Universal MinUDS in Duplex Network Control

• MinUDS is a duplex-network control objective that identifies the smallest feasible union of driver nodes while preserving fixed controllability budgets.
• It leverages the non-uniqueness of maximum matchings, optimizing the overlap between layer-specific driver sets via cross-layer augmenting paths to reduce redundancy.
• The CLAP-S algorithm efficiently computes MinUDS, demonstrating significant driver savings and improved performance in both synthetic and real-world networks.

Universal Minimum Union Driver Set (MinUDS) denotes a duplex-network control objective in which two directed layers $G_1=(V,E_1)$ and $G_2=(V,E_2)$ share a common node set, and the task is to find the smallest set of physical driver nodes that can simultaneously control both layers while preserving each layer’s structural-controllability budget (Zheng et al., 26 Sep 2025). In the formulation introduced for duplex networks, MinUDS is not merely the union of two independently chosen minimum driver sets (MDSs): it is the minimum possible union over all feasible pairs of layerwise matchings with the prescribed driver counts, so the optimization acts on the non-uniqueness of maximum matchings in each layer rather than on the driver budgets themselves (Zheng et al., 26 Sep 2025).

1. Duplex-network setting and structural-controllability basis

A duplex network is a two-layer multiplex on a common node set $V$, with layers differing only in their intra-layer edges. The control model is the standard linear time-invariant system

$$\frac{d\mathbf{x}(t)}{dt} = A\,\mathbf{x}(t) + B\,\mathbf{u}(t),$$

with structural controllability determined by the zero/non-zero pattern of $A$ rather than by exact edge weights (Zheng et al., 26 Sep 2025).

For a directed graph $G=(V,E)$, structural controllability is mapped to maximum matching on a bipartite representation

$\mathcal{B}=(V^+\cup V^-,E_{\mathcal B}),$

where

$$V^+ = \{v^+ : v\in V\},\qquad V^- = \{v^- : v\in V\},$$

and every directed edge $(u,v)\in E$ becomes a bipartite edge $(u^+,v^-)\in E_{\mathcal B}$ (Zheng et al., 26 Sep 2025). If $G_2=(V,E_2)$0 is a matching, the associated driver set is

$G_2=(V,E_2)$1

so that $G_2=(V,E_2)$2, and for a maximum matching $G_2=(V,E_2)$3 the corresponding driver set $G_2=(V,E_2)$4 is an MDS. In each duplex layer $G_2=(V,E_2)$5, one obtains a maximum matching $G_2=(V,E_2)$6, an MDS $G_2=(V,E_2)$7, and a fixed driver budget

$G_2=(V,E_2)$8

The MinUDS problem keeps these budgets fixed and optimizes only the overlap structure of the two layerwise driver sets (Zheng et al., 26 Sep 2025).

This formulation depends critically on the fact that an MDS is generally not unique. Different maximum matchings of the same layer can induce different unmatched $G_2=(V,E_2)$9-vertices and therefore different driver-node configurations of identical cardinality. MinUDS exploits that internal combinatorial freedom across both layers simultaneously (Zheng et al., 26 Sep 2025).

2. Formal definition of MinUDS and the difference-mass objective

Given fixed budgets $V$0, the feasible matching sets are

$V$1

and the search space is $V$2. For any feasible state $V$3, the union driver set is

$V$4

The budget-preserving optimization problem is

$V$5

and any minimizing union is a MinUDS $V$6 (Zheng et al., 26 Sep 2025).

The paper rewrites this objective through a four-way partition of $V$7. For a state with driver sets $V$8, define the Consistently Driven Set

$V$9

the Consistently Matched Set

$$\frac{d\mathbf{x}(t)}{dt} = A\,\mathbf{x}(t) + B\,\mathbf{u}(t),$$0

and the difference-driver sets

$$\frac{d\mathbf{x}(t)}{dt} = A\,\mathbf{x}(t) + B\,\mathbf{u}(t),$$1

The inefficient part of a naive union lies precisely in $$\frac{d\mathbf{x}(t)}{dt} = A\,\mathbf{x}(t) + B\,\mathbf{u}(t),$$2: these are nodes that must be actuated in one layer but not in the other (Zheng et al., 26 Sep 2025).

