Related work

Consensus algorithms are widely used in distributed algorithms for coordination and cooperation in multi-agent systems [14]. They are often employed in auction protocols, clock synchronization, sensor fusion, etc. In the context of topology management, consensus is mainly used for connectivity maintenance, i.e. preserving the network’s initial structure or ensuring that the communication graph remains connected even when environmental perturbations occur. For example, consider a robotic team exploring an unknown area. During exploration, their formation must adapt to fit corners and tight spaces while remaining connected. In such cases, connectivity control laws are typically incorporated within the motion controllers [15–19]. A similar approach is used in sensor networks, where stationary nodes maintain connectivity by adjusting the communication radius in case of failing sensors [20]. However, these methods use only Euclidean distances between agents to model network state and ignore other unpredictable phenomena that affect wireless network performance. In contrast, our method does not rely on distance criteria but on a generalized channel quality measure.

Another important aspect of topology management is connectivity determination. Distributed methods are used to gain insight into the overall topology of the multi-agent system, using only limited local knowledge of individual agents. In our work, we use a variation of a trust-inspired consensus algorithm [21] to estimate the network structure through information exchange between agents. Most distributed methods for determining connectivity rely on power iteration [22–24], where algebraic connectivity is computed from iteratively updated normalized eigenvectors corresponding to the Laplacian matrix of the graph. Wheeler et al. [25] combined a switching signal that determines the network topology with the consensus algorithm to rule out fake nodes and achieve agreement. Zhang et al. [26] extended a well-known centralized method by estimating two unavailable terms using high-pass consensus filters. They theoretically proved the globally asymptotic convergence, but the technique applies only to static graphs. An approach for dealing with time-varying directed graphs in underwater sensor networks was presented in [27]. The authors proposed a Q-learning method with an adaptive learning rate to estimate the probabilistic adjacency matrix and the corresponding generalized algebraic connectivity.

Determining the connectivity of a graph can be achieved through various metrics. Algebraic connectivity is often utilized in networked system research as it is a fundamental gauge of system performance [28] and has a direct impact on the speed of convergences in the consensus protocol [29, 30]. Furthermore, a higher value of algebraic connectivity improves the graph’s robustness to node and link failures [31]. Algebraic connectivity can be trivially increased by adding more links in the graph, but determining which links lead to the quickest increase in connectivity is a challenging problem. To address this problem, Ghosh and Boyd [32] proposed a simple greedy heuristic based on the first-order approximation of the derivative of algebraic connectivity. Their approach demonstrated that the connectivity can be maximized by adding a link between two nodes with the largest difference between their corresponding elements of the Laplacian eigenvector. The second-order approximation was later presented in [33]. Kim and Mesbahi [34] presented a computationally efficient bisection algorithm that yields better results than the simple greedy approach in a similar running time. Another heuristic involves connecting the two nodes with the minimum degrees and maximum distance from each other [35]. Authors in [36] provided necessary and sufficient conditions to determine redundant links in undirected graphs. In certain cases, removing these links improves the convergence rate of the consensus algorithm. Finally, a convolutional neural network was used in [37] to find the optimal placement of relay agents that maximizes the connectivity.

In real networks, increasing the number of connections increases the frequency of communication medium usage, leading to a potentially higher number of transmission collisions, transmission delays, degradation of throughput, and excessive energy consumption [38]. Balancing the need for maximum connectivity and robustness with minimum network cost is a challenging convex optimization problem [39], and selecting the right set of links to modify to achieve a desired connectivity level is NP-hard [40].

In [41], the authors investigated this problem in the context of optimizing cooperative multi-vehicle localization. They formulated the constrained maximization problem as a Mixed Integer Semidefinite Program (MISDP). Using the novel maximum cost heuristic, their method can quickly find good solutions with tight upper bounds. Alenazi et al. [42] focused on topology optimization in the context of telecommunication operators, where the cost of adding a link is mainly associated with its length. They developed an iterative algorithm that selects the desired number of links to add while balancing between improving network connectivity and minimizing the total cost. A trade-off between area coverage and connectivity was considered in [43]. The problem is divided into parts, each with a cost function that is jointly optimized using a gradient descent controller. Algebraic connectivity and its partial derivatives were calculated distributively. Some works also explore minimizing connectivity while keeping the graph connected, for instance, to slow the spread of disease [44] or cascading power failure [45].

