Topology Control of Low-Connection UAV Laser Network with Virtual Nodes

Abstract

Space laser communication technology has advantages such as high bandwidth and low interference, but the limitation of the load capacity of an unmanned aerial vehicle (UAV) makes it difficult for UAV clusters to have sufficient communication links to build a stable communication network. Aiming to address the problem of the low upper limit of the node degree of the self-organized network after the introduction of space laser communication technology for UAVs, a virtual node topology control (VNTC) algorithm is proposed for constructing an efficient and stable communication network structure. The algorithm is based on the unit disc graph model and achieves a lower-node-degree upper bound while maintaining connectivity through node pairing and virtual graph construction. The theoretical analysis and experimental results of the algorithm show that the algorithm can effectively deal with the topology control problem in UAV laser networks. In contrast to the XTC algorithm (exceptional topology control), the VNTC algorithm maintains the connectivity and most of the planarizability of the network, and, at the same time, it performs better in reducing the node degree and number of edges, thus providing an effective solution for low-connectivity topology control.

1. Introduction

With the widespread application of drone technology in the military, civilian, and commercial sectors, unmanned aerial vehicle (UAV) networks are facing unprecedented challenges, such as the growing demand for transmission bandwidth and increasingly prominent data security issues. These challenges pose significant tests to the existing radio frequency communication technologies [1,2,3]. Laser communication technology utilizes lasers as carriers to achieve information transmission in free space, featuring high bandwidth, high directionality, low intercept rates, and low interference [2,3,4,5]. These advantages demonstrate great potential for high data transmission rates, resistance to electromagnetic interference, and enhanced communication security. As a result, space laser communication technology is gradually becoming an important choice for information transmission in space information networks.

However, the introduction of laser communication technology has also presented new issues to UAV ad hoc networks, particularly in the critical research area of network topology control. Network topology control is a crucial aspect of UAV networking, involving the construction and maintenance of an efficient and stable communication network structure in a dynamically changing spatial environment. The high directionality of laser communication demands precise alignment and tracking during the communication process, thus significantly increasing the cost of switching between communication nodes. This becomes a critical issue that must be considered in the dynamic communication process among UAV nodes. Furthermore, this characteristic of laser communication renders traditional neighbor discovery processes inapplicable as they typically rely on omnidirectional signal propagation.

Despite the significant progress achieved by many research institutions in the field of UAV laser communication technology [6,7,8,9], constructing a proximity graph that is suitable for the characteristics of laser networks remains an urgent problem to be solved. Due to the high cost of alignment and acquisition in laser communication, the time-division multiplexing technology traditionally used in radio frequency communication is not easily applicable to laser communication. This means that, within a certain period, a communication module can only communicate with one other module. Consequently, the number of connections a UAV can have is dependent on the number of laser communication modules it is capable of carrying. Regarding UAV laser networks, the power and payload capacity of the UAVs themselves are limited, making it difficult to connect multiple UAVs simultaneously by simply adding more laser communication modules. This imposes an upper limit on the degree of the nodes in the neighbor graph.

Most past researchers have focused on reducing network redundancy while attempting to maintain the overall performance of the original graph. Consequently, many methods have not effectively constrained the node degree limits in worst-case scenarios. Although researchers worldwide have achieved numerous advancements aimed at increasing the connection limits of laser communication modules [10,11,12], they still struggle to meet the stringent node degree requirements imposed by traditional neighbor graph planning algorithms.

The main contributions of this paper are as follows:

• To address the issue of low node degree limits in laser networks, we designed a virtual node topology control algorithm (VNTC) that constrains the upper limit of the node degree to 4.
• Sufficient conditions necessary for the operation of the algorithm on two-dimensional topological graphs are provided.
• An analysis and resolution of the issues related to unmatched nodes and boundary conditions within the algorithm are presented, and the effectiveness of the algorithm is verified through simulations.

