The Effect of Network Properties on Hysteresis Structure in - PowerPoint PPT Presentation

The Effect of Network Properties on Hysteresis Structure in Socio-Ecological System Hendrik Santoso SUGIARTO School of Physical and Mathematical Sciences

Human/Nature Interaction

Tragedy of the Commons G. Hardin, The Tragedy of The Commons, Science 163 (3859):344-348

Tragedy of the Commons Social Aspect Ecological Aspect Ecological variable Ecological variable Ecological variable G. Hardin, The Tragedy of The Commons, Science 163 (3859):344-348

Implications Will CPR users self-organize? Hardin said never Many policies based on that conclusion • Governments must impose certain rules on all forests, or fisheries, or water systems • Or Privatizations of properties • Results: Many failures

Self-Organization Many empirical studies show that a lot of CPRs survive by self-management of local communities

Models Ecological Aspect Renewable natural resource SocialAspect Preserve CPR vs exploit CPR • Sethi et al & Noailly et al : Costly Mechanism punishment • Tavoni et al: Equity-driven ostracism R. Sethi, E. Somanathan, A Simple Model of Collective Action. J. Noailly, C. Withagen, J. van den Bergh, Spatial Evolution of Social Norms in a Common-Pool Resource Game. Tavoni et al., The survival of the conformist: equity-driven ostracism and renewable resource management.

Network In fact social interaction is constrained by social network Will network properties affect overall cooperation and its stability?

C 𝑑𝑝𝑝𝑞𝑓𝑠𝑏𝑢𝑝𝑠 → 𝑛𝑏𝑦𝑗𝑛𝑏𝑨𝑗𝑜𝑕 𝑢𝑝𝑢𝑏𝑚 𝑑𝑝𝑛𝑛𝑣𝑜𝑗𝑢𝑧 𝑞𝑏𝑧𝑝𝑔𝑔 → 𝑒𝑈𝜌 𝑒𝐹 = 0 → 𝑓 𝑑 𝑒𝑓𝑔𝑓𝑑𝑢𝑝𝑠 → 𝑛𝑏𝑦𝑗𝑛𝑏𝑨𝑗𝑜𝑕 ℎ𝑗𝑡 𝑝𝑥𝑜 𝑞𝑏𝑧𝑝𝑔𝑔 → 𝑒𝜌 𝑒 D 𝑒𝐹 = 0 → 𝑓 𝑒 D C C 𝑈𝑝𝑢𝑏𝑚 𝑓𝑔𝑔𝑝𝑠𝑢 → 𝐹 = 𝑂 𝑑 𝑓 𝑑 + 𝑂 𝑒 𝑓 𝑒 D D 𝐷𝑝𝑐𝑐 − 𝐸𝑝𝑣𝑕𝑚𝑏𝑡 𝑞𝑠𝑝𝑒𝑣𝑑𝑢𝑗𝑝𝑜 𝑔𝑣𝑜𝑑𝑢𝑗𝑝𝑜 → 𝐺 = 𝛿𝐹 𝛽 𝑆 𝛾 C D C 𝜌 𝑑 = 𝑓 𝑑 𝐹 𝐺 − 𝑥𝑓 𝑑 𝑤𝑡 𝜌 𝑒 = 𝑓 𝑒 𝐹 𝐺 − 𝑥𝑓 𝑒 D C C C 𝑀𝑝𝑑𝑏𝑚 𝑃𝑡𝑢𝑠𝑏𝑑𝑗𝑡𝑛 𝑛𝑓𝑑ℎ𝑏𝑜𝑗𝑡𝑛 𝑉 𝑑 = 𝜌 𝑑 𝑤𝑡 𝑉 𝑒 𝑜 𝑑 = 𝜌 𝑒 − 𝑃(𝑜 𝑑 ) 𝜌 𝑒 − 𝜌 𝑑 D D D 𝜌 𝑒 C 𝑓 𝑑 𝑓 𝑒 𝑓 𝑑 𝑓 𝑑 Exploitation by 𝑓 𝑒 𝑓 𝑒 𝑓 𝑒 𝑓 𝑑 Human 𝑙 𝑓 𝑒 𝑆 𝑢 𝑓 𝑒 𝑆 𝑢 𝑓 𝑑 𝑓 𝑑 𝑆 𝑢+1 = 𝑆 𝑢 + 𝑑 − 𝑒 − 𝑟𝐹 𝑢 𝑆 𝑢 𝑓 𝑑 𝑓 𝑒 𝑆 𝑛𝑏𝑦 𝑓 𝑒 𝑓 𝑑 Resource Inflow Natural Depreciation

