The means of electrifying households and the resulting electricity networks are rapidly evolving. Traditionally, an extension of existing centralized grids was the only prominent technique, but now electrification is seeing massive expansion via decentralized solar home systems (SHSs). These systems consist of a low-wattage photovoltaic (PV) panel (typically 5-100W), a battery, a collection of energy-efficient DC appliances, and a charge controller. Spurred by significant advances and reduced costs in solar, batteries, energy-efficient appliances, and mobile money-driven business models, SHSs have proliferated rapidly, with tens of millions of systems now deployed, primarily in regions with otherwise low rates of electricity access. In this work, we profile a large deployment of solar home systems in Western Kenya to ascertain the dominant generation and consumption patterns. We note that there are often substantial mismatches between generation and consumption, and that PV overgeneration presents an opportunity via networking of households. We consider the opportunity to leverage system interconnection to enable increased connectivity among households, challenging typical electricity system architecture by effectively creating ad hoc electricity grids at the edges of the overall electricity network. Further, we consider the potential to integrate households without SHSs (passive nodes) into these electricity networks, as a low-cost opportunity to increase electrification rates. Considering energy curtailment, the spatial distribution of households, and infrastructure costs, we build a decision problem for interconnecting existing SHSs with passive nodes. Our analysis shows that compared to the all-SHS solutions that are presently achieving widespread deployment, we show that interconnecting existing SHSs can increase electrification rates by more than 25% and reduce average costs by up to 30% per household.