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 (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 thefunction of a specific network.

Motivations(1) An edge linking two nodes: correlation/interactionbetween them. Effective/ineffective for realizing thefunction 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 allLinks which may be time-dependent and arrives at dynamic equilibrium by co-evolution.

• 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)globi j ij

EN N l

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( )• Effective links:• The flow-producing efficiency of dynamic net

works with binary node states:

,

1E

Defi ne the effi ci ency of a fl ow-network:

3. Inverse voter model

3. Inverse voter model (IVM)

,1 0

( ) 2[(1 )( 2 ) ]

1/

(1)

n nkk k

n kk n

d p k nB 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 describing its evolution reads

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)

3. Inverse voter model (IVM)

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(IVM)

• Stationary solution to master equation.

Let in formula (2).0dt

d

4.Results and physical meanings

• Continuous phase transition is verified by numerical simulation. (keeping and )

• Compared with W-S solution to Ising model (anti-ferromagnetic)

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

4.Results for random graphs

• Scaling relation of SRO , coevolution from initial random graph with the mechanism of IVM.

•

• (5)

~ ( / ), 0f k

1.27, 0.15

4.Results for small world networks

4.Results for small world networks

• (6)

~ ( / ), 0g k

Scaling relation of SRO , coevolution from initial small-world with the mechanism of IVM.

0.92

4.Results and physical meanings

4.Results and physical meanings

• For the case of ， another scaling relation is found for the coevolution from initial SW

• (7)

• where ， c= -0.5 is a constant, ， and is the critical average degree, h(x) is universal function.

• It reflects the correlation between evolution processes starting from different initial rewiring probability which related to effective dimension of SWNs.

It is impossible to see E=1 no matter coevolution starting from RG or SW when .

ck

1

~ ( ) /c h k

c

c

k

kk

05.0

0

4.Results and physical meanings

• Simulation results demonstrate that global efficiency of coevolutionary network with binary node states is enhanced with the flipping and rewiring mechanism of IVM.

• -dependent critical degree exists, below which global efficiency arrive at unity.

• Sparsity of practically existing networks can be explained as a result of co-evolution mechanism of IVM .

)(ck

Calculation of Eglob based on SPLs in the the same coevolution process gives drastically opposite results

with the stationary efficiency.

• which can not be correct since it violates the sp

arsity of real systems.

E

globglob EkEk ,;,

• Actually, the other solution got from eq(3) can behave as

globglob EkEk ,;,

• The definition of global efficiency of networks should no longer be applied to co-

evolutionary networks since it merely describes topological connectivity of static graphs.

• Efficiency of a network should be relevant to its function, and to its node states. It should be the ratio of successful interactions between individuals.

5.Discussion and conclusions

5.Discussion and conclusions

• A physical inverse voter model is presented to describe flow networks with binary node states. Different from previous models, we have found out new absorbing state of network evolution, continuous phase transition and scaling relations corresponding to it.

• We have explained possible origin of sparsity of real networks based on IVM. Small average degree of a network could be accounted as the result of co-evolution into high efficient state.

Acknowledgement

• Nation natural science foundation

70471084 ， 10775071 ， 10635040 ， 10874071

• 973 Project ： 2006CB705500

Thanks for your attention!

Outline

• 1.Motivations• 2.A coevolutionary voter model• 3.Our inverse voter model (IVM)• 4.Results and their physical meanings• 5.Possible applications• 6.Discussion and Conclusions

2.A coevolutionary voter model

F. Vazquez et al. PRL 100,108701(2008)

2. A coevolutionary Voter model

2.A coevolutionary Voter model

Motivations

1. Most of colleagues in physical community get used to describing complex networks with graphic variables. The number of parameters is increasing. However, we could not tell out, which ones are independent with each other, while which ones overlap with others even unnecessary.

In particular, thinking about global efficiency pertaining to any network, it is always calculated by counting inverse shortest path lengths (SPLs) between all possible pairs of nodes. It is irrelevant to either the function of a network nor to the state of any node, which is abnormal from the viewpoint of statistical physics.

Motivations

2. The definition of above mentioned global efficiency Eglob is not applicable to many coevolutionary dynamic networks. Examples: (1) SIR epidemics with an immune node on a SPL (2) nodes inside the transition circle of a speaking node in a MANET with MAC mechanism; /alternative working mechanism for energy-saving in a MANET. (3) neural networks with synaptic plasticity and refractory period of neurons. Connections along SPLs works intermittently due to state flipping of bin

ary nodes. They could be changed even by rewired from a node due to coevolution with its dynamic states.

Eglob based on SPLs is a measure of connectivity of a static network,It should not be misused to dynamic networks.

Motivations

3. An edge linking two nodes: correlation/interaction between them.

Effective/ineffective for realization of function?

Variation of node states< ---- >efficacy of links,

even rewires, correspondingly.

Global efficiency should be the ratio of income/pay-off.

It should refers to the density of effective ones among all

links, which may be time-dependent and arrives at dynamic

equilibrium by coevolution.

