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We propose a novel network measure of topological invariants, calledsmall-worldness, for identifying topological phase transitions of quantum andclassical spin models. Small-worldness is usually defined in the study ofsocial networks based on the best known discovery that one can find a shortchain of acquaintances connecting almost any two people on the planet. Here wedemonstrate that the small-world effect provides a useful description todistinguish topologically trivial and non-trivial phases in the Kitaev chainand accurately capture the Kosterlitz-Thouless transition in the rotor plane.Our results further suggest that the small-worldness containing both localityand non-locality of the network topology can be a practical approach to extractcharacteristic quantities of topological states of matter.

Network topology: detecting topological phase transitions in the Kitaev chain and the rotor plane