2. Related Works

In 2022, Dabiri and colleagues conducted a study on unmanned-aerial-vehicle-based free space optics (FSO) communication using modulating retro-reflector technology [13]. They developed models that consider atmospheric turbulence and fluctuations in the vehicle’s orientation and created analytical expressions for signal-to-noise ratio, outage probability, and bit error rate. Then, in 2023, Hayal et al. developed a new FSO channel model to enhance the performance of UAV-based FSO relay systems under pointing errors and atmospheric turbulence, finding that a higher signal-to-noise ratio is needed to maintain low error rates as wind variance increases [14]. The study concluded that relay-assisted UAV-FSO systems perform better than conventional FSO systems in terms of error rate and signal-to-noise ratio. The two papers concentrated on the channel issues of UAV nodes during FSO communication, and, as another important aspect of UAV laser communication, the connectivity capabilities of UAVs are also a topic that requires research.

In 2021, the team led by You Quan at the Wuhan National Information Optoelectronics Innovation Center designed an eight-channel silicon-based optical phased array chip capable of producing 1–5 output beams but with a transmission distance limited to only 15 cm [10]. Similarly, Paul Searcy and Barry A. Matsumori utilized a managed optical communication array based on liquid crystal phased array devices, along with time-division multiple access technology, to achieve a laser communication terminal capable of one-to-four-point communication [11]. While these studies enhance the connectivity of laser terminals, they still fail to meet the high connectivity demands of traditional topology control.

Therefore, developing a topology control algorithm to effectively reduce the upper limit on node degree to the threshold achievable with the current laser communication technology is undoubtedly a feasible and forward-looking solution.

The research on self-organizing network topology control originated in 1986 [15]. The Gabriel Graph (GG), proposed by K. R. Gabriel in Ref. [16], is constructed based on the following principle: for any two nodes in the graph, if the circle with its diameter equal to the distance between these nodes contains no other nodes, then there exists a connection between the two nodes. The upper node degree limit in a GG graph itself is ${\mathrm{degree}}_{+}=n-1$.

The Relative Neighborhood Graph (RNG), introduced by Godfried et al. in Ref. [17], is constructed based on the criterion that, if no other node is closer to both of the two nodes in question than the distance between them, a connection exists between the two nodes. Similarly, the upper node degree limit for an RNG graph itself is also ${\mathrm{degree}}_{+}=n-1$.

R. Wattenhofer and colleagues, in Ref. [18], developed a construction algorithm known as XTC (exceptional topology control) that achieves a node degree upper limit of six by sorting and comparing neighboring nodes. Their algorithm’s main idea is that, if there exists a node whose ranking is higher than the rankings of these two nodes in each other’s lists, then these two nodes will not connect. If the distances between any node and its neighboring nodes are all distinct, then the outcome of the XTC algorithm is equivalent to an RNG graph. While these approaches can effectively construct neighbor graphs, the node degree limits they impose still do not meet the requirements of UAV laser networks.

3. Model

The UAV laser network discussed in this paper consists of n nodes distributed in a two-dimensional plane. Each node is equipped with four laser communication modules, all of which have equal maximum communication ranges. Each module can establish a communication link with another module, and the coverage area of a single laser communication module is a circle with a radius equal to its maximum communication distance. Additionally, each node can determine its own location through GPS or other means.

Since the actual size of the nodes is much smaller than the maximum communication distance of the laser communication modules, the maximum communication distance can be considered as the communication range of the nodes. For convenience in description, we normalize the maximum communication distance of the nodes to 1. Thus, we can model this network using the commonly used unit disk graph (UDG) in ad hoc networks.

For the $UDG=\left(V,E\right)$, let

$$V=\left\{v_i|v_i\in R^2,i=1,2\cdots n\right\}$$

represent the set of nodes and

$$E=\left\{e_k|\forall u,v\in V,u\ne v,{\parallel uv\parallel }_2⩽1\right\}$$

represent the set of edges. Clearly, since each UAV is equipped with only four laser communication modules, the degree of each node is limited to a maximum of 4.

4. Virtual Node Topology Control Algorithm

This section introduces the proposed virtual node topology control (VNTC) algorithm.

4.1. Algorithm Construction

Given an arbitrary unit disk graph denoted as G, we propose the following lemma as the foundation for constructing the VNTC algorithm.

Lemma 1.

If the graph G is connected, then, for any node v in G, the number of connected components in the graph $G-v$ (the graph obtained by removing node v) is at most 5.

