Transition due to preferential cluster growth of collective emotions in online communities.

We consider a preferential cluster growth in a stochastic model describing the dynamics of a binary Markov chain with an additional long-range memory. The model is driven by data describing emotional patterns observed in online community discussions with binary states corresponding to emotional valences. Numerical simulations and approximate analytical calculations show that the pattern of frequencies depends on a preference exponent related to the memory strength in our model. For low values of this exponent in the majority of simulated discussion threads both emotions are observed with similar frequencies. When the exponent increases an ordered phase emerges in the majority of threads, i.e., only one emotion is represented from a certain moment. Similar changes are observed with increase of a single-step Markov memory value. The transition becomes discontinuous in the thermodynamical limit when discussions are infinitely long and even an infinitely small preference exponent leads to ordered behavior in each discussion thread. Numerical simulations are in a good agreement with the approximated analytical formula. The model resembles a dynamical phase transition observed in other Markov models with a long memory where persistent dynamics follows from a transition to a superdiffusion phase. The ordered patterns predicted by our model have been found in the Blog06 dataset although their number is limited by fluctuations and sentiment classification errors.