Levy geometric graphs

We present a family of graphs with remarkable properties. They are obtained by connecting the points of a random walk when their distance is smaller than a given scale. Their degree (number of neighbors) does not depend on the graph's size but only on the considered scale. It follows a gamma distribution and thus presents an exponential decay. Levy flights are particular random walks with some power-law increments of infinite variance. When building the geometric graphs from them, we show from dimensional arguments that the number of connected components (clusters) follows an inverse power of the scale. The distribution of the size of their components, properly normalized, is scale invariant, which reflects the self-similar nature of the underlying process. This allows to test if a graph (including nonspatial ones) could possibly result from an underlying Levy process. When the scale increases, these graphs never tend towards a single cluster, the giant component. In other words, while the autocorrelation of the process scales as a power of the distance, they never undergo a phase transition of percolation type. The Levy graphs may find applications in community detection and in the analysis of collective behaviors as in face-to-face interaction networks.