# Interaction Strength Analysis to Model Retweet Cascade Graphs
## 5 Evaluation Metrics for Refweet Cascade Validation  
Evaluating the models ability in deriving the retweet cascade graph is not a trivial task. The absence of ground truth information prevents the use of standard evaluation metrics such as  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Set nr** & **Retweet Weight** & **Quote Weight** & **Reply Weight** \\ \hline
0.1 & 1.0 & 0.0 & 0.0 \\
0.2 & 0.0 & 1.0 & 0.0 \\
0.3 & 0.0 & 0.0 & 1.0 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Weight sets—guideline setups.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Set nr** & **Retweet Weight** & **Quote Weight** & **Reply Weight** \\ \hline
1 & 0.0 & 0.3 & 0.6 \\
2 & 0.0 & 0.6 & 0.3 \\
3 & 0.3 & 0.0 & 0.6 \\
4 & 0.3 & 0.6 & 0.0 \\
5 & 0.6 & 0.0 & 0.3 \\
6 & 0.6 & 0.3 & 0.0 \\
7 & 0.1 & 0.4 & 0.5 \\
8 & 0.1 & 0.5 & 0.4 \\
9 & 0.2 & 0.3 & 0.5 \\
10 & 0.2 & 0.5 & 0.3 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Weight sets—experimental setups.  
accuracy. Following [11] we evaluate the retweet information graph considering all the cascade forest, i.e., including the unconnected components, and computing the following metrics:  
1. Cascade average strength (CAS): given the \(IS\) assigned to each edge \((u,y)\in\) E we derive the CAS as the average of the maximum IS between each pair of edges in E such as: \[CAS=\frac{\sum(\max(IS_{u,y}\forall u,y\in\text{E}))}{|\text{E}|}\] (3) The aim is to maximise CAS for each cascade graph.
2. Connected components count (CCC): returns the number of the connected components in the network. A connected component is a subgraph in which any two vertices \(v\in V\) are connected to each other by paths. This metric provide a description of the graph shape.
3. Root fan ratio (RFR): it assesses whether there is a path to the \(root_{author}\) from every other user. In other words, it measures the percentage of nodes directly or indirectly connected to the root. In the ISN-AF model, the RFR asses the percentage of \(root_{author}\) followers. A higher RFR reflects a very concentrated graph around the \(root_{author}\) determining the typical star shape of the cascade graph [25].
4. Giant component size (GCS): the size, expressed in percentage of the cascade nodes, of the nodes present in the giant component (GC) which is the connected component with biggest size. The GCS is computed as follow: \[GCS=\frac{|u\in\text{V}\longrightarrow u\in GC|}{|\text{V}|}\] (4) This metric provides a description of the graph shape in terms of node dispersion: The lower the GCS, the higher the dispersion of nodes in the graph, which can be sparse or connected. To further investigate the nodes' dispersion we investigate the global reaching centrality and sparse node incidence.
5. Global reaching centrality (GRC): It is the average over all nodes of the difference between the node local reaching centrality and the greatest local reaching centrality of any node in the graph. The local reaching centrality, \(C_{R}(i)\), of node \(i\) is the proportion of all nodes in the graph that can be reached from node \(i\) via outgoing edges [35]. \[GRC=\frac{\sum_{i\in\text{V}}(C_{R}^{max}-C_{R}(i))}{|\text{V}|-1}\] (5)
6. Sparse node incidence (SNI): it measures the incidence (in percentage) of sparse nodes (i.e., nodes without links) with respect to the total number of nodes in the cascade. A lower SNI determines more connected and realistic retweet cascades.  
Those metrics are computed in order to identify the best IS weights set among the proposed ones in Table 3. Moreover, the metrics are evaluated in order to compare the proposed ISN and ISN-AF methods with respect to the baseline approach proposed by [11].