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Clustering $unicode{x2013}$ the tendency for neighbors of nodes to be connected $unicode{x2013}$ quantifies the coupling of a complex network to its underlying latent metric space. In random geometric graphs, clustering undergoes a continuous phase transition, separating a phase with finite clustering from a regime where clustering vanishes in the thermodynamic limit. We prove this geometric-to-nongeometric phase transition to be topological in nature, with atypical features such as diverging free energy and entropy as well as anomalous finite size scaling behavior. Moreover, a slow decay of clustering in the nongeometric phase implies that some real networks with relatively high levels of clustering may be better described in this regime.
As we add rigid bars between points in the plane, at what point is there a giant (linear-sized) rigid component, which can be rotated and translated, but which has no internal flexibility? If the points are generic, this depends only on the combinato
The transition from topologically nontrivial to a trivial state is studied by first-principles calculations on bulk zinc-blende type (Hg$_{1-x}$Zn$_x$)(Te$_{1-x}$S$_x$) disordered alloy series. The random chemical disorder was treated by means of the
The recently discovered three dimensional or bulk topological insulators are expected to exhibit exotic quantum phenomena. It is believed that a trivial insulator can be twisted into a topological state by modulating the spin-orbit interaction or the crystal lattice via odd number of band
Real networks are finite metric spaces. Yet the geometry induced by shortest path distances in a network is definitely not its only geometry. Other forms of network geometry are the geometry of latent spaces underlying many networks, and the effectiv
We investigate the possibility of using structural disorder to induce a topological phase in a solid state system. Using first-principles calculations, we introduce structural disorder in the trivial insulator BiTeI and observe the emergence of a top