The tsNET algorithm utilizes t-SNE to compute high-quality graph drawings, preserving the neighborhood and clustering structure. We present three fast algorithms for reducing the time complexity of tsNET algorithm from O(nm) time to O(n log n) time and O(n) time. To reduce the runtime of tsNET, there are three components that need to be reduced: (C0) computation of high-dimensional probabilities, (C1) computation of KL divergence gradient, and (C2) entropy computation. Specifically, we reduce the overall runtime of tsNET, integrating our new fast approaches for C0 and C2 with fast t-SNE algorithms for C1. We first present O(n log n)-time BH-tsNET, based on (C0) new O(n)-time partial BFS-based high-dimensional probability computation and (C2) new O(n log n)-time quadtree-based entropy computation, integrated with (C1) O(n log n)-time quadtree-based KL divergence computation of BH-SNE. We next present faster O(n log n)-time FIt-tsNET, using (C0) O(n)-time partial BFS-based high-dimensional probability computation and (C2) quadtree-based O(n log n)-time entropy computation, integrated with (C1) O(n)-time interpolation-based KL divergence computation of FIt-SNE. Finally, we present the O(n)-time L-tsNET, integrating (C2) new O(n)-time FFT-accelerated interpolation-based entropy computation with (C0) O(n)-time partial BFS-based high-dimensional probability computation, and (C1) O(n)-time interpolation-based KL divergence computation of FIt-SNE. Extensive experiments using benchmark data sets confirm that BH-tsNET, FIt-tsNET, and L-tsNET outperform tsNET, running 93.5%, 96%, and 98.6% faster while computing similar quality drawings in terms of quality metrics (neighborhood preservation, stress, edge crossing, and shape-based metrics) and visual comparison. We also present a comparison between our algorithms and DRGraph, another dimension reduction-based graph drawing algorithm.