Constrained Ising storage model on a time-varying graph

Definitions and mathematical statement

Aldous's Problem B asks to study the constrained Ising storage model defined on the source page when its underlying graph changes with time. Denote the evolving graph by $G_t=(V_t,E_t)$ and the storage-model configuration by $X_t$. A mathematical specification must give a joint evolution $(G_t,X_t)$, including the original storage state space and constraints, the original update law on each fixed graph, and the interaction between graph updates and storage updates.

None of those fixed-graph model rules is included in the supplied excerpt. Nor is the graph evolution specified: time may be discrete or continuous; graph changes may be deterministic or random, independent of the storage process or state-dependent, and may affect edges alone or also vertices. If a graph update invalidates an admissible configuration, a rule for that event is needed. No particular convention is imposed here.

The request is to study this specific model on a dynamic graph. It does not state a theorem, limiting regime or observable to determine, and cannot be replaced by an arbitrary time-dependent Ising Hamiltonian or generic hard-core process. The research summary describes links with hard-core models and load balancing, but provides no transition rules or exact equivalence establishing those descriptions.

Short English statement

Extend and analyze Aldous's specified constrained Ising storage model when its graph evolves in time. Both the original storage dynamics and the graph-update mechanism are missing from the extract.

Sources

• Local UnsolvedMath registry, permanent ID 9700006, source code catalogue entry; David Aldous; no proposal year supplied; update 2026-08-06.
• Aldous, Open problems, cbrw.html, Problem B: https://www.stat.berkeley.edu/~aldous/Research/OP/index.html. Registry access date 2026-07-29; HTML text with linked primary-PDF text. The background reports 23 active pages, with explicit delegated subproblems separated and compound bullets retained. Present primary-literature status and redistribution terms require review.
• The supplied summary mentions other dynamic-graph, hard-core and load-balancing models but gives no exact reference or verified result for this source-specific model. Its reports of literature searches were not repeated for this draft.