Strategy Selection 𝑉 𝑒 D D D C C C 𝑉 𝑑 C C C C C 𝑉 𝑑 C C C C 𝑉 𝑑 𝑉 𝑒 C D ? C D C D C D C C C 𝑉 𝑒 𝑉 𝑑 𝑉 𝑒 D D D D D C C D C 𝑉 𝑒 D D D D D D Every time step, a random player selects new strategy

Strategy Mutation D D D C C C C C C C C C C C C C C C C D ? D D D C C C D D D D D D C C C D D D D D D Every mutation period, a random player’s strategy is changed to the opposite strategy

Numerical Result Complete Network 1 realization  increase c, decrease c

Network Degree High degree Low degree

Social Hysteresis 𝑙 ⇒ 𝑏𝑤𝑓𝑠𝑏𝑕𝑓 𝑒𝑓𝑕𝑠𝑓 𝑂 = 50

Ecological Hysteresis

Social Hysteresis

Ecological Hysteresis

Network Topology Erdos-Renyi Network Scale-Free Network

Effect of Topology

Network Community 𝜈 =0.4, modularity=0.35 𝜈 =0.2, modularity=0.542

Effect of Community 2 𝑑𝑝𝑛𝑛𝑣𝑜𝑗𝑢𝑗𝑓𝑡 2 𝑑𝑝𝑛𝑛𝑣𝑜𝑗𝑢𝑗𝑓𝑡 𝑙 = 5 𝑙 = 45

Effect of Community 2 𝑑𝑝𝑛𝑛𝑣𝑜𝑗𝑢𝑗𝑓𝑡 4 𝑑𝑝𝑛𝑛𝑣𝑜𝑗𝑢𝑗𝑓𝑡 𝑙 = 15 𝑙 = 15

Test Case: 1 Realization, 5 communities

Analytical Approximation • 𝑄 𝐸 → probability choose a defector • 𝑄 𝐷 𝐸 → conditional probability choose a co- operator that connected to defector D D C C C C C C C C D D C D C D C C D D D D C C D D D D

Transition probability • Probability of 𝑂 𝑑 increases by 1 1 𝑈 + 𝑂 𝑑 = 𝑄 𝐸 𝑄 𝐷 𝐸 𝑣 𝑑 − 𝑣 𝑒 + 𝑄 𝐸 𝑛𝑞 • Probability of 𝑂 𝑑 decreases by 1 1 𝑈 − 𝑂 𝑑 = 𝑄 𝐷 𝑄 𝐸 𝐷 (𝑣 𝑒 − 𝑣 𝑑 ) + 𝑄 𝐷 𝑛𝑞

Master Equation 𝑄 𝜐+1 𝑂 𝑑 − 𝑄 𝜐 𝑂 𝑑 𝑄 𝜐 𝑂 𝑑 − 1 𝑈 + 𝑂 𝑑 − 1 − 𝑄 𝜐 𝑂 𝑑 𝑈 − 𝑂 𝑑 = + 𝑄 𝜐 𝑂 𝑑 + 1 𝑈 − 𝑂 𝑑 + 1 − 𝑄 𝜐 𝑂 𝑑 𝑈 + (𝑂 𝑑 )

