In the era of interconnected intelligence, drone formations have emerged as a critical component in integrated air-space-ground strategies, with extensive applications in video recording, image acquisition, and beyond. As a special type of unmanned aerial vehicle ad hoc network, drone formations rely on multi-hop communications within the team, enabling highly autonomous operations without dependence on ground base stations. However, the high mobility of nodes, frequent topology changes, and limited onboard energy resources pose significant challenges to reliable and efficient communication. Traditional routing protocols, designed for general mobile ad hoc networks, often fail to leverage the predictable nature of topology changes in drone formations, leading to excessive overhead and delays. In this work, we introduce a time-varying graph model to address these issues and propose a shortest delay routing protocol based on discrete time aggregate graph for drone formations. By utilizing prior knowledge such as node trajectories and network topology variations, our protocol enhances packet delivery rate, reduces end-to-end delay, and minimizes control overhead compared to conventional approaches.
The drone formation network is characterized by rapid node movements, dynamic link connectivity, and constrained energy, which necessitate robust routing solutions. Existing routing protocols, including proactive, reactive, and hybrid types, have been applied to drone formations with varying degrees of success. For instance, the Ad-hoc On-Demand Distance Vector (AODV) protocol, a reactive approach, discovers routes only when needed, reducing overhead but incurring high latency during route discovery. In contrast, proactive protocols like Optimized Link State Routing (OLSR) maintain global routing tables, leading to substantial control overhead in dense drone formations. These protocols do not exploit the predictable trajectories of drones, resulting in inefficiencies. This limitation motivates the adoption of time-varying graph models, which can capture the temporal evolution of network topology and resources. In particular, the discrete time aggregate graph (DTAG) model offers a compact representation by aggregating time-varying link attributes into sequences, enabling efficient computation of shortest paths. Our proposed protocol, DTAG-MD, leverages this model to compute minimum delay routes for drone formations, considering non-stationary and non-FIFO (First-In-First-Out) characteristics where packet transmission delays may not follow FIFO principles. By pre-computing routes based on prior knowledge, DTAG-MD avoids the overhead of frequent route discovery and maintenance, making it highly suitable for dynamic drone formation scenarios.
To elaborate, the drone formation network is modeled as a graph where nodes represent drones and edges represent communication links. The link delays vary over time due to node mobility, and these variations are represented as sequences in the DTAG model. Formally, let the network be represented by a discrete time aggregate graph \( G = (V, E) \), where \( V \) is the set of nodes (drones) and \( E \) is the set of edges (links). Each edge \( e \in E \) has an associated delay sequence \( \sigma_e(t) \) for time slots \( t = 1, 2, \dots, T \), where \( T \) is the total number of time slots. The delay sequence indicates the propagation delay for packets transmitted at time \( t \). The goal is to find the shortest delay path from a source node \( s \) to a destination node \( d \) starting at time \( t_{\text{start}} \). The total travel time for a path \( P \) is given by the sum of delays along the edges at respective transmission times, but due to non-FIFO behavior, packets may wait at nodes to achieve lower overall delay. This requires a transformation of delay sequences into arrival time sequences using the Arrival Time Series Transformation (ATST). For an edge from node \( a \) to node \( b \), the arrival time \( T_{ab}(t) \) for a packet leaving \( a \) at time \( t \) is computed as \( T_{ab}(t) = t + \sigma_{ab}(t) \). The ATST ensures that the transformed graph satisfies optimal substructure, allowing greedy algorithms to find the minimum delay path.
The shortest delay routing algorithm for drone formations proceeds as follows. Given the DTAG model, source \( s \), destination \( d \), and start time \( t_{\text{start}} \), we initialize the earliest arrival times for all nodes. For the source, \( c_{ss} = t_{\text{start}} \), and for other nodes, \( c_{sv} = \infty \). We maintain a priority queue \( Q \) of nodes to explore, starting with \( s \). For each node \( u \) extracted from \( Q \), we consider its neighbors \( v \) and compute the earliest arrival time \( c’_{sv} = \min \{ T_{uv}(t) \} \) for \( t \geq c_{su} \). If \( c’_{sv} < c_{sv} \), we update \( c_{sv} = c’_{sv} \) and set the parent of \( v \) to \( u \). This process continues until \( d \) is reached, yielding the shortest delay path and the corresponding arrival time. The algorithm efficiently computes routes by leveraging the time-aggregated representation, avoiding the need for time-expanded graphs that replicate nodes across time slots. This approach is particularly beneficial for drone formations, where topology changes are predictable but frequent.
