  This paper studies a general class of stochastic population processes in
which agents interact with one another over a network. Agents update their
behaviors in a random and decentralized manner according to a policy that
depends only on the agent's current state and an estimate of the macroscopic
population state, given by a weighted average of the neighboring states. When
the number of agents is large and the network is a complete graph (has
all-to-all information access), the macroscopic behavior of the population can
be well-approximated by a set of deterministic differential equations called a
{\it mean-field approximation}. For incomplete networks such characterizations
remained previously unclear, i.e., in general whether a suitable mean-field
approximation exists for the macroscopic behavior of the population. The paper
addresses this gap by establishing a generic theory describing when various
mean-field approximations are accurate for \emph{arbitrary} interaction
structures.
  Our results are threefold. Letting $W$ be the matrix describing agent
interactions, we first show that a simple mean-field approximation that
incorrectly assumes a homogeneous interaction structure is accurate provided
$W$ has a large spectral gap. Second, we show that a more complex mean-field
approximation which takes into account agent interactions is accurate as long
as the Frobenius norm of $W$ is small. Finally, we compare the predictions of
the two mean-field approximations through simulations, highlighting cases where
using mean-field approximations that assume a homogeneous interaction structure
can lead to inaccurate qualitative and quantitative predictions.