Motivations4. A sort of real networks consisting of nodes with

binary states:(a) SIS epidemic networks(b) Stock market with buyers and sellers(c) Neural networks abstracted with firing and quiescent nodes(d) Communication networks with transmitting and receiving nodes.

Generalized message-flow yields between possible pairs of nodes in the opposite states.

The function of flow networks: producing flow

Only links between the nodes in the opposite states are effective for thefunction of a specific network.

2.A coevolutionary voter model

F. Vazquez et al. PRL 100,108701(2008)

2.A coevolutionary Voter model

2.A coevolutionary Voter model

3.Our inverse voter model(IVM)

• Flow of generalized message• Flow networks: SIS epidemics, stock

markets, MANET• Nodes with binary states( )• Effective links:• The flow-producing efficiency of dynamic

networks with binary node states:

1E

Defi ne the effi ci ency of a fl ow-network:

3.Our inverse voter model (IVM)

3.Our inverse voter model (IVM)

,1 0

( ) 2[(1 )( 2 ) ]

1/

(1)

n nkk k

n kk n

d p k nB 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 describing its evolution reads

3.Our inverse voter model (IVM)

• dual meanings of parameter :• probabilistic modifying factor to nearest neighbor

interactions, randomness.• Normalized inverse-temperature, =1/(T+1)

3.Our inverse voter model (IVM)

3.Our 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.Our inverse voter model(IVM)

• Stationary solution to master equation.

Let in formula (2).0dt

d

3.Our inverse voter model (IVM)

• Distinct states as the analytic result of coevolution of a network, efficiency

• Continuous phase transition represented by order parameter

• ( ) is the absorbing state of coevolution.

1

0 0

( 1 )

0

1

1

4.Results and physical meanings

• Continuous phase transition is verified by numerical simulation. (keeping and )

• Compared with Ising model (anti-ferromagnetic)

We define

• Long-range order(LRO):

• Short-rang order(SRO): • Both relies on two independent variables( , )

N

N

k ( )p k

k

4.Results and physical meanings

4.Results and physical meanings

• Scaling relation of SRO , coevolution from initial random graph with the mechanism of IVM.

•

• (5)

~ ( / ), 0f k

1.27, 0.15

4.Results and physical meanings

4.Results and physical meanings

• (6)

~ ( / ), 0g k

Scaling relation of SRO , coevolution from initial small-world with the mechanism of IVM.

0.92

4.Results and physical meanings

• Simulation results demonstrate that global efficiency of coevolutionary network with binary node states is enhanced with the flipping and rewiring mechanism of IVM.

• -dependent critical degree exists, below which global efficiency arrive at unity.

• , . is independent of only when .

• Sparsity of practically existing networks can be thought as a result of coevolution mechanism .

)(ck

)(ck 0, ( 0)but

4.Results and physical meanings

4.Results and physical meanings

• For the case of ， another scaling relation is found for the coevolution from initial SW

• (7)

• where ， c= -0.5 is a constant, ， and is the critical average degree, h(x) is universal function.

• It reflects the correlation between evolution processes starting from different initial rewiring probability which related to effective dimension of small world networks.

It is impossible to see E=1 no matter coevolution starting from RG or SW when .

ck

1

~ ( ) /c h k

c

c

k

kk

05.0

0

4.Results and physical meanings

• Calculation of Eglob based on SPLs in the the same coevolution process gives drastically opposite results with the efficiency

•

• which can not be correct since it violates the sparsity of real systems. At least, it provides us with a reversal evidence that we need to measure global efficiency in a different way.

globglob EkEk ,;,

E

5.Possible applications

• In principle, both VM and IVM could be modified to describe various systems consisting of nodes with binary states, such as epidemics, stock market, MANET, sensor networks, neural networks, evolutionary games and possibly some problems in systems biology.

5.Possible applications

• An example: friction in social networks.• Combining SRO with LRO in IVM, we can

measure direct friction and indirect friction in socioeconomic systems quantitatively:

N

NLRO

df

2

2

/

/(1 )

( )

id

id

f LRO

f LRO

moment closure approximation

or,

5.Discussion and conclusions

• Popularly misused definition of global efficiency of networks should no longer be applied to co-evolutionary networks since it merely describe topological connectivity of static graphs.

• Efficiency of a network should be relevant to its function, and to its node states. It should be the ratio of successful interactions between individuals.

• A physical inverse voter model is presented to describe flow networks with binary node states. We defined long-range-order and short-range-order, respectively. Different from previous models, we have found out new absorbing state of network evolution, continuous phase transition and scaling relations corresponding to it.

• Explained possible origin of sparsity of real networks based on IVM. Small average degree of a network could be accounted as the result of co-evolution into high efficient state. In addition, we may define ground state of a network by our parameter SRO=0,. The efficiency E=1for ground state, while in exciting states E<1.

Acknowledgement

• Nation natural science foundation ：

70471084 ， 10775071 ， 10635040 ， 10874071

• 973 Project ： 2006CB705500

• Thanks for your patience!