q-Voter model

q-Voter model is a generalization of the voter model [1]. The main difference is that in the q-voter model agent interacts with \( q \) randomly selected neighbors. While also external noise is also present in this model.

The model

As in the voter model at each time tick a random agent is selected. This agent will interact with his neighbors, yet unlike in the voter model it will interact with his \( q \) randomly selected neighbors. If we would set that \( q=1 \), then this model would be equivalent to the voter model.

As it is possible that on random networks \( q \) would be larger than the degree of some nodes, it is often allowed for the same neighbors to be selected multiple times. E.g., on regular two dimensional lattice (as used in the app below) the degree of all nodes is \( 4 \), hence otherwise it would hard to implement \( q > 4 \) cases.

So, if all \( q \) neighbors share the same opinion, the agent also switches to (copies) that opinion. Yet, if at least one neighbor has different opinion, then the agent flips his original opinion with probability \( \epsilon \).

This q-voter model is often referred to as non-linear q-voter model as the probability for a randomly selected agent to change his opinion is non-linear for \( q > 1 \) [1]:

\begin{equation} p_q(x)= x^q + \epsilon [1 - x^q - (1-x)^q] . \end{equation}

In the above \( x \) is a fraction of disagreeing neighbors.

It is more or less evident that the q-voter model is similar to other models of opinion dynamics besides the already mentioned connection to the voter model. This model also bears resemblance the Sznajd model (with \( q=2 \) and \( \epsilon=0 \)) and the vacillating voter model (with \( q=2 \) and \( \epsilon=1 \)).

Quite a lot is known analytically about the dynamics of this model. In [1] it was shown that model for \( q>1 \) has two critical values:

\begin{equation} \epsilon_{c1} = \frac{q-1}{2^q-2} , \quad \epsilon_{c2} = \frac{\frac{q^3}{3}-2 q^2 + \frac{17}{3} q -4}{2^{q+2} -2 ( q^2 - q +4)} . \end{equation}

These critical values determine the observed dynamics:

• If \( \epsilon > \epsilon_{c2} \), then the paramagnetic phase is observed (\( M \) fluctuates around zero, but the configuration of opinions constantly changes).
• If \( \epsilon < \epsilon_{c1} \), then the ferromagnetic phase with absorbing states \( \pm 1 \) is observed (given long enough time agents will converge towards a single opinion).
• For intermediate \( \epsilon \), \( M \) will keep fluctuating. "Competition" between the paramagnetic and the ferromagnetic phases will be observed, but no phase should take over in the long run.

Note that for \( q = 2 \) and \( q =3 \) both critical exponents are equal. If in this case \( \epsilon \) is set to critical value, then given enough time \( p(M) \) should become flat.

HTML 5 app

Explore the rich dynamics of the q-voter model using the app below. Try out different values of \( q \). By changing \( \epsilon \) observe all of the different model's phases. Note that for you convenience the app reports critical values for selected (active) \( q \). As PDF figure will not always be useful, you may hide this plot so it wouldn't distract you.

Acknowledgment. This post was written while reviewing literature relevant to the planned activities in postdoctoral fellowship ''Physical modeling of order-book and opinion dynamics'' (09.3.3-LMT-K-712-02-0026) project. The fellowship is funded by the European Social Fund under the No 09.3.3-LMT-K-712 ''Development of Competences of Scientists, other Researchers and Students through Practical Research Activities'' measure.

References

• C. Castellano, M. A. Munoz, R. Pastor-Satorras. The non-linear q-voter model. Physical Review E 80: 041129 (2009). doi: 10.1103/PhysRevE.80.041129. arXiv: 0907.1775 [physics.soc-ph].