The corresponding scalar objective is the difference mass

$$\frac{d\mathbf{x}(t)}{dt} = A\,\mathbf{x}(t) + B\,\mathbf{u}(t),$$3

Because

$$\frac{d\mathbf{x}(t)}{dt} = A\,\mathbf{x}(t) + B\,\mathbf{u}(t),$$4

the union size satisfies

$$\frac{d\mathbf{x}(t)}{dt} = A\,\mathbf{x}(t) + B\,\mathbf{u}(t),$$5

With fixed $$\frac{d\mathbf{x}(t)}{dt} = A\,\mathbf{x}(t) + B\,\mathbf{u}(t),$$6 and $$\frac{d\mathbf{x}(t)}{dt} = A\,\mathbf{x}(t) + B\,\mathbf{u}(t),$$7, minimizing $$\frac{d\mathbf{x}(t)}{dt} = A\,\mathbf{x}(t) + B\,\mathbf{u}(t),$$8 is therefore equivalent to minimizing $$\frac{d\mathbf{x}(t)}{dt} = A\,\mathbf{x}(t) + B\,\mathbf{u}(t),$$9 (Zheng et al., 26 Sep 2025).

Several consequences follow immediately. First, MinUDS exists because the feasible space is finite. Second, it need not be unique, since multiple feasible matching pairs can attain the same minimum union size. Third, the paper notes two lower bounds: the formula above yields the weaker bound $A$0, while the absolute theoretical lower bound is $A$1, attainable only if the topology permits one layer’s driver set to nest perfectly inside the other (Zheng et al., 26 Sep 2025).

3. Cross-Layer Augmenting Paths and the CLAP-S algorithm

The operational mechanism behind MinUDS optimization is a cross-layer generalization of alternating-path exchanges. In a single layer, the Driver Exchange Principle states that if $A$2 and $A$3, then an $A$4-alternating path from $A$5 to $A$6 exists if and only if the symmetric-difference update

$A$7

preserves matching size and swaps the driver status,

$A$8

This gives budget-preserving mobility within one layer’s feasible MDS space (Zheng et al., 26 Sep 2025).

A Cross-Layer Augmenting Path (CLAP) is built from admissible segments. In layer 1, an ordered pair $A$9 is an admissible segment if $G=(V,E)$0, $G=(V,E)$1, and there exists an $G=(V,E)$2-alternating path enabling the exchange that removes $G=(V,E)$3 from $G=(V,E)$4 and adds $G=(V,E)$5. In layer 2, the polarity is reversed: $G=(V,E)$6, $G=(V,E)$7, and an $G=(V,E)$8-alternating path removes $G=(V,E)$9 from $\mathcal{B}=(V^+\cup V^-,E_{\mathcal B}),$0 and adds $\mathcal{B}=(V^+\cup V^-,E_{\mathcal B}),$1 (Zheng et al., 26 Sep 2025).

A CLAP is a sequence

$\mathcal{B}=(V^+\cup V^-,E_{\mathcal B}),$2

such that $\mathcal{B}=(V^+\cup V^-,E_{\mathcal B}),$3, $\mathcal{B}=(V^+\cup V^-,E_{\mathcal B}),$4, adjacent segments alternate layers, and all nodes are distinct. The relay nodes are constrained by type: if the incoming segment is in layer 1, the relay must lie in $\mathcal{B}=(V^+\cup V^-,E_{\mathcal B}),$5; if the incoming segment is in layer 2, the relay must lie in $\mathcal{B}=(V^+\cup V^-,E_{\mathcal B}),$6 (Zheng et al., 26 Sep 2025).

The central gain theorem states that any feasible CLAP preserves both budgets, reduces the difference mass by 2,

$\mathcal{B}=(V^+\cup V^-,E_{\mathcal B}),$7

and therefore contracts the union by exactly one node,

$\mathcal{B}=(V^+\cup V^-,E_{\mathcal B}),$8

Shortest CLAPs are automatically feasible, because any same-layer witness overlap would imply the existence of a shorter CLAP (Zheng et al., 26 Sep 2025).

CLAP-S, the Shortest Cross-Layer Augmenting Path Search algorithm, iteratively applies these reductions. It starts from one feasible matching in each layer, computes the initial driver sets, and then repeatedly invokes a shortest-CLAP search. The search is implemented as a layer-alternating breadth-first search over states $\mathcal{B}=(V^+\cup V^-,E_{\mathcal B}),$9: from a node $$V^+ = \{v^+ : v\in V\},\qquad V^- = \{v^- : v\in V\},$$0 and layer $$V^+ = \{v^+ : v\in V\},\qquad V^- = \{v^- : v\in V\},$$1, the algorithm computes the alternating-reachable set $$V^+ = \{v^+ : v\in V\},\qquad V^- = \{v^- : v\in V\},$$2, checks whether $$V^+ = \{v^+ : v\in V\},\qquad V^- = \{v^- : v\in V\},$$3 is nonempty, and otherwise restricts the next frontier to valid relay nodes—$$V^+ = \{v^+ : v\in V\},\qquad V^- = \{v^- : v\in V\},$$4 when $$V^+ = \{v^+ : v\in V\},\qquad V^- = \{v^- : v\in V\},$$5, and $$V^+ = \{v^+ : v\in V\},\qquad V^- = \{v^- : v\in V\},$$6 when $$V^+ = \{v^+ : v\in V\},\qquad V^- = \{v^- : v\in V\},$$7 (Zheng et al., 26 Sep 2025). If a CLAP is found, its witness paths are recovered and applied via symmetric difference to the relevant layer matchings; if none exists, the state is declared CLAP-stable.

