# Network Visualization of Cooperative Games
## 4 Network Visualization of a Cooperative Game  
In social analysis, the network visualization of individuals' interactions stands as a cornerstone for exploring the complex dynamics of collective behaviors. To use this instrument, we introduce the formal definition of a cooperative game representation as a network. **Definition 2** (Network visualization of a cooperative game). : _Given a game \((N,v)\in G^{N}\), we define the network visualization of a cooperative game as \((\Psi,R)\), where the following hold:_  
* \(\Psi=\{\psi_{i}\in\mathbb{R}\mid i\in N\}\) _is the vector representing each player's relevance in the game. In this context, the player relevance_ \(\psi_{i}\) _is computed as an aggregated measure that combines the contributions of player_ \(i\) _over all coalitions in which player_ \(i\) _participates._
* \(R=\{r_{i,j}\in\mathbb{R}\mid i,j\in N\}\) _is the matrix that collects the information about the relationship between pairs of players._  
Here, we propose the network visualization for a real cooperation situation where the size of a node corresponding to a player \(i\in N\) will be represented by \(\psi_{i}\) belonging to the relevance vector \(\Psi=Sh\). The value \(\psi_{i}\) is distinct from the characteristic function \(v\), because \(v(\{i\})\) measures the wealth or payoff of player \(i\) when acting individually. In contrast, \(\psi_{i}\) aggregates the various contributions player \(i\) makes when participating in different coalitions, thereby capturing the synergy and complementary effects inherent in joint cooperation. As the magnitude of the share for a player increases, the size of the node representing the player will increase as its relevance increases. Depending on the sign of this value, the nodes will be colored differently, black for positive values of \(\psi_{i}\) and red for negative ones. The links of the network will show the relationship \(r_{i,j}\) between player \(i\) and player \(j\). To represent it, we will use the Matrix of the Interaction Index \(R=I\). It is also possible to use the Positive Matrix of the Interaction Index \(R=I^{+}\), the Negative Matrix of the Interaction Index \(R=I^{-}\), or the Matrix of Absolute Values of the Interaction Index \(R=I^{|\cdot|}\) when needed. Negative relationships will be painted in red, while black color will represent positive ones. We will depict the intensity of each pair of players' relationship by the thickness of each link \((i,j)\) in the network. It will be proportional to the corresponding value \(r_{i,j}\in R\), leading to thicker links in stronger relationships, and thinner for weaker connections. A potential problem with the network representation of a cooperative game is that the number of links defined in the player relationship matrix may increase exponentially with the addition of more players to the game. To address this issue, we have incorporated a feature that filters the potential number of links and nodes in the network representation of the game. This is achieved through the use of the threshold parameters \(\alpha^{+}\), \(\alpha^{-}\), \(\beta^{+}\), and \(\beta^{-}\), which prevent an overabundance of links and nodes in the network and enable the visualization of significant relationships. **Definition 3** (Filtered representation of a cooperative game). : _Given a network representation \((\Psi,R)\) of a game \((N,v)\in G^{N}\), and four threshold limit values named \(\alpha^{+},\alpha^{-},\beta^{+}\), and \(\beta^{-}\in\mathbb{R}^{+}\), we define the filtered representation of the cooperative game \((N,v)\in G^{N}\) as \((\Psi_{\alpha^{+},\alpha^{-}},R_{\beta^{+},\beta^{-}})\) in such a way that the following hold:_  
1. _Nodes with positive sign in_ \(\Psi\) _are assigned a value of zero if their absolute value does not exceed_ \(\alpha^{+}\)_._
2. _Nodes with negative sign in_ \(\Psi\) _are assigned a value of zero if their absolute value does not exceed_ \(\alpha^{-}\)_._
3.