A cognitive network refers to the one where two overlaid structures, called primary and secondary networks coexist. The primary network consists of primary nodes who are licensed spectrum users while the secondary network comprises unauthorized users that have to access the licensed spectrum opportunistically. In this paper, we study the percolation degree of the secondary network to achieve k-percolation in large scale cognitive radio networks. The percolation degree is defined as the number

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... nearest neighbors for each secondary user when there are at least k vertex-disjoint paths existing between any two secondary relays in the percolated cluster. The percolated cluster is formed when there are an infinite number of mutually connected secondary users spanning the whole network. Each user in the cluster is possibly connected to several neighbors, inducing more communication links between any two of them. Since nodes located near the boundary have fewer neighbors, the boundary effect becomes a bottleneck in determining the percolation degree. For cognitive networks, when the primary node density becomes considerably large, the boundary effect spreads inside the network. The transmission area of most secondary users who are located near the primary nodes decreases due to the restriction of the primary network. Therefore, to ensure kconnectivity in the percolated cluster, each secondary user must be connected to more neighbors, and the percolation degree of the secondary network yields a function of the primary node density. We specify the relationship into three regimes regarding the topology variation of the cognitive network. A closed-form expression of the percolation degree under different primary node densities is presented. The expression characterizes the connectivity strength in the secondary percolated cluster, therefore providing analytical insight on fault tolerance improvement in cognitive networks.