A Fractional Cyber–Biological Model for Energy-Aware Trust-Based Mobile-Agent Wireless Sensor Networks: Secure Plant-Disease Surveillance

Technical Summary: A Fractional Cyber–Biological Model for Energy-Aware Trust-Based Mobile-Agent Wireless Sensor Networks

Problem Statement
Low-power Wireless Sensor Networks (WSNs) increasingly serve as the measurement layer for biological decision systems, such as plant-disease surveillance. A critical vulnerability exists where communication-security failures do not merely degrade network performance but propagate downstream to distort biological inferences. Specifically, compromised nodes performing selective forwarding or on–off attacks can falsify microclimate data (e.g., leaf wetness, humidity), leading to erroneous disease-risk estimates. Existing trust mechanisms often treat node behavior as memoryless or rely on independent energy and trust rankings, failing to capture the complex temporal interplay between residual energy, trust degradation, and compromise probability under bounded attack evidence. Furthermore, the integration of network security states with biological decision-making remains largely unmodeled as a coupled dynamical system.

Methodology
The authors propose a coupled two-layer framework: a Cyber Layer and a Biological Layer, linked by a mobile-agent routing decision and secure sensor fusion.

1. Fractional Cyber Model:
   The core of the cyber layer is a three-state Caputo fractional differential system modeling normalized residual energy ($E$), trust ($T$), and compromise probability ($C$) under an attack signal $A(t)$. The system is defined as:

   • $D^\alpha_t E = -E(\epsilon_0 + \epsilon_1\rho + \epsilon_2C + \epsilon_3A(t))$
   • $D^\alpha_t T = \lambda(1-T) - \mu CT - \xi A(t)T$
   • $D^\alpha_t C = \kappa A(t)(1-C) - \delta TC$
     Here, $0 < \alpha \leq 1$ represents the fractional order, introducing a power-law weighted memory of past states. The model assumes bounded selective-forwarding or on–off attacks.

2. Mathematical Analysis:
   The paper establishes the well-posedness of the system, proving finite-horizon existence and uniqueness of solutions. It demonstrates that the physical state space $\Omega = [0, 1]^3$ is positively invariant. Crucially, the authors analyze the equilibrium of the trust–compromise subsystem under constant attack intensity, proving the existence of a unique admissible equilibrium that is independent of the fractional order $\alpha$. Local asymptotic stability is confirmed for all $0 < \alpha \leq 1$ via the Matignon sector condition.

3. Mobile-Agent Decision Rule:
   An energy-aware utility function $U(t) = w_E E(t) + w_T T(t) - w_C C(t)$ is defined to determine node eligibility for mobile-agent itineraries. Hard guards are applied to prevent high-energy nodes with unacceptable security states from being selected.

4. Biological Integration:
   The network model is coupled to a plant-disease biosurveillance module. Microclimate variables (relative humidity, leaf wetness, thermal suitability) are fused into a suitability index. Unlike standard averaging, the fusion weights are dynamically generated based on the cyber reliability states ($E, T, C$) of the sensor nodes. This fused signal drives a bounded disease-pressure differential equation.

5. Numerical Implementation:
   Simulations utilize a fractional Adams–Bashforth–Moulton predictor–corrector scheme. Experiments compare integer-order ($\alpha=1$) and fractional-order ($\alpha \in \{0.95, 0.85, 0.75\}$) responses to persistent and on–off attacks in a synthetic five-node greenhouse scenario.

Key Contributions

• Coupled Fractional State Model: Introduction of a unified dynamical system where energy, trust, and compromise probability evolve simultaneously under fractional calculus, rather than being estimated independently.
• Analytical Characterization of Memory: Proof that while the fractional order $\alpha$ alters the transient approach to equilibrium (memory depth and speed), it does not shift the constant-attack equilibrium point. This distinguishes fractional models as memory-scale mechanisms rather than curve-fitting tools for changing steady states.
• Secure Sensor Fusion Mechanism: Development of a trust-aware fusion rule where cyber reliability states directly modulate the weight of biological sensor data, quantifying the propagation of cyber compromise into biological risk estimation.
• Reproducible Synthetic Benchmark: Provision of a complete synthetic experimental setup, including parameter sets, attack profiles, and a packet-to-state mapping, to evaluate the trade-offs between detection speed and memory retention.

Results

• Transient Dynamics: In on–off attack scenarios, lower fractional orders ($\alpha < 1$) attenuate the instantaneous depth of trust collapse and compromise probability spikes compared to the integer-order model. However, this comes at the cost of slower recovery once the attack ceases.
• Routing Decisions: In a five-node routing case study, the fractional model ($\alpha=0.85$) delayed the exclusion of a compromised node by one integration step compared to the integer model, illustrating the "price of memory" as a finite detection delay.
• Biological Impact: In the synthetic greenhouse stress test, where a compromised node injected falsified microclimate data:
  • Trust-aware fusion reduced the Root Mean Square Error (RMSE) of the environmental suitability estimate during the attack window from 0.07162 (standard average) to 0.02818 (60.7% reduction).
  • The downstream disease-pressure state RMSE decreased from 0.02059 to 0.00645 (68.7% reduction).
  • Post-attack, the trust-aware fusion maintained a significantly lower error (0.00130 vs. 0.01480) due to the hysteresis effect of the trust state.
• Trade-off Identification: The results highlight a genuine security–biology trade-off: while fractional memory smooths transient noise, it can cause a previously trusted compromised node to retain excessive influence briefly after an attack begins, potentially delaying necessary biological interventions.

Significance and Claims
The paper positions itself as a mechanism study rather than a field-validation or performance-optimization claim. Its primary significance lies in:

1. Decoupling Equilibrium from Transient Memory: Explicitly demonstrating that fractional calculus changes how a system reacts to history (transient memory) without altering the final steady-state outcome under constant attacks.
2. Quantifying Cyber-Biological Propagation: Providing a mathematical framework to trace how packet-level evidence translates into biological decision errors, showing that trust-aware fusion can significantly mitigate these errors.
3. Defining the Cost of Memory: Arguing that fractional models should not be viewed as inherently superior; rather, they offer a tunable parameter to balance the speed of anomaly detection against the stability provided by historical evidence.
4. Reproducibility: Offering a fully specified synthetic benchmark for future research to test fractional cyber-biological models against integer-order baselines using out-of-sample data.

The authors conclude that while the framework successfully connects packet evidence, fractional memory, and biological risk estimation, future work requires joint calibration with real-world field data (packet traces, battery telemetry, and disease outcomes) to estimate the fractional order $\alpha$ and validate the model against specific crop-pathogen systems.