3. Inverse voter model(VM) It is averaged over nodes with the same degree k for bi- nomial distribution Bn.k, then they are summarized over degree distribution p(k) for the global density p. Ex ponential function in formula(1)is linearized for easier analysis before averaging. The master equation is now derived into dp 2Sp(k) dt ∑ N不[B(k+1)<n2>-28<n23>一k< 习 (2) where <. are the moments of the binomial distribu
3. Inverse voter model (IVM)
3. Inverse voter model(VM Mean-field approximation (7)=-(k2+35k-)+0 n2)=p3k(k-1)+pk n)=2k
3. Inverse voter model (IVM) • Mean-field approximation 3 3 2 2 2 ( 1)( 2) 3 ( 1) ( 1) n k k k k k k n k k k n k = − − + − + = − + =
3. Inverse voter model(VM) Stationary solution p to master equation Let dt o in formula (2) 2(1 -2)p k-)-61-)-(1--)=0 Or 0
3. Inverse voter model(IVM) • Stationary solution to master equation. Let in formula (2). = 0 dt d
4. Results and physical meanings Continuous phase transition is verified by numerical simulation ( keeping k and p(k)) Compared with W-s solution to Ising model (anti- ferromagnetic We define Long-range order(LRo) Short-rang order(SRO): P Both relies on two independent variables(k, B
4.Results and physical meanings • Continuous phase transition is verified by numerical simulation. (keeping and ) • Compared with W-S solution to Ising model (antiferromagnetic) We define • Long-range order(LRO): • Short-rang order(SRO): • Both relies on two independent variables( , ) N N + k p k( )k
4. Results for random graphs N/N 0.7 (a) (b) e50 0.75 e55 0.85 电65 analytic simnlation 1121416 D.0020406810 : (d) 046 035 =-127 0l5 σ=0.15 0.7D10 0.6 ■D.5 k/B? FIG. 2:(color online)(a)stationary efficiency p+-=l-p.(b) ratio N+ as LRO.(c)primitive p(upper panel)and rescaled Bp(k/B(lower panel).(d)global efficiency Eglob. Results e averaged over 100(5 for E
4.Results for random graphs