# Statistical Properties of SIS Processes with Heterogeneous Nodal Recovery Rates in Networks
## 3 Methodology
### Mean Value and Variance of Steady-State Infection Fraction  
In this work, we confined ourselves to the scenario demonstrated in Figure 2, where the heterogeneous SIS process spreads in the network with the topology of a complete graph. A complete graph \(K_{N}\) of size \(N\) is defined as a graph where there exists a link between any nodal pair. Specifically, the infection rate of each node is identical, such that \(\beta_{i}=\beta\) holds for any \(i\). The nodal recovery rates are different, meaning that \(\delta_{i}\
eq\delta_{j}\) if \(i\
eq j\). We divided the node set of the whole network into two groups, represented by \(G_{a}\) and \(G_{b}\). The recovery rates of the nodes in groups \(G_{a}\) and \(G_{b}\) are denoted by \(\delta_{a}\) and \(\delta_{b}\), respectively. In other words, the infection rate as well as the recovery rate of any node \(i\) is either the setup \(C_{a}=(\beta,\delta_{a})\) or \(C_{b}=(\beta,\delta_{b})\). This scenario was previously proposed to model a flooding attack in an NDN network [15]. For the considered scenario in this work, we first prove that one node is the same as the other nodes with respect to the expectation of the steady-state nodal probability of being in the infected state. The proof is depicted as Lemma 1. **Lemma 1**. : _All nodes in a complete graph are divided into two groups, \(G_{a}\) and \(G_{b}\), with the same infection rate but heterogeneous recovery rates. A node \(i\) is configured either with the parameters \(C_{a}=(\beta,\delta_{a})\) or \(C_{b}=(\beta,\delta_{b})\). The steady-state infection probability \(v_{i}\) of node \(i\) of group \(G_{a}\) is equal to the probability for any other node \(j\) of the same group \(G_{a}\)._  
Proof of Lemma 1.: A proof by contradiction is used, which is detailed in Appendix A. Regarding the steady-state infection probability \(v_{i}\) as a function of the nodal recovery rate, we denote it by \(v_{i}:=v_{i}(\delta_{i})\). Before proceeding, we introduce \(N_{a}\) and \(N_{b}\) to represent the numbers of nodes in \(G_{a}\) and \(G_{b}\), respectively. Figure 2: Schematic diagram of the scenario of interest in this work. The topology of the network of interest is a complete graph. Any node is configured with either the parameters \(C_{a}=(\beta,\delta_{a})\) or with \(C_{b}=(\beta,\delta_{b})\), as shown by the circles with horizontal lines and the circles with dots, respectively. **Lemma 2**. : _For the considered scenario shown in Figure 2, the steady-state infection probability \(v_{i}\) of group \(G_{a}\) is not equal to the probability for any other node \(j\) of the same group \(G_{b}\). Specifically, we denote by \(v_{a}=v_{i}(\delta_{a})\) and \(v_{b}=v_{i}(\delta_{b})\) the probabilities for the groups \(G_{a}\) and \(G_{b}\), respectively. Then, we arrive at \(v_{a}\leq v_{b}\) if \(\delta_{a}\geq\delta_{b}\)._  
Proof of Lemma 2.: Using Lemma 1, one can derive this lemma. The details of the proof are shown in Appendix B. **Lemma 3**. : _We denote \(y_{a}=v_{i\infty}(\delta_{a})\), \(y_{b}=v_{i\infty}(\delta_{b})\), \(N_{a}=pN\), and \(N_{b}=(1-p)N\). For the scenario of interest as shown in Figure 2, the steady-state expectation of the fraction \(y_{\infty}\) of infected nodes is_  
\[y_{\infty}=py_{a}+(1-p)y_{b}, \tag{14}\]  
_where \(y_{a}\) and \(y_{b}\) can be derived from the equations_  
\[\beta(N_{a}-1)y_{a}^{2}+(\delta_{a}-\beta(N_{a}-1)+\beta N_{b}y_{ b})y_{a}-\beta N_{b}y_{b}=0 \tag{15}\] \[\beta(N_{b}-1)y_{b}^{2}+(\delta_{b}-\beta(N_{b}-1)+\beta N_{a}y_ {a})y_{b}-\beta N_{a}y_{a}=0 \tag{16}\]  
Proof of Lemma 3.: Using Lemma 1, one can derive this lemma. The details of the proof are shown in Appendix C. A similar brief proof was previously provided ([15], (24-26) on p. 1867). Furthermore, we can obtain the expression of the steady-state variance of the number of infected nodes as follows. **Proposition 1**.