Centrality-Based Security Allocation in Networked Control Systems

1. Problem Statement

The paper addresses the security allocation problem in Networked Control Systems (NCS), modeled as a continuous-time system consisting of multiple interconnected subsystems (vertices) represented by an undirected connected graph. The scenario involves two strategic agents:

• The Adversary: A malicious actor aiming to maximize the worst-case impact on the global system performance by launching stealthy false data injection (FDI) attacks on a limited number of vertices (attack budget $n_a$). The attack must remain undetected by the defender's monitoring system.
• The Defender: An operator aiming to allocate limited security resources (monitor budget $n_s$) to a subset of vertices to detect attacks and mitigate their impact.

Key Constraints & Objectives:

• Stealthiness: The attack is considered "stealthy" if the energy of the monitor outputs remains below a predefined alarm threshold $\delta$.
• Resource Limitation: Both agents have finite budgets (number of attack nodes and number of monitor nodes).
• Optimization Goal: The defender seeks to minimize the Worst-Case Attack Impact (WCAI), defined as the maximum possible system performance degradation (measured by $L_2$ norm of state outputs) the adversary can achieve while satisfying stealth and energy constraints.

2. Methodology

A. Mathematical Formulation

The system dynamics are modeled using a Laplacian matrix $L$ derived from the graph topology. The closed-loop system under attack is:
$$\dot{x}(t) = -Lx(t) + B(A)\zeta(t)$$
where $B(A)$ is a binary vector indicating attacked nodes and $\zeta(t)$ is the attack signal.

The Worst-Case Attack Impact (WCAI), denoted as $J(A, M)$, is formulated as a semi-infinite optimization problem. The authors prove (Theorem 1) that this non-convex problem can be transformed into a Semidefinite Programming (SDP) problem using dissipative system theory and duality.

B. Optimal Security Allocation

The ideal solution involves solving a Minimax problem where the defender chooses a monitor set $M$ to minimize the maximum impact the adversary can cause on any attack set $A$.

• Formulation: $\min_{M} \max_{A} J(A, M)$.
• Complexity: This is a Mixed-Integer SDP (MISDP) problem. Solving it requires enumerating all possible attack scenarios (combinatorial explosion), making it computationally prohibitive for large-scale networks.

C. Proposed Centrality-Based Approach

To overcome the computational burden of the optimal MISDP, the authors propose a heuristic approach using graph centrality measures to select monitor vertices. The logic is that the most "influential" or "connected" nodes are the most critical to monitor.

• Metrics Used:
  1. Degree Centrality ($C_D$): Number of direct connections.
  2. Closeness Centrality ($C_C$): Inverse of the average shortest path to all other nodes.
  3. Betweenness Centrality ($C_B$): Number of shortest paths passing through a node.
• Combined Approach: The authors also propose a method where monitor sets are generated for each of the three metrics, and the set yielding the lowest WCAI (calculated via SDP) is selected.
• Advantage: This reduces the problem from a complex combinatorial optimization to a simple sorting and selection process, followed by a standard SDP calculation (without integer variables) to verify the impact.

3. Key Contributions

1. Game-Theoretic Framework: The paper formulates the NCS security problem as a non-cooperative game between a defender and an adversary, explicitly modeling the trade-off between attack impact and stealth.
2. SDP Reformulation: The authors provide a rigorous proof that the worst-case impact of stealthy FDI attacks can be computed efficiently via SDP, avoiding the need for exhaustive simulation of attack signals.
3. Centrality Heuristic: The primary contribution is the demonstration that centrality measures (specifically Betweenness and a Combined approach) provide a near-optimal security allocation strategy with significantly reduced computational complexity compared to the exact optimal solution.
4. Scalability: The method is designed to be scalable for large networks where solving the optimal MISDP is infeasible.

4. Experimental Results

The methodology was validated using two types of networks:

• Erdős–Rényi Random Graphs: Simulations were run on graphs with $N \in \{10, \dots, 20\}$ vertices.
  • Performance: The Combined Centrality and Betweenness Centrality approaches achieved a median WCAI gap of <10% compared to the optimal solution.
  • Efficiency: The centrality-based methods reduced solving time by 70% to 90% compared to the optimal MISDP approach.
  • Trend: As network size increased, the gap for Degree and Closeness centrality widened, but Betweenness and Combined approaches remained robust (gap <15%).
• IEEE 14-Bus Power System: A real-world benchmark was used.
  • The optimal solution identified Bus 2 as the best monitor.
  • The Betweenness Centrality approach identified Bus 4.
  • Result: The WCAI gap between the centrality-based choice (Bus 4) and the optimal choice (Bus 2) was <2%, demonstrating high practical efficacy.

5. Significance and Conclusion

The paper concludes that while finding the exact optimal security allocation is theoretically superior, it is computationally expensive and often impractical for real-time or large-scale applications.

• Practical Utility: The proposed centrality-based approach offers a "good enough" solution (within 10% of optimal performance) that can be computed orders of magnitude faster.
• Best Metric: Among the individual metrics, Betweenness Centrality consistently performed the best, likely because it captures the structural importance of nodes in information flow, which is critical for detecting stealthy attacks propagating through the network.
• Recommendation: For large-scale NCS (e.g., power grids, transportation), defenders should utilize Betweenness or Combined Centrality measures to allocate monitoring resources, especially when hardware constraints limit the ability to solve complex mixed-integer optimization problems.

This work bridges the gap between theoretical game-theoretic security and practical, scalable implementation in cyber-physical systems.