Reconstructing the macroscopic human cortical connectome by Diffusion Weighted Imaging (DWI) is a challenging research topic that has recently gained a lot of attention. In the present work, we investigate the effects of intra-voxel fiber direction modeling and tractography algorithm on derived structural network indices (e.g. density, small-worldness and global efficiency). The investigation is centered on three semi-independent distinctions within the large set of available diffusion models and tractography methods: i) single fiber direction versus multiple directions in the intra-voxel diffusion model, ii) deterministic versus probabilistic tractography and iii) local versus global measure-of-fit of the reconstructed fiber trajectories. The effect of algorithm and parameter choice has two components. First, there is the large effect of tractography algorithm and parameters on global network density, which is known to strongly affect graph indices. Second, and more importantly, there are remaining effects on graph indices which range in the tens of percent even when global density is controlled for. This is crucial for the sensitivity of any human structural network study and for the validity of study comparisons. We then investigate the effect of the choice of tractography algorithm on sensitivity and specificity of the resulting connections with a connectome dissection quality control (QC) approach. In this approach, evaluation of Tract Specific Density Coefficients (TSDCs) measures sensitivity while careful inspection of tractography path results assesses specificity. We use this to discuss interactions in the combined effects of these methods and implications for future studies.