Laplacian spectrum gives a lot of useful information about complex structural properties and relevant dynamical aspects, which has attracted the attention of mathematicians. We introduced the weighted scale-free network inspired by the binary scale-free network. First, the weighted scale-free network with a weight factor is constructed by an iterative way. In the next step, we use the definition of eigenvalue and eigenvector to obtain the recursive relationship of its eigenvalues and multiplicities at two successive generations. Through analysis of eigenvalues of transition weight matrix we find that multiplicities of eigenvalues 0 of transition matrix are different for the binary scale-free network and the weighted scale-free network. Then, we obtain the eigenvalues for the normalized Laplacian matrix of the weighted scale-free network by using the obtained eigenvalues of transition weight matrix. Finally, we show some applications of the Laplacian spectrum in calculating eigentime identity and Kirchhoff index. The leading term of these indexes are completely different for the binary and the weighted scale-free network.
The eigentime identity for random walks on the weighted networks is the expected time for a walker going from a node to another node. Eigentime identity can be studied by the sum of reciprocals of all nonzero Laplacian eigenvalues on the weighted networks. In this paper, we study the weighted [Formula: see text]-flower networks with the weight factor [Formula: see text]. We divide the set of the nonzero Laplacian eigenvalues into three subsets according to the obtained characteristic polynomial. Then we obtain the analytic expression of the eigentime identity [Formula: see text] of the weighted [Formula: see text]-flower networks by using the characteristic polynomial of Laplacian and recurrent structure of Markov spectrum. We take [Formula: see text], [Formula: see text] as example, and show that the leading term of the eigentime identity on the weighted [Formula: see text]-flower networks obey superlinearly, linearly with the network size.
In a traditional electronic auction, the centralized auctioneer and decentralized bidders are in an asymmetric structure, where the auctioneer has more ability to decide the auction result. This asymmetric auction structure is not fair to the participants and not suitable for data auctions in the Internet of Things (IoT). The blockchain-based auction system, with participant equality and fairness, is typically symmetrical and particularly suitable for IoT data sharing. However, when applied to IoT data sharing in reality, it faces privacy and efficiency problems. In this context, how to guarantee privacy and break the inherent performance bottleneck of blockchain is still a major challenge. In this paper, a consensus-based distributed auction scheme is proposed for data sharing, which enforces privacy preservation and collusion resistance. A reverse auction-based decentralized data trading model is introduced to solve the trust problem without a centralized auctioneer, where bidders reach consensus on the auction result. Specifically, we devise a differentially private auction mechanism to incentivize data owners to participate in data sharing. An effective hybrid consensus algorithm is constructed among bidders to reach consensus on the auction result with improved security and efficiency. Theoretical analysis shows that the proposed scheme ensures the properties of privacy preservation, incentive compatibility and collusion resistance. Experimental results reveal that the proposed mechanism guarantees the data sharing efficiency and has certain scalability.
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