  We investigate the emergence of different effective geometries in stochastic
Clifford circuits with sparse coupling. By changing the probability
distribution for choosing two-site gates as a function of distance, we generate
sparse interactions that either decay or grow with distance as a function of a
single tunable parameter. Tuning this parameter reveals three distinct regimes
of geometry for the spreading of correlations and growth of entanglement in the
system. We observe linear geometry for short-range interactions, treelike
geometry on a sparse coupling graph for long-range interactions, and an
intermediate fast scrambling regime at the crossover point between the linear
and treelike geometries. This transition in geometry is revealed in
calculations of the subsystem entanglement entropy and tripartite mutual
information. We also study emergent lightcones that govern these effective
geometries by teleporting a single qubit of information from an input qubit to
an output qubit. These tools help to analyze distinct geometries arising in
dynamics and correlation spreading in quantum many-body systems.