This study does not aim to solve the problem of optimal network design, since the definitions of cost and optimal connectivity depend on the particular use case. Instead, we focus on addressing the challenge of distributively estimating network topology and controlling it in time to track the desired connectivity while rejecting disturbances. This challenge is not very well addressed in the literature. One of the few examples compares methods from previous studies [32, 34] and proposes a hybrid algorithm based on the bagging random forest classifier to select the best method depending on the current topology [46]. While this approach successfully exploits the advantages of both methods, it does not account for node failures and struggles to provide reliable results in a real distributed system. Dutta et al. [47] investigate connectivity tracking as part of controller design for unmanned aerial vehicle formation. They propose an offline method for computing the controller gains necessary to maintain a set connectivity level. This offline method is then repeatedly re-run to update the gains for the changing reference connectivity. In another study, the authors present a distributed stochastic power iteration method for estimating algebraic connectivity in realistic scenarios [48]. The estimated connectivity is used in a simple connectivity controller that adjusts the transmission power of the nodes. However, it assumes equal power transmission for all nodes, limiting its efficiency. A recent study focuses on controlling algebraic connectivity by adding and removing links until an approximate k-regular graph is constructed [49]. Two distributed algorithms are proposed: one relies on agreement between agents, while the other lets nodes modify their edges independently. The second algorithm is faster than state-of-the-art methods, and both offer resilience to node attacks by preserving connectivity. However, their applicability is limited to the algebraic connectivity values and structure of k-regular graphs.

Our previous research also tackled this problem by distributively estimating MAS topology through consensus and modifying links using a probabilistic approach [50].

Paper contributions

In this work, we propose a novel network connectivity control method for multi-agent systems that enables them to accurately track the reference value of connectivity over time. By assuming an external reference generator, our method allows the system to balance the need for better performance or lower energy consumption depending on the user’s requirements. We measure connectivity using the second smallest eigenvalue of the communication graph’s Laplacian matrix, also known as algebraic connectivity. Connectivity is estimated in a decentralized manner using a trust-inspired consensus algorithm applied to the adjacency matrix. Our method relies solely on local information about the immediate neighbors of the nodes and does not require global information about the initial communication graph. We also consider the actual time-varying quality of the communication channels between agents, which can be modeled with a generalized function. Building on our prior work [51], we enhance connectivity estimation with a novel adaptation mechanism and a convergence detection method to improve convergence speed.

To achieve the desired connectivity value, our proposed connectivity feedback controller analytically selects which links in the network to add or remove based on the Fiedler vector approximation [32]. This approach is additionally adapted for reconfiguring the communication graph by removing connections that consume energy but do not contribute to the overall connectivity. Unlike existing work, our method is robust to interference and agent failures. As we show in the results section, a multi-agent system can detect these adverse events, exclude failing agents from communication, and quickly reconfigure itself to maintain the desired connectivity. The proposed method is successfully validated in simulations with different scenarios and system sizes. In summary, the main contributions of this paper are:

1. An adaptive consensus-based algorithm for distributed estimation of the communication network in multi-agent systems.
2. An analytical connectivity feedback controller based on Fiedler vector approximation for managing active and redundant links.
3. A distributed topology control method resilient to disturbances and agent failures for dynamically tracking the desired algebraic connectivity.

Paper structure

The remainder of the paper is structured as follows. First, we give basic notations and background on the graph theory required to formulate our approach. The following section introduces a definition of a multi-agent system, necessary assumptions, and a detailed description of the distributed connectivity control problem. Then, we present our solution for estimating the graph topology and controlling the connectivity by modifying appropriate links, followed by a description of the experiments and results of the proposed approach. We conclude with a brief summary and some comments on possible future research directions.

