# A Generalized Framework for the Estimation of Edge Infection Probabilities
## 3 Properties of GIIM
### Observations  
In the definition above, \(O\) is defined as a set of vectors \(\vec{a}_{t}\in O\) for all \(t\in T\), and each vector assigns an observation to all vertices of the graph at a specific time step \(t\). Since \(T\) represents the set of sample times, it is safe to assume that it can be ordered; thus, the set of observations can also be ordered according to \(T\). The cardinality of \(T\) is not specified, but at least two time samples are needed to make a meaningful prediction for \(W\). The upper limit is application-dependent. In the case of a discrete-time infection model, \(T\) usually contains natural numbers.  
We define \(i_{0}=0\ldots i_{max}\) as the time period of the reference infection process. We assume this process to be finite, as described in Section 2.1. If the infection model is finite, i.e., it finishes at iteration \(i_{max}\), it is meaningless to make observations after \(t_{max}=i_{max}\) because the process does not change after \(i_{max}\).  
From a practical perspective, the first and last observations--at \(t_{0}\) and \(t_{max}\), respectively--give bounds, since we do not have other observations apart from the ones at \(T\). Therefore, if we want to explain the witnessed infection scenario, we should consider \(t_{0}\) and \(t_{max}\) to be the beginning and the end of the infection process.  
The task of the GIIM is to find an edge weight configuration that explains the original observed infection process. Therefore, the set of observations \(O\) should contain information that represents the behavior of nodes in the infection process. In this paper, we suggest two extreme cases of observation levels, but the proposed framework is able to handle other types as well. Realistic cases may be in between these two extremes, all of which can be accommodated within the modeling framework.  
1. Two observations are available at \(t_{0}=i_{0}\) and \(t_{1}=t_{max}=i_{max}\).
2. Observations for all iterations of the discrete infection process are available \(T\): \(\{t_{0}=i_{0},\ldots,t_{max}=i_{max}\}\).  
The set of observations \(O\) should contain information that represents the behavior of nodes in the infection process. In this paper, we suggest two specific _types_ of observations that can be made.  
The first observation type defines the observations as real values between \(0\) and \(1\), which are assigned to each node at the time steps where observations are available. For this type of observation, for time step \(t\) and node \(v\), the value \(o_{v_{t}}\) represents the likelihood that node \(v\) is infected at time step \(t\), i.e., the cumulative infection probability of \(v\) up to time step \(t\), while \(\vec{o}_{t}\) represents the vector of observations for all nodes at time step \(t\). The initial infections, the probability that \(v\in V(G)\) is infected before the process, are stored in \(\vec{o}_{t_{0}}\), while the probability that a node becomes infected measured at the end of the process is in \(\vec{o}_{t_{max}}\), corresponding to the a priori and a posteriori distributions assumed known in the two-observation case. For finite infection models, if we limit our viewpoint to a single node,the series of observations, \(o_{v_{t}},t=0\ldots t_{max}\), will be a monotonically increasing function since \(o_{v_{t}}\) starts at the initial infection probability \(o_{v_{0}}\), and the infection process may never decrease this value.  
Another way to make observations is to simply consider flags on the nodes signaling whether they have been infected or not. In this case, for a node \(v\in V(G)\), \(o_{v_{t}}=0\) indicates that it has not been infected up to time step \(t\), and \(o_{v_{t}}=1\) means that it is already infected at \(t\). This approach is the most common in the literature [15; 16; 17; 24; 25]. It can be used directly, or if time stamps are available on the nodes, we can simply consider \(o_{v_{t}}=0\) if \(t<t_{inf}\) and \(o_{v_{t}}=1\) if \(t\geq t_{inf}\), where \(t_{inf}\) denotes the time at which a node was infected.