Leaky Zero Forcing: Theory & Applications

• Leaky Zero Forcing is a leakage-aware extension of traditional zero forcing used for fault tolerance in graphs and controlled interference in beamforming.
• It adapts the classical color-change rule by introducing leaks, ensuring robust propagation even when some vertices, edges, or forces are restricted.
• In communications, the approach relaxes strict interference nulling to allow controlled leakage, optimizing feasibility in coordinated beamforming and massive MIMO.

Leaky zero forcing is a term used in two distinct technical literatures. In graph theory, it denotes a fault-tolerant extension of zero forcing in which some vertices are designated as leaks and cannot perform forces; the objective is to find an initial blue set that forces all vertices blue regardless of the placement of up to $\ell$ leaks, yielding the $\ell$-leaky forcing number $Z_{(\ell)}(G)$, $z_\ell(G)$, or $Z_\ell(G)$ depending on notation (Dillman et al., 2019). In multiuser beamforming and massive MIMO, the expression appears in the closely related sense of relaxed or regularized zero forcing, where strict interference nulling is relaxed and controlled interference leakage is allowed or explicitly penalized in the precoder design (Park et al., 2012). The two usages are mathematically unrelated, but both replace exact zero forcing by a leakage-aware formulation that trades rigidity for robustness or feasibility.

1. Graph-theoretic definition and basic formalism

Classical zero forcing begins with a graph $G=(V,E)$, an initial blue set $B\subseteq V$, and the color change rule: if a blue vertex has exactly one white neighbor, it can force that neighbor to blue. A zero forcing set is an initial blue set that eventually colors all vertices blue, and the minimum size of such a set is the zero forcing number $Z(G)$ (Dillman et al., 2019).

Leaky forcing generalizes this process by introducing leaks. In one standard formulation, a vertex leak is a vertex that is not allowed to perform any force during the process; in another, a leak is defined by adding a new vertex adjacent to only one existing vertex, creating a persistent uncolored neighbor that disables forcing at that location (Alameda et al., 2020). For a fixed nonnegative integer $\ell$, an $\ell$-leaky forcing set is a set $\ell$0 such that, for any choice of $\ell$1 leaks, the color change rule still colors all vertices blue. When $\ell$2, one recovers ordinary zero forcing: $\ell$3 (Herrman, 2022).

Several equivalent formalizations appear in the literature. One uses a leaky derived set $\ell$4, defined as the set of black nodes produced by exhaustively applying the zero forcing rule while treating the leaks in $\ell$5 as always refusing to force; then $\ell$6 is an $\ell$7-leaky forcing set if $\ell$8 for every $\ell$9 with $Z_{(\ell)}(G)$0 (Abbas, 2023). The associated parameters are monotone in $Z_{(\ell)}(G)$1: $Z_{(\ell)}(G)$2 This monotonicity expresses the increasing redundancy required to survive additional leaks (Dillman et al., 2019).

A related notion is $Z_{(\ell)}(G)$3-resilience: a graph is $Z_{(\ell)}(G)$4-resilient if its $Z_{(\ell)}(G)$5-leaky forcing number equals its zero forcing number, that is,

$Z_{(\ell)}(G)$6

In this case, the minimum zero forcing set is already fault tolerant with respect to $Z_{(\ell)}(G)$7 leaks (Alameda et al., 2020).

2. Characterizations, equivalences, and structural constraints

A basic structural constraint is degree-based. Any $Z_{(\ell)}(G)$8-leaky forcing set must contain all vertices of degree at most $Z_{(\ell)}(G)$9; equivalently, if $z_\ell(G)$0, then every vertex must be initially blue, so $z_\ell(G)$1 (Dillman et al., 2019). This condition is sharp for several graph families, most notably trees (Alameda et al., 2020).

The case $z_\ell(G)$2 admits a particularly useful characterization. A set $z_\ell(G)$3 is a $z_\ell(G)$4-leaky forcing set if and only if, for every $z_\ell(G)$5, there exist two distinct blue vertices that can each force $z_\ell(G)$6 in some zero forcing process starting at $z_\ell(G)$7. In the notation of possible forces, this is the requirement that for every $z_\ell(G)$8, there exist $z_\ell(G)$9 with $Z_\ell(G)$0 (Alameda et al., 2020). Intuitively, every nonblue vertex must have two distinct potential forcers so that one leak cannot block it.

For general $Z_\ell(G)$1, the recursive characterization has the same redundancy flavor. A set $Z_\ell(G)$2 is an $Z_\ell(G)$3-leaky forcing set if and only if $Z_\ell(G)$4 is an $Z_\ell(G)$5-leaky forcing set and, for every set $Z_\ell(G)$6 of $Z_\ell(G)$7 leaks and every $Z_\ell(G)$8, there are two distinct possible forces $Z_\ell(G)$9 with $G=(V,E)$0 (Alameda et al., 2020). This recursion underlies much of the later theory.