Fokker-Planck Equation 𝑑 = 𝑂 𝑑 𝑂 , 𝑢 = 𝜐 • Let 𝑔 𝑂 • By using Taylor series, expand up to 1 𝑂 2 𝑒 𝑒𝑢 𝑄 𝑔 𝑑 , 𝑢 = − 𝑒 𝑈 + 𝑔 𝑑 − 𝑈 − 𝑔 𝑄 𝑔 𝑑 , 𝑢 𝑑 𝑒𝑔 𝑑 𝑒 2 + 1 𝑑 , 𝑢 1 𝑂 𝑈 + 𝑔 𝑑 + 𝑈 − (𝑔 2 𝑄 𝑔 𝑑 ) 2 𝑒𝑔 𝑑

Langevin Equation 𝑒𝑔 𝑒𝑢 = 𝑄 𝐷𝐸 𝜌 𝑒 − 𝜌 𝑑 𝑑 𝜍 𝑗 𝑃 𝑜 𝑑 𝑗 − 𝜌 𝑒 + 𝜃(𝑢) 𝜌 𝑒 𝑗

Ecological Differential Equation 2 d𝑆 𝑆 d𝑢 = 𝑑 − 𝑒 − 𝑟𝐹𝑆 𝑆 𝑛𝑏𝑦

Equilibrium • The condition for stable and unstable manifold: 𝑆 = 0 & 𝑔 = 0 𝑑 2 𝑒 𝑆 𝑛𝑏𝑦 𝑆 ∗ = 𝐹 2 + 4𝑑 −𝐹 + 2𝑒 𝑆 𝑛𝑏𝑦 𝑙 𝑄(𝐷|𝐸) 𝑃 𝑗 = 𝜌 𝑒 𝑔 𝑑 𝑗

Random Connection Assumption Assume the connection between cooperators and defectors and random 𝑄 𝐷 𝐸 = 𝑙 𝐷 𝑗 1 − 𝐷 𝑙−𝑗 𝑗

Analytical vs Numerical

Improvement on the Assumption • Assume the distribution of cooperator lies between random distribution and clustered distribution 𝑙 𝑑 , 𝑆 ∗ 𝑄(𝐷|𝐸) 𝑃 𝑗 = 𝜌 𝑒 𝑔 𝑗 𝑙 𝑑 , 𝑆 ∗ ) (1 − 𝑞) 𝑄 𝐷 𝐸 𝑠𝑏𝑜𝑒𝑝𝑛 + 𝑞 𝑄 𝐷 𝐸 𝑃 𝑗 = 𝜌 𝑒 (𝑔 𝑑𝑚𝑣𝑡𝑢𝑓𝑠𝑓𝑒 𝑗 𝑑 , 𝑆 ∗ ) (1 − 𝑞) ( 𝑄 𝐷 𝐸 𝑠𝑏𝑜𝑒𝑝𝑛 𝑃(𝑗) + 𝑞 ( 𝑄 𝐷 𝐸 𝑑𝑚𝑣𝑡𝑢𝑓𝑠𝑓𝑒 𝑃(𝑗) = 𝜌 𝑒 (𝑔 𝑗 𝑗 𝑞 = 1 − 𝑔 𝑑 𝑔 𝑑

Analytical vs Numerical

2 Communities • For every nodes, 1 − 𝜈 probability connected to its own community 𝜈 probability connected to other community

Pair Approximation for 2 Communities • 𝑄 𝐷 2 𝐸 1 → conditional probability choose a co-operator in community 2 that connected to defector in community 1 • 𝑄 𝐷 1 𝐸 1 → conditional probability choose a co-operator in community 1 that connected to defector in community 1

