DiFlip: Directed Flip-Chain Operation for Regular Directed Graph

We introduce the directed Flip-Chain algorithm, which transforms the regular directed graph into another regular directed graph by swapping two adjacent vertices in the trail of directed edges. Repeatedly applying the directed Flip-Chain operation to an arbitrary regular directed graph will build every regular directed graph with the same probability in the limit. The directed Flip-Chain algorithm preserves the connectivity and regularity of a regular directed graph. The operations of the directed Flip-Chain will transform any regular directed graph into an expander and introduce randomness to the regular directed graph, which is particularly useful in dynamic graphs. Cheeger's inequality determines whether a directed graph is an expander-directed graph. The simulation results indicate that the directed vertex expansion and the directed edge conductance of regular directed graphs converge after performing directed Flip-Chain operations. In addition, the directed Flip-Chain operation with the Push-Sum communication protocol algorithm for randomized rumor spreading on a regular directed graph is observed. The simulation results indicate that the Push-Sum protocol algorithm with directed Flip-Chain operations improves the Mean Squared Error compared to the Push-Sum protocol algorithm without Flip-Chain operations regarding time complexity for rumor dissemination.