BesiegeField: IoBT Battlefield Dynamics

• BesiegeField is defined as adversarial IoBT environments where targeted node attacks challenge network connectivity and sensor coverage.
• The dynamic multistage Stackelberg game framework models attacker and defender strategies, optimizing connectivity recovery and cost trade-offs.
• Numerical evaluations demonstrate that Feedback Stackelberg Equilibrium strategies reduce disconnected sensors by up to 46%, enhancing network resilience.

BesiegeField refers to a class of adversarial battlefield environments characterized by persistent efforts to disrupt, isolate, or paralyze distributed Internet of Battlefield Things (IoBT) networks through targeted attacks on network nodes. In such scenarios, the success of field operations is highly contingent upon the dynamic resilience of the IoBT infrastructure—defined by its capacity to maintain connectivity, sensor coverage, and latency thresholds even as a capable adversary attempts systematic compromise of critical communication components. The analytical foundation for understanding and mitigating threats in BesiegeField contexts derives from the game-theoretic approach, specifically via dynamic multistage connectivity games formulated to model both the attack and defense strategies under battlefield constraints.

1. Dynamic Multistage Stackelberg Connectivity Game

In BesiegeField scenarios, IoBT networks are composed of heterogeneous nodes—sensors, cluster heads (CH), local sinks (LS), and a general sink (GS)—interconnected to support situational awareness, data aggregation, and mission command. The adversarial context is modeled as a discrete-time, dynamic multistage Stackelberg connectivity game in which, at every time epoch $t \in \{1, 2, \ldots, T\}$, the state $\psi_t$ (encoding node roles, connectivity, and sensor distribution) is observed by both attacker and defender.

The attacker, acting as the game leader, selects a single IoBT node (sensor, active CH, or LS) to compromise, with the intention to maximize the expected disruption—quantified as the weighted disconnection of nodes from the GS minus the attack cost. In response, the defender implements restorative actions: deploying new nodes of required types, reassigning cluster heads, or replacing failed critical nodes, with the dual objective of restoring full connectivity and maintaining information-theoretic thresholds such as minimum sensor counts per cluster ($N_{\text{th},jh}$).

This framework generalizes classical connectivity games by accounting for heterogeneity of node types and roles, multicast traffic flows, latency, and weighted node contributions, all of which are central to adversarial battlefield environments.

2. Attacker and Defender Strategy Spaces

The attacker's strategy consists of selecting, at each epoch, which node to destroy to induce maximal downstream disconnection effects at minimal resource expenditure. The payoffs for the attacker are formalized as: $$P_{a,t}(a_t, b_t, \psi_t) = S_{D,t}(a_t, b_t, \psi_t) - \nu C_{a,t}(a_t, b_t, \psi_t),$$ where $S_{D,t}$ is the weighted sum of nodes disconnected, $C_{a,t}$ is the cost (with cost parameters $c_\tau$ for device type $\tau$, $c_L$ for LS, $c_{CH}/c_{aL}$ for determining CH or activated LS), and $\nu$ is a weight parameter.

The defender's action set, conditioned on the post-attack network state, comprises:

• $b_{d,\tau h}$: deploying a new IoBT device of type $\tau$ in cluster $h$
• $b_{c,ijh}$: assigning a new CH in cluster $h$
• $b_{a,lh}$, $b_{L,h}$: activating/switching local sinks or deploying a new LS

If an attack reduces a cluster's sensors below $N_{\text{th},jh}$, the defender's action set contracts to prioritize restoration. The payoff to the defender is: $$P_{d,t}(a_t, b_t, \psi_t) = U_{d,t}(a_t, b_t, \psi_t) - \eta S_{D,t}(a_t, b_t, \psi_t) - \mu \Lambda_t(a_t, b_t, \psi_t) - \lambda C_{d,t}(a_t, b_t, \psi_t),$$ with terms capturing network utility, the penalty for disconnections, latency $\Lambda_t$, and deployment cost $C_{d,t}$.

Both sets of actions, iteratively applied across stages, define the evolution of network state under adversarial pressure.