Communication quality measurements

At each iteration k of the consensus algorithm for topology estimation, the agents exchange the local estimates of the global adjacency matrix with their neighbors. The measured quality of the communication channel between agents i and j, as observed by the agent l, is defined as a ratio of successfully received messages over a rolling window of size N*, (5) where indicates whether agent j received the message from agent i in step q, as observed by agent l.

This model is easy to implement and considers the real-world challenges of message transmission, including channel noise, dropouts, multi-pathing, and delays that exceed the specified communication period. To reduce sensitivity to the number of dropped messages, the window size N* can be increased, and vice versa. Additionally, the model can be expanded to handle multiple messages per communication period without difficulty. The consensus protocol for updating the local estimate of the adjacency matrix described in the next section does not depend on the actual measurement model as long as so other models can be easily used as well.

When it comes to allowing agent l to access the measurements between agents i and j, it’s important to note that in a realistic scenario, only the agents involved in the data exchange can measure the communication quality. Additionally, an agent can only measure the success rate of incoming messages and must assume that the channel is symmetric. As a result, it is typically the case that l = i (or l = j) and (6)

Adjacency matrix updates

Consensus protocol is a commonly used tool for reaching an agreement in distributed systems without global knowledge. General discrete consensus protocol takes the form of (7) or in the iterative version: (8) The state vector x(k) represents the current state of the system, e.g., a robot’s position or the ambient temperature measured by the agent. The matrix G is a control law matrix and it should be designed so that the system achieves the desired behavior. For the average consensus, which we use to gather communication link qualities that cannot be directly measured, G must be designed such that for any initial value x(0), x(k) converges to the average vector [56]. In other words, G should follow (9) A good choice for G that satisfies (9) is G = I − σ L [56]. Since L = D − A, it can be shown that substituting G in (8) results in (10) where σ is an update step size, and a_ij, the element of the adjacency matrix, represents the level of relation between two agents, e.g., trust, attraction, or in our case, communication quality. To capture the dynamic changes in link qualities, general control law (10) needs to be extended with an additional term to update the current estimate based on quality measurements: (11) Referring to (6) and (11), if i and j do not communicate (l ≠ i ≠ j), the measurements of the transmitted and received data packets are not available and we set .

Note that our goal is to estimate the topology of a multi-agent system. Therefore, we assign the quality of the communication link to both the relationship property and the state in the consensus protocol, since it simultaneously represents our interest and affects the consensus performance. This is in line with the research on a trust-based consensus algorithm presented in [21]. The authors assigned a trust value to each link in the network and introduced an algorithm that ensures the convergence of both the agents’ states and trusts to a common value. From a mathematical point of view, the communication link quality can be treated similarly to trust in that algorithm.

Using the elements of the adjacency matrix as state variables and combining the average consensus protocol (10) with additional measurements (11) we derive the following control law: (12) where is the agent l’s own view on the quality of the communication between it and its neighbor, and and are the current estimates of the edge weight between agents i and j, by agents p and l respectively. In general, we allow each agent to have its independent time-varying value of σ_l(k). Assuming the initial underlying communication graph contains a spanning tree, the update equation guarantees that (3) holds. For a detailed derivation of the control law and proof of convergence, we refer the reader to [50].

With the up-to-date estimate of the adjacency matrix A^l(k), the agents can calculate the corresponding Laplacian matrix and the current algebraic connectivity .

Update step adaptation

From the control law (12), it is clear that the parameter σ has an important influence on the convergence of the system. If σ is very small, the future estimate of the edge weight a_ij(k + 1) will mainly depend on the previous value, and convergence will be slow. On the other hand, making σ too large may lead to erratic changes and divergence. The optimal value of σ depends heavily on the number of nodes in the graph and how they are connected. Therefore, to ensure the stable behavior of the algorithm in all situations, the initial σ must be small which would limit the performance. We propose an adaptation mechanism that monitors the current network of the multi-agent system and adjusts the value of σ to keep it at the optimal level.

