Guard Vector: Applications & Methods

• Guard Vector is a binary or parameter difference representation used to configure optimal guard placement across disciplines such as graph theory, computational geometry, and network security.
• It underpins algorithms in dynamic routing and universal guard problems by formalizing coverage, minimality, and robust adaptation under adversarial conditions.
• In machine learning and network security, Guard Vectors facilitate efficient safety alignments and hierarchical clustering, enhancing protection and performance.

A Guard Vector is a mathematical or algorithmic construct representing the assignment, placement, or combination of guards for protective or surveillance tasks in diverse domains such as graph theory, computational geometry, network security, dynamic routing, and LLM safety. The term typically denotes either a binary vector that encodes which agents (guards, sensors, policies, etc.) are active or a parameter difference encoding behavioral modification in neural models. The Guard Vector formalism serves as a foundational framework to paper minimality, coverage, security, and adaptability under adversarial conditions or uncertainty.

1. Guard Vector: Definitions and Mathematical Formulations

The precise definition of a Guard Vector depends on context:

• Graph Protection: A Guard Vector may encode a set of vertices (or edges) in a graph where guards are stationed such that, under dynamics of attack and movement, the set maintains a covering, dominating, or related property. In combinatorial form, it is represented by an indicator vector $g \in \{0,1\}^n$ for $n$ vertices, with $g_i=1$ signifying a guard at vertex $i$ (Klostermeyer et al., 2014).
• Universal Guard Problems: Here, the Guard Vector is a binary vector over a planar point set $S$, with each entry mapping to a vertex in $S$. The vector encodes a selection $G \subset S$ that guards all polygonalizations over $S$; i.e., for all polygons $P$ with vertices $S$, every point in $P$ is visible from some $g \in G$ (Fekete et al., 2016).
• LLM Safety Alignment: The Guard Vector is a parameter difference, $V_{GV}[t]=\theta_{GM}[t]-\theta_{PLM}[t]$, for parameter indices $t$, representing the safety behavior modifications between a Guard Model and pretrained LM. This enables transfer via task-vector arithmetic: $\theta_{TGM}[t]=\theta_{CP}[t]+V_{GV}[t]$, yielding a Target Guard Model with enhanced safety characteristics (Lee et al., 27 Sep 2025).
• Network Security: The Guard Vector can also encode multidimensional features—metrics such as reliability, location distance, and risk—supporting hierarchical clustering and adversary containment strategies in anonymizing systems (Imani et al., 2017).

The Guard Vector thus abstracts the principle of optimal or robust deployment, parametrization, or adaptation of security and surveillance agents.

2. Guard Vector in Graph Protection and Algorithmic Dynamics

In graph-protection models, Guard Vectors represent dynamic or static placements of mobile guards that must respond to adversarial attacks on the graph’s structure:

• Eternal Domination: Guards occupy a dominating set $D_i$, and after each attack at $r_i$, a guard adjacent to $r_i$ moves there, yielding the update $D_{i+1}=(D_i-\{v\})\cup\{r_i\}$ with $vr_i\in E$. The minimum eternal domination number $\gamma^\infty(G)$ is bounded as $\alpha(G)\leq\gamma^\infty(G)\leq\theta(G)$, tying the Guard Vector’s minimal density to independence and clique cover parameters (Klostermeyer et al., 2014).
• m-Eternal Domination: All guards can move simultaneously in response to an attack, often reducing the minimal required guard set. The guard vector changes to maintain a dominating set after each step, with $\gamma(G)\leq\gamma^\infty_m(G)\leq\alpha(G)$ (Klostermeyer et al., 2014).
• Vertex Cover Dynamics: The Guard Vector must ensure that the set $C$ always covers all edges, with eternal vertex cover bounds $\tau(G)\leq\tau^\infty_m(G)\leq 2\tau(G)$.

The strategic selection and movement encoded by the Guard Vector enforce robust protection against infinite attack sequences, often requiring high coverage density and careful tracking of private neighborhoods or induced subgraph connectivity.

3. Universal Guard Problems: Guard Vector for Geometric Coverage

In computational geometry, the Universal Guard Problem investigates the construction of a Guard Vector that guarantees visibility coverage across all polygonalizations of a fixed vertex set:

• The algorithm decomposes $S$ into convex shells $(B_1,\ldots,B_m)$, constructing $G$ by guarding all points on the inner shell or sparsifying appropriately depending on the shell population.
• For two-shell instances, if $n_2\geq\frac12\sqrt{n_1}$, set $G=B_1$; otherwise, the Guard Vector is built by avoiding tangent-induced chambers and leaving every second point in a clean sequence unguarded, with $|G|\leq(1-\frac{1}{\sqrt{6|S|}})|S|$ (Fekete et al., 2016).
• In generalized $m$-shell scenarios, the Guard Vector’s 0–1 density adapts to maximize coverage while minimizing redundancy, with bounds such as $$|G|\leq \left(1-\frac{1}{16|S|^{1-\frac{1}{2m}}}\right)|S|$$.
• Lower-bound constructions demonstrate that near-complete coverage (high-density Guard Vector) is required for adversarial input sets.