Proof. 

Since only the node v is deleted, we need to consider only its neighbors. The removal of v may create multiple connected components, meaning there are neighbor nodes without a path between them. This implies that certain neighbor nodes are not within the communication range of other neighbor nodes. As shown in Figure 1, if a certain neighbor node is outside the range of other neighbors, the central angle corresponding to the arc between this neighbor and other neighbor nodes exceeds $\pi /3$. To maximize the number of connected components, each angle between neighbor nodes should be greater than $\pi /3$; hence, the maximum number of connected components is 5.    □

By Lemma 1, to maintain the connectivity of the graph, the lower bound of the degree of node u, denoted as ${\mathrm{degree}}_{-}\left(u\right)$, is 5.

Given the requirement that the upper bound of the node degree ${\mathrm{degree}}_{+}=4$, it is evident that, under this condition, the graph G may not be connected. This means that maintaining the overall connectivity of the graph within the original communication range of the nodes is challenging. Under the condition of keeping the communication range unchanged, if the maximum distance between a node and its neighbors is reduced, then the corresponding distances between neighbor nodes will also decrease accordingly. Under the condition of maintaining the communication range unchanged, if the maximum distance between a node and its neighbors is reduced, the corresponding distances between neighbor nodes will also decrease accordingly. Furthermore, when this distance falls below a certain threshold, it is possible to achieve a node degree of $max\left\{{\mathrm{degree}}_{-}\left(V\right)\right\}\le 4$. This leads to the following theorem:

Theorem 1.

Suppose that graph $G\left(V\right)$ remains connected when the communication radius of the nodes is 0.850. For any node v in graph G, after removing v from G, the number of connected components in G is at most 4.

Proof. 

For any neighbor v of node u, it is assumed that ${∥uv∥}_2\le d_{\max }$. When any two neighbors of node u are not neighbors of each other, meaning the distance between any two neighbors is greater than 1, to ensure that the number of connected components is at most 4, the central angle of the arc corresponding to the coverage range of the neighboring nodes must be at least ${\displaystyle \frac{2\pi }{5}}$. This leads to the following calculation for the maximum distance $d_{\max }$:

$$d_{\max }={\displaystyle \frac{sin\left({54}^{\circ }\right)}{sin\left({72}^{\circ }\right)}}={\displaystyle \frac{1+\sqrt{5}}{\sqrt{10+2\sqrt{5}}}}\approx 0.850$$

Therefore, for node u, if the graph G remains connected when the communication radius is 0.850, then, after removing u from G, the number of connected components in G will be at most 4. More generally, for any node v in graph G, if graph $G\left(V\right)$ remains connected when the communication radius is 0.850, then, after removing v from G, the number of connected components in G will be at most 4.    □

The XTC algorithm proposed in Ref. [18] can reduce the upper bound of the node degree to 6, whereas the required upper bound for the node degree is 4. Considering connecting two nodes and treating them as a single node, each node would then have a remaining degree of 3, making the degree upper bound of this new node exactly 6. Therefore, based on the aforementioned approach, one can utilize the XTC algorithm to construct a subgraph $G^{\prime }\left(V\right)$ with a degree upper bound of 4.

Assume that the communication radius of the new node (hereafter referred to as the virtual node) is $r^{*}$ and graph G remains connected under the condition that the communication radius $r<1$. Let the new graph formed by virtual nodes be denoted as the virtual graph $G^{*}(V^{*},E^{*})$, and let the subgraph of graph G at communication radius r be denoted as $G_r^{\prime }$. Graphs G, $G_r^{\prime }$, and virtual graph $G^{*}$ need to satisfy the following conditions:

• If subgraph $G_r^{\prime }$ is connected, then the new graph obtained after pairing nodes should also be connected. That is, for the virtual graph $G^{*}$, at a communication radius of $r^{*}$, $G^{*}$ should be a connected graph. This implies that the process of creating virtual nodes by pairing existing nodes in $G_r^{\prime }$ does not disrupt the overall connectivity of the graph, and the virtual graph $G^{*}$ inherits the connectedness property from $G_r^{\prime }$.
• For every edge in the virtual graph $G^{*}$, there must be at least one corresponding edge in the original graph G. This ensures that the virtual connections in $G^{*}$ are grounded in actual connections within G, maintaining a reflection of the original graph’s structure and connectivity within the virtual framework.

