# A Generalized Framework for the Estimation of Edge Infection Probabilities
## 4 Evaluation
### Complex Networks  
In addition to trees and DAGs, we also examine the performance of the GIIM on a more general complex network structure and under several different infection scenarios. The graph in question is undirected, and it has 250 nodes and 897 edges. The graph was created with the forest fire model [29]. The network itself is relatively small in network science terms, but our task here is to find the correct value for each edge, which is moderately challenging for the optimization algorithm. We examine the scalability of the method in the next section. Our goal in this section is to compare the performance of the method in terms of accuracy and running time under a number of fundamentally different infection scenarios. The scenarios differ according to the infection model, the type and number of observations,  
Figure 3: The RMSE at different distances from the source in the fully observed IC model for (**a**) the second DAG and (**b**) the second tree in Table 1.  
the fraction of initially infected nodes and the size of the reference edge infection values. We measure the RMSE of real-valued observations and the ROC AUC of binary ones and consider minimally and fully observed cases as defined in Section 4.1. The scenarios examined are summarized as follows:  
* Four different infection models: IC, SI, SIR with \(\tau_{i}=2\), and SEIR with \(\tau_{e}=2\) and \(\tau_{i}=3\). * Three sets of initially infected vertices: 33%, 10% or 2% of the nodes are initially infected with node infection probabilities ranging between 0 and 0.5 for the real-valued case and uniformly 1 for the binary case. * Three sets of reference edge infection values as defined in Section 4.1. The values were drawn from a uniform random distribution between 0 and 0.25, 0 and 0.5, and 0 and 0.75. If we only consider the infection scenarios, we have \(4\times 3\times 3=36\) tasks, and if we add the observation types, this goes up to 108. Each of these tasks was computed ten times, and the results were averaged. The standard deviation measured between results was below 10% for all infection scenarios. The RMSE and the AUC values are presented for the real-valued and binary infection tasks, respectively, for the two-observation case (\(t_{0}=i_{0}\), \(t_{1}=t_{max}=i_{max}\)) in Table 2. The results can be seen for all four of the infection models and all possible initial infection states and infection probability combinations. In the first column, the first number denotes the percentage of initially infected nodes, and the second one indicates the maximum value of the edge infection probability. In the SI model, the second observation was selected so that the estimated average node infection value at the observation was close to the estimated average node infection value at the last of observation of the SEIR model. As we saw in the previous section, the binary tasks are much easier to compute than the real-valued scenarios. Based on the results in Table 2, the larger infections are easier to predict. The reason for this is that the infection probability of a node is bounded: no matter how dense the infections are, the node infection probability cannot be greater than one. This means that if the infections are dense--either because there are many initially infected nodes, or the edge infection probabilities are large--a sizable fraction of the nodes are going to have an infection value close to one. This makes their distribution more homogeneous, and this also makes the optimization task easier, since there are fewer distinct values to estimate. The difference between the infection models is deceptive.