It's a small world! (And how complex networks help us understand why)

by Dr Subodh Patil (Leiden University)

Tuesday 19 Nov 2024, 16:00 → 17:00 Asia/Kolkata

69 (AG)

Networks are a useful mathematical abstraction to model a variety of real and virtual systems spanning biology, social science, economics, and informatic systems, where the relations between any two elements or agents are as significant to our understanding as the elements themselves. The formal study of such systems – network theory (really, applied graph theory) -- is tasked with understanding the properties and collective behavior of any system that can be inferred from this relational structure alone. 

A remarkable feature emerges in a large class of networks: on average, one can hop between any two elements with a very small number of hops -- the so called small world property. In this talk I will review this idea illustrated with some familiar examples, and present simple renormalization group and mean field derivations that apply when there is an approximate underlying lattice structure to the network. Many real world networks, of course, do not possess such an underlying structure. To complicate matters, notions of locality are a derived, rather than an intrinsic property of a random network, suggesting the need to rethink standard mean field and renormalization group approaches. In this talk, I will speak on recent and ongoing attempts at both, presenting a novel mean field method to study the small world properties of complex networks applicable where significant heterogeneity and disorder may be present. I will also present exactly renormalizable cases of a random graph model where one can introduce dependence on the probability for any link to appear conditioned on other links (i.e. interactions), concluding with some applications of the work presented towards social, biological, and epidemiological systems.