Assuming that the virtual node is located at the midpoint of the line segment connecting the paired nodes, the maximum distance from the virtual node to either of the two nodes would be $r/2$. To ensure the connectivity of the virtual graph, it is assumed that the communication range of the virtual node is contained within the common communication range of the paired nodes. Additionally, the paired nodes corresponding to neighboring virtual nodes should also be located within the common communication range, as illustrated in Figure 2. Under these conditions, Condition 2 is necessarily satisfied. Given the conditions from Figure 2, where $max\left\{{∥m{m}^{\prime }∥}_2\right\}=r^{*}$, $max\left\{{∥mu∥}_2\right\}=max\left\{{∥mv∥}_2\right\}=r/2$, and the paired node corresponding to $m^{\prime }$ is located within a circle of radius $r/2$ centered at $m^{\prime }$, we can derive the following inequality:

$${\displaystyle \frac{r}{2}}+r^{*}+{\displaystyle \frac{r}{2}}\le 1$$

This simplifies to

$$r^{*}\le 1-r$$

(1)

This inequality ensures that the communication range of the virtual node $r^{*}$, when combined with the half-distance to the paired nodes $r/2$, does not exceed the overall communication radius of 1. This is a necessary condition to maintain the connectivity of the virtual graph $G^{*}$ within the constraints of the original graph G.

Proof. 

In the subgraph $G_r^{\prime }$, for all $u,v\in V$, if there exists a path $\{u,v_1,v_2,\cdots ,v\}$ between u and v, where any two adjacent nodes on this path are neighbors and the corresponding virtual nodes of these nodes are also neighbors, then there exists a corresponding path $\{u^{*},v_1^{*},v_2^{*},\cdots ,v^{*}\}$ in the virtual graph $G^{*}$, that is, a path between the virtual nodes corresponding to u and v.Therefore, if graph G is connected, then, for all $u,v\in V$, there exists a path between them, and the corresponding virtual nodes $u^{*}$ and $v^{*}$ also have a path between them, implying that the virtual graph $G^{*}$ is connected.    □

Considering two adjacent pairs of paired nodes, the maximum distance between the nodes in these pairs is $3r$. Since the virtual node is located at the center of the line connecting the paired nodes, the communication distance of the virtual node is

$$r^{*}\ge 3r-2\times {\displaystyle \frac{r}{2}}=2r$$

(2)

From Equations (1) and (2), we obtain $2r\le r^{*}\le 1-r$, which yields $r\le 1/3$, and $r^{*}=2/3$. Therefore, if the subgraph $G_r^{\prime }$ remains connected when $r=1/3$, then a connected subgraph with a maximum node degree of 4 can be constructed using the aforementioned method.

The previous discussion has established the feasibility of node pairing. Clearly, a node can only be paired once, which means that, in the subgraph $G_r^{\prime }$, a pairing method is needed to ensure that as many nodes as possible can be successfully paired, which is essentially the maximum matching problem in general graphs. For this particular challenge, a well-established solution exists, namely the blossom algorithm.

4.2. Algorithm Implementation

This section provides a summary of the algorithmic process, with the caveat that the algorithm is predicated on the subgraph $G_r^{\prime }$ remaining connected when $r=1/3$. The overall algorithm can be divided into three main steps:

• Execute the blossom algorithm on subgraph $G_r^{\prime }$ to obtain matched node pairs.
• From the matched node pairs, calculate the virtual nodes to form the virtual graph $G^{*}$, and then run the XTC algorithm on this virtual graph to obtain the virtual subgraph $G^{{*}^{\prime }}$.
• Map the virtual subgraph back to the original graph G to obtain the resulting graph.

For a detailed description of the algorithmic computation process, refer to Algorithm 1.

