Uniqueness of steady-state throughputs

Determine whether a splitter network can admit two steady-states that yield different throughputs on the input and output arcs. Prove or refute the conjecture that steady-states are unique up to minor modifications (changing membership of some arcs in the fluid set F and adding or subtracting a circulation on the residual graph that leaves the inputs and outputs unchanged), thereby establishing whether output and input throughputs are invariant across all steady-states of a given splitter network.

Background

The paper introduces splitter networks and defines steady-states as pairs consisting of a throughput function and a set of fluid arcs subject to fairness and maximization constraints. While steady-states can be non-unique—e.g., directed cycles can carry arbitrary constant flow—the authors highlight that in their examples the input and output throughputs remained unchanged across different steady-states.

This observation motivates a sharper question: can different steady-states ever produce different aggregate throughputs at inputs and outputs? The authors formulate a conjecture asserting such differences cannot occur, suggesting a restricted form of uniqueness that allows only minor internal adjustments without changing terminal throughputs.

Establishing this would clarify how variability in internal arc status (fluid vs. saturated) and residual circulations impacts terminal flows, and would strengthen the theoretical foundation of algorithms for computing steady-states and for designing balancing networks.