To determine the optimal value of σ (here we omit the agent index l and step iteration k for brevity), we start by writing the proposed control law in the standard form using x_i as the state variable of agent i. Measurement τ_i is considered as an input to the system (u_i): (13)

In matrix form, the update rule can be written as (14)

The transfer function of the system with difference Eq (14) is (15)

The stability of discrete-time LTI (linear time-invariant) systems such as (15) depends on the locations of system poles which are the roots of the characteristic equation, in this case, (16)

Since the Laplacian matrix is real and symmetric, it can be diagonalized, and the system can be separated into independent subsystems according to the eigenvalues of L. The characteristic equation corresponding to the eigenvalue λ_i is then (17)

In addition to stability, the maximum magnitude of the system poles determines the convergence rate of the consensus: (18)

Using the root locus technique common in the analysis of LTI systems applied to (17), we can observe that the pole starts at z = 1 and moves left as λ = λ_i, i = 1, …, n increases. The minimum ρ (fastest convergence rate) is achieved when the poles related to the smallest and the largest eigenvalue are equally distant from the center of the unit circle, i.e., when (19)

Substituting λ_i and z in (17) with conditions in (19) and solving for σ and ρ, we get the optimal convergence rate ρ* and optimal σ* for which that convergence rate is achieved (20)

Using the same set of equations and a condition that ρ = max_p=1,…,n |z_p| < 1, we can additionally determine the allowed range of values for σ that ensure the stability of the consensus: (21)

We can calculate the value of together with in every step, which allows us to continuously update the optimal update step . As a result, we are able to achieve fast convergence independent of the topology size and structure. Fig 3 illustrates the effect of the adaptation procedure by showing the estimated algebraic connectivity of each agent over time. In the beginning, the agents are only aware of their immediate neighbors, which do not form a connected graph, and the estimated connectivity is zero. Until k = 150, they continue to estimate the initial communication graph defined as a line. Afterward, the connectivity controller is enabled, and the agents collectively add and remove links to track the reference value of connectivity. Lines with stronger colors indicate that σ adaptation is enabled, and demonstrate much faster performance compared to the case when the update step is set to the fixed value of σ = 0.15.

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Fig 3. Effectiveness of the proposed σ adaptation method.

Top: Comparison of the effects of different update step sizes in a group of 6 agents. Lighter colors correspond to a fixed σ = 0.15, while darker colors represent the case with adaptive σ. Bottom: Value of update step size σ with enabled adaptation.

https://doi.org/10.1371/journal.pone.0314642.g003

The proposed complete algorithm for adjacency matrix estimation is summarized in Algorithm 1 and Algorithm 2.

Algorithm 1: The pseudo-code for calculating the algebraic connectivity and optimal update step value.

Input : A^l(k)

Output: , ,

L^l(k) = D^l(k) − A^l(k)

Algorithm 2: The pseudo-code of the proposed adjacency matrix estimation on agent l.

Input :

Output:

foreach i ∈ [1, n] do

 foreach j ∈ [1, n] do

  Calculate quality measurements:

  if l ≠ i or l ≠ j then

  else

  end

  Calculate the update term:

  Calculate the new estimates of adjacency matrix elements:

 end

end

Adding and removing communication links

After deciding to add or remove a link, each agent should try to select the optimal link to minimize the time required to adopt the required topology and the energy spent on establishing new connections. To determine the effect the modified link will have on the network, we use a heuristic rule based on the elements of the Fiedler vector. The control period κ is set to a larger value than the convergence time of the consensus. This allows the agents to have the same state and can deterministically select the link in a decentralized manner. The relation between the Fiedler vector and algebraic connectivity change under perturbation (such as adding a link) is given in the following lemma.

Lemma 1. The first-order approximation of the algebraic connectivity change under perturbation is: (24) where f is a normalized eigenvector of the Laplacian, also known as Fiedler vector, such that Lf = λ₂f, and l_ij is a changing element of the Laplacian matrix.

Proof. A small change in edge weight a_ij will cause a change in l_ij. We can denote the corresponding change in the Laplacian matrix as L′ = ∂L/∂l_ij. From the perturbation theory of symmetric matrices [57, Theorem 2.3], we have (25)

Since the Fiedler vector is normalized, f^Tf = 1. From [54, Lemma 13.1.5], we have that (26) Combined, we get that the change in algebraic connectivity due to the change in the link a_ij is proportional to the difference of the corresponding elements of the Fiedler vector: (27)

We can use this result to introduce the rule for selecting the appropriate links for addition and removal.

