractal dimension of percolating cluster in the inset measured by the number of boxes of size

needed to cover the cluster:

, and fractal dimension of the boundary measured by the number of boxes needed to cover the hull:

. (e) Probability distribution of the area of the obesity clusters,

*P*(

*A*) ~

*A*^{−τ}, at percolation

averaged from 2004–2008. This scaling law generalizes Zipf's law from urban to obesity clusters.

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