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. 
Massimo Ostilli and Ginestra Bianconi
cond-mat.dis-nn cond-mat.stat-mech math.CO