Theorem 2 (Analytic model for adding and removing links). To maximize the likelihood of achieving the desired configuration in the shortest possible time with minimal oscillations and energy expenditure, links should be added according to (28) and removed according to (29)

Proof. To minimize the number of links and the time required to reach reference connectivity, each added link should cause the largest possible increase in λ₂. From Theorem 1 we know that (24) approximates the change in λ₂ caused by changes in l_ij (a_ij). Thus, by choosing the link for which (f_i − f_j)² has the largest value, we maximize the likelihood that connectivity increases as much as possible. It should be noted that adding more terms of the Taylor series expansion of (24) could sometimes improve the approximation and lead to better connectivity, but it would also significantly increase the computational complexity.

When removing a link, we look for the minimum non-zero change in (f_i − f_j)² to avoid the possibility of accidentally disconnecting the graph and to minimize oscillations around the reference value.

To further minimize the likelihood of accidentally disconnecting the link that would reduce connectivity too much or even disconnect the graph, the controller additionally checks the minimum condition for desired connectivity.

Corollary 2.1. To achieve the topology with desired algebraic connectivity λ_2,ref, the minimum degree of the graph should be (30)

Proof. The bound for the minimum degree of the graph directly follows from the lower and upper bounds of the algebraic connectivity defined in Table 2: (31)

Finally, we return to the definition of the controller parameter .

Theorem 3 (Lower bound of ). For (23) to hold, the relay parameter should satisfy (32) where δ_m ∈ [0, 1] is a user-defined maximum allowable variation in link quality, and f_i and f_j are the i-th and j-th elements of the normalized Fiedler vector f.

Proof. We know from Lemma 1 that for a small change δ in l_ij (a_ij), the change in λ₂ is proportional to the difference in the corresponding Fiedler vector components, . Thus, by finding the maximum difference between pairs of all entries of f corresponding to existing edges, we determine the maximum (worst-case) change in λ₂ caused by a sudden but small change δ_m in the quality of the most sensitive connection (33)

Thus, to avoid oscillations in the algebraic connectivity caused by external disturbances of the connections, i.e., to guarantee that (23) is satisfied, the parameter of the relay controller should be larger than the expected Δλ_2,max. The value of δ_m depends on the user-defined tolerance for variations in the connectivity level. We set it to 20% of the maximum connectivity quality, δ_m = 0.2. If the quality change is greater than this value, new links should be added or removed.

Energy conservation

Depending on the network structure, some links in the graph may not contribute to the total algebraic connectivity. For example, adding a link to a node whose degree is already larger than the or in the case that λ₂ has multiple repeating values (λ₂ = λ₃ = …) will not immediately increase the connectivity. Such links still consume energy and possibly add noise to the channel.

When the estimated value of λ₂ reaches the commanded reference value and enters a stationary state, the agents can activate a strategy for removing energy-consuming redundant links. The strategy uses the result from Lemma 1 by finding the link for which (f_i − f_j) = 0.

Convergence detection

The connectivity controller works best when all agents in the system have approximately the same estimation of their current communication network. That way, they can reach a common decision without excess communication using a deterministic rule for modifying links. Ensuring that all agents reach a consensus before running the connectivity controller can be achieved by allowing a long enough control period κ or by detecting when the agents have reached a consensus.

First, every agent calculates the maximum difference between its own and neighboring adjacency matrices in the form of local Frobenius norms (34)

If the local norm is smaller than the set threshold e_min, that means that the agent l has converged to a common value with its immediate neighbors and would be ready to start another round of connectivity control. However, as other agents may not yet be ready, the local information needs to be propagated through the network.

Each agent tracks its own and other agents’ readiness in a boolean flag vector r^l ∈ {0, 1}ⁿ and updates it with an OR consensus running in parallel to the main procedure: (35) where ∨ and ⋁ represent element-wise OR operations between the vectors, and 1 and 0 are true and false values respectively. Agents can communicate their readiness vectors along with adjacency matrices in each iteration. When all values in the readiness vector become true, (36) the agent l performs the necessary connectivity control procedure.

