From Control to Opinion Dynamics: Signum Consensus Protocol on Arbitrary Weighted Directed Graphs

Abstract: We study the signum consensus protocol in continuous-time systems over arbitrary weighted directed graphs with bounded disturbances. The right-hand side of the differential equation is discontinuous on a codimension-$n$ manifold ($n > 1$). On such a manifold, the Filippov sliding vector is not uniquely determined. This results in non-unique solutions and makes the analysis of the system challenging. We define the Polarization Index as the supremum of the growth rate of the difference between the maximum and minimum agent states in the system, derive its closed-form expression, and show that some solution attains this supremum at all forward times except during consensus. From this result, we derive necessary and sufficient conditions for consensus and provide a least upper bound on consensus time. To address the high computational complexity of evaluating the Polarization Index, we propose a low-average-complexity algorithm. Finally, we develop an opinion dynamics model grounded in the signum consensus protocol, revealing a fundamental link between dissensus and community structures.