Non-consensus Opinion Models on Complex Networks

Qian Li, Lidia A. Braunstein, Huijuan Wang, Jia Shao, H. Eugene Stanley, Shlomo Havlin

Research output: Contribution to journalArticlepeer-review

37 Scopus citations


Social dynamic opinion models have been widely studied to understand how interactions among individuals cause opinions to evolve. Most opinion models that utilize spin interaction models usually produce a consensus steady state in which only one opinion exists. Because in reality different opinions usually coexist, we focus on non-consensus opinion models in which above a certain threshold two opinions coexist in a stable relationship. We revisit and extend the non-consensus opinion (NCO) model introduced by Shao et al. (Phys. Rev. Lett. 103:01870, 2009). The NCO model in random networks displays a second order phase transition that belongs to regular mean field percolation and is characterized by the appearance (above a certain threshold) of a large spanning cluster of the minority opinion. We generalize the NCO model by adding a weight factor W to each individual's original opinion when determining their future opinion (NCOW model). We find that as W increases the minority opinion holders tend to form stable clusters with a smaller initial minority fraction than in the NCO model. We also revisit another non-consensus opinion model based on the NCO model, the inflexible contrarian opinion (ICO) model (Li et al. in Phys. Rev. E 84:066101, 2011), which introduces inflexible contrarians to model the competition between two opinions in a steady state. Inflexible contrarians are individuals that never change their original opinion but may influence the opinions of others. To place the inflexible contrarians in the ICO model we use two different strategies, random placement and one in which high-degree nodes are targeted. The inflexible contrarians effectively decrease the size of the largest rival-opinion cluster in both strategies, but the effect is more pronounced under the targeted method. All of the above models have previously been explored in terms of a single network, but human communities are usually interconnected, not isolated. Because opinions propagate not only within single networks but also between networks, and because the rules of opinion formation within a network may differ from those between networks, we study here the opinion dynamics in coupled networks. Each network represents a social group or community and the interdependent links joining individuals from different networks may be social ties that are unusually strong, e. g., married couples. We apply the non-consensus opinion (NCO) rule on each individual network and the global majority rule on interdependent pairs such that two interdependent agents with different opinions will, due to the influence of mass media, follow the majority opinion of the entire population. The opinion interactions within each network and the interdependent links across networks interlace periodically until a steady state is reached. We find that the interdependent links effectively force the system from a second order phase transition, which is characteristic of the NCO model on a single network, to a hybrid phase transition, i. e., a mix of second-order and abrupt jump-like transitions that ultimately becomes, as we increase the percentage of interdependent agents, a pure abrupt transition. We conclude that for the NCO model on coupled networks, interactions through interdependent links could push the non-consensus opinion model to a consensus opinion model, which mimics the reality that increased mass communication causes people to hold opinions that are increasingly similar. We also find that the effect of interdependent links is more pronounced in interdependent scale free networks than in interdependent Erdo{double acute}s Rényi networks.

Original languageEnglish
Pages (from-to)92-112
Number of pages21
JournalJournal of Statistical Physics
Issue number1-2
StatePublished - Apr 2013

Bibliographical note

Funding Information:
Acknowledgement We acknowledge support from the DTRA, ONR, NSF CDI program, the European EPIWORK, LINC and MULTIPLEX projects, the Israel Science Foundation, the PICT 0293/00 and UNMdP, NGI and CONGAS.


Acknowledgement We acknowledge support from the DTRA, ONR, NSF CDI program, the European EPIWORK, LINC and MULTIPLEX projects, the Israel Science Foundation, the PICT 0293/00 and UNMdP, NGI and CONGAS.

FundersFunder number
National Science Foundation
Office of Naval Research
Defense Threat Reduction Agency
Israel Science FoundationPICT 0293/00
Universidad Nacional de Mar del Plata


    • Complex networks
    • Coupled networks
    • Non-consensus
    • Opinion model
    • Percolation


    Dive into the research topics of 'Non-consensus Opinion Models on Complex Networks'. Together they form a unique fingerprint.

    Cite this