# Distributed Average Consensus Algorithms in d-Regular Bipartite Graphs: Comparative Study

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## Abstract

**:**

## 1. Introduction

- ☐
- Data Aggregation in Multi-Agent Systems: this subsection explains the importance of data aggregation, contains its definition and four-step process, discusses the benefits of its application, provides insight into multi-agent systems (MASs), and justifies the application of data aggregation mechanisms in these systems;
- ☐
- Distributed Consensus-Based Data Aggregation: in this subsection, we explain the general meaning of the term consensus, its meaning in the context of MASs, and the requirements for distributed consensus algorithms;
- ☐
- Theoretical Insight into d-Regular Bipartite Graphs: this subsection provides the definition of d-regular bipartite graphs, their graphical example, and examples of their applications;
- ☐
- Our Contribution: this subsection specifies our contribution presented in this paper and justifies the benefit of this manuscript compared to related papers;
- ☐
- Paper Organization: here, the paper structure is provided.

#### 1.1. Data Aggregation in Multi-Agent Systems

- ■
- Phase 1: extracting data from independent sources;
- ■
- Phase 2: storing the extracted data;
- ■
- Phase 3: interpretation of the stored data;
- ■
- Phase 4: presenting the processed data in an appropriate form.

#### 1.2. Distributed Consensus-Based Data Aggregation

- ■
- Agreement: all the non-faulty agents of MASs are required to agree on the same value (resp. on a precise estimate of this value);
- ■
- Validity: All the agents of MASs have to agree on a value suggested by the values of these agents. In other words, none of the non-faulty agents in MASs can decide on a value that is not suggested by the value of the agents in the system;
- ■
- Termination: the distributed consensus is achieved, provided that each non-faulty agent in MASs is in the agreement with all the other ones.

#### 1.3. Theoretical Insight into d-Regular Bipartite Graphs

**Definition**

**1.**

- ■
- Definition of d-regular graphs: the degree of each vertex from
**V**is the same and equal to d; - ■
- Definition of bipartite graphs: its vertex set
**V**is splittable into two disjoint subsets such that two vertices from the same disjoint subset are not linked to one another.

#### 1.4. Our Contribution

- ■
- Maximum-Degree weights (MD);
- ■
- Local-Degree weights (LD);
- ■
- Metropolis–Hastings algorithm (MH);
- ■
- Best-Constant weights (BC);
- ■
- Convex Optimized weights (OW);
- ■
- Constant weights (CW);
- ■
- Generalized Metropolis–Hastings algorithm (GMH).

#### 1.5. Paper Organization

- ☐
- Section 2—Related Work: this is divided into two subsections and consists of topical and frequently cited papers addressing either consensus-based data aggregation in subjected/closely related graph topologies or the algorithms chosen for evaluation in non-regular non-bipartite graphs;
- ☐
- Section 3—Theoretical Background: this is divided into three subsections and provides the used mathematical model of MASs, a general definition of distributed consensus algorithms, and the weight matrices of the examined distributed average consensus algorithms;
- ☐
- Section 4—Experiments and Discussion: this is formed by three subsections again, consisting of the applied research methodology, experimental results, and a comparison of our conclusions with conclusions presented in papers where the selected algorithms are examined in non-regular non-bipartite graphs;
- ☐
- Section 5—Conclusions: this provides a brief summary of the contribution presented in this paper;
- ☐
- Appendix A—Appendix: this contains tables with the experimental results in numerical form.

## 2. Related Work

- ☐
- Distributed Consensus Algorithms in d-Regular Bipartite Graphs: this subsection introduces papers addressing consensus-based algorithms for data aggregation in regular bipartite graphs and related graphs;
- ☐
- Distributed Consensus Algorithms in non-Regular non-Bipartite Graphs: in this subsection, we provide an overview of papers concerned with a comparison of the chosen algorithms in non-regular non-bipartite graphs.

