| Summary: | We present a synchronization transition study of the locally coupled Kuramoto model on extremely large graphs. We compare regular <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>40</mn><mn>5</mn></msup></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>100</mn><mn>4</mn></msup></semantics></math></inline-formula> lattice results with those of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>12</mn><mo>,</mo><msup><mn>000</mn><mn>2</mn></msup></mrow></semantics></math></inline-formula> lattice substrates with power-law decaying long links (ll). The latter heterogeneous network exhibits <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>d</mi><mi>s</mi></msub><mo>></mo><mn>4</mn></mrow></semantics></math></inline-formula> spectral dimensions. We show strong corrections to scaling and mean-field type of criticality at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><mn>5</mn></mrow></semantics></math></inline-formula>, with logarithmic corrections at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>=</mo><mn>4</mn></mrow></semantics></math></inline-formula> Euclidean dimensions. Contrarily, the ll model exhibits a non-mean-field smeared transition, with oscillating corrections at similarly high spectral dimensions. This suggests that the network heterogeneity is relevant, causing frustrated synchronization akin to Griffiths effects.
|
|---|
