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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,

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Hendrik Santoso SUGIARTO School of Physical and Mathematical Sciences

The Effect of Network Properties on Hysteresis Structure in Socio-Ecological System

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Human/Nature Interaction

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Tragedy of the Commons

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

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

Ecological variable Ecological variable Ecological variable

Social Aspect Ecological Aspect

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Implications

Will CPR users self-organize? Many policies based on that conclusion

Hardin said never

  • Governments must impose certain rules on all forests,
  • r fisheries, or water systems
  • Or Privatizations of properties
  • Results: Many failures
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Self-Organization

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

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Models

  • 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.

Renewable natural resource Preserve CPR vs exploit CPR

  • Sethi et al & Noailly et al : Costly

punishment

  • Tavoni et al: Equity-driven ostracism

Ecological Aspect SocialAspect Mechanism

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Network

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

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C D C C C C C C D D D D D D D C

𝑆𝑒

𝑓𝑑 𝑓𝑑 𝑓𝑑 𝑓𝑑 𝑓𝑑 𝑓𝑑 𝑓𝑑 𝑓𝑑 𝑓𝑒 𝑓𝑒 𝑓𝑒 𝑓𝑒 𝑓𝑒 𝑓𝑒 𝑓𝑒 𝑓𝑒

Resource Inflow Natural Depreciation Exploitation by Human 𝑆𝑒+1 = 𝑆𝑒 + 𝑑 βˆ’ 𝑒 𝑆𝑒 𝑆𝑛𝑏𝑦

𝑙

βˆ’ π‘ŸπΉπ‘’π‘†π‘’ π‘ˆπ‘π‘’π‘π‘š 𝑓𝑔𝑔𝑝𝑠𝑒 β†’ 𝐹 = 𝑂𝑑𝑓𝑑 + 𝑂𝑒𝑓𝑒 𝐷𝑝𝑐𝑐 βˆ’ πΈπ‘π‘£π‘•π‘šπ‘π‘‘ π‘žπ‘ π‘π‘’π‘£π‘‘π‘’π‘—π‘π‘œ π‘”π‘£π‘œπ‘‘π‘’π‘—π‘π‘œ β†’ 𝐺 = 𝛿𝐹𝛽𝑆𝛾 πœŒπ‘‘ = 𝑓𝑑 𝐹 𝐺 βˆ’ π‘₯𝑓𝑑 𝑀𝑑 πœŒπ‘’ = 𝑓𝑒 𝐹 𝐺 βˆ’ π‘₯𝑓𝑒 π‘€π‘π‘‘π‘π‘š 𝑃𝑑𝑒𝑠𝑏𝑑𝑗𝑑𝑛 π‘›π‘“π‘‘β„Žπ‘π‘œπ‘—π‘‘π‘› 𝑉𝑑 = πœŒπ‘‘ 𝑀𝑑 𝑉𝑒 π‘œπ‘‘ = πœŒπ‘’ βˆ’ 𝑃(π‘œπ‘‘) πœŒπ‘’ βˆ’ πœŒπ‘‘ πœŒπ‘’

C π‘‘π‘π‘π‘žπ‘“π‘ π‘π‘’π‘π‘  β†’ π‘›π‘π‘¦π‘—π‘›π‘π‘¨π‘—π‘œπ‘• π‘’π‘π‘’π‘π‘š π‘‘π‘π‘›π‘›π‘£π‘œπ‘—π‘’π‘§ π‘žπ‘π‘§π‘π‘”π‘” β†’ π‘’π‘ˆπœŒ 𝑒𝐹 = 0 β†’ 𝑓𝑑 D 𝑒𝑓𝑔𝑓𝑑𝑒𝑝𝑠 β†’ π‘›π‘π‘¦π‘—π‘›π‘π‘¨π‘—π‘œπ‘• β„Žπ‘—π‘‘ 𝑝π‘₯π‘œ π‘žπ‘π‘§π‘π‘”π‘” β†’ π‘’πœŒπ‘’ 𝑒𝐹 = 0 β†’ 𝑓𝑒

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Strategy Selection

D D D D D D D C C C C C C C D D D D D D C C C C C C C C 𝑉𝑑 D D D D D D ? C C C C C C C 𝑉𝑒 𝑉𝑒 𝑉𝑒 𝑉𝑒 𝑉𝑑 𝑉𝑑 𝑉𝑑 𝑉𝑒

Every time step, a random player selects new strategy

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Strategy Mutation

Every mutation period, a random player’s strategy is changed to the opposite strategy

