Effects of quadrilateral clustering on complex contagion

Clustering is one of the most important properties that determine the function of complex networks. But the conventional clustering coefficient considers only triangles without a clear basis. To examine the role of higher-order clustering beyond the conventional triangular clustering, we propose the quadrilateral clustering coefficient that counts the number of cycles of length 4. We also present algorithms to generate quadrilateral clustered networks with regular and scale-free degree distributions. We study the complex contagion model, where clustering promotes spreading. We show that quadrilateral clustered networks have a significant clustering effect, despite negligible conventional clustering coefficient. Moreover, we demonstrate that the clustering effect is stronger in the square lattice with zero conventional clustering coefficient than in the kagome lattice with a sizable conventional clustering coefficient, counterintuitively. Therefore, we conclude that the clustering by quadrilaterals is critical as well as the classical triangular clustering at least in complex contagion.