Let Gₙ be the random graph on [n] = 1,...,n with the probability of i,j being an edge decaying as a power of the distance, specifically the probability being $p_{|i-j|}=1/{|i-j|}^{\alpha }$, where the constant α ∈ (0,1) is irrational. We analyze this theory using an appropriate weight function on a pair (A,B) of graphs and using an equivalence relation on B∖A. We then investigate the model theory of this theory, including a “finite compactness”. Lastly, as a consequence, we prove that the zero-one law (for first order logic)...

Let Gₙ be the random graph on [n] = 1,...,n with the possible edge i,j having probability $p_{|i-j|}=1/{|i-j|}^{\alpha }$ for j ≠ i, i+1, i-1 with α ∈ (0,1) irrational. We prove that the zero-one law (for first order logic) holds..