This geometric Guard Vector framework enables analytical and algorithmic quantification of coverage requirements and supports efficient computation (e.g., $O(|S|\log|S|)$ for hulls and tangents).

4. Guard Vector Concepts in Dynamic Routing and Distributed Operations

The Guard Vector formalism also appears in dynamic boundary-guarding settings, such as vehicle interception of intruders or adaptive patrol strategies:

• A Guard Vector encodes the vehicle’s policy or scheduling for intercepting radially incoming targets generated by a Poisson process on the perimeter of a protected region.
• Objectives include maximizing the long-run capture fraction, $\mathcal{F}_{cap}(\mathcal{P})$, subject to kinematic and causal constraints, with bounds:

$$\mathcal{F}_{cap}(\mathcal{P})\leq\min\left\{1,(1+v)\sqrt{2/(v\lambda\pi\rho)}\right\}$$

where $v$ is target speed, $\lambda$ is arrival rate, and $\rho$ is perimeter radius (Bajaj et al., 2019).

• Policy-specific Guard Vectors are instantiated as scheduling plans (e.g., FCFS, Look-Ahead, RMHP-fraction), with graph-theoretic tools (longest path in DAGs, EMHP approximations) guiding dynamic guard assignments.

The underlying principle is that effective Guard Vectors lead to provable and near-optimal interception rates, balancing computational complexity with real-time responsiveness.

5. Guard Vector in Data Security and LLM Safety Alignment

Guard Vector methodology has been generalized to the context of machine learning safety:

• Safety Task Vector: In LLM guardrails, the Guard Vector is the parameter difference $V_{GV}$ between a safety-aligned Guard Model and a pretrained LM (Lee et al., 27 Sep 2025).
• Task-Vector Composition: This Guard Vector is combined element-wise with a target LLM’s parameters, $\theta_{TGM}[t]=\theta_{CP}[t]+V_{GV}[t]$, facilitating the production of safety-enhanced models (Target Guard Models) even in non-English or low-resource settings.
• Streaming-Aware Prefix SFT: Further adaptation uses cumulative prefix-based supervised fine-tuning, aligning behavior for streaming content via single-token classifiers. Output tokens such as <SAFE>, <UNSAFE> yield efficient streaming safety decisions:

$$p_\theta(\mathrm{UNSAFE}\mid p(r_{1:K}))=\frac{\exp(z_{\mathrm{UNSAFE}})}{\exp(z_{\mathrm{SAFE}})+\exp(z_{\mathrm{UNSAFE}})}$$

(Lee et al., 27 Sep 2025)

• The transfer mechanism requires no target-language labels, supporting extensibility and portability (e.g., ChineseSafe F1 improvement when composing an English-derived Guard Vector with a Chinese LM). Reported throughput gains are 51–100% QPS improvement, with ~34–50% latency reduction.

In this paradigm, the Guard Vector is a modular, resource-efficient means of propagating robust safety alignment across architectures and languages.

6. Guard Vector in Network Security and Anonymity Systems

• Hierarchical Guard Set Formation: In network anonymization, notably within the Tor infrastructure, Guard Vector methodology is used to form guard sets via hierarchical clustering over Internet location attributes, enhancing security and performance (Imani et al., 2017).
• The multidimensional Guard Vector incorporates metrics such as node reliability, network distance, and adversarial risk, with the risk function:

$R=\sum_{i\in S}\frac{w_i}{d_i}$

where $w_i$ is reliability, $d_i$ is topology-derived distance. A high $d_i$ for adversaries restricts harmful infiltration to a small number of guard sets.

• Empirical evaluation demonstrates 20–30% decrease in adversary capture probability, bandwidth stability, and reduced client exposure times.

This operationalizes the Guard Vector concept as a composite instrument for security-aware resource allocation and attack confinement.

7. Practical Implications and Theoretical Significance

Guard Vectors unify disparate domains under a formal approach to optimal security agent assignment, reconfiguration, and behavioral modification:

• Mathematical bounds link Guard Vector density and coverage to fundamental invariants (e.g., domination numbers, clique covers, coverage ratios).
• Algorithmic construction of Guard Vectors is efficient in geometric cases ($O(n \log n)$), modular in neural contexts (parameter arithmetic), and adaptive in dynamic scenarios (policy scheduling).
• Lower bounds reveal inherent minimality constraints in adversarial or uncertain environments, compelling high-density or nearly-complete Guard Vectors for robust protection.
• In machine learning, the Guard Vector technique democratizes safety by eliminating the need for supplementary data/labeling, enabling real-time and cross-lingual deployment.
• In network systems, hierarchically enriched Guard Vectors foster resilience and operational efficiency.

The cross-disciplinary significance of Guard Vector methodology lies in its capacity to encode, transmit, and enforce robust protection, coverage, and safety across settings characterized by adversarial dynamics, uncertainty, and real-time operational constraints.