This work develops a set of mathematical tools that map the topology of an economic network to a probability distribution of possible outcomes for the economy. We can apply these tools to analyze complex economic systems in closed form and to construct error bounds about the paths of aggregated networked economies. To generate this mapping from network topology to probability distribution, we focus on a class of economies that has the following three features: (1) a population of N agents, each with a binary-valued attribute, (2) a network on which these N agents are organized, and (3) decision-making by each networked agent that depends on the local relative frequency of the attribute. This class of economies also has an aggregate feature: the global relative frequency of the attribute. Given the system's aggregate feature, underlying network, and population size, we construct in closed form the distribution of possible local relative frequencies of the attribute. The topology of the network determines the extent to which the local relative frequency of the attribute can deviate from its global relative frequency, thereby determining the extent to which the outcome of the economy can deviate from a benchmark outcome. Given this distribution and agents' decision-making behavior, we then construct the distribution of possible outcomes for the economy. For realistic agent interaction structures featuring a very large population of agents, the distribution of outcomes is meaningfully non-degenerate. We adapt the theoretical framework and mathematical tools developed in this work to study locally formed macroeconomic sentiment and how agents' interaction structure shapes the capacity for there to exist non-fundamental swings in aggregate sentiment, with implications for election outcomes and for our understanding of animal spirits.