耦合演化动态网络的稳定效率与反选举人模型（Stationary efficiency of co-evolutionary networks：an inverse voter model）

Stationary efficiency of co-evolutionary networks: an inverse voter model Chen-Ping Zhu12, Hui Kong1, Li Li3 Zhi-Ming Gul, Shi-Jie Xiong4

Stationary efficiency of co-evolutionary networks: an inverse voter model Chen-Ping Zhu12 ，Hui Kong1 ，Li Li3 ， Zhi-Ming Gu1 ， Shi-Jie Xiong4

Outline 1. Motivations of our work 2. Inverse voter model (vM)for Co-evolutionary networks 3. Analytic and numerical results 4. Discussion and conclusions

Outline 1. Motivations of our work 2. Inverse voter model (IVM) for co-evolutionary networks 3. Analytic and numerical results 4. Discussion and Conclusions

Motivations (1)A sort of real networks composed of nodes with binary states (a)sIS epidemic networks: susceptible/infected (b Stock market: buyers/sellers ( c) Neural networks abstracted with firing and quiescent nodes ( d) Communication networks with transmitters and receivers Generalized information-flow yields between linked pairs of nodes in two opposite states The function of such networks is to produce generalized information flow So we call them flow networks Only links between the nodes in the opposite states are effective for the function of a specific network

Motivations (1) A sort of real networks composed of nodes with binary states: (a) SIS epidemic networks: susceptible/infected (b) Stock market: buyers /sellers (c) Neural networks abstracted with firing and quiescent nodes (d) Communication networks with transmitters and receivers. Generalized information-flow yields between linked pairs of nodes in two opposite states. The function of such networks is to produce generalized information flow. So we call them flow networks. Only links between the nodes in the opposite states are effective for the function of a specific network

Motivations (1)An edge linking two nodes: correlation/interaction between them. Effective/ineffective for realizing the function of a network? In a co-evolutionary network, node states vary with time, it changes the efficacy of links between them, even cause links to rewire correspondingly Two actions feed back with each other We advocate Global efficiency should be the ratio of income/ pay-off It should refers to the density of effective links among all Links which may be time-dependent and arrives at dynamic equilibrium by co-evolution

Motivations (1) An edge linking two nodes: correlation/interaction between them. Effective/ineffective for realizing the function of a network? In a co-evolutionary network, node states vary with time, it changes the efficacy of links between them, even cause links to rewire correspondingly. Two actions feed back with each other. We advocate : Global efficiency should be the ratio of income/pay-off. It should refers to the density of effective links among all Links which may be time-dependent and arrives at dynamic equilibrium by co-evolution

The primitive definition of global efficiency for a complex network 2 b (N-1)分l It is irrelevant to either the function of a network or to the state of any node, which is abnormal from the viewpoint of statistical physICS

• The primitive definition of global efficiency for a complex network • It is irrelevant to either the function of a network or to the state of any node, which is abnormal from the viewpoint of statistical physics 2 1 ( 1) glob i j ij E N N l  = − 

Motivations (3)Most practically existing networks are sparse k << Inx<< m O k≤lnN<<N Rarely could I find papers interpreting sparseness

Motivations (3) Most practically existing networks are sparse: Rarely could I find papers interpreting sparseness。 ln ln k N N or k N N    

3. Inverse voter model(IVM) Flow of generalized message, and coevolution Flow networks: SIS epidemics, stock markets, mAnET and neural networks Nodes with binary states( t) Effective links:+<> The flow-producing efficiency of dynamic networks with binary node states p,p=p+p p++ tp+tp Define the efficiency of a flow-network E

3. Inverse voter model (IVM) • Flow of generalized message, and coevolution • Flow networks: SIS epidemics, stock markets, MANET, and neural networks. • Nodes with binary states( ) • Effective links: • The flow-producing efficiency of dynamic networks with binary node states:  +  − , E 1           + − +− ++ −− ++ +− −− + − = = + + + = = − Define the efficiency of a flow-network:

3. Inverse voter model --n inert links n-1 inert links k-n active links k-n+l active links rewire e k-n inert links n active links flir 2(k-2n) FIG. 1: Flipping or rewiring and associated changes in the global density of inert links in the inverse voter model

3. Inverse voter model

3. Inverse voter model(IVM) Define the ratio of edges linking both nodes in the same state as p, then master equation describing its evolution reads =2Ph)k ∑B B"2 dt k=11/N n=0 , k )(k-2m) N where n and k represent the number of links connecting the same state from any node and the degree of it, respectively u is average degree under the distribution p(k), and Bnk is binary distribution. Adjustable parameter BE(0, 1.0) controls average rewiring probability of nodes, reflecting randomness of interaction among nodes which modifies the deterministic nearest neighbor correlations

3. Inverse voter model (IVM) , 1 0 ( ) 2 [(1 )( 2 ) ] 1 / (1) n n k k k n k k n d p k n B e k n e dt N k N      − − = = =   − − − n,k where n and k represent the number of links connecting the same state from any node and the degree of it, respectively, is average degree under the distribution p(k), and B is binary distributi  on. Adjustable parameter (0,1.0) controls average rewiring probability of nodes, reflecting randomness of interaction among nodes which modifies the deterministic nearest neighbor correlations.   Define the ratio of edges linking both nodes in the same state as , then master equation describin  g its evolution reads

3. Inverse voter model(VM dual meanings of parameter p probabilistic modifying factor to decide flipping/rewiring Random factor describing effects caused by non-deterministic factors other nearest interactions ie normalized inverse-temperature β=1/(T+1

3. Inverse voter model (IVM) • dual meanings of parameter : • probabilistic modifying factor to decide flipping/rewiring; • Random factor describing effects caused by non-deterministic factors other nearest interactions, i.e., normalized inverse-temperature, =1/(T+1)  