Fragile Power Domination in Stochastic PMUs

• Fragile power domination is a stochastic model that integrates independent PMU failures into classical power domination, producing a polynomial expected coverage metric.
• It employs both subset-sum and vertex-sum formulations to aggregate observation probabilities over surviving PMU sets, enabling precise robustness analysis.
• The framework supports tuning of sensor robustness via graph-gadget constructions, bridging stochastic and worst-case fault tolerance in network observability.

Fragile power domination is a stochastic extension of graph-theoretic power domination in which each phasor measurement unit (PMU) fails independently with probability $q\in[0,1]$ before the usual domination and zero-forcing propagation begins. Its central object is the expected number of observed vertices, denoted $\mathcal E(G;S,q)$, which is a polynomial in $q$ of degree at most $|S|$. The framework was introduced as “fragile power domination” in “Power domination with random sensor failure” (Bjorkman et al., 2023) and was subsequently developed in “On Fragile Power Domination” (Bjorkman et al., 19 Jul 2025), where the expected-coverage polynomial is characterized algebraically and shown to be controllable through graph-gadget constructions.

1. Formal model

Let $G=(V,E)$ be a simple graph and let $S\subseteq V$ be a set, or more generally a multiset, of PMU locations. In classical power domination, the process begins with the domination step

$B:=N[S]=\bigl\{\,v\}\cup\{u:uv\in E,\;v\in S\},$

or equivalently $B_0=\bigcup_{v\in S}N[v]$, and then repeats the zero-forcing step: whenever an observed vertex has exactly one neighbor outside the currently observed set, that unique neighbor is added. At termination, the observed set is $\Obs(G;S)$, and the power-domination number is

$\gamma_P(G)=\min\{\lvert S\rvert:S\subseteq V,\;\Obs(G;S)=V\}.$

This is the standard deterministic baseline against which fragile power domination is defined (Bjorkman et al., 2023).

Fragile power domination inserts an independent sensor-failure step before observation begins. Given $\mathcal E(G;S,q)$0, each PMU survives with probability $\mathcal E(G;S,q)$1, independently, producing a random surviving set $\mathcal E(G;S,q)$2. The classical power-domination process is then run from $\mathcal E(G;S,q)$3. In the notation of (Bjorkman et al., 2023), the terminal random observed set is $\mathcal E(G;S,q)$4 and the associated random variable is

$\mathcal E(G;S,q)$5

In the notation emphasized in (Bjorkman et al., 19 Jul 2025), the focus is the expected observed cardinality rather than the full distribution.

The model is motivated by power-grid monitoring, where PMUs may break or otherwise fail. The classical objective of full observation is therefore replaced by a stochastic coverage objective, while preserving the same domination and zero-forcing dynamics after failures occur.

2. Expected-coverage polynomial

The expected-coverage polynomial is

$\mathcal E(G;S,q)$6

in the notation of (Bjorkman et al., 19 Jul 2025), and equivalently

$\mathcal E(G;S,q)$7

in the notation of (Bjorkman et al., 2023). Two equivalent formulas are fundamental: $\mathcal E(G;S,q)$8 and

$\mathcal E(G;S,q)$9

The first is a subset-sum formula obtained by conditioning on the surviving PMU set; the second is a vertex-sum formula expressing expectation as a sum of observation probabilities (Bjorkman et al., 19 Jul 2025).

It follows immediately that $q$0 is a polynomial in $q$1 of degree at most $q$2. If one writes

$q$3

then the coefficient of $q$4, for $q$5, is

$q$6

This coefficient formula makes explicit that each polynomial coefficient is an inclusion-exclusion aggregate of classical observed-set sizes over all surviving subsets (Bjorkman et al., 2023).

Several boundary cases are immediate. The constant term is

$q$7

so at $q$8 the model collapses to classical power domination. At the opposite extreme, large values of $q$9 emphasize placements whose surviving PMUs still provide local coverage, even when the original set $|S|$0 was chosen to guarantee full domination only in the no-failure regime (Bjorkman et al., 2023).

3. Algebraic characterization and robustness criteria

A central structural result in (Bjorkman et al., 19 Jul 2025) characterizes when two placements have identical expected-coverage polynomials. If $|S|$1 and $|S|$2 satisfy $|S|$3, define

$|S|$4

Then

$|S|$5

as polynomials in $|S|$6 if and only if

$|S|$7

The proof expands $|S|$8 in the Bernstein-type basis

$|S|$9

which is a basis for the space of polynomials of degree $G=(V,E)$0 (Bjorkman et al., 19 Jul 2025). The consequence is that equality of expected coverage is determined by the aggregated observed-set sizes over all $G=(V,E)$1-element surviving subsets, not by graph isomorphism or by the fine-grained chronology of zero-forcing.

