# Variance of the Infection Number of Heterogeneous Malware Spread in Network
## 3 Methodology
### Variance of Infection Fraction for Bipartite Graph  
The main topologies concerned are the complete bipartite graph and the star graph. A complete bipartite graph \(K_{m,n}\) is composed of two groups of nodes, where any nodal pair in one group does not connect and a node in one group is fully connected to all nodes in another group. As shown in Figure 2a, the nodes on the left side belong to a group of \(m=2\) nodes, while the nodes on the right side are of another group of \(n=3\) nodes. As shown in Figure 2b, a star graph of \(n+1\) is a tree structure composed of one special node of degree \(n\) and the other \(n\) nodes of degree \(1\). The special node, referred to as the center or the root, is connected to each of the other nodes. A star graph is isomorphic to a complete bipartite graph \(K_{m,n}\) with \(m=1\). **Lemma 1**. : _The metastable-state average number \(y\) of infected nodes for the heterogeneous SIS process in a complete bipartite network \(K_{m,n}\) with the setting \((\beta_{i}=\beta_{m},\delta_{i}=\delta_{m})\) for \(i=1,\cdots,m\) and \((\beta_{i}=\beta_{n},\delta_{i}=\delta_{n})\) for \(i=m+1,\cdots,m+n\), can be expressed as_  
\[E[I(t)]=m\omega_{m}+n\omega_{n}, \tag{12}\]  
_where_  
\[\omega_{m} =\frac{\beta_{m}\beta_{n}mn-\delta_{m}\delta_{n}}{(\delta_{m}+ \beta_{n}n)\beta_{m}m}, \tag{13}\] \[\omega_{n} =\frac{\beta_{m}\beta_{n}mn-\delta_{m}\delta_{n}}{(\delta_{n}+ \beta_{m}m)\beta_{n}n}. \tag{14}\]  
Proof of Lemma 1.: The expression for the metastable-state fraction \(y\) of the number of infected nodes was previously proved and shown as Theorem 1 in [7]. Substituting \(N=m+n\) and using \(E[I(t)]=y\cdot N\) leads to the expression for \(E[I(t)]\). **Proposition 1**. : _The variance \(\mathrm{Var}[I]\) of the metastable-state average number of infected nodes for the heterogeneous SIS process in a complete bipartite network \(K_{m,n}\) with the setting \((\beta_{i}=\beta_{m},\delta_{i}=\delta_{m})\) for \(i=1,\cdots,m\) and \((\beta_{i}=\beta_{n},\delta_{i}=\delta_{n})\) for \(i=m+1,\cdots,m+n\), can be expressed as_  
\[\mathrm{Var}[I]=m(1-\omega_{m})\omega_{m}+n(1-\omega_{n})\omega_{n}, \tag{15}\]  
_where \(\omega_{n}\) and \(\omega_{n}\) are shown in (13) and (14)_  
Proof of Proposition 1.: The nodes in a complete bipartite graph are divided into two sets. Within each set, the nodes are identical for the configuration concerned, which was proven in [7]. For the node \(i\) of the complete bipartite graph with \(i=1,\cdots,m\), the metastable-state  
Figure 2: Topology of networks concerned in this work. Two sorts of networks are chosen to investigate the heterogeneous SIS epidemic dynamics: (**a**) a complete bipartite graph \(K_{m,n}\) with two disjoint node sets, \(m=2\) and \(n=3\). (**b**) A star graph. fraction \(v_{i}\) of infection is identical, e.g., \(v_{i}=\omega_{m}\) as shown in (13). Similarly, \(v_{i}=\omega_{n}\) for \(i=m+1,\cdots,m+n\). Substituting \(\omega_{m}\) in (13) and \(\omega_{n}\) in (14) into (11) leads to  
\[ \begin{split}\text{Var}[I(t)]&=\sum_{i=1}^{m}(1- \omega_{i})\omega_{i}+\sum_{j=m+1}^{m+n}(1-v_{j})v_{j}\\ &=m(1-v_{m})v_{m}+n(1-\omega_{n})\omega_{n}.\end{split} \]  
A complete bipartite graph \(K_{m,n}\) is isomorphic to a star graph with the setting \(m=1\). A star graph of size \(1+n\) is a tree with one rooted node as the center, having degree \(n\) and the other having degree \(1\) as siblings connecting to the center. **Corollary 1**. : _The variance \(\text{Var}[I]\) of the metastable-state average number of infected nodes for the heterogeneous SIS process in a star graph of \(n+1\) nodes with the setting \((\beta_{i}=\beta_{m},\delta_{i}=\delta_{m})\) for \(i=1\) and \((\beta_{i}=\beta_{n},\delta_{i}=\delta_{n})\) for \(i=2,\cdots,n+1\) can be expressed as_  
\[