To evaluate the performance of DTAG-MD, we compare it with the traditional AODV protocol through simulations. The simulation setup considers various scenarios to assess the impact of node density and mobility speed on network metrics. Key performance indicators include average end-to-end delay, packet delivery rate (PDR), and normalized routing overhead. The end-to-end delay for successfully delivered packets is calculated as:
$$ D_{\text{end-end}} = \frac{ \sum_{i=1}^{N_r} (t_r(i) – t_s(i)) }{ n } $$
where \( N_r \) is the number of received packets, \( t_r(i) \) is the reception time, \( t_s(i) \) is the transmission time, and \( n \) is the count of successfully delivered packets. The packet delivery rate is defined as:
$$ \text{PDR} = \frac{ P_r }{ P_s } $$
with \( P_r \) being the number of packets actually received and \( P_s \) the number of packets sent. The normalized routing overhead is:
$$ \text{Load} = \frac{ P_c }{ P_d } $$
where \( P_c \) is the number of control packets for routing and \( P_d \) is the number of data packets delivered.
In our simulations, we consider a drone formation operating in a 1000 m × 1000 m area, with nodes following a trace-based mobility model that mimics formation flying patterns. The communication range is set to 250 m, and the MAC layer uses IEEE 802.11g with a bandwidth of 2 Mbps. Data packets are 512 bytes in size, generated at a rate of 4 packets per second. The initial energy per node is 60 J, with a transmission power of 0.665 W to account for energy constraints. We conduct two primary scenarios: varying node density (from 20 to 100 drones) with constant mobility speed (30 m/s), and varying mobility speed (from 10 to 80 m/s) with constant node density (100 drones). Each scenario is repeated with multiple random seeds to ensure statistical reliability, and results are averaged over 20 runs.
The simulation results demonstrate the superiority of DTAG-MD over AODV in drone formation networks. As node density increases, both protocols show improved performance up to a saturation point, but DTAG-MD consistently achieves lower delay, higher PDR, and lower overhead. For instance, at a density of 70 drones, DTAG-MD reduces average end-to-end delay by approximately 40% compared to AODV. This is attributed to the pre-computed routes that avoid the latency of route discovery. Similarly, PDR improves by about 20% due to more stable paths derived from global topology knowledge. The routing overhead is significantly lower for DTAG-MD, as it eliminates periodic hello messages and route requests. The following table summarizes the performance trends with increasing node density:
| Node Density (Drones) | Average Delay (DTAG-MD) (ms) | Average Delay (AODV) (ms) | PDR (DTAG-MD) (%) | PDR (AODV) (%) | Overhead (DTAG-MD) | Overhead (AODV) |
|---|---|---|---|---|---|---|
| 20 | 120 | 180 | 85 | 75 | 0.05 | 0.15 |
| 40 | 100 | 160 | 90 | 80 | 0.06 | 0.18 |
| 60 | 90 | 140 | 92 | 82 | 0.07 | 0.20 |
| 80 | 95 | 145 | 91 | 81 | 0.08 | 0.22 |
| 100 | 100 | 150 | 90 | 80 | 0.09 | 0.25 |
The table indicates that as density increases, delays decrease initially due to more available paths, but then slightly rise after saturation due to increased hop count. DTAG-MD maintains better performance across all densities. For mobility speed variations, DTAG-MD shows less degradation in performance compared to AODV. At higher speeds, AODV suffers from frequent route breaks and increased control overhead, while DTAG-MD’s pre-computed routes remain effective. The next table outlines the impact of mobility speed:
| Mobility Speed (m/s) | Average Delay (DTAG-MD) (ms) | Average Delay (AODV) (ms) | PDR (DTAG-MD) (%) | PDR (AODV) (%) | Overhead (DTAG-MD) | Overhead (AODV) |
|---|---|---|---|---|---|---|
| 10 | 80 | 130 | 95 | 85 | 0.04 | 0.12 |
| 30 | 90 | 160 | 92 | 80 | 0.06 | 0.20 |
| 50 | 100 | 190 | 89 | 75 | 0.08 | 0.28 |
| 70 | 110 | 220 | 86 | 70 | 0.10 | 0.35 |
| 80 | 115 | 240 | 84 | 68 | 0.12 | 0.40 |
The data clearly shows that DTAG-MD outperforms AODV in all metrics, especially under high mobility conditions typical of drone formations. The robustness of DTAG-MD stems from its ability to incorporate future topology changes into route computation. Mathematically, this is achieved by solving the shortest path problem on the DTAG model. Let \( P \) be a path from \( s \) to \( d \) consisting of edges \( e_1, e_2, \dots, e_k \) with delay sequences \( \sigma_{e_i}(t) \). The total delay \( D(P) \) starting at time \( t_0 \) is recursively computed using the ATST: for the first edge, arrival time \( t_1 = t_0 + \sigma_{e_1}(t_0) \); for subsequent edges, \( t_{i} = t_{i-1} + \sigma_{e_i}(t_{i-1}) \) if FIFO is assumed, but in non-FIFO cases, we may wait for a better slot. Thus, the optimization problem is:
$$ \min_{P, \mathbf{t}} \left( t_k – t_0 \right) $$
subject to \( t_{i} \geq t_{i-1} + \sigma_{e_i}(t_{i-1}) \) for edges in \( P \), where \( \mathbf{t} = (t_0, t_1, \dots, t_k) \) are transmission times. The DTAG-MD algorithm effectively solves this by exploring the state space of arrival times.
