Probabilistic Flooding Performance Analysis Exploiting Graph Spectra Properties

Abstract

Probabilistic flooding is an efficient information dissemination policy capable of spreading information to the network nodes by sending <i>information messages</i> according to a fixed <i>forwarding probability</i> in a per-hop manner starting from an <i>initiator</i> node. It is a suitable approach, especially in topologies where the number of information messages sent under traditional approaches is significantly increased. The analysis presented in this paper considers graph spectra properties such as the <i>largest eigenvalue</i> <inline-formula> <tex-math notation="LaTeX">$\lambda_1$</tex-math> </inline-formula> of the adjacency matrix, and the <i>eigenvector centrality</i>. Both are analytically investigated and <inline-formula> <tex-math notation="LaTeX">$\frac{4}{\lambda_1}$</tex-math> </inline-formula> is derived as a <i>lower bound</i> of the forwarding probability that allows for <i>global coverage</i>, i.e., all network nodes receive the information message, under certain conditions also investigated here (e.g., the condition of the binomial approximation). It is shown that for any value of the forwarding probability equal to or larger than <inline-formula> <tex-math notation="LaTeX">$\frac{4}{\lambda_1}$</tex-math> </inline-formula>: (i) coverage is proportional to the initiator node&#x2019;s eigenvector centrality; (ii) the probability a node receives the information message is proportional to the node&#x2019;s eigenvector centrality; (iii) <i>termination time</i> decreases as the initiator node&#x2019;s eigenvector centrality increases. If knowledge of <inline-formula> <tex-math notation="LaTeX">$\lambda_1$</tex-math> </inline-formula> is not available, then the <i>average node degree</i> <inline-formula> <tex-math notation="LaTeX">$\bar{d}$</tex-math> </inline-formula> can be used for ensuring global coverage. If knowledge of both <inline-formula> <tex-math notation="LaTeX">$\lambda_1$</tex-math> </inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\bar{d}$</tex-math> </inline-formula> is not available, a dissemination policy is proposed that forwards messages to <inline-formula> <tex-math notation="LaTeX">$m$</tex-math> </inline-formula> (randomly selected) neighbor nodes. It is analytically shown that any value of <inline-formula> <tex-math notation="LaTeX">$m \geq 4$</tex-math> </inline-formula> allows for global coverage. Simulation results demonstrate the effectiveness of the considered analytical approach and the introduced policy.