#### 2.1. Distributed Consensus Algorithms in d-Regular Bipartite Graphs

#### 2.2. Distributed Consensus Algorithms in Non-Regular Non-Bipartite Graphs

## 3. Theorethical Background

- ☐
- Applied Mathematical Model of Multi-Agent Systems: this subsection is concerned with the applied mathematical model of MASs;
- ☐
- General Definition of Average Consensus Algorithms: here, we provide general update rules of distributed average consensus algorithms and their convergence conditions;
- ☐
- Examined Distributed Consensus Algorithms: in this subsection, we introduce all the algorithms chosen for evaluation in d-regular bipartite graphs.

#### 3.1. Applied Mathematical Model of Multi-Agent Systems

**V**and the edge set

**E**(i.e., G = (

**V**,

**E**)) [71,72]. We assume that both sets are of finite size and time-invariant in our experiments. The first set (

**V**) contains all the graph vertices (each with a unique index), which represent agents in MASs (i.e.,

**V**= {v${}_{1}$, v${}_{2}$, … v${}_{n}$,}). The cardinality of

**V**(∣

**V**∣) is determined by the graph order n or, in other words, by the size of MAS. The edge set

**E**consists of all the graph edges, which are direct connections between two vertices (i.e., the distance between two vertices is one hop, and these two nodes are referred to as neighbors). An edge linking v${}_{i}$ and v${}_{j}$ is unique and labeled as e${}_{ij}$ later in this paper. The parameter d${}_{i}$ represents the degree of v${}_{i}$, which is the number of all the edges incident to v${}_{i}$. The maximum degree of G is labeled as [73]:

**L**: a diagonally symmetric squared matrix with n${}^{2}$ entries that is defined as follows [76]:

**$\mathbf{L}$**) is the largest Laplacian eigenvalue, while, ${\lambda}_{n-1}$(

**$\mathbf{L}$**) represents the second smallest eigenvalue of the Laplacian spectrum and is known as the algebraic connectivity of G.

#### 3.2. General Definition of Average Consensus Algorithms

**x**(k) is a column variant vector formed by the inner states of all the agents in MAS at k-th iteration. The value of k = 0 represents the initial inner states.

**W**is the weight matrix of an algorithm whose spectrum is defined as follows [52]:

**W**, ${\lambda}_{2}\left(\mathbf{W}\right)$ represents its second largest eigenvalue, and ${\lambda}_{n}\left(\mathbf{W}\right)$ is its smallest eigenvalue. Distributed consensus algorithms differ from each other by the entries of this matrix. As discussed in the mentioned paper, the weight matrix

**W**conditions many algorithm aspects such as the convergence/the divergence, the convergence rate, the robustness to potential threads, the distortion of the final estimates caused by various noises, etc.

**1**is an all-ones vector, whose entries are all equal to one, and

**1**${}^{\mathrm{T}}$ represents its transpose [52]. The existence of this limit means that all the inner states asymptotically converge to the value of the wanted aggregated function. Thus, a precise estimate of (for example) the arithmetic mean from the initial inner states of all the agents in MASs can be obtained by applying distributed consensus algorithms. As identified in [62], meeting the three convergence conditions provided in (8)–(10) guarantees that this limit is sure to exist.

#### 3.3. Examined Distributed Consensus Algorithms

**W**can be, thus, expressed as follows:

**W**) and the smallest eigenvalue ${\lambda}_{n}$(

**W**) of the weight matrix

**W**are always located on the unit circle in d-regular bipartite graphs if (12) is applied. Furthermore, it is mathematically proven in our previous paper that all the other eigenvalues of the weight matrix

**W**, except for the two previously mentioned ones, are situated inside the unit circle. These facts cause the algorithm to diverge (as the third convergence condition is broken) in such a way that the values of the inner states oscillate between two values (more specifically, between values approaching the averaged initial inner states of the agents forming both the disjointed subsets). Further details about the divergence of MD in regular bipartite graphs can be found in the mentioned manuscript.