D D D D D D C C C C C C C C D D D D D D C C C C C C C ? D D D D D D C C C C C C C D

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Numerical Result

Complete Network

1 realization οƒ  increase c, decrease c

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Network Degree

High degree Low degree

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Social Hysteresis

𝑙 β‡’ 𝑏𝑀𝑓𝑠𝑏𝑕𝑓 𝑒𝑓𝑕𝑠𝑓 𝑂 = 50

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Ecological Hysteresis

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Social Hysteresis

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Ecological Hysteresis

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Network Topology

Erdos-Renyi Network Scale-Free Network

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Effect of Topology

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Network Community

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

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Effect of Community

2 π‘‘π‘π‘›π‘›π‘£π‘œπ‘—π‘’π‘—π‘“π‘‘ 𝑙 = 5 2 π‘‘π‘π‘›π‘›π‘£π‘œπ‘—π‘’π‘—π‘“π‘‘ 𝑙 = 45

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Effect of Community

2 π‘‘π‘π‘›π‘›π‘£π‘œπ‘—π‘’π‘—π‘“π‘‘ 𝑙 = 15 4 π‘‘π‘π‘›π‘›π‘£π‘œπ‘—π‘’π‘—π‘“π‘‘ 𝑙 = 15

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Test Case: 1 Realization, 5 communities

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Analytical Approximation

D D D D D D D C C C C C C C

  • 𝑄 𝐸 β†’probability choose a defector
  • 𝑄 𝐷 𝐸 β†’conditional probability choose a co-
  • perator that connected to defector

D D D D D D D C C C C C C C

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Transition probability

  • Probability of 𝑂𝑑 increases by 1

π‘ˆ+ 𝑂𝑑 = 𝑄 𝐸 𝑄 𝐷 𝐸 𝑣𝑑 βˆ’ 𝑣𝑒 + 𝑄 𝐸 1 π‘›π‘ž

  • Probability of 𝑂𝑑 decreases by 1

π‘ˆβˆ’ 𝑂𝑑 = 𝑄 𝐷 𝑄 𝐸 𝐷 (𝑣𝑒 βˆ’ 𝑣𝑑) + 𝑄 𝐷 1 π‘›π‘ž

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Master Equation

π‘„πœ 𝑂𝑑 βˆ’ 1 π‘ˆ+ 𝑂𝑑 βˆ’ 1 βˆ’ π‘„πœ 𝑂𝑑 π‘ˆβˆ’ 𝑂𝑑 + π‘„πœ 𝑂𝑑 + 1 π‘ˆβˆ’ 𝑂𝑑 + 1 βˆ’ π‘„πœ 𝑂𝑑 π‘ˆ+(𝑂𝑑)

π‘„πœ+1 𝑂𝑑 βˆ’ π‘„πœ 𝑂𝑑 =

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Fokker-Planck Equation

  • Let 𝑔

𝑑 = 𝑂𝑑 𝑂 , 𝑒 = 𝜐 𝑂

  • By using Taylor series, expand up to 1

𝑂2

𝑒 𝑒𝑒 𝑄 𝑔

𝑑, 𝑒

= βˆ’ 𝑒 𝑒𝑔

𝑑

𝑄 𝑔

𝑑, 𝑒

π‘ˆ+ 𝑔

𝑑 βˆ’ π‘ˆβˆ’ 𝑔 𝑑

+ 1 2 𝑒2 𝑒𝑔

𝑑 2 𝑄 𝑔 𝑑, 𝑒 1

𝑂 π‘ˆ+ 𝑔

𝑑 + π‘ˆβˆ’(𝑔 𝑑)

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Langevin Equation

𝑒𝑔

𝑑

𝑒𝑒 = 𝑄 𝐷𝐸 πœŒπ‘’ βˆ’ πœŒπ‘‘ πœŒπ‘’ πœπ‘—π‘ƒ π‘œπ‘‘π‘—

𝑗

βˆ’ πœŒπ‘’ + πœƒ(𝑒)

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Ecological Differential Equation

d𝑆 d𝑒 = 𝑑 βˆ’ 𝑒 𝑆 𝑆𝑛𝑏𝑦

2

βˆ’ π‘ŸπΉπ‘†

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Equilibrium

  • The condition for stable and unstable manifold:

𝑆 = 0 & 𝑔

𝑑

= 0

π‘†βˆ— = βˆ’πΉ + 𝐹2 + 4𝑑 𝑒 𝑆𝑛𝑏𝑦 𝑆𝑛𝑏𝑦

2

2𝑒

𝑄(𝐷|𝐸) 𝑃 𝑗

𝑙 𝑗

= πœŒπ‘’ 𝑔

𝑑

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Random Connection Assumption

