  We study a Markovian agent-based model (MABM) in this paper. Each agent is
endowed with a local state that changes over time as the agent interacts with
its neighbours. The neighbourhood structure is given by a graph. In a recent
paper [Simon et al. 2011], the authors used the automorphisms of the underlying
graph to generate a lumpable partition of the joint state space ensuring
Markovianness of the lumped process for binary dynamics. However, many large
random graphs tend to become asymmetric rendering the automorphism-based
lumping approach ineffective as a tool of model reduction. In order to mitigate
this problem, we propose a lumping method based on a notion of local symmetry,
which compares only local neighbourhoods of vertices. Since local symmetry only
ensures approximate lumpability, we quantify the approximation error by means
of Kullback-Leibler divergence rate between the original Markov chain and a
lifted Markov chain. We prove the approximation error decreases monotonically.
The connections to fibrations of graphs are also discussed.