**W**of this algorithm is identical to the weight matrix

**W**of MD in d-regular bipartite graphs, whereby this algorithm also diverges in this critical graph topology. See (13) for its weight matrix

**W**[62].

**W**of this algorithm is identical to the weight matrix

**W**of GMH with the mixing parameter $\u03f5$ = 1 [51]. Mathematically, its weight matrix

**W**is defined as follows:

**W**is defined as [83]:

**W**is optimized according to eigenvalues, i.e.:

**W**, where

**S**is defined as follows:

**W**is defined as (20) [51]. Its mixing parameter $\u03f5$ is a variable again, taking the values from (21):

**W**for all the examined algorithms in 3-regular bipartite graphs with n = 30. As seen from Figure 7, the smallest eigenvalue ${\lambda}_{n}\left(\mathbf{W}\right)$ of the weight matrix

**W**is greater than minus one for each examined algorithm, except MD and LD, as stated above in this paper.

## 4. Experiments and Discussion

- ☐
- Research Methodology and Applied Metric for Performance Evaluation: in this subsection, we introduce the simulation tool used, we specify the used d-regular bipartite graphs, and we provide the applied metric, the used stopping criterion, the method of generating the initial inner states, and the examined setups of CW and GMH;
- ☐
- Experimental Results and Discussion about Observable Phenomena: this subsection consists of the experimentally obtained results depicted in 15 figures and a subsequent discussion;
- ☐
- Comparison with Papers Concerned with Examined Algorithms in Non-Regular Non-Bipartite Graphs: here, we compare the contributions presented in this paper with manuscripts addressing the examined algorithms in non-regular non-bipartite graphs.

#### 4.1. Research Methodology and Applied Metric for Performance Evaluation

**W**contain the mixing parameter $\u03f5$ [51]. In our experiments, it took the values from (26) in the case of CW and (27) if GMH was applied.

#### 4.2. Experimental Results and Discussion about Observable Phenomena

**W**of OW) and d = 10 (i.e., in the densest examined topologies), its performance was relatively pretty good. In both these cases, its performance was the second best among the analyzed algorithms. Overall, OW was the best-performing approach according to the presented experimental results since it outperformed all the other algorithms for each precision of the used stopping criterion except for the graphs with d = 2 (the performance was lower if the low precision of the stopping criterion was applied). CW was one of the worst algorithms in terms of the number of iterations required for consensus achievement. In addition, it was necessary to know the value of d to identify the optimal configuration of this algorithm, which was its other significant drawback. However, it performed relatively well in sparsely connected graphs with the low precision of the stopping criterion. GMH outperformed all its competitors in the sparsest graphs (i.e., d = 2), but its performance was significantly degraded with an increase in connectivity (for d = 10, this was the worst algorithm among those examined). Similar to CW, knowing d was also required for the optimal configuration of this algorithm. What is most important is that all the examined algorithms except for MD and LD can achieve the consensus in each d-regular bipartite graph regardless of the precision of the applied stopping criterion. Thus, except for MD and LD, no other algorithm required the implementation of the mechanism presented in [52] or required to be reconfigured.

#### 4.3. Comparison with Papers Concerned with Examined Algorithms in Non-Regular Non-Bipartite Graphs

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BC | Best constant weights |

CW | Constant weights |

GMH | Generalized Metropolis–Hastings algorithm |

IID | Independent and identically distributed |

LD | Local-degree weights |

MAS | Multi-agent system |

MD | Maximum-degree weights |

MH | Metropolis–Hastings algorithm |

OW | Convex optimized weights |

QoS | Quality of service |

UAV | Unmanned aerial vehicle |

## Appendix A

MH | BC | OW | CW${}_{\mathit{best}}$ | GMH${}_{\mathit{best}}$ | |
---|---|---|---|---|---|

