## 3 Mathematical Model of Multi-Agent Systems  
In this paper, MASs are modeled as simple finite undirected unweighted graphs labeled as \(G\) and determined by two time-invariant sets, namely the vertex set \(\mathbf{V}\) and the edge set \(\mathbf{E}\) (\(G=(\mathbf{V},\mathbf{E})\)) [43; 44]. The vertex set \(\mathbf{V}\) consists of all the graph vertices representing the agents in MAS. The cardinality of this set determines the graph order (labeled as \(n\)), i.e., the number of agents in MAS. Each vertex from \(\mathbf{V}\) is allocated a unique index, a positive integer value from the range [1, \(n\)], i.e., \(\mathbf{V}=\{v_{1},\,v_{2},\,...\,\,v_{n}\}\). The other set, the edge set \(\mathbf{E}\subset\mathbf{V}\times\mathbf{V}\), is formed by all the graph edges, which represent a direct link between two vertices. The cardinality of the edge set \(\mathbf{E}\) determines the size of the corresponding graph \(G\), i.e., the overall number of direct connections in MAS. The direct link between two agents \(v_{i}\) and \(v_{j}\) (i.e., their distance is one hop) is indicated by the existence of an edge in \(G\) labeled as \(e_{ij}\). Every two vertices directly linked to one another are said to be neighbors. Subsequently, the set containing all the neighbors of an agent \(v_{i}\) can be defined as:  
\[\mathcal{N}_{i}=\{v_{j}:e_{ij}\in\mathbf{E}\} \tag{1}\]  
In this paper, we examine the chosen algorithms over RGGs, graphs that represent spatial MASs with \(n\) randomly deployed agents [43; 44] (see Figure 4 for an example of RGGs (one of 100 RGGs applied in our research is shown in this figure to illustrate what the network topology of the used graphs looks like) and Table 1 containing the graph parameters (the average value of each parameter of 100 applied graphs is shown)). In these graphs, the agents are placed uniformly at random over a square area of finite size. Subsequently, two agents are directly linked to one another, provided their distance is not greater than the transmission radius (each agent has the same transmission radius). In the case that there is a path between each pair of two vertices, we say that this graph is connected. If not, the graph is disconnected, thereby being composed of two or more components. Different connectivity in a graph can be ensured by modifying the value of the transmission radius. As shown in the literature [22; 34; 45; 46], this graph type is often applied to modeling real-world systems such as wireless sensor networks, ad-hoc IoT networks, social networks, etc. The algorithm used to generate RGGs applied in our research can be described as follows: in the beginning, we have a blank working area formed by 150 points on both the x- and the y-axis. Thus, there are \(150^{2}\) coordinates (x,y) in the working area. Then, the first vertex is placed uniformly at random on one of these coordinates (i.e., it can be placed on each coordinate with the same probability equal to \(1/150^{2}\)). Afterward, the other vertices (one by one) are placed on free coordinates--one coordinate can be allocated to only one vertex. Once all the vertices are distributed over the working area (200 vertices in overall in our case), edges linking all the adjacent vertices are added to the graph. As mentioned earlier, two vertices are linked to one another, provided that their distance is not greater than their transmission radius.