The same polynomial also encodes worst-case robustness notions. Let $G=(V,E)$2 with $G=(V,E)$3. Then $G=(V,E)$4 is $G=(V,E)$5-PMU-defect-robust if and only if

$G=(V,E)$6

for some polynomial $G=(V,E)$7. The same factorization characterizes $G=(V,E)$8-fault-tolerant sets: $G=(V,E)$9 Equivalently, the vanishing of the first $S\subseteq V$0 coefficients of $S\subseteq V$1 is exactly the certificate that $S\subseteq V$2 is resilient to any $S\subseteq V$3 PMU failures (Bjorkman et al., 2023).

This relationship prevents a common conflation of models. Fragile power domination is not merely a reformulation of worst-case $S\subseteq V$4-fault tolerance: it replaces the worst-case assumption by an independent-failure model and studies expected coverage as a $S\subseteq V$5-dependent polynomial. The factorization above shows precisely how the stochastic model bridges to exactly-$S\subseteq V$6-fault formulations.

4. Full observation, coefficient constraints, and polynomial synthesis

Beyond expectation, (Bjorkman et al., 2023) studies the probability of complete observation. Writing

$S\subseteq V$7

and

$S\subseteq V$8

one has

$S\subseteq V$9

If $B:=N[S]=\bigl\{\,v\}\cup\{u:uv\in E,\;v\in S\},$0 is $B:=N[S]=\bigl\{\,v\}\cup\{u:uv\in E,\;v\in S\},$1-PMU-defect-robust, then

$B:=N[S]=\bigl\{\,v\}\cup\{u:uv\in E,\;v\in S\},$2

with equality when $B:=N[S]=\bigl\{\,v\}\cup\{u:uv\in E,\;v\in S\},$3 (Bjorkman et al., 2023). These formulas isolate the distinction between expected partial coverage and the event of total observability.

At the same time, (Bjorkman et al., 19 Jul 2025) shows that many coverage polynomials can be engineered by local graph modifications. For each integer $B:=N[S]=\bigl\{\,v\}\cup\{u:uv\in E,\;v\in S\},$4 and each subset $B:=N[S]=\bigl\{\,v\}\cup\{u:uv\in E,\;v\in S\},$5, one can attach a small “star-plus-path” or “bipartite-plus-path” gadget at $B:=N[S]=\bigl\{\,v\}\cup\{u:uv\in E,\;v\in S\},$6. By varying the appended path length $B:=N[S]=\bigl\{\,v\}\cup\{u:uv\in E,\;v\in S\},$7, one can control the coefficient of $B:=N[S]=\bigl\{\,v\}\cup\{u:uv\in E,\;v\in S\},$8 arbitrarily in the total polynomial without affecting higher powers. This leads to the induced-any-polynomial theorem: if $B:=N[S]=\bigl\{\,v\}\cup\{u:uv\in E,\;v\in S\},$9, then for any integers $B_0=\bigcup_{v\in S}N[v]$0, there exists a graph $B_0=\bigcup_{v\in S}N[v]$1 as an induced subgraph such that

$B_0=\bigcup_{v\in S}N[v]$2

for suitable integers $B_0=\bigcup_{v\in S}N[v]$3. In particular, the coefficients of all powers $B_0=\bigcup_{v\in S}N[v]$4 for $B_0=\bigcup_{v\in S}N[v]$5 can be prescribed (Bjorkman et al., 19 Jul 2025).

This flexibility is not unconstrained. If

$B_0=\bigcup_{v\in S}N[v]$6

then $B_0=\bigcup_{v\in S}N[v]$7, $B_0=\bigcup_{v\in S}N[v]$8, and all other $B_0=\bigcup_{v\in S}N[v]$9 with $\Obs(G;S)$0 sensor-multiplicities}}) vanish (Bjorkman et al., 19 Jul 2025). This suggests a sharp asymmetry: intermediate coefficients are highly tunable, while the constant and linear terms retain strong structural restrictions.

5. Complexity and canonical graph families

The algorithmic landscape is explicitly intractable in general. Computing $\Obs(G;S)$1 is NP-complete, and by reduction, checking $\Obs(G;S)$2-fault-tolerance or $\Obs(G;S)$3-PMU-defect-robustness is NP-hard. A naïve computation of $\Obs(G;S)$4 via the subset-sum formula requires

$\Obs(G;S)$5

time, where $\Obs(G;S)$6 is the cost of one power-domination run. For trees, and more generally bounded-treewidth graphs, $\Obs(G;S)$7 can be computed in polynomial time by dynamic programming over a tree decomposition. Even selecting the best set $\Obs(G;S)$8 of a given size to maximize $\Obs(G;S)$9 is combinatorial and intractable in general (Bjorkman et al., 2023).

