A Dynamic Graph Fluid System Toward a Graph-Theoretic Analog of Fluid Flow, Self-Assembly, and Terminal Circulation

We introduce the Dynamic Graph Fluid System (DGFS) — a mathematical framework in which vertices behave as particles that bond, break, and rebond dynamically, flowing through a self-assembling graph topology under an intrinsic driving force. Particles bond with their lowest-resistance neighbors subject to a uniform valence constraint and a type-affinity function, carving out graph structure as they flow. Persistent bonds among same-type particles give rise to coherent subgraphs called eigenstructures, characterized by a dominant leading eigenvalue. All particles ultimately drain into a terminal basin — the lake — whose boundary is dynamically expanding and whose interior supports perpetual, stratified circulation with no escape.
We prove that when the lake stabilizes to a complete graph on m particles, inflow and internal circulation satisfy an exact relation. We further prove a Necessary Growth Condition: a static lake under nonzero inflow produces upstream stagnation. In the continuous limit the system recovers the porous medium equation with exponent m=2. The framework requires no background geometry and applies universally to any system whose components interact and accumulate according to least-resistance bonding principles.