Seminarium "Coherence-Correlations-Complexity" (KFT, PWr)
Sala 320a bud. A-1, Politechnika Wrocławska
q-neighbor Ising model on complex networks
dr Anna Chmiel
Katedra Fizyki Teoretycznej WPPT PWr
We modify the kinetic Ising model with Metropolis dynamics, allowing each spin to interact only with q spins randomly chosen from the whole system, which corresponds to the topology of a complete graph. We show that the model with q >= 3 exhibits a phase transition between ferromagnetic and paramagnetic phases at temperature T^*, which linearly increases with q. Moreover, we show that for q=3 the phase transition is continuous and discontinuous for larger values of q. For q>3 the hysteresis exhibits oscillatory behavior -- expanding for even values of q and shrinking for odd values of q. If only simulation results were taken into account, this phenomenon could be mistakenly interpreted as switching from discontinuous to continuous phase transitions or even as evidence of the so-called mixed phase transitions. Due to the mean-field like nature of the model we are able to derive the analytical form of transition probabilities and, therefore, calculate not only the probability density function of the order parameter but also precisely determine the hysteresis and the effective potential showing stable, unstable and metastable steady states. Our results show that a seemingly small modification of the kinetic Ising model leads not only to the switch from a continuous to a discontinuous phase transition but also to an unexpected oscillating behavior of the hysteresis.
The analysis of the q-neighbor Ising model on complete graph is presented in [1]. In the case of a random regular graph, where the degree of node k is sufficiently larger than q, we observe the same behavior as for the complete graph (oscillating behavior of the hysteresis). In a spirit of [2] we generalize the q-neighbor Ising model for multiplex networks. A node will change the state only if both levels suggest the change (AND -rule). In case of duplex networks we observe only continuous phase transition, for all values of q.
[1] A. Jędrzejewski, A. Chmiel, K. Sznajd-Weron, Oscillating hysteresis in the q-neighbor Ising model, Phys. Rev. E 92, 052105 (2015).
[2] A. Chmiel, K. Sznajd-Weron ,Phase transitions in the q-voter model with independence on a duplex clique, Phys. Rev. E f92 052812 (2015).