4. Optimality certificate, meta-graph structure, and complexity

The global optimality proof proceeds through a layer-labeled meta-graph constructed from the symmetric differences $$V^+ = \{v^+ : v\in V\},\qquad V^- = \{v^- : v\in V\},$$8 between the current state and any comparator state $$V^+ = \{v^+ : v\in V\},\qquad V^- = \{v^- : v\in V\},$$9. For each layer, path components of the symmetric-difference subgraph induce label-$(u,v)\in E$0 edges in a meta-graph $(u,v)\in E$1, while cycles induce no edge. Each connected component of $(u,v)\in E$2 is a simple path or cycle, and along any path the labels alternate $(u,v)\in E$3 (Zheng et al., 26 Sep 2025).

After orienting each label-$(u,v)\in E$4 edge according to the admissible exchange direction, one obtains a directed multigraph $(u,v)\in E$5. Any directed label-alternating path in $(u,v)\in E$6 from a node in $(u,v)\in E$7 to a node in $(u,v)\in E$8 induces a CLAP. Conversely, if there exists any feasible comparator state with strictly smaller difference mass, then $(u,v)\in E$9 contains such a path. This yields the CLAP-or-Optimal theorem:

1. there is no CLAP from $(u^+,v^-)\in E_{\mathcal B}$0 to $(u^+,v^-)\in E_{\mathcal B}$1;
2. $(u^+,v^-)\in E_{\mathcal B}$2 is minimal over the feasible space;
3. $(u^+,v^-)\in E_{\mathcal B}$3 is minimal over the feasible space (Zheng et al., 26 Sep 2025).

This theorem makes the absence of a CLAP an optimality certificate, analogous in spirit to Berge’s augmenting-path characterization of maximum matching. CLAP-S therefore terminates exactly at a global optimum of the fixed-budget MinUDS problem (Zheng et al., 26 Sep 2025).

The computational profile follows the same decomposition. Initial maximum matchings are computed with Hopcroft–Karp in

$(u^+,v^-)\in E_{\mathcal B}$4

If a shortest CLAP in one iteration has $(u^+,v^-)\in E_{\mathcal B}$5 segments, the search cost is

$(u^+,v^-)\in E_{\mathcal B}$6

and since each successful iteration reduces $(u^+,v^-)\in E_{\mathcal B}$7 by 2, the number of iterations is at most $(u^+,v^-)\in E_{\mathcal B}$8. The resulting worst-case total is

$(u^+,v^-)\in E_{\mathcal B}$9

Empirically, the average CLAP length is reported as $G_2=(V,E_2)$00, which makes the search phase close to linear in the total edge count per iteration (Zheng et al., 26 Sep 2025).

5. Empirical behavior, baselines, and application domains

The empirical evaluation compares CLAP-S with three baselines: Random Sample Union (RSU), which samples $G_2=(V,E_2)$01 randomized maximum matchings per layer and chooses the best among the $G_2=(V,E_2)$02 resulting unions; CLAP-G, a greedy length-1 local search that repeatedly applies any single improving segment; and ILP-Exact, an exact integer linear program used as a gold standard when feasible (Zheng et al., 26 Sep 2025).

Method	Mechanism	Role
RSU	Sample randomized maximum matchings in each layer and keep the smallest of $G_2=(V,E_2)$03 unions	Sampling baseline
CLAP-G	Repeatedly apply any single admissible improving segment	Greedy baseline
ILP-Exact	Exact integer linear programming formulation minimizing $G_2=(V,E_2)$04	Optimal reference

The synthetic experiments use $G_2=(V,E_2)$05 duplexes of types ER-ER, BA-BA, and ER-BA, with average degree $G_2=(V,E_2)$06 varied from 2 to 10 and edge-overlap Jaccard similarity $G_2=(V,E_2)$07 varied from 0.1 to 0.9. The reported driver-node savings are measured by

$G_2=(V,E_2)$08

where $G_2=(V,E_2)$09 is the naive initial union, and by the net gain over RSU,

$G_2=(V,E_2)$10

with relative optimization rate

$G_2=(V,E_2)$11

CLAP-S consistently improves on RSU. In BA-BA duplexes with moderate sparsity, it saves nearly 70 drivers over naive initial unions, and across synthetic families it is typically one order of magnitude faster than RSU: for $G_2=(V,E_2)$12, ER+ER timing is $G_2=(V,E_2)$13 s for RSU versus $G_2=(V,E_2)$14 s for CLAP-S; BA+BA timing is $G_2=(V,E_2)$15 s versus $G_2=(V,E_2)$16 s; ER+BA timing is $G_2=(V,E_2)$17 s versus $G_2=(V,E_2)$18 s (Zheng et al., 26 Sep 2025).

