Precisely Sculpting Complex Attractor Landscapes through Topological Design - A Resource-Conserving Share-and-Plunder Graph Dynamical Equation

Attractor models in neuroscience and cognitive dynamics sometimes use deliberately designed topology to produce discrete choices, continuous manifolds, memory states, or temporal patterns. These examples show that topology can shape the attractor landscape. In many existing models, however, the intended functional regime still depends on detailed tuning of coupling strengths, gains, timescales, and nonlinearities, and the resulting designs usually address relatively simple attractor structures such as binary selection or a single continuous manifold. Here we introduce a resource-conserving share-and-plunder network in which signed weighted topology is given a direct resource-flow semantics. Positive edges implement sharing among compatible supports, negative edges implement plundering between incompatible supports, and, after initialization, the total resource remains conserved in the absence of external sources, sinks, or interventions that change the total amount. Nonnegative boundary cutoff changes the effective flow when suppressed supports reach the boundary. This formulation makes topology a more explicit design language for attractor structure: once compatibility and incompatibility relations are specified, many qualitative regimes are realized over broad parameter ranges rather than at isolated tuned points. We define a minimal motif dictionary including citadels, multi-citadel camps, three-node circulation motifs, neutralizers, and far-plundering rings, and show how these motifs organize selection, coexistence, periodic dynamics, shared support, and node-to-motif substitution. The framework provides a compact way to design increasingly refined attractor landscapes through signed resource-flow topology.