  We revisit the topic of self-organized criticality (SOC) in simple
statistical graph models, with the purpose of capturing essential processes
leading to the emergence of macroscopic spacetime from the microscopic dynamics
in loop quantum gravity (LQG). We performed a large set of simulations based on
extensions of the frozen trivalent spin network (TSN) model explored previously
by Ansari and Smolin. Their model mimicked the sandpile dynamics by the
application of random vertex propagation rules in the TSN, leading to a SOC
behavior in the distribution of the avalanche sizes, as well as a slowly
expanding, $2$-dimensional dual (triangulated) space. Here we show that a
growth scheme for the stochastic, slow external driving force, differing from
the classical sandpile model, also resulted in power-law distributed avalanche
sizes. Our simulations also produced expanding dual spaces, with two basic
classes of evolution: one with power-law correlations in "space" and "time",
and the other with "loitering" and exponential phases. Our work expands the
range of models in which critical states in the TSN may lead to expansion
effects in the dual space, without fine-tuning.