Algorithm 1 VNTC Algorithm

Input: G Output:  $G_{result}$   compute $G_r^{\prime }$   $match$ = blossom($G_r^{\prime }$)    $virtual=\left\{\right\}$   for$(u,v)$ in $match$ do     m = midpoint of $(u,v)$     $virtual$=$virtual\cup \left\{m\right\}$   end for   compute virtual graph $G^{*}$ with radius of $2r\left(r^{*}\right)$   $G^{{*}^{\prime }}$=XTC($G^{*}$)    $E=\left\{\right\}$    $G^{\prime }=(V,E)$   for$edge=(u^{\prime },v^{\prime })$ in $G^{{*}^{\prime }}$ do     $u_1$,$u_2$=$match\left[u^{\prime }\right]$     $v_1$,$v_2$=$match\left[v^{\prime }\right]$     $u_m$=$min\{degre{e}_{G^{\prime }}\left(u_1\right),degre{e}_{G^{\prime }}\left(u_2\right)\}$     $v_m$=$min\{degre{e}_{G^{\prime }}\left(v_1\right),degre{e}_{G^{\prime }}\left(v_2\right)\}$     $E=E\cup \{(u_m,v_m),(u_1,u_2),(v_1,v_2)\}$ $\hspace{1em}{G}^{\prime }=(V,E)$   end for

5. Problems and Optimization

5.1. Mapping Algorithm

In the process of mapping the virtual subgraph $G^{{*}^{\prime }}$ back to the graph G, Algorithm 1 employs a strategy that prioritizes connecting the two nodes with the lowest degree. This approach effectively prevents any single node from handling an excessive number of virtual node connections, thus avoiding the node degree from exceeding the acceptable threshold. However, this method does not take into account the weight of the edges, which means that the resulting graph $G^{\prime }$ may not be the optimal solution in terms of edge weights.

Let $f:E^{*}\to E$ represent the mapping of any edge from the virtual graph $G^{*}$ to an edge in the graph G, and let $E_f=\{e\mid e=f\left(e^{*}\right),e^{*}\in E^{{*}^{\prime }}\}$ be the set of all mapped edges from the virtual subgraph $G^{{*}^{\prime }}$ to the graph G. The problem can be formulated as finding a subset $E_f^{\prime }$ of $E_f$ such that

$$\forall e^{*}\in E^{{*}^{\prime }},\left|E_f^{\prime }\cap \{e\mid e=f\left(e^{*}\right)\}\right|=1$$

and

$$\forall u\in V,\mathrm{degree}\left(v\right)\le 3$$

while minimizing

$$\underset{E_f^{\prime }\subseteq E_f}{argmin}\sum _{e\in E_f^{\prime }}{w}_e$$

where $w_e$ represents the weight of edge e. This formulation ensures that each edge in the virtual subgraph $G^{{*}^{\prime }}$ maps to exactly one edge in the graph G, maintains the degree of each node in G to be no more than 3, and seeks to find the subset $E_f^{\prime }$ of $E_f$ that minimizes the total weight of the edges included in the mapping.

Let $x_e=1$ indicate that edge e is included in the edge set $E_f^{\prime }$. Then, the problem can be transformed into a 0–1 integer programming problem as follows:

$$\begin{array}{cc}\hfill \mathrm{Minimize}& z=\sum _{e\in E_f}{x}_e{w}_e,w_e>0\hfill \\ \hfill \mathrm{Subject}\mathrm{to}& \sum _{e\in \{e\mid e=f\left(e^{*}\right)\}}{x}_e=1,\forall e^{*}\in E^{{*}^{\prime }}\hfill \\ \hfill & \sum _{e\in E_u}{x}_e\le 3,\forall u\in V,E_u=\{(u,v)\mid v\in V,(u,v)\in E_G^{*}\}\hfill \\ \hfill & x_e=0\mathrm{or}1\hfill \end{array}$$

The optimal mapping can be obtained by solving this 0–1 integer programming problem. Since 0–1 integer programming is an NP-complete problem, as the number of nodes in the graph increases significantly, the number of edges also grows exponentially. This makes finding the optimal solution to the problem very challenging.

In light of this issue, an alternative heuristic approach has been proposed. The method involves sorting all the mapped edges and sequentially selecting the edge with the minimum weight to be included in $G^{\prime }$ while eliminating other edges that share the same mapping. For details of the algorithm, refer to Algorithm 2.

