Finite-Time Consensus of Opinion Dynamics and its Applications to Distributed Optimization Over Digraph

Area of Science:

• Control Theory
• Network Science
• Distributed Systems

Background:

• Nonsmooth dynamical systems present challenges in consensus analysis.
• Understanding opinion dynamics in networks is crucial for social and technological applications.

Purpose of the Study:

• To establish efficient criteria for finite-time consensus in nonsmooth opinion dynamics over digraphs.
• To analyze and compare Carathéodory and Filippov solutions for these dynamics.
• To investigate leaderless and leader-based consensus, including bipartite consensus in signed digraphs.

Main Methods:

• Utilizing tools from nonsmooth analysis and algebraic graph theory.
• Analyzing Carathéodory and Filippov solutions of nonsmooth opinion dynamics.
• Developing continuous-time protocols for distributed optimization problems.

Main Results:

• Efficient criteria for finite-time consensus are established.
• Lower and upper bounds on settling time are derived using digraph cut capacities.
• Leaderless dynamic consensus and finite-time bipartite consensus are demonstrated.
• Convergence to optimal solutions in distributed optimization is guaranteed.

Conclusions:

• The proposed criteria and protocols ensure finite-time consensus in nonsmooth opinion dynamics.
• The methods are effective for solving distributed optimization problems over unbalanced digraphs.
• Numerical examples validate the effectiveness of the developed protocols.