3. Feedback Stackelberg Equilibrium (FSE) Solution

Solving the network defense problem in a BesiegeField requires that the strategies of both attacker and defender account for immediate and future consequences. The Feedback Stackelberg Equilibrium (FSE) formalism is adopted, with recursive expected payoffs: $$\Omega_{a,t} = \sum_{a_t \in S_{a_t}} q_{a_t} \left[ P_{a,t}(a_t, b_t, \psi_t) + \Omega_{a,t+1}(\psi_{t+1}) \right],$$

$$\Omega_{d,t} = \sum_{a_t \in S_{a_t}} q_{a_t} \left[ P_{d,t}(a_t, b_t, \psi_t) + \Omega_{d,t+1}(\psi_{t+1}) \right],$$

where $q_{a_t}$ denotes the attacker's mixed strategy and $b_t$ is the defender's best response given $a_t$ and $\psi_t$.

The equilibrium strategies $(q^*_t, b^*_t)$ for each state $\psi_t$ satisfy: $$\Omega_{a,t}(q^*_t, b^*_t, \psi_t) = \max_{q_t \in \mathcal{M}_{a,t}} \max_{b_t \in \mathcal{R}_d(q_t)} \Omega_{a,t}(q_t, b_t, \psi_t)$$ where $\mathcal{M}_{a,t}$ and $\mathcal{R}_d(q_t)$ are the feasible strategy sets.

A key result is the establishment of sufficient conditions under which FSE ensures connectivity, by comparing future expected payoffs for different attack/defense decision tuples. Practically, system operators may tune cost, latency, and utility parameters to operate within these connectivity-preservation conditions.

4. Numerical Evaluation and Strategic Insights

Comprehensive simulations compare the FSE approach against a No-Feedback Stackelberg Equilibrium (NFSE, single-stage) and a baseline where the attacker selects nodes uniformly at random. Under FSE, the expected number of disconnected sensors is reduced by up to 46% compared to NFSE and up to 43% compared to uniform random targeting.

The analysis reveals critical tradeoffs:

• When the cost to attack LS nodes is low, optimally, attackers concentrate on LSs. As this cost increases, attackers diversify (e.g., towards CHs), with defender responses involving deployment/reassignment.
• The anticipatory nature of FSE yields improved long-term resilience; by considering future state transitions, the defender may proactively reinforce key nodes or reallocate cluster roles.
• The attacker’s mixed-strategy optimization, formulated as a linear programming problem over reduced action sets (e.g., nodes whose compromise most impacts connectivity), provides an efficient solution mechanism in practical BesiegeField instantiations.

5. Mathematical Structure and Solution Algorithms

The dynamic Stackelberg connectivity game is fully characterized by recursive Bellman-type equations for each player. The attacker’s and defender’s payoffs are parameterized by:

• Disconnection penalties and weights
• Latency and information flow delays
• Cost structures for various node actions
• Node thresholds for cluster integrity

The FSE is computed by recursively evaluating expectations over action sets and network state transitions, employing mixed strategy selection via linear programming for the attacker and deterministic role/node deployment for the defender.

A table summarizing key payoff components:

Symbol	Attacker	Defender
$S_{D,t}$	Weighted disconnected nodes	Penalty in payoff
$C_{a,t}$	Attack cost	—
$U_{d,t}$	—	Network utility
$\Lambda_t$	—	Data delivery latency
$C_{d,t}$	—	Defender's deployment/reassignment cost

The sufficient conditions for connectivity, detailed in the original proposition, can be operationalized to guarantee field network resilience by regime parameter tuning.

6. Application to BesiegeField Scenarios

In environments described by the BesiegeField concept, the tactical and operational effectiveness of a command structure is linked to its capacity to prevent or rapidly restore network partitioning under sustained attack. The FSE-driven dynamic connectivity defense affords several operational advantages:

• Preemptive deployment and adaptive reconfiguration of sensor, LS, and CH assignments in response to observed attack patterns
• Quantitative risk assessment and simulation via explicit models of adversarial cost, payoff, and network response parameters
• Parameter-based guarantees for minimum-service levels, wherein certain cost/latency settings assure persistent connectivity

The demonstrated numerical reductions in disconnected sensor rates (up to 46% over NFSE, 43% over uniform policies) indicate significant improvements to communication robustness and, by extension, situational awareness in live BesiegeField operations.

7. Strategic Significance and Future Directions

The dynamic connectivity game-based approach provides an extensible analytical and practical foundation for the protection of IoBT systems under adversarial siege. Extensions may involve:

• More granular models of adversarial learning/adaptation
• Incorporation of communication jamming and physical mobility
• Integration with distributed fusion and command layers

A plausible implication is that future BesiegeField research may leverage these game-theoretic defense models to drive autonomic, real-time reconfiguration of battlefield IoT assets—significantly bolstering the resilience of military operations in contested and degraded information environments.