Algorithm 2 Node Mapping Algorithm

Input: $E_f$,W Output:  $newE$   E=sort $E_f$ with W,$newE=\left\{\right\}$   for$edge=(u,v)$ in $E.pop$ do     if $degree\left(u\right)<3$ and $degree\left(v\right)<3$ then       $newE=newE\cup \left\{edge\right\}$       $E=E-f\left(f^{-1}\left(edge\right)\right)$     end if   end for

5.2. Unmatched Nodes

In the previous subsection, the issue of the mapping algorithm was addressed, but there is still a significant problem that has not been resolved prior to that. Specifically, the generation of virtual nodes depends on the node matching algorithm, which implicitly requires that the number of nodes in the graph be even. However, in practical applications, it is not uncommon to encounter cases where the total number of nodes is odd, and perfect matchings do not always exist in general graphs. These issues can lead to the presence of unmatched nodes in the graph G. Therefore, effectively dealing with these unmatchable nodes has become an urgent problem to address. Below, a solution to this problem is presented.

Theorem 2.

After removing the unmatchable nodes from the virtual graph, the number of connected components is at most 4.

Proof. 

As shown in Figure 3, the furthest distance between the unpaired node v and the virtual nodes corresponding to its neighboring nodes in the subgraph $G_r^{\prime }$ is $3r/2$. That is, the subgraph formed by v and its neighboring virtual nodes is connected within a radius of $3r/2$, while the connectivity radius when it acts as a virtual node is $r^{*}\ge 2r$. Since $3r/2÷r^{*}\le 0.75<0.850$, by Theorem 1, we know that the number of connected components in the virtual graph after removing the unmatchable nodes is at most 4.    □

Hence, unmatchable nodes only require up to four edges to maintain the connectivity of the graph, fulfilling the condition ${\mathrm{degree}}_{+}\le 4$. Due to the maximum degree of a node v in the virtual subgraph $G^{{*}^{\prime }}$ being 6, no additional work is required when $\mathrm{degree}\left(v\right)\le 4$. When $\mathrm{degree}\left(v\right)>4$, it implies that there are at least two neighboring nodes that have a path between them. When the degree of a node v is 5, select the pair of neighboring nodes that correspond to the smallest central angle of the circle. Then, determine if there are any nodes other than v within the neighborhood defined by the distance between these two neighboring nodes as the radius. If there are, remove the connection between v and one of these neighboring nodes. If not, connect the two neighboring nodes and remove the connection between v and one of them. If the degree is 6, the same method can be applied twice to address the situation. Let $N_G\left(v\right)$ denote the set of neighbors of node v in graph G and $UMN\left(V\right)$ denote the set of all unmatched nodes. The algorithm is described as Algorithm 3.

Algorithm 3 Unmatched Node Handling Algorithm

Input: $UMN\left(V\right)$,$G^{{*}^{\prime }}$ Output:  $G^{{*}^{\prime }}$   for v in $UMN\left(V\right)$ do     $n=degree\left(v\right)$     while $n>4$ do       $n=n-1$       $edges=N_{G^{{*}^{\prime }}}\left(v\right)\times N_{G^{{*}^{\prime }}}\left(v\right)$       $(u_1,u_2)$=min distance in $edges$       $r= \parallel u_1,u_2{\parallel }_2$       if $N_{G_r^{*}}\left(u_1\right)\cap N_{G_r^{*}}\left(u_2\right)-\left\{v\right\}=\varnothing$ then           add $(u_1,u_2)$ to $G^{{*}^{\prime }}$       end if       remove $(u_1,v)$ from $G^{{*}^{\prime }}$     end while   end for

5.3. Planarity

In topology control of ad hoc networks that use radio frequency as a means of communication, it is often required that the generated topology graph is planar. This facilitates the operation of some geometry-based routing algorithms and reduces communication interference between nodes. Although the high directivity of laser communication significantly reduces communication interference between nodes, the excellent properties of planar graphs still make it a worthwhile goal to pursue.

