Scaling Laws for Connectivity in Random Threshold Graph Models with Non-Negative Fitness Variables

We explore the scaling properties for graph connectivity in random threshold graphs. In the many node limit, we provide a complete characterization for the existence and type of the underlying zero-one laws, and identify the corresponding critical scalings. These results are consequences of well-known facts in Extreme Value Theory concerning the asymptotic behavior of running maxima on i.i.d. random variables. In the important special case of exponentially distributed fitness, we show that the (essentially unique) critical scaling which ensures a power-law degree distribution, does not result in graph connectivity in the asymptotically almost sure (a.a.s.) sense.