Low | 261.69 it. | 196.75 it. | 196.75 it. | 193.47 it. | 182.78 it. |

Medium | 418.54 it. | 301.86 it. | 301.86 it. | 309.40 it. | 292.16 it. |

High | 575.45 it. | 407.27 it. | 407.27 it. | 425.36 it. | 401.74 it. |

MH | BC | OW | CW${}_{\mathit{best}}$ | GMH${}_{\mathit{best}}$ | |
---|---|---|---|---|---|

Low | 43.52 it. | 39.03 it. | 34.35 it. | 36.31 it. | 36.11 it. |

Medium | 70.10 it. | 60.39 it. | 50.88 it. | 58.07 it. | 57.67 it. |

High | 97.18 it. | 81.85 it. | 67.66 it. | 80.48 it. | 79.70 it. |

MH | BC | OW | CW${}_{\mathit{best}}$ | GMH${}_{\mathit{best}}$ | |
---|---|---|---|---|---|

Low | 19.47 it. | 19.43 it. | 17.50 it. | 19.31 it. | 18.38 it. |

Medium | 31.02 it. | 30.08 it. | 25.44 it. | 30.08 it. | 28.83 it. |

High | 42.99 it. | 40.68 it. | 33.33 it. | 41.16 it. | 39.61 it. |

MH | BC | OW | CW${}_{\mathit{best}}$ | GMH${}_{\mathit{best}}$ | |
---|---|---|---|---|---|

Low | 13.12 it. | 13.54 it. | 12.40 it. | 13.32 it. | 13.19 it. |

Medium | 20.07 it. | 20.58 it. | 17.72 it. | 20.69 it. | 20.10 it. |

High | 27.27 it. | 27.79 it. | 23.11 it. | 28.35 it. | 27.27 it. |

MH | BC | OW | CW${}_{\mathit{best}}$ | GMH${}_{\mathit{best}}$ | |
---|---|---|---|---|---|

Low | 14.56 it. | 7.12 it. | 7.05 it. | 7.33 it. | 16.11 it. |

Medium | 25.57 it. | 10.21 it. | 10.02 it. | 10.82 it. | 28.37 it. |

High | 36.86 it. | 13.46 it. | 13.03 it. | 14.31 it. | 41.03 it. |

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**Figure 4.**Network with three broken conditions required to be met by consensus algorithms and error-free execution—min-consensus is applied.

**Figure 6.**Comparison of evolution of inner state in regular bipartite/non-regular non-bipartite graph.

**Figure 9.**Convergence rate expressed as number of iterations required for consensus achievement in 2-regular bipartite graphs and for three precisions of applied stopping criterion.

**Figure 10.**Convergence rate expressed as number of iterations required for consensus achievement in 3-regular bipartite graphs and for three precisions of applied stopping criterion.

**Figure 11.**Convergence rate expressed as number of iterations required for consensus achievement in 4-regular bipartite graphs and for three precisions of applied stopping criterion.

**Figure 12.**Convergence rate expressed as number of iterations required for consensus achievement in 5-regular bipartite graphs and for three precisions of applied stopping criterion.

**Figure 13.**Convergence rate expressed as number of iterations required for consensus achievement in 10-regular bipartite graphs and for three precisions of applied stopping criterion.

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Kenyeres, M.; Kenyeres, J.
Distributed Average Consensus Algorithms in d-Regular Bipartite Graphs: Comparative Study. *Future Internet* **2023**, *15*, 183.
https://doi.org/10.3390/fi15050183

**AMA Style**

Kenyeres M, Kenyeres J.
Distributed Average Consensus Algorithms in d-Regular Bipartite Graphs: Comparative Study. *Future Internet*. 2023; 15(5):183.
https://doi.org/10.3390/fi15050183

**Chicago/Turabian Style**

Kenyeres, Martin, and Jozef Kenyeres.
2023. "Distributed Average Consensus Algorithms in d-Regular Bipartite Graphs: Comparative Study" *Future Internet* 15, no. 5: 183.
https://doi.org/10.3390/fi15050183