In the initial algorithm, the use of random initialization in graph matching algorithms often leads to intersecting pairs of nodes, which is one of the reasons the topology control graph may not be planar. Furthermore, during the mapping of the virtual subgraph back to the original graph, the mapping algorithm does not consider the positional relationships between nodes, causing the mapped edges to intersect. This section will address these issues with optimizations, with the goal of obtaining a superior topological structure.

Firstly, addressing the issue of nodes crossing due to random initialization in the graph matching algorithm, let us consider the intersecting graph (Figure 4). According to the pigeonhole principle, in a quadrilateral with vertices A, B, C, and D, there must be at least one angle that is not less than 90 degrees, which implies that at least one diagonal is longer than any of the sides of the quadrilateral. If we select the shorter edges as the objects for initialization, we can minimize the occurrence of intersections.

Secondly, regarding the mapping process, paired nodes can divide the range of virtual nodes into two semicircles. Crossing edges often occur when two nodes are connected to nodes outside their respective semicircles. Since, after topology control, the angle between the edges connected to the virtual nodes in the virtual subgraph is no less than $\pi /3$ [18], the number of connections within any semicircle, without considering the boundary of the semicircle, is always no more than 3. Therefore, by only considering the virtual nodes within their respective semicircles for pairing, the occurrence of crossing edges can be prevented.

5.4. Radius Limit Value

In the algorithm reasoning from the previous section, sufficient conditions were provided to meet condition (2), but they are not necessary conditions. This condition is satisfied by preserving all edges from paired nodes to their adjacent paired nodes. To fully leverage the long-distance transmission advantages of laser communication and to explore the maximum radius that the algorithm can extend while satisfying condition (2), this is indeed a research-worthy question. In the section on planarity, a method was proposed to divide the range of virtual nodes into two parts for consideration. Below, we will further analyze and expand on this method.

Consider any pair of nodes u and v, with their corresponding virtual node m. Due to the symmetry of the paired nodes, we only need to consider the semicircle where node u is located. Consider the extreme case where a node $m^{\prime }$ connected to the virtual node m is located at the boundary of m’s range. Since the paired nodes are symmetric with respect to $m^{\prime }$, the maximum distance between u and its paired node is achieved when the line connecting the paired nodes is perpendicular to $u{m}^{\prime }$. Let us assume the length of this maximum distance is $r_m$, as shown in Figure 5. It can be easily derived that

$$\begin{array}{cc}\hfill r_m^2& =r_2^2+{r^{\prime }}^2\hfill \\ \hfill & =r_2^2+{(\sqrt{r^{*2}-h^2}-r_1)}^2+h^2\hfill \\ \hfill & =r_2^2+r_1^2+r^{*2}-2r_1\sqrt{r^{*2}-h^2}\hfill \end{array}$$

It is evident that $r_m$ is directly proportional to h. When h takes its maximum value $r^{*}$, $r_m$ reaches its maximum value $\sqrt{r_2^2+r_1^2+r^{*2}}$. Since $r_1,r_2\le r/2$, it follows that $r_m\le \sqrt{r^2/2+r^{*2}}$. Because the point of maximum distance should be within the maximum communication range of node u, that is, $max{r}_m\le 1$, it follows that $r^{*}\le \sqrt{1-r^2/2}$. 

In the previous section, it was determined that a sufficient condition to meet requirement (1) is $r^{*}\ge 2r$. Based on the arguments presented above, it can be concluded that $r\le {\displaystyle \frac{\sqrt{2}}{3}}$, and, consequently, $r^{*}={\displaystyle \frac{2\sqrt{2}}{3}}$.

6. Experiment

This section conducts a brief experiment on the proposed algorithm and compares it with the XTC algorithm. First, it is assumed that there are 5 nodes uniformly distributed around a given node, with each distance being ${\displaystyle \frac{\sqrt{2}}{3}}$. The results for both the XTC and VNTC algorithms are shown in Figure 6. It can be observed that the node degree in the proximity graph constructed by the XTC algorithm is 5, while the VNTC algorithm maintains a node degree of 4.

