## 4 Methodology
### Problem Statement, System Model, and Correlated Failures Model  
As mentioned earlier, the goal of this paper is to devise an optimal multiculture network capable of improving the network resilience when correlated failures impair simultaneously several nodes. The communication network topology is mathematically represented by the undirected graph \(G=(V,E)\), where \(V=\{1,2,\ldots,n\}\) is the set of communication nodes and \(E=\{(u,v):\text{nodes }u\text{ and }v\text{ are connected}\}\) is the set of communication links between nodes. Suppose that a network architect must carry out a technological update of network nodes. To do so, a multiculture set of nodes may be used to replace the existing ones, thereby different technologies, i.e., vendors, OSs, and protocol stack implementations, can be introduced in the topology. We denote here by \(K=\{1,2,\ldots,N\}\) the set of \(N\) different networking node technologies available to the architect. We assume also that the \(N\) available technologies implement a joint protocol stack with a set of \(L\) different communication protocols, where \(Y_{il}=1\) (correspondingly, \(Y_{il}=0\)) denotes that a node, equipped with technology \(i\), is able to (correspondingly, incapable of) communicate to other nodes through protocol \(l\). Next, let us assume that the set \(K^{\prime}=\{1,2,\ldots,\kappa\}\) represents the technologies to be implemented in the network devices during the upgrade. (This set is defined by solving the problem in Section 4.2.1). We are interested here in modeling correlated failures triggered by breakdowns or attacks which diminish the connectivity of the infrastructure by impairing several nodes at the same time and for a long period. Thus, we assume here both that each regarded technology is prone to fail or to be attacked without recovery and that they share common risks. **Definition 3**. : A SRNG in a data communication network is a set of nodes that may be affected by a common failure to the infrastructure under the condition that they share a common failure risk. Suppose now that there exists a set \(\mathcal{A}=\{A_{1},A_{2},\ldots,A_{M}\}\) of \(M\) different SRNG events that may induce correlated failures to the network. Furthermore, assume that each event \(A_{r}\in\mathcal{A}\) has a probability of occurrence of \(P_{r}\). Consequently, when the SRNG event \(A\) happens, the set of nodes \(V\) can be partitioned into two sets: \(V_{A_{r}}\) and \(V_{A_{r}}^{c}\), where the former set denotes the collection of all networking nodes sharing the common risk associated to the SRNG event \(A_{r}\) and the latter set denotes all those nodes unaffected by the event. **Definition 4**. : A Probabilistic Shared Risk Node Group (PSRNG) in a data communication network is a set of nodes that fail with a positive failure probability, in the event of a SRNG failure. More precisely, the failure probability of the \(i\)th node, conditional on the SRNG failure event \(A_{r}\), is denoted as \(P_{i,r}\) and satisfies: \(P_{i,r}>0\) for all \(i\in V_{A}\) and \(P_{i,r}=0\) otherwise. **Definition 5**. : We say that the nodes \(i\) and \(j\) belonging to a data communication network are correlated if \(P_{i,r}\) and \(P_{j,r}\) are both positive for the \(A_{r}\) PSRNG. Moreover, upon the occurrence of the \(A_{r}\) SRNG event, these probabilities are identical and mutually exclusive for all the pairs of nodes in \(V_{A_{r}}\), that is: \(P_{i,r}=P_{j,r}=P_{r}\) for all \(i,j\). Following [2], we assume that only one PSRNG event may occur at a time. This means that the \(M\) shared risks defining the PSRNG events are mutually exclusive; therefore: \(\sum_{r=1}^{M}P_{r}=1\). This otherwise arbitrary definition has been effectively used in the networking community and makes sense in the context of the class of failures considered in this paper [2; 3; 4]. We note also that from Definitions 1 to 5, both monoculture and multiculture data network technologies can be affected by more than one SRNG event. **Definition 6**.