Year: 2007
Type: article
Abstract: We consider triad dynamics as it was recently considered by Antal et al. [Phys. Rev. E 72, 036121 (2005)] as an approach to social balance. Here we generalize the topology from all-to-all to the regular one of a two-dimensional triangular lattice. The driving force in this dynamics is the reduction of frustrated triads in order to reach a balanced state. The dynamics is parametrized by a so-called propensity parameter $p$ that determines the tendency of negative links to become positive. As a function of $p$ we find a phase transition between different kinds of absorbing states. The phases differ by the existence of an infinitely connected (percolated) cluster of negative links that forms whenever $p\ensuremath{\leqslant}{p}_{c}$. Moreover, for $p\ensuremath{\leqslant}{p}_{c}$, the time to reach the absorbing state grows powerlike with the system size $L$, while it increases logarithmically with $L$ for $p>{p}_{c}$. From a finite-size scaling analysis we numerically determine the static critical exponents $\ensuremath{\beta}$ and ${\ensuremath{\nu}}_{\ensuremath{\perp}}$ together with $\ensuremath{\gamma}$, $\ensuremath{\tau}$, $\ensuremath{\sigma}$, and the dynamical critical exponents ${\ensuremath{\nu}}_{\ensuremath{\Vert}}$ and $\ensuremath{\delta}$. The exponents satisfy the hyperscaling relations. We also determine the fractal dimension ${d}_{f}$ that satisfies a hyperscaling relation as well. The transition of triad dynamics between different absorbing states belongs to a universality class with different critical exponents. We generalize the triad dynamics to four-cycle dynamics on a square lattice. In this case, again there is a transition between different absorbing states, going along with the formation of an infinite cluster of negative links, but the usual scaling and hyperscaling relations are violated.
Source: Physical Review E
Institution: Constructor University
Language: English
Subfield: Condensed Matter Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2048994185