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Random geometric graphs (RGG) can be formalized as hidden-variables modelswhere the hidden variables are the coordinates of the nodes. Here we develop ageneral approach to extract the typical configurations of a generichidden-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 nontrivial satisfaction problem: Given $N$ nodes, a domain $\mathcal{D}$, and adesired average connectivity $\langle k\rangle$, find - if any - thedistribution of nodes having support in $\mathcal{D}$ and average connectivity$\langle k\rangle$. We find out that, in the thermodynamic limit, nodes areeither uniformly distributed or highly condensed in a small region, the tworegimes 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 phasetransition is observed as a function of a parameter $a\in[0,1]$ that tunes theunderlying geometry. In particular, $a=1$ indicates a rigid geometry where onlyclose nodes are connected, while $a=0$ indicates a rigid anti-geometry whereonly distant nodes are connected. Consistently, when $a=1/2$ there is nogeometry and no phase transition. After discussing the numerical analysis, weprovide a combinatorial argument to fully explain the mechanism inducing thisphase transition and recognize it as an easy-hard-easy transition. Our resultshows that, in general, ad hoc optimized networks can hardly be designed,unless to rely to specific heterogeneous constructions, not necessarily scalefree.

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