For a node graph distributed uniformly according to a grid, with the distance between nodes remaining ${\displaystyle \frac{\sqrt{2}}{3}}$, the resulting graph in Figure 7 shows that the XTC algorithm produced cycles of length 4, while the VNTC algorithm predominantly generated cycles of length 6. This is because forming a cycle of length 4 requires either 2, 3, or 4 virtual nodes. After node pairing, only one edge between virtual nodes is retained. Therefore, it is impossible for 2 virtual nodes to form a cycle of length 4. Additionally, since the proximity graph generated by the XTC algorithm does not contain cycles of length 3, 3 virtual nodes also cannot form a cycle of length 4. Consequently, only when 4 virtual nodes form a cycle of length 4 can there be a possibility of forming a cycle of length 4 in the actual graph. Due to the sequential arrangement of nodes in the graph and the fact that the node pairing process is also carried out in the order of edge lengths, the graph exhibits regular cycles of length 6. When using a pairing algorithm with random initialization, as shown in Figure 8, it can be observed that the majority of cycles in the graph have a length greater than 4.

Since the virtual graph employs the XTC algorithm for topology control, its performance in the virtual graph is identical to that of the XTC algorithm. However, after the graph mapping, one virtual node corresponds to two actual nodes. Therefore, when a virtual node acts as an intermediary, it may trigger communication between the two actual nodes, resulting in a slight degradation in actual performance compared to the XTC performance. By randomly distributing nodes within a $5\times 5$ area and progressively increasing the node density, the average minimum jump count between nodes is calculated at various densities. The results are depicted in Figure 9, which shows that the average minimum jump count of the VNTC algorithm is slightly greater than that of the XTC algorithm.

Finally, by randomly and uniformly distributing 1200 points within a range of $10\times 10$, the actual performance of the algorithm was tested 2000 times, yielding results similar to those shown in Figure 10, Figure 11 and Figure 12. Through statistical analysis of the test data,

Table 1. Comparison of algorithm performance.

Statistical Items	Original	XTC	VNTC
Number of graphs with maximum node degree greater than 4	2000	127	0
Average node degree	34.42	2.51	2.27
Maximum node degree	72	5	4
Longest shortest path (average distance) for any node	29.27	36.51	44.44
Longest shortest path (average jumps) for any node	15.91	68.91	78.53
Average shortest distance	11.15	14.14	17.35
Average minimum jumps	6.3	26.88	30.65
Average number of edges	20,649.27	1503.32	1362.57

was obtained.

Based on the fourth and fifth items in the table, the performance gap between the VNTC algorithm and the XTC algorithm in the worst-case scenario is less than 20%, while their performance in average cases is similar. Therefore, the VNTC algorithm is able to maintain a topology graph with performance close to that of the XTC algorithm while reducing the upper limit of the degree of individual nodes.

7. Conclusions

This paper addresses the issue of network topology control in UAV laser communication networks by proposing a novel algorithm—the VNTC algorithm. The VNTC algorithm is designed to tackle the upper node degree limit caused by the limited power and payload capacity of UAVs, aiming to construct an efficient, stable communication network structure that aligns with the characteristics of laser communication.

Through the analysis of the UAV laser network model, it was found that, under the condition where each node is equipped with only four laser communication modules, the traditional topology control algorithms struggle to meet the requirements. To address this, the VNTC algorithm employs node pairing and the construction of virtual nodes to reduce the upper node degree limit while ensuring network connectivity. The theoretical foundation and derivation process of the algorithm demonstrate that, under certain radius conditions, if the graph remains connected, it can ensure that any node has no more than four neighboring nodes. Furthermore, in response to potential unmatched nodes and the planarity of the topology graph, this paper proposes corresponding strategies to further optimize the network structure, enhancing the practicality and flexibility of the algorithm.

The experimental results indicate that, compared to the XTC algorithm, the VNTC algorithm effectively reduces the upper node degree limit and the number of edges while maintaining network connectivity, enhancing the sparsity of the network, and ensuring to some extent the planarity of the topology graph. This not only validates the effectiveness of the VNTC algorithm but also provides a new solution for topology control in UAV laser communication networks.

The research presented in this paper still has certain limitations, such as the efficiency and performance of the algorithm when dealing with large-scale networks, as well as its adaptability in dynamically changing environments, requiring further research and exploration.