Several graph families admit explicit formulas. For paths $\gamma_P(G)=\min\{\lvert S\rvert:S\subseteq V,\;\Obs(G;S)=V\}.$0, cycles $\gamma_P(G)=\min\{\lvert S\rvert:S\subseteq V,\;\Obs(G;S)=V\}.$1, and complete graphs $\gamma_P(G)=\min\{\lvert S\rvert:S\subseteq V,\;\Obs(G;S)=V\}.$2, one PMU suffices classically, and a single surviving PMU observes all $\gamma_P(G)=\min\{\lvert S\rvert:S\subseteq V,\;\Obs(G;S)=V\}.$3 vertices. Consequently,

$\gamma_P(G)=\min\{\lvert S\rvert:S\subseteq V,\;\Obs(G;S)=V\}.$4

with

$\gamma_P(G)=\min\{\lvert S\rvert:S\subseteq V,\;\Obs(G;S)=V\}.$5

in each case (Bjorkman et al., 2023). For paths, the same formula holds whether the PMU is placed at an endpoint or at an interior vertex.

For the grid $\gamma_P(G)=\min\{\lvert S\rvert:S\subseteq V,\;\Obs(G;S)=V\}.$6, the behavior is more delicate. The source notes that $\gamma_P(G)=\min\{\lvert S\rvert:S\subseteq V,\;\Obs(G;S)=V\}.$7 is nontrivial but known to be $\gamma_P(G)=\min\{\lvert S\rvert:S\subseteq V,\;\Obs(G;S)=V\}.$8 for large grids. If two PMUs are placed in opposite corners, full observation occurs whenever at least one PMU survives, yielding

$\gamma_P(G)=\min\{\lvert S\rvert:S\subseteq V,\;\Obs(G;S)=V\}.$9

More general formulas can be obtained by conditioning on which of the two PMUs survives, but in general computing $\mathcal E(G;S,q)$00 on grids is exponential (Bjorkman et al., 2023).

6. Worked example, interpretive significance, and extensions

A worked example in (Bjorkman et al., 19 Jul 2025) illustrates how complicated zero-forcing dynamics can collapse to a very simple expected-coverage polynomial. For the graph in Figure 1 of that paper, with three PMUs at $\mathcal E(G;S,q)$01,

$\mathcal E(G;S,q)$02

$\mathcal E(G;S,q)$03

and

$\mathcal E(G;S,q)$04

Using the subset-sum formula,

$\mathcal E(G;S,q)$05

The example demonstrates that nontrivial cancellation among subset contributions can compress a high-dimensional family of classical observed-set sizes into a linear expectation polynomial (Bjorkman et al., 19 Jul 2025).

In application terms, the fragile model changes the optimization target. In real power-grid monitoring, one places a minimum number of PMUs so that, in the absence of failure, the grid is fully observed, as measured by $\mathcal E(G;S,q)$06. Fragile power domination replaces the “worst-case $\mathcal E(G;S,q)$07 faulty sensors” assumption by a stochastic failure probability $\mathcal E(G;S,q)$08 and asks instead for the expected coverage $\mathcal E(G;S,q)$09. One can then optimize sensor placement to maximize expected coverage, or ensure $\mathcal E(G;S,q)$10 up to a given $\mathcal E(G;S,q)$11. The algebraic characterization further shows that only the multiset of values

$\mathcal E(G;S,q)$12

for $\mathcal E(G;S,q)$13 matters, which can guide heuristics and integer-programming formulations. Gadget constructions imply that one can inflate or deflate intermediate polynomial coefficients, which is useful in designing hard instances or in reverse-engineering required robustness profiles (Bjorkman et al., 19 Jul 2025).

The extensions identified in the sources are broad but technically coherent. They include non-uniform sensor-failure probabilities $\mathcal E(G;S,q)$14, time-dependent propagations, continuous-time failure models, and combinations with classical $\mathcal E(G;S,q)$15-fault-tolerant power domination (Bjorkman et al., 19 Jul 2025). Closely related open directions include weighted or colored graphs where PMU reliability $\mathcal E(G;S,q)$16 depends on the vertex, dynamic failures that occur in stages during the zero-forcing process, sharper formulas for Cartesian-product graphs such as grids and tori, random graph models, structural parameters such as degree sequences and connectivity that control the shape of $\mathcal E(G;S,q)$17, and approximation algorithms for selecting $\mathcal E(G;S,q)$18 of fixed size to maximize expected coverage (Bjorkman et al., 2023). Together these directions indicate that fragile power domination is both a robustness formalism for PMU placement and a polynomial-valued graph invariant with substantial combinatorial structure.