Alladi Ramakrishnan Hall

Partially bipartite networks are infra-small world

Aradhana Singh

IMSc Chennai

Partially bipartite networks characterized by connections primarily occurring between two groups of nodes forming distinct subnetworks (modules) allow investigating the effect of strong inter-dependence between them on the collective properties of the system. We show that the anti-modular character of these networks, which is controlled by the ratio of inter- to intra-group connection density, can result in large-scale structural properties that do not fall within the well-known regular-small world-random network paradigm. In particular, anti-modular networks having strong heterogeneity in their module sizes (e.g., networks defined by adjacent occurrence of characters in different natural languages that are written using an alphabetic system) can exhibit co-occurrence of higher efficiency and lower clustering compared to their randomized counterparts, that we term as infra-small world. The distinct properties of such networks is also reflected in their spectral properties, viz., in the distribution of their Laplacian eignvalues. Strong module size heterogeneity in anti-modular networks is manifested in a prominent degeneracy at unity eigenvalue, with the corresponding eigenmodes exhibiting high degree of localization. We also show that the fastest eigenmode (corresponding to the largest eignevalue of the Laplacian) undergoes a localization-delocalization transition with the meso-scale nature of the network changing from modular to anti-modular. Further, the distribution of the components of this eigenmode has a strong bimodal nature that increases as the network comes closer to being bipartite and we propose this as a signature for identifying anti-modular organization in complex systems that occur in the natural, technological and social domains.