# Research

This work develops a set of mathematical tools that allows us to map the topology of an economic network to a probability distribution of possible outcomes for the economy. 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. We begin by constructing in closed form the distribution of possible local relative frequencies of the attribute given the topology of the underlying network and the attribute's global relative frequency. The topology of the underlying 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. Then, given this distribution and agents’ decision-making behavior, we 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 macroeconomic sentiment, with implications for our understanding of animal spirits. We can moreover apply these tools to analyze complex economic systems in closed form and to construct error bounds about the paths of aggregated networked economies.

We have a collection of *N* agents and an outside entity that is interested in the population's aggregate action. The outside entity would like to enact a policy with the intention of increasing the aggregate action; for example, if the outside entity were a national government, it might be interested in enacting a policy that increases aggregate output and stimulates economic growth. The policy of the outside entity allocates e > 0 units of additional wealth to *n* *≤* *N* agents, funded either by internal transfer or by an external source. Our outside entity would like to know the exact effect of its planned policy on the aggregate action, and it would like to know the corresponding economic multiplier, that is, the change in the aggregate action from the e shock. Now, even though the setting is simple, the effect is complicated: the population of agents is networked; agents' actions are interdependent, so depending on which subset of agents actually receives the positive shock, the change in the aggregate action and the corresponding economic multiplier can both widely differ. In this work, we consider three broad settings with network-based interaction: (1) networked environments with strategic complements and substitutes, (2) networked environments with coordination and anti-coordination, and (3) networked environments with production. We show how there is an entire distribution of possible aggregate actions and economic multipliers associated with a particular policy: given *n*, for each environment, we map the topology of agents' interaction network to the distribution of possible resulting aggregate actions and economic multipliers. The mathematics is the same across all three environments. We can compute the features of these distributions in closed-form, including the maximum and minimum possible aggregate actions and economic multipliers for a particular network topology. We can also rank networks so that the outside entity's policy is more effective the higher ranked the network. We show how non-trivial network topologies generate negative multipliers. Across all three settings, there is a non-zero probability that the enacted policy will reduce the aggregate action below its no-intervention level, and we can compute this probability in closed form.