Fig. 1

Example of Betti number calculation for a simple network. A simple network with red nodes and blue edges consists of two four-node cycles connected via a three-node cycle (Panel a). The Betti number algorithm adds the yellow edges in calculation of filtration index effectively “killing” the four-node cycles by eliminating the cycles. The numerical value for the Betti number is the number of bars lasting from index i to index j in a Betti barcode. Each bar starts when a homology class is born and ends when this class is killed due to appearance in the filtration of a higher dimensional chain whose boundary is the underlying cycle. Thus, Betti0 is equal to 9 and represents the number of nodes (Panel b), and Betti1, is equal to 2, and represents the number of rings of four nodes (Panel c)