Assume the connection between cooperators and defectors and random 𝑄 𝐷 𝐸 = 𝑙 𝑗 𝐷 𝑗 1 βˆ’ 𝐷 π‘™βˆ’π‘—

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Analytical vs Numerical

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Improvement on the Assumption

  • Assume the distribution of cooperator lies between random distribution and

clustered distribution 𝑄(𝐷|𝐸) 𝑃 𝑗

𝑙 𝑗

= πœŒπ‘’ 𝑔

𝑑, π‘†βˆ—

(1 βˆ’ π‘ž) 𝑄 𝐷 𝐸

π‘ π‘π‘œπ‘’π‘π‘› + π‘ž 𝑄 𝐷 𝐸 π‘‘π‘šπ‘£π‘‘π‘’π‘“π‘ π‘“π‘’

𝑃 𝑗

𝑙 𝑗

= πœŒπ‘’(𝑔

𝑑, π‘†βˆ—)

(1 βˆ’ π‘ž) ( 𝑄 𝐷 𝐸

π‘ π‘π‘œπ‘’π‘π‘›π‘ƒ(𝑗) 𝑗

+ π‘ž ( 𝑄 𝐷 𝐸

π‘‘π‘šπ‘£π‘‘π‘’π‘“π‘ π‘“π‘’π‘ƒ(𝑗) 𝑗

= πœŒπ‘’(𝑔

𝑑, π‘†βˆ—)

π‘ž = 1 βˆ’ 𝑔

𝑑 𝑔 𝑑

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Analytical vs Numerical

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2 Communities

  • For every nodes,

1 βˆ’ 𝜈 probability connected to its own community 𝜈 probability connected to other community

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Pair Approximation for 2 Communities

  • 𝑄 𝐷2 𝐸1 β†’conditional probability choose a co-operator in

community 2 that connected to defector in community 1

  • 𝑄 𝐷1 𝐸1 β†’conditional probability choose a co-operator in

community 1 that connected to defector in community 1

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Transition probability

π‘ˆ+ 𝑂𝑑 = 𝑄 𝐸1 1 βˆ’ 𝜈 𝑄 𝐷1 𝐸1 + πœˆπ‘„ 𝐷2 𝐸1 [𝑣𝑑 βˆ’ 𝑣𝑒] + 𝑄 𝐸1 1 π‘›π‘ž + 𝑄 𝐸2 1 βˆ’ 𝜈 𝑄 𝐷2 𝐸1 + πœˆπ‘„ 𝐷1 𝐸2 [𝑣𝑑 βˆ’ 𝑣𝑒] + 𝑄 𝐸2 1 π‘›π‘ž Probability of 𝑢𝒅 increases by 1 Probability of 𝑂𝑑 decreases by 1

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Transition probability

π‘ˆβˆ’ 𝑂𝑑 = 𝑄 𝐷1 1 βˆ’ 𝜈 𝑄 𝐸1 𝐷1 + πœˆπ‘„ 𝐸2 𝐷1 [𝑣𝑒 βˆ’ 𝑣𝑑] + 𝑄 𝐷1 1 π‘›π‘ž + 𝑄 𝐷2 1 βˆ’ 𝜈 𝑄 𝐸2 𝐷1 + πœˆπ‘„ 𝐸1 𝐷2 [𝑣𝑒 βˆ’ 𝑣𝑑] + 𝑄 𝐷2 1 π‘›π‘ž Probability of 𝑂𝑑 increases by 1 Probability of 𝑢𝒅 decreases by 1

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Social Equilibrium

𝑄 𝐸1 1 βˆ’ 𝜈 𝑄 𝐷1 𝐸1 + πœˆπ‘„ 𝐷2 𝐸1 𝑃 𝑗 βˆ’ πœŒπ‘’ 𝑔

𝑑 𝑙 𝑗

+ 𝑄 𝐸2 1 βˆ’ 𝜈 𝑄 𝐷2 𝐸2 + πœˆπ‘„ 𝐷1 𝐸2 𝑃 𝑗 βˆ’ πœŒπ‘’ 𝑔

𝑑 𝑙 𝑗

= 0

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Community’s Cooperation

Let 𝑄 𝐷1 = 𝐷1, 𝑄 𝐸1 = 1 βˆ’ 𝐷1and 𝑄 𝐷2 = 𝐷2, 𝑄 𝐸2 = 1 βˆ’ 𝐷2 Assume the cooperation spread in one community first before spread to other community 𝑗𝑔 𝑔

𝑑 < 0.5 𝐷1 = 2𝑔 𝐷

𝐷2 = 0 𝑗𝑔 𝑔

𝑑 > 0.5

