## 2 Network Tomography Model and Classification Overview  
### Linear Measurement Model and Basic Definitions  
Network tomography falls in the category of _statistical inverse problems_. More precisely, network tomography problems can be approximately described by the linear model \(\mathbf{Y}_{t}=\mathbf{A}\mathbf{X}_{t}+\mathbf{\epsilon}\), where \(\mathbf{Y}_{t}\) is an _I-dimensional_ vector of measurements taken at a specific time \(t\) at various sites/nodes, \(\mathbf{A}\) is the \(I\times J\) routing matrix (with \(I\) rows corresponding to measurement paths and \(J\) columns corresponding to links of which the unknown parameters are of interest) that represents the topology of the network, \(\mathbf{X}_{t}\) is a _J-dimensional_ vector of time-dependent performance parameters like mean delays and logarithms of loss probabilities, and \(\mathbf{\epsilon}\) is a noise vector. In some cases, the random vector \(\mathbf{X}_{t}\) has an underlying parameterized distribution, and we are interested in those parameters. The routing matrix is usually binary (entries are 0 or 1, indicating the participation of a particular link--denoted by column index--in a routing/measurement path--denoted by row index), but its entries can also be probabilities in the case of multiple paths in a network due to load balancing considerations.  
Network inference refers to the inverse problem of estimating the statistics of the unobserved random vector \(\mathbf{X}_{t}\) given the linear measurements model and a set of assumptions regarding the statistical distribution of the noise \(\mathbf{\epsilon}\) or the introduction of some form or regularization to induce identifiability. From the perspective of linear algebra, the components of \(\mathbf{X}_{t}\) are uniquely identifiable if and only if the number of linear independent measurement paths equals the number of \(\mathbf{X}_{t}\)-components. The challenge lies in the fact that \(\mathbf{A}\) is usually an ill-posed matrix (i.e., the number of observations \(I\) is much smaller than the number of variables \(J\)) and, hence, non-invertible. Thus, the number of variables to be estimated is much larger than the number of equations, resulting in the non-uniqueness of solutions.  
Network tomography problems require the estimation of a possibly large number of instances of parameters such as link loss rates, delay distributions, and traffic intensity, which are spatially distributed. To face these estimation tasks, researchers nowadays abandon intricate and complex statistical models of network traffic and adopt simpler models that do not introduce significant estimation errors and reduce the complexity. The most common relevant assumptions made are the spatial independence of the link states (Assumption 1) and the stationary behavior of network traffic (Assumption 2).  
**Assumption 1** (Spatial Independence).: _The link states are independent from each other (but not identically distributed)._  
**Assumption 2** (Temporal Stationarity).: _The link states are stationary during the measurement period._  
Although the aforementioned simpler models are not suitable for studying network queuing dynamics and for making detailed analyses of the network behavior, they are sufficient for the inference of general performance statistics.