First Betti num­ber of the path ho­mol­ogy of ran­dom di­rected graphs

Posted to site: 17/05/22

Path ho­mol­ogy is a topo­log­i­cal in­vari­ant for di­rected graphs, which is sen­si­tive to their asym­me­try and can dis­cern be­tween di­graphs which are in­dis­tin­guish­able to the di­rected flag com­plex. In Erdös-Rényi di­rected ran­dom graphs, the first Betti num­ber un­der­goes two dis­tinct tran­si­tions, ap­pear­ing at a low-den­sity bound­ary and van­ish­ing again at a high-den­sity bound­ary. In this video, I briefly de­scribe tech­niques for study­ing these tran­si­tions, with more de­tails on the arXiv pre-print.


This work was com­pleted whilst I was a mem­ber of the Centre for Topological Data Analysis, which is funded by the EPSRC grant New Approaches to Data Science: Application Driven Topological Data Analysis’ EP/R018472/1.