# On the Role of Matrix-Weights Elements in Consensus Algorithms for Multi-Agent Systems
## 2 Problem Formulation
### Graph Theory  
The communication topology of the \(n\) agents is modeled by an \(n\)_-partite_ graph \(\mathcal{G}(\mathcal{V},\mathcal{E},\mathcal{W})\)[19]. The vertex set \(\mathcal{V}\) is partitioned into \(n\) classes \(\mathcal{V}=\{v_{1},\ldots,v_{n}\}\), where each class represents an agent. Every partition class contains exactly \(m\) vertices, which represent the \(m\)-dimensional state of an agent. Every edge in \(\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}\) has its ends in different classes: vertices in the same partition class are not adjacent. The matrix weight  
\[\mathcal{W}=\{W_{ij}|i=1,\ldots,n,j=1,\ldots,n\}\]  
models the connections between every two different classes, where \(W_{ij}\succeq 0\in\mathbb{R}^{mx\times m}\) shows the connections of the states between agents \(v_{i}\) and \(v_{j}\). We use \(v_{j}^{[l]}v_{i}^{[k]}\) to represent an edge from the \(l\)th state in agent \(v_{j}\) to the \(k\)th state in agent \(v_{i}\), which means that \(x_{i}^{[k]}\) of agent \(v_{i}\) is affected by \(x_{j}^{[l]}\) of agent \(v_{j}\). The strength of this connection is characterized by \([W_{ij}]_{kl}=w_{kl}^{ij}\), the \((k,l)\) entry of \(W_{ij}\). If \(w_{kl}^{ij}=0\), \(x_{j}^{[l]}\) of agent \(v_{j}\) has no influence on \(x_{i}^{[k]}\) of agent \(v_{i}\). If there is at least one entry in \(W_{ij}\) in which \(w_{kl}^{ij}>0\), we say agent \(v_{j}\) is a neighbor of agent \(v_{i}\) and \(v_{j}v_{i}\in\mathcal{E}\). The set of neighbors of agent \(v_{i}\) is defined as \(\mathcal{N}_{i}\triangleq\{v_{j}\in\mathcal{V}:\exists w_{kl}^{ij}>0\}\). An induced subgraph \(\mathcal{G}^{[k]}\) of \(\mathcal{G}\) with the vertex set  
\[\mathcal{V}^{[k]}=\left\{v_{1}^{[k]},\ldots,v_{n}^{[k]}\right\}\]  
is called the _k-state_ graph. A path in \(\mathcal{G}^{[k]}\) is given by a sequence of distinct vertices in \(\mathcal{V}^{[k]}\) connected by an edge in \(\mathcal{E}\). If the induced subgraph \(\mathcal{G}^{[k]}\) contains a spanning arborescence, then we say the _k-_state graph is _connected_. If the _k-_state graph is not connected, _k_th-states of all agents can be partitioned into distinct connected components. The graph is connected in the agent level if for any two distinct agents, there is a path to connect at least one state in each agent.  
The degree or valency of a given vertex \(d(v_{i})\in\mathbb{R}^{m\times m}\) is defined as  
\[d(v_{i})\triangleq\sum_{v_{j}\in\mathcal{  
\[[A_{w}]_{ij}=\left\{\begin{array}{ll}w_{ij}&\text{if }\ v_{j}v_{i}\in\mathcal{E}\\ 0&\text{otherwise}\end{array}\right. \tag{7}\]  
_The matrix-weighted Laplacian of the graph, \(\mathcal{L}_{W}(\mathcal{G})\in\mathbb{R}^{nm\times nm}\) is given by_  
\[\mathcal{L}_{W}(\mathcal{G})=\Delta_{W}-A_{W}=\mathcal{L}_{w}\otimes l_{m}, \tag{8}\]  
_where \(\mathcal{L}_{w}=\Delta_{w}-A_{w}\)._  
**Remark 3**.: _Matrix weighted graphs are closely related to signed graphs [20]. Let us use a simple example of two agents, where each agent has two states, to illustrate the similarity. The matrix weight is given by \(W_{12}=W_{21}=l_{2}\), and the Laplacian matrix is_  
\[\mathcal{L}=\left[\begin{array}{cc}l_{2}&-I_{2}\\ -l_{2}&l_{2}\end{array}\right]. \tag{9}\]  
_If we use the definition of Laplacian matrix for scalar weighted graphs, it is equivalent to say that the weight between \(x_{1}^{[1]}\) and \(x_{1}^{[2]}\) is \(-1\)._