LIE OPERATORS FOR TRUST DYNAMICS ON THE UNIT SPHERE IN COMPLEX NETWORKS

Keywords: Hilbert space, Lie algebra, Embedding, Operator, Trust propagation.

This paper is to introduce a geometric framework for trust modeling

in complex networks in which trust states are embedded as unit vectors in

a Hilbert space. Trust evolution is formulated as a continuous dynamical

system generated by Lie operators acting on the unit sphere, ensuring

norm preservation and stability of the representation. We show that

network interactions naturally induce skew-adjoint operators, leading to

flows that correspond to rotations on the trust sphere. This geometric

structure guarantees that trust propagation follows geodesic trajecto-

ries, providing both mathematical consistency and interpretability. The

proposed framework establishes a connection between graph-based inter-

actions, Lie algebraic dynamics, and geometric deep learning. It oﬀers a

principled foundation for modeling trust propagation with stability, in-

variance, and continuous-time dynamics in complex networked systems.