Rumor Dynamics and Inoculation of Nodes in Weighted Scale Free Networks with Degree-Degree Correlation

The degree-degree correlation has considerable effect in the real world networks. In the classic rumor spreading model also known as susceptible-infected-refractory (SIR) model, all the edges have uniform tie strength between the nodes. Therefore, for the real world complex networks, SIR model need to be modified to consider the dependence of rumor spread rate on the weight based on the tie strength between the nodes. Main objective of this work is to study the effect of degree-degree correlation on the rumor spreading and on the effectiveness of inoculation strategies in the real world networks. The weight of edges, node strength and degree degree correlation affect the threshold for rumor spreading. The weight of an edge between two nodes is proportional to (k_ik_j)^β, where k_i and k_j are degrees of node i and j. By using the exponent β in any degree-degree correlated scale free network, rumor threshold turns out to be some finite value. It was not observed in the previous models. Inoculation of a certain fraction of nodes against rumors is required to control the rumor spread. The inoculation of nodes is done randomly or in targeted fashion. It has been observed that rumor threshold in random inoculation scheme is greater than the rumor threshold in the model without any inoculation scheme. In targeted inoculation scheme, rumor threshold for correlated networks is found to be larger than the threshold in uncorrelated networks. But random inoculation is not much effective. It is found that degree-degree correlation in real world networks does not play any role in increasing the effectiveness of random inoculation. However, it is not true for targeted inoculation. The proposed hypothesis has also been verified by simulation results.