# Statistical Properties of SIS Processes with Heterogeneous Nodal Recovery Rates in Networks
## 3 Methodology  
### Notation and Model  
We first introduce notation before describing the general HNIMFA model. The dynamics of interest are those of the SIS process with heterogeneous nodal infection and recovery rates in a network represented by the undirected graph \(G(N,L)\), composed of \(L\) links and \(N\) nodes. The topology of the network (graph) is denoted by a symmetric \(N\times N\) adjacency matrix \(A:=\{a_{ij}|0,1\}\), where \(a_{ij}\) is equal to \(1\) if there is a link between nodes \(i\) and \(j\); and \(0\) otherwise. We denote by \(d_{i}\) the degree of the node \(i\), such that \(d_{i}=\sum_{j=1}^{N}a_{ij}=\sum_{i=1}^{N}a_{ij}\) for the undirected graph. Regarding the SIS process in a network as a system, the state of the system can be specified by the viral state of each node and the topology. At any time, the viral state of a node can only be one of two states: infected or susceptible. In order to specify the viral state of a node \(i\), the Bernoulli random variable \(X_{i}(t)\in\{0,1\}\) is introduced. Specifically, the random variable \(X_{i}(t)=1\) means node \(i\) is in the infection state at time \(t\). The random variable \(X_{i}(t)=0\) means node \(i\) is in the susceptible state at time \(t\). The probability of node \(i\) being in the infected or susceptible state is represented by \(v_{i}(t)=\Pr[X_{i}(t)=1]\)or \(1-v_{i}(t)=\Pr[X_{i}(t)]=0\), respectively. Using the properties of the Bernoulli random variable provides the expectation and variance of the nodal viral state \(X_{i}\), which are  
\[E[X_{i}] =1\cdot\Pr[X_{i}=1]+0\cdot\Pr[X_{i}=0]=v_{i}, \tag{1}\] \[\Var[X_{i}] =(1-v_{i})v_{i}. \tag{2}\]  
Figure 1 provides instances of two Poisson processes constituting the SIS process in a network of a pairwise type. Any process between a nodal pair is independent of others. We consider a scenario where node \(i\) in the infected state is connected with node \(j\) in the susceptible state, as shown in Figure 1a. There exists an infection process between node \(i\) and node \(j\), where the infected node \(i\) tries to infect the susceptible node \(j\). This infection is a Poisson process with a rate of \(\beta_{i}\), such that node \(j\) will probably become infected soon, changing its state to susceptible. On the other hand, any infected node will recover and change to a susceptible state, which is also characterized by a Poisson process, as shown in Figure 1b. Node \(i\) in the infected state will recover following a Poisson process at a rate of \(\delta_{i}\). All the infection and recovery processes simultaneously existing in a finite-size network of interest constitute a heterogeneous SIS process. Additionally, we introduce the notation \(\tau_{i}=\beta_{i}/\delta_{i}\), which is named the effective infection rate and used to simplify the differential equations of the dynamics in the following. The general HNIMFA model is elaborated in the following. One can depict the SIS process with heterogeneous infection and recovery rates in a network using a Markov chain. The exact Markov differential equation for the viral state of node \(i\) is expressed as follows:  
\[\frac{dE[X_{i}(t)]}{dt}=E\left[-\delta_{i}X_{i}+\sum_{j=1}^{N}a_{ij}\beta_{j}X _{j}(1-X_{i})\right]. \tag{3}\]  
The first-order NIMFA model [16] employs the assumption of independence approximation:  
\[\Pr[X_{i}=1,X_{j}=1]=\Pr[X_{i}=1]\cdot\Pr[X_{j}=1]. \tag{4}\]  
In other words, the approximation means that the random variables \(X_{i}\) and \(X_{j}\) are assumed to be independent. Using the property of the Bernoulli random variable (1) and assuming independence, we arrive at  
\[E[X_{i}=1,X_{j}=1]=E[X_{i}=1]\cdot E[X_{j}=1]=v_{i}v_{j}. \tag{5}\]  
Figure 1: Demonstration of Poisson processes constituting the SIS process in a pairwise network. (**a**) The infection process between nodes \(i\) and \(j\), depicting the infected node \(i\) infecting its neighbor \(j\), which is in a susceptible state, at a rate of \(\beta_{i}\).