Statistical mechanics of random geometric graphs: Geometry-induced first order phase transition

Random geometric graphs (RGG) can be formalized as hidden-variables models where the hidden variables are the coordinates of the nodes. Here we develop a general approach to extract the typical configurations of a generic hidden-variables model and apply the resulting equations to RGG. For any RGG, defined through a rigid or a soft geometric rule, the method reduces to a non trivial satisfaction problem: Given $N$ nodes, a domain $\mathcal{D}$, and a desired average connectivity $\langle k\rangle$, find - if any - the distribution of nodes having support in $\mathcal{D}$ and average connectivity $\langle k\rangle$. We find out that, in the thermodynamic limit, nodes are either uniformly distributed or highly condensed in a small region, the two regimes being separated by a first order phase transition characterized by a $\mathop{O}(N)$ jump of $\langle k\rangle$. Other intermediate values of $\langle k\rangle$ correspond to very rare graph realizations. The phase transition is observed as a function of a parameter $a\in[0,1]$ that tunes the underlying geometry. In particular, $a=1$ indicates a rigid geometry where only close nodes are connected, while $a=0$ indicates a rigid anti-geometry where only distant nodes are connected. Consistently, when $a=1/2$ there is no geometry and no phase transition. After discussing the numerical analysis, we provide a combinatorial argument to fully explain the mechanism inducing this phase transition and recognize it as an easy-hard-easy transition. Our result shows that, in general, ad hoc optimized networks can hardly be designed, unless to rely to specific heterogeneous constructions, not necessarily scale free.