The literature also shows that several apparently different leak models are equivalent. For all graphs $G=(V,E)$1 and all $G=(V,E)$2,

$G=(V,E)$3

where $G=(V,E)$4 is the $G=(V,E)$5-edge-leaky forcing number and $G=(V,E)$6 is the specified $G=(V,E)$7-leaky forcing number; moreover, the mixed $G=(V,E)$8-leaky forcing number also equals the vertex-leaky forcing number (Alameda et al., 2020). Thus, robustness to vertex leaks, edge leaks, specified forbidden forces, or mixtures of these is governed by the same minimum cardinality parameter.

These constraints have consequences for resilient graphs. If $G=(V,E)$9 is $B\subseteq V$0-resilient, then $B\subseteq V$1 (Alameda et al., 2020). More generally, resilience forces a graph to avoid local bottlenecks that can be blocked by a small number of leaks. A useful upper bound follows from reversal constructions: if $B\subseteq V$2 is a zero forcing set and $B\subseteq V$3 is a reversal, then $B\subseteq V$4 is always a $B\subseteq V$5-leaky forcing set, implying

$B\subseteq V$6

(Alameda et al., 2020).

3. Exact values and bounds for standard graph families

The earliest systematic computations established exact leaky forcing numbers for many classical families. For paths,

$B\subseteq V$7

while for cycles,

$B\subseteq V$8

For complete graphs,

$B\subseteq V$9

and for wheel graphs,

$Z(G)$0

For trees with $Z(G)$1 leaves, excluding $Z(G)$2, one has $Z(G)$3 (Dillman et al., 2019). This was later generalized: for every tree $Z(G)$4,

$Z(G)$5

(Alameda et al., 2020).

Grid graphs played a central role in the early development of the subject. The zero forcing number of $Z(G)$6 is $Z(G)$7, and the $Z(G)$8-leaky forcing number satisfies

$Z(G)$9

for all $\ell$0; this resolved a question posed by Dillman and Kenter (Alameda et al., 2020). The bound is tight for sufficiently wide grids: $\ell$1 Earlier bounds also included $\ell$2 and $\ell$3 for square grids (Dillman et al., 2019).

Hypercubes exhibit a strong form of resilience. The zero forcing number is $\ell$4, and the same value persists for several leak budgets: $\ell$5 (Dillman et al., 2019). A later result showed

$\ell$6

and conjectured the bound to be tight; this equality is known for $\ell$7 (Herrman, 2022).

For the prism graph $\ell$8, the $\ell$9-leaky forcing number satisfies

$\ell$0

with additional bounds $\ell$1 and $\ell$2 for $\ell$3 (Herrman, 2022). More recently, leaky forcing numbers were completely determined for all unicyclic graphs and all $\ell$4, and connected graphs with extremal $\ell$5-leaky forcing number were characterized: $\ell$6 if and only if $\ell$7 is a path or a cycle, while $\ell$8 if and only if $\ell$9 (Bjorkman et al., 4 Aug 2025).

4. Forts, PSD forcing, and controllability extensions

Forts provide a structural obstruction theory for leaky forcing. In the standard setting, the constraint-generation integer programming approach uses forts as unavoidable leftover sets: if the current coloring fails, then the remaining uncolored set is a fort and must be intersected by any feasible initial blue set (Dillman et al., 2019). Recent work on Hopi rectangle graphs sharpened this idea by defining $\ell$00-leaky forts and proving that a set $\ell$01 is an $\ell$02-leaky forcing set if and only if it intersects every $\ell$03-leaky fort (Jr et al., 25 Sep 2025).

For Hopi rectangle graphs $\ell$04, this fort-based analysis yields a complete formula: $\ell$05 The same work also shows that $\ell$06, where $\ell$07 denotes the maximum nullity (Jr et al., 25 Sep 2025).

A parallel generalization is leaky positive semidefinite forcing. The $\ell$08-leaky PSD forcing number $\ell$09 combines leaks with the PSD color change rule and satisfies

$\ell$10

It is monotone in $\ell$11, inherits the minimum-degree constraint, and admits a fort characterization: a set is an $\ell$12-leaky PSD forcing set if and only if it intersects every $\ell$13-leaky PSD fort (Elias et al., 2023). Exact values were obtained for paths, cycles, complete graphs, wheel graphs, complete bipartite graphs, trees, hypercubes, and prisms; for example,

$\ell$14

Leaky forcing also has a direct interpretation in network control. In strong structural controllability, a leader set renders a network SSC if and only if it is a zero forcing set; replacing zero forcing by leaky forcing produces a resilience model for misbehaving nodes and edges (Abbas, 2023). The central equivalence theorem states that for a candidate leader set $\ell$15 and integer $\ell$16,

$\ell$17

Thus resilience to leak nodes, non-forcing edges, and removable edges is equivalent. The corresponding minimum leader-selection problem is NP-hard, but greedy heuristics were numerically evaluated on Erdős–Rényi and Barabási–Albert graphs with $\ell$18 and found to be very close to optimal (Abbas, 2023).

