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Preferential attachment networks are a type of random network where new nodesare connected to existing ones at random, and are more likely to connect tothose that already have many connections. We investigate further a family ofmodels introduced by Antunovi\'{c}, Mossel and R\'{a}cz where each vertex in apreferential attachment graph is assigned a type, based on the types of itsneighbours. Instances of this type of process where the proportions of eachtype present do not converge over time seem to be rare.  Previous work found that a "rock-paper-scissors" setup where each new node'stype was determined by a rock-paper-scissors contest between its two neighboursdoes not converge. Here, two cases similar to that are considered, one which islike the above but with an arbitrarily small chance of picking a random typeand one where there are four neighbours which perform a knockout tournament todetermine the new type.  These two new setups, despite seeming very similar to the rock-paper-scissorsmodel, do in fact converge, perhaps surprisingly.

Fragility of non-convergence in preferential attachment graphs with three types