A Variant Voter Model in Assortative Networks 在同配网络上的一个改进投票模型的研究 潘黎明 荣智海 王直杰 Donghua University October 2010 Presented in CCCN2010, Suzhou

Contents Review of the Voter Models Voter Model Invasion Process Link Dynamics Introduction to a variant voter model Degree correlations The variant voter model in assortative networks Conclusions

Review of the Voter models The origin voter model (VM): (i) randomly pick a node; (ii) the node adopts the state of a random neighbor. The invasion process (IP): (i) pick a random node; (ii) the node exports its state to a random neighbor. link dynamics (LD) (i) pick a random link; (ii) one of the nodes on the link, adopts the state of the other end node. V. Sood, T. Antal, and S. Redner, Phys. Rev. E 77, (2008) T. M. Liggett, Interacting Particle Systems, (Springer-Verlag, Berlin, 2005). P. L. Krapivsky, Phys. Rev. A 45, 1067 (1992). C. Castellano, AIP Conference Proceedings 779, ).

Review of the Voter models In uncorrelated networks, consider all the nodes of like degree to be indistinguishable, the probability of node i adopts node j’s opinion is: (C. M. Schneider-Mizell and L. M. Sander, J. Stat. Phys. 136, 59 (2009) ) VM: LD: IP: n i,n j : the fraction of nodes with degree i and j k i,k j : the degree of node i j μ1: the average degree

Review of the Voter models In uncorrelated networks, consider all the nodes of like degree to be indistinguishable, the probability of node i adopts node j’s opinion is: (C. M. Schneider-Mizell and L. M. Sander, J. Stat. Phys. 136, 59 (2009) ) VM: LD: IP: n i,n j : the fraction of nodes with degree i and j k i,k j : the degree of node i j μ1: the average degree The differences seems to be trivial, but they are fundamental.

Discussion of a Variant Voter Model A variant of the voter model with respect to the heterogeneous influence of individuals. The nodes update their state in the following two steps: (i) Select a pair of nodes using node selection (ii) The pair of nodes choose of the opinions as their common opinion. The probability of choosing i’s opinion as the common opinion is Han-Xin Yang, Zhi-Xi Wu, Changsong Zhou, Tao Zhou and Bing-Hong Wang Phys. Rev. E 80, (2009)

Discussion of a Variant Voter Model On uncorrelated networks, the probability of node i adopting node j’s opinion is: When the parameter α=1, The model reduces to the origin voter model

Consensus time as a function of the parameter α for different G Han-Xin Yang, Zhi-Xi Wu, Changsong Zhou, Tao Zhou and Bing- Hong Wang Phys. Rev. E 80, (2009) A.-L. Barabasi and R. Albert, Science 286, 509 (1999).

Degree-mixing patterns Many networks show ‘‘assortative mixing’’ on their degrees, i.e., a preference for high degree vertices to attach to other high-degree vertices. Whereas, others show disassortative mixing: high-degree vertices attach to low-degree ones. Most social networks have assortative mixing, while technological and biological networks are disassortative. M. E. J. Newman, Phys. Rev. Lett. 89,208701,2002;

The variant voter model in assortative scale-free networks Simulations in assortative scale-free networks with N=5000, and G=2 Maslov, S. and Sneppen, K. (2002). Science,296, R. Xulvi-Brunet and I. M. Sokolov, Phys. Rev. E 70, (2004)

Discussion of the situation when α=0 Comparison of the α =0 situation and the link dynamics LD: Symmetric random walks on integers. α =0: When an active link is selected, the probability for increasing or decreasing the 1 nodes are the same for networks of different degree-mixing coefficients, but the probability of selecting an active link are different.

A plausible explanation for α=0

Average degree of the final common opinion during the time process when α=4.(G=2)

The consensus time T c when G=5000 when

The number of opinion clusters during the time process when G=5000

The fraction of the final common opinion for different degree regions Node selection

The fraction of the final common opinion for different degree regions Link Selection

Conclusions We studied a variant of the voter model on assortative scale-free networks. We find that under node selection, the assortative degree mixing will resulting faster convergence speed due to that the opinion clusters around high-degree individuals will easier to vanish. The slight difference of the way selecting a pair of nodes will change the dynamical behavior dramatically. When using the link selection, assortative-degree mixing will inhibit the consensus for some region of the parameter alpha. Further study is in progress.

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