Further Results for Edge Convergence of Directed Signed Networks

Area of Science:

• Network Science
• Graph Theory
• Systems Dynamics

Background:

• Previous research on edge convergence in directed signed networks (Du et al., 2019) relied on strong connectivity assumptions.
• The limitations of existing models necessitate exploring convergence under relaxed connectivity constraints.

Purpose of the Study:

• To determine if and how edge convergence can be achieved in directed signed networks lacking strong connectivity.
• To analyze the conditions under which edge states converge to constant or time-varying vector trajectories.

Main Methods:

• Analysis of edge convergence for spanning tree network topologies.
• Investigation of network properties including structural balance and r-structural imbalance.
• Derivation of dynamic edge behavior to inform node convergence analysis.

Main Results:

• Edge states converge to a constant vector if the signed network is structurally balanced or r-structurally unbalanced.
• Edge states converge to a time-varying vector trajectory with constant speed if the signed network is structurally unbalanced and r-structurally balanced.
• Established edge convergence dynamics provide a basis for addressing node convergence problems.

Conclusions:

• Edge convergence in directed signed networks is achievable even without strong connectivity, depending on structural properties.
• The findings offer new insights into the dynamic behavior of signed networks and their convergence properties.
• Simulation examples validate the theoretical results on edge convergence.