Incomplete ordering of the voter model on small-world networks

¹ Dipartimento di Fisica, Università di Roma “La Sapienza” P. le A. Moro 2, I-00185 Roma, Italy
² INFM, Unità di Roma 1 - P. le A. Moro 2, I-00185 Roma, Italy
³ Laboratoire di Physique Théorique (UMR du CNRS 8627) Bâtiment 210 Université de Paris-Sud - 91405 Orsay Cedex, France

Corresponding author: castella@pil.phys.uniroma1.it

Received: 24 October 2002
Accepted: 2 May 2003

Abstract

We investigate how the topology of small-world networks affects the dynamics of the voter model for opinion formation. We show that, contrary to what occurs on regular topologies with local interactions, the voter model on small-world networks does not display the emergence of complete order in the thermodynamic limit. The system settles in a stationary state with coexisting opinions whose lifetime diverges with the system size. Hence the nontrivial connectivity pattern leads to the counterintuitive conclusion that long-range connections inhibit the ordering process. However, for networks of finite size, for which full uniformity is reached, the ordering process takes a time shorter than on a regular lattice of the same size.