Transition probability Probability of 𝑶 𝒅 Probability of 𝑂 𝑑 increases by 1 decreases by 1 𝑈 + 𝑂 𝑑 = 𝑄 𝐸 1 1 − 𝜈 𝑄 𝐷 1 𝐸 1 + 𝜈𝑄 𝐷 2 𝐸 1 [𝑣 𝑑 − 𝑣 𝑒 ] 1 + 𝑄 𝐸 1 𝑛𝑞 + 𝑄 𝐸 2 1 − 𝜈 𝑄 𝐷 2 𝐸 1 + 𝜈𝑄 𝐷 1 𝐸 2 [𝑣 𝑑 − 𝑣 𝑒 ] 1 + 𝑄 𝐸 2 𝑛𝑞

Transition probability Probability of 𝑂 𝑑 Probability of 𝑶 𝒅 increases by 1 decreases by 1 𝑈 − 𝑂 𝑑 = 𝑄 𝐷 1 1 − 𝜈 𝑄 𝐸 1 𝐷 1 + 𝜈𝑄 𝐸 2 𝐷 1 [𝑣 𝑒 − 𝑣 𝑑 ] 1 + 𝑄 𝐷 1 𝑛𝑞 + 𝑄 𝐷 2 1 − 𝜈 𝑄 𝐸 2 𝐷 1 + 𝜈𝑄 𝐸 1 𝐷 2 [𝑣 𝑒 − 𝑣 𝑑 ] 1 + 𝑄 𝐷 2 𝑛𝑞

Social Equilibrium 𝑙 𝑄 𝐸 1 1 − 𝜈 𝑄 𝐷 1 𝐸 1 + 𝜈𝑄 𝐷 2 𝐸 1 𝑃 𝑗 − 𝜌 𝑒 𝑔 𝑑 𝑗 𝑙 + 𝑄 𝐸 2 1 − 𝜈 𝑄 𝐷 2 𝐸 2 + 𝜈𝑄 𝐷 1 𝐸 2 𝑃 𝑗 − 𝜌 𝑒 𝑔 𝑑 𝑗 = 0

Community’s Cooperation Let 𝑄 𝐷 1 = 𝐷 1 , 𝑄 𝐸 1 = 1 − 𝐷 1 and 𝑄 𝐷 2 = 𝐷 2 , 𝑄 𝐸 2 = 1 − 𝐷 2 Assume the cooperation spread in one community first before spread to other community 𝑑 < 0.5 𝐷 1 = 2𝑔 𝐷 𝑗𝑔 𝑔 𝐷 2 = 0 𝐷 1 = 1 𝑗𝑔 𝑔 𝑑 > 0.5 𝐷 − 1 𝐷 2 = 2𝑔

Random Connection Assumption Assume the connection between cooperators and defectors and random 𝐷 1 𝑗 1 − 𝐷 1 𝑙 𝑑 −𝑗 𝑙 𝑜𝑑 • 𝑄 𝐷 1 𝐸 1 = 𝑙 𝑑 𝐷 2 𝑘 1 − 𝐷 2 𝑙 𝑜𝑑 −𝑘 𝑘 𝑗 • 𝑄 𝐷 2 𝐸 1 = 𝑙 𝑜𝑑 𝐷 2 𝑘 1 − 𝐷 2 𝑙 𝑜𝑑 −𝑘 𝑙 𝑑 𝐷 1 𝑗 1 − 𝐷 1 𝑙 𝑑 −𝑗 𝑘 𝑗 • 𝑄 𝐷 1 𝐸 2 = 𝑙 𝑜𝑑 𝐷 1 𝑘 1 − 𝐷 1 𝑙 𝑜𝑑 −𝑘 𝑙 𝑑 𝐷 2 𝑗 1 − 𝐷 2 𝑙 𝑑 −𝑗 𝑘 𝑗 𝐷 2 𝑗 1 − 𝐷 2 𝑙 𝑑 −𝑗 𝑙 𝑜𝑑 • 𝑄 𝐷 2 𝐸 2 = 𝑙 𝑑 𝐷 1 𝑘 1 − 𝐷 1 𝑙 𝑜𝑑 −𝑘 𝑘 𝑗

Analytical vs Numerical