5. Relaxed zero forcing in coordinated beamforming

In the communications literature, “leaky ZF” refers to relaxed zero forcing rather than graph coloring. In coordinated beamforming for $\ell$19 transmitter–receiver MISO interference channels, conventional ZF enforces zero interference to all undesired receivers, but relaxed zero forcing allows predetermined interference leakage: $\ell$20 Here $\ell$21 is the allowed leakage factor normalized to the noise power at receiver $\ell$22; $\ell$23 reduces to ZF (Park et al., 2012).

This relaxation yields a separable per-transmitter design. If $\ell$24, then the achievable rate admits the lower bound

$\ell$25

so each transmitter can solve

$\ell$26

subject to the leakage constraints and the power constraint $\ell$27 (Park et al., 2012). The optimal beam vector for fixed leakage levels can be constructed by sequential orthogonal projection combining (SOPC), which allocates power first in the matched-filter direction and then along progressively projected directions as interference constraints become active.

The same paper extends the framework to MIMO interference channels by replacing scalar leakage constraints with

$\ell$28

and deriving the lower bound

$\ell$29

A projected gradient method then solves the distributed beam design problem (Park et al., 2012).

A related leakage-based perspective appears in signal and interference leakage minimization (SILM), which minimizes a weighted sum of inter-cell interference powers and signal power leaked outside receive subspaces. The weight $\ell$30 controls the trade-off between pure interference minimization and a more “leaky” design that preserves desired signal dimensions; intra-cell interference is then handled by MMSE or ZF precoding (Elkourdi et al., 2014). In this sense, the communications meaning of leaky zero forcing is not failure tolerance but controlled nonzero interference.

6. Regularization, quantization, and secrecy-aware leakage control

Regularized zero forcing (RZF) is frequently interpreted as a leakage-aware generalization of strict ZF. In one-bit quantized massive MIMO for mmWave downlink, an SLNR-based precoder was derived using the Bussgang model

$\ell$31

with $\ell$32 the linear approximation matrix and $\ell$33 the covariance of quantization distortion. The resulting user-$\ell$34 precoder is

$\ell$35

and the paper states explicitly that “RZF can be seen as a form of leaky ZF”; when quantization impairments are ignored, the design collapses to standard RZF (Yapıcı et al., 2019).

In massive MIMO with conventional RZF precoding, the asymptotic signal-to-leakage-plus-noise ratio (SLNR) converges to a deterministic value as the numbers of antennas and users grow with fixed ratio, and in symmetric uncorrelated channels the SLNR is asymptotically equal to the SINR (Park et al., 2016). This permits optimization of user loading for spectral efficiency. The optimal user loading equals one in both the low and high SNR regimes, although it can drop below one at intermediate SNR (Park et al., 2016).

Leakage can also be incorporated for physical-layer security. Secure regularized zero forcing (SRZF) augments the standard RZF design by penalizing the received signal power at eavesdroppers: $\ell$36 leading to

$\ell$37

The resulting achievable per-user secrecy rate is

$\ell$38

where $\ell$39 is the SINR at the legitimate user and $\ell$40 is the SINR at the eavesdroppers under full cooperation (Asaad et al., 2019). Large-system analysis yields closed-form asymptotic expressions, and numerical investigations show robustness against the quality of eavesdroppers’ channels (Asaad et al., 2019).

A further robustness layer appears in MU-MISO beamforming with per-antenna power constraints and quantized CDI. There, non-robust ZF beamforming under quantized CDI is proved to minimize the average inter-user leakage, a closed-form CDF of the leakage power is derived, and two leakage-threshold schemes are proposed: minimum average leakage control (MALC) and relaxed average leakage control (RALC). Their thresholds are

$\ell$41

and

$\ell$42

with per-user powers updated by geometric programming under per-antenna constraints (Ding et al., 2020). In this literature, leakage is therefore both a performance metric and a design variable.

Across these two domains, leaky zero forcing consistently denotes a structured relaxation of classical zero forcing. In graphs, the relaxation is adversarial and combinatorial: forcing must succeed despite disabled forcing agents. In beamforming, the relaxation is analytic and optimization-driven: exact interference nulling is replaced by controlled leakage, regularization, or explicit leakage penalties.