For a better overview, the pseudocode of the whole proposed connectivity estimation and control method can be found in Algorithm 3.

Algorithm 3: The pseudo-code of the proposed connectivity control procedure running on agent l.

Initialize adjacency matrix estimate A^l(0) with known local connections.

while True do

 Exchange communication link qualities with neighbors .

 Update the adjacency matrix estimate A^l according to Algorithm 2.

 Calculate from A^l using Algorithm 1.

 Calculate from (22).

 if Condition (36) is satisfied then

  if then

   Add a link according to (28).

  else if then

   Remove a link according to (29).

  else if then

   Remove unnecessary link for energy conservation.

  end

 end

end

Adjacency matrix estimation

In the first example, illustrated in Fig 5, we demonstrate the proposed method’s ability to correctly estimate the global network topology and its algebraic connectivity. Each agent starts with a local view of the initial topology shown in Fig 4. As these local connections do not initially form a connected graph, the estimated algebraic connectivity in the first step is zero. Upon exchanging information about their connections, the local adjacency matrices converge quickly to the true value of the global adjacency matrix, thereby satisfying (3).

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Fig 5. Results of topology estimation experiment.

Top: Time responses of local algebraic connectivity estimates for the initial topology shown in Fig 4 with bounds of the allowable tracking error. Bottom: Local estimates of the link quality between agents 1 and 4.

https://doi.org/10.1371/journal.pone.0314642.g005

At step k = 150, we simulate a degradation in the communication channel between agents 1 and 4. Specifically, each message has a probability of 90% to arrive successfully at its destination. The affected agents can directly measure the change in the link quality () and are the first to update the estimate of that link to a₁₄ = a₄₁ = 0.9 (shown in the bottom part of Fig 5). Other agents converge at different rates depending on their distance from agent 1, as the information propagates through the network using the consensus protocol.

The reduction in quality of the link 1-4 causes a small change in the estimated algebraic connectivity. However, since the change is within the expected δ_m = 0.2 of the link quality, the connectivity tracking error remains within bounds and no corrective action is necessary. When the quality of the link 1-4 is further reduced to a₁₄ = a₄₁ = 0.7 at step k = 350, the tracking error exceeds the allowable limit. Consequently, agents decide to reconfigure the network by adding and removing links, resulting in λ₂ returning to the desired reference value.

Connectivity tracking

In the following experiments, we demonstrate our method’s ability to adjust the network connectivity based on user-defined reference, recover from unexpected agent failures, and manage redundant links to provide better energy efficiency.

Following the connectivity profile.

Fig 6 shows the estimated values of the algebraic connectivity while tracking the given reference (dashed line). After the first 100 calculation steps, the agents reach a consensus on their initial topology (see Fig 4) and converge to a common value of λ₂ = 0.6571. At k = 100 the reference tracking begins by setting λ_2,ref = 2. Although several edges need to be added to the graph, the group eventually achieves the commanded connectivity. This process is repeated and the connectivity increases further to λ₂ = 3. At k = 500, the reference value is reduced to 1.6, prompting agents to remove links until the final reference is reached. Snapshots of the network topology at certain steps during the experiment can be found in Fig 7.

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Fig 6. Time responses of algebraic connectivity estimates for the system of 6 agents and changing λ_ref.

https://doi.org/10.1371/journal.pone.0314642.g006

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Fig 7. Snapshots of the multi-agent topologies during the experiment with changing algebraic connectivity reference.

The thickness of the edges represents the link quality.

https://doi.org/10.1371/journal.pone.0314642.g007

Resilience to agent failures.

Depending on the topology structure, agent failures can cause significant changes in connectivity or result in a disconnected communication graph. To prevent this, the proposed method enables the multi-agent system to detect unresponsive neighbors and reconfigure to maintain connectivity. The experiment depicted in Fig 8 starts with a period of initial topology estimation and reaching the reference connectivity λ_2,ref = 2. At k = 350, a simulated failure of agent 3 leads to a sharp drop in algebraic connectivity. The other agents detect this unresponsiveness and exclude agent 3 from future calculations while establishing additional links to maintain connectivity. At k = 600, the failed agent becomes responsive again and rejoins the group. Adding a new, poorly connected agent initially reduces the overall λ₂, but after several rounds of connectivity control, the commanded reference value is restored.