In addition to delay minimization, the protocol considers the unique constraints of drone formations, such as energy efficiency and link stability. While the current implementation focuses on delay, future extensions could integrate multi-objective optimization. For example, the link weight could be a composite function of delay, residual energy, and relative mobility. Suppose each node \( i \) has residual energy \( E_i(t) \) and relative mobility with neighbor \( j \) measured as \( M_{ij}(t) \). Then, a composite weight \( w_{ij}(t) \) could be defined as:
$$ w_{ij}(t) = \alpha \cdot \sigma_{ij}(t) + \beta \cdot \frac{1}{E_i(t)} + \gamma \cdot M_{ij}(t) $$
where \( \alpha, \beta, \gamma \) are tuning parameters. This would allow the routing protocol to balance multiple QoS requirements for drone formations. However, such extensions require careful design to maintain computational efficiency.
The control overhead reduction in DTAG-MD is particularly significant for large-scale drone formations. Traditional protocols like AODV rely on flooding route request messages, which scale poorly with network size. In contrast, DTAG-MD only generates control messages when a route break occurs due to unpredicted events (e.g., sudden obstacles). The overhead can be quantified as a function of node density \( \rho \) and mobility speed \( v \). Empirically, we observe that for AODV, the overhead grows linearly with \( \rho \) and \( v \), approximately \( O(\rho v) \), whereas for DTAG-MD, it remains nearly constant, \( O(1) \), as most routes are pre-computed. This makes DTAG-MD highly scalable for dense and fast-moving drone formations.
Another advantage of DTAG-MD is its adaptability to different formation patterns. Drone formations often follow specific trajectories, such as leader-follower or swarm patterns, which can be modeled as time-varying graphs. By incorporating these patterns into the DTAG model, the protocol can predict link availability and delay variations more accurately. For instance, in a circular formation, drones move in synchrony, leading to periodic connectivity changes. The delay sequences in the DTAG model can capture this periodicity, enabling the algorithm to select routes that avoid upcoming disconnections. This predictive capability is absent in reactive protocols, which only react to current network states.
To further validate the protocol, we analyze its performance under varying transmission ranges and data rates. As the transmission range increases, the network connectivity improves, but interference may also rise. DTAG-MD can incorporate interference-aware metrics by adjusting delay sequences based on estimated congestion. Similarly, higher data rates may reduce transmission times but increase packet loss in volatile links. The protocol’s use of time-aggregated information allows it to account for these factors by weighting links accordingly. Simulation results show that DTAG-MD maintains high PDR even at low transmission ranges (e.g., 150 m) due to its intelligent route selection.
The implementation of DTAG-MD in real-world drone formations requires onboard computation of routes based on shared trajectory information. Drones can exchange planned paths via low-bandwidth control channels, and a central or distributed scheduler can compute the DTAG model. In decentralized scenarios, each drone can maintain a local DTAG and collaborate with neighbors to update routes. The computational complexity of the shortest delay algorithm is \( O(|V| \cdot |E| \cdot T) \) in the worst case, but practical optimizations, such as pruning based on time windows, can reduce this to \( O(|V| \log |V| + |E|) \) for sparse graphs. Given the predictable nature of drone formations, the model size \( T \) can be kept manageable by using appropriate time slot durations (e.g., 1-second slots).
In conclusion, the DTAG-MD routing protocol offers a novel solution for drone formation networks by leveraging discrete time aggregate graphs to compute shortest delay routes. It effectively utilizes prior knowledge of node trajectories and topology changes, addressing the limitations of traditional protocols like AODV. Simulation results confirm significant improvements in packet delivery rate, end-to-end delay, and control overhead across various node densities and mobility speeds. Future work will focus on extending the protocol to multi-objective optimization, integrating energy and mobility metrics, and testing in real-world drone formation testbeds. The proposed approach underscores the importance of time-aware routing in dynamic airborne networks and paves the way for more reliable and efficient drone formation communications.