Weighted (Co)homology and Weighted Laplacian

In this paper, we generalize the combinatorial Laplace operator of Horak and Jost by introducing the $\phi$-weighted coboundary operator induced by a weight function $\phi$. Our weight function $\phi$ is a generalization of Dawson's weighted boundary map. We show that our above-mentioned generalizations include new cases that are not covered by previous literature. Our definition of weighted Laplacian for weighted simplicial complexes is also applicable to weighted/unweighted graphs and digraphs.Comment: 22 page

Multi-Pair Two-Way Relay Network with Harvest-Then-Transmit Users: Resolving Pairwise Uplink-Downlink Coupling

While two-way relaying is a promising way to enhance the spectral efficiency of wireless networks, the imbalance of relay-user distances may lead to excessive wireless power at the nearby-users. To exploit the excessive power, the recently proposed harvest-then-transmit technique can be applied. However, it is well-known that harvest-then-transmit introduces uplink-downlink coupling for a user. Together with the co-dependent relationship between paired users and interference among multiple user pairs, wirelessly powered two-way relay network suffers from the unique pairwise uplink-downlink coupling, and the joint uplink-downlink network design is nontrivial. To this end, for the one pair users case, we show that a global optimal solution can be obtained. For the general case of multi-pair users, based on the rank-constrained difference of convex program, a convergence guaranteed iterative algorithm with an efficient initialization is proposed. Furthermore, a lower bound to the performance of the optimal solution is derived by introducing virtual receivers at relay. Numerical results on total transmit power show that the proposed algorithm achieves a transmit power value close to the lower bound