Opinion Consensus with Delay When the Zero Eigenvalue of the Connection Matrix is Semi-simple

Two or more groups with different initial opinions can develop into a consensus with appropriate inter- and intra group interactions and communication delay. Here we consider the case when zero is a semi-simple eigenvalue of the normalized connection matrix, and we expand the concept of consensus where opinions of all individual agents must converge to the common (consensus) value to the weak consensus where opinions are clustered and the consensus value is no longer a scalar constant but a vector in a certain consensus subspace. The linearity of the system enables us to adopt an argument of Atay, using the phase space decomposition in terms of the standard bilinear form for functional differential equations, to derive sharp conditions for the system to reach a weak consensus, and to calculate the consensus vector. This calculation also informs how the consensus value of the coupled systems depends on the consensus of the subgroups, initial conditions and communication delay. We also point out the challenge of extending these results to nonlinear versions of the opinion dynamical systems with delay.