The real-world evaluation covers genetic, neuronal, social, and human-relationship duplexes. On all tested networks where ILP is feasible, CLAP-S matches the ILP optimum exactly. Representative cases include Drosophila, where the initial union is $G_2=(V,E_2)$19, CLAP-S reaches $G_2=(V,E_2)$20, RSU reaches $G_2=(V,E_2)$21, and ILP also gives $G_2=(V,E_2)$22; YeastLandscape, where $G_2=(V,E_2)$23, CLAP-S reaches $G_2=(V,E_2)$24, RSU reaches $G_2=(V,E_2)$25, and ILP gives $G_2=(V,E_2)$26; and CelegansConnectome, where $G_2=(V,E_2)$27, CLAP-S reaches $G_2=(V,E_2)$28, RSU reaches $G_2=(V,E_2)$29, CLAP-G reaches $G_2=(V,E_2)$30, and ILP gives $G_2=(V,E_2)$31 (Zheng et al., 26 Sep 2025).

The strongest predictor of optimization potential is the initial disagreement between layerwise MDSs. The paper reports strong correlations between initial difference-set size and CLAP-S savings, with $G_2=(V,E_2)$32 in synthetic ER duplexes, $G_2=(V,E_2)$33 in synthetic BA duplexes, and $G_2=(V,E_2)$34 in real-world networks. Sparse networks and scale-free duplexes often provide the largest absolute savings because they tend to have large initial MDSs and large difference-driver sets; high layer overlap can further increase the relative optimization rate (Zheng et al., 26 Sep 2025).

In applied terms, MinUDS is presented as a way to reduce intervention cost in biological, neuronal, social, and organizational systems. A plausible implication is that the principal benefit does not come from lowering each layer’s controllability budget—those remain fixed—but from removing cross-layer redundancy in where control inputs are physically placed.

6. Related notions, terminological boundaries, and current limitations

The term MinUDS is specific to duplex-network control in the cited formulation. It should be distinguished from the Minimum $G_2=(V,E_2)$35-Union problem, where one is given a set system and asked to select $G_2=(V,E_2)$36 sets minimizing the size of their union. That problem is equivalent to Small Set Bipartite Vertex Expansion, and the cited work gives an $G_2=(V,E_2)$37-approximation for Minimum $G_2=(V,E_2)$38-Union and an $G_2=(V,E_2)$39-approximation for SSBVE (Chlamtáč et al., 2016). The name resemblance is close, but the objective is different: Minimum $G_2=(V,E_2)$40-Union optimizes over subsets of sets, whereas MinUDS optimizes over pairs of structurally controllable matching states in a duplex network.

A different line of work uses “driver set” only interpretively. In union-closed-family theory, an FC-family is a family $G_2=(V,E_2)$41 such that every union-closed family containing $G_2=(V,E_2)$42 satisfies Frankl’s conjecture; the source explicitly identifies this with a driver-set interpretation. For 3-sets on $G_2=(V,E_2)$43, the smallest universal driver size in that setting is

$G_2=(V,E_2)$44

for all $G_2=(V,E_2)$45 (Pulaj, 2019). In the infinite-breadth setting, no single universal driver exists; instead, three canonical configurations $G_2=(V,E_2)$46, $G_2=(V,E_2)$47, and $G_2=(V,E_2)$48 form a minimal universal catalog of unavoidable subprojections (Choi et al., 2017). These are conceptual analogies rather than direct uses of the MinUDS terminology.

Within duplex-network control itself, the current formulation has explicit limitations. It assumes linear time-invariant dynamics, structural controllability, and no inter-layer couplings; the two layers share only a node set, not cross-layer edges. The budgets $G_2=(V,E_2)$49 are fixed and usually minimal, so the problem does not address budget-tradeoff variants in which slightly increasing one layer’s driver count might further reduce the union. The paper also notes that extension beyond two layers is nontrivial: the basic ideas of difference sets and alternating segments may generalize, but the combinatorial complexity of multi-layer paths would increase significantly and would require new algorithmic approaches to maintain tractability (Zheng et al., 26 Sep 2025).

Under these assumptions, MinUDS has a precise graph-theoretic meaning: it is the minimum union of two budget-preserving driver sets induced by feasible layerwise matchings, and CLAP-S computes it exactly by iterated shortest cross-layer augmenting paths (Zheng et al., 26 Sep 2025).