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Fig 8. Time responses of algebraic connectivity estimates in case of an agent failure.

In the system of 6 agents a failure of agent 3 is simulated at step k = 350. The agent rejoins at step k = 600.

https://doi.org/10.1371/journal.pone.0314642.g008

Energy conservation.

In certain topologies, some links may not contribute to the overall algebraic connectivity but still consume energy. When reference connectivity is reached in steady-state situations, such links can be removed to improve energy efficiency without compromising connectivity. Fig 9 illustrates the time responses of algebraic connectivity for a well-connected group of 6 agents along with the average values of their local estimates of individual link qualities. At k = 200, the group reaches the desired connectivity and enters a steady state. The agents collectively determine that links 1-4 and 2-5 are currently unnecessary, leading to their removal. When the reference changes to λ_2,ref(k = 300) = 4, the previously removed links must be added before connectivity is further increased.

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Fig 9. Results of energy conservation experiment.

Top: Time responses of algebraic connectivity estimates for the system of 6 agents and changing λ_ref. Bottom: Average link qualities showing how removing some edges (e.g., 1-4 and 2-5) does not affect the connectivity but reduces energy consumption.

https://doi.org/10.1371/journal.pone.0314642.g009

Extended scenario

The extended scenario explores a group of 15 agents, showcasing the effectiveness of the proposed approach for larger systems. The initial topology used in experiments is depicted in Fig 10. We assess the system’s capability to track the reference connectivity profile and its resilience to agent failures. Fig 11 illustrates the time responses of the estimated algebraic connectivity in the given scenario with 15 agents.

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Fig 10. Initial topology of a system with 15 agents.

https://doi.org/10.1371/journal.pone.0314642.g010

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Fig 11. Time responses of algebraic connectivity estimates for the system of 15 agents.

A failure of agents 3, 9, and 12 is simulated at step k = 350. Agent 12 rejoins at step k = 1400.

https://doi.org/10.1371/journal.pone.0314642.g011

Initially, the agents reach a consensus on their starting topology and add a link to increase connectivity to λ_2,ref = 1.4. At step k = 350, a simulated failure of agents 3, 9, and 12 occurs. Similar to the case with a group of 6 agents, this failure results in a sharp decrease in connectivity. Subsequently, depending on the number of neighbors, the failed agents are excluded one by one in rapid succession until the connectivity stabilizes. The remaining agents then reconfigure their topology to restore the commanded connectivity value.

Starting from k = 1000, the agents begin following the new reference λ_2,ref = 1.2. Agent 12 resumes communication in step k = 1400 and rejoins the group, causing a change in connectivity. Finally, additional links are created to maintain the desired connectivity.

Possible extensions and future work

In this study, we addressed the challenge of distributed connectivity tracking in multi-agent systems through reconfiguration of communication links, offering key insights into the self-organization and resilience of such systems. We based our approach on the trust-inspired average consensus protocol, chosen for its simplicity and demonstrated effectiveness—crucial properties for implementation on distributed devices with limited hardware resources.

A potential avenue for future research would be to extend these methods to more complex real-world scenarios, such as networks with limited or quantized communication, as discussed in [50], or communication with long delays. Our current analysis is limited to undirected networks, where bidirectional communication between agents is assumed. A natural extension of this work would involve applying our methods to directed networks and exploring generalized algebraic connectivity, with [27] offering a potential starting point.

The use of the Fiedler vector to select links for modifying the network topology provides an analytical, efficient, and straightforward approach. However, predicting connectivity changes after link modifications remains a challenge, given the non-linear relationship between algebraic connectivity and link weights, which can result in overshooting in the connectivity controller. Our future research will explore alternative link selection strategies, including the use of artificial intelligence and machine learning techniques specifically designed for graph structures, to better predict and manage connectivity dynamics.