The further back we look in time, the less we know about the course of life's history. Genomics, through phylogenomics, will eventually resolve the evolution of macroscopic life, whose phylogeny can be properly modeled in the mathematical image of a bifurcating tree; where the evolutionary process is fundamentally tree-like in nature, we only have to collect enough data to bring the structure of the tree into focus. But when we look back into the evolution of microscopic life and early evolution that is, prokaryotic evolution, the prokaryote-to-eukaryote transition, and the origin of life genome sequences are only of limited help. That is because neither the evolutionary process linking the evolution of genes across prokaryotic genomes nor the process linking prokaryotes to eukaryotes is strictly tree-like in nature. In prokaryote genome evolution, lateral gene transfer (LGT) is an important mechanism of natural variation, while the prokaryote-to-eukaryote transition involved the wholesale merger of prokaryotic genomes via endosymbiosis. This proposal aims to deliver a quantum advance in our understanding of early evolution. Prokaryotic genome evolution and the prokaryote-to-eukaryote transition will be investigated with mathematical tools that better approximate the process as it occurs in nature, by using the graph theoretical tools of networks rather than that of trees. For understanding the origin of life, genome data is inapplicable, because genes cannot be compared to inorganic compounds from which life ultimately arose. When it comes to linking microbial life to geochemical processes, the comparison of chemical reaction sequences in living things to those geochemistry is all with which we have to work. Some forms of hydrothermal vents harbour newly discovered chemical reaction sequences with striking overall similarity to that used by methanogens and acetogens, findings that bear upon the nature of the deepest evolutionary divide among modern microbes.