Understanding Blackholes in large-scale Cognitive Radio Networks under generic failures

It has been demonstrated that in wireless networks, Blackholes, which are typically generated by isolated node failures, and augmented by failure correlations, can easily result in devastating impact on network performance. Therefore, many solutions, such as routing protocols and restoration algorithms, are proposed to deal with Blackholes by identifying alternative paths to bypass these holes such that the effect of Blackholes can be mitigated. These advancements are based on an underlying premise that there exists at least one alternative path in the network. However, such a hypothesis remains an open question. In other words, we do not know whether the network is resilient to Blackholes or whether an alternative path exists. The answer to this question can complement our understanding of designing routing protocols, as well as topology evolution in the presence of random failures. In order to address this issue, we focus on the topology of Cognitive Radio Networks (CRNs) because of their phenomenal benefits in improving spectrum efficiency through opportunistic communications. Particularly, we first define two metrics, namely the failure occurrence probability p and failure connection function g(·), to characterize node failures and their spreading properties, respectively. Then we prove that each Blackhole is exponentially bounded based on percolation theory. By mapping failure spreading using a branching process, we further derive an upper bound on the expected size of Blackholes. With the observations from our analysis, we are able to find a sufficient condition for a resilient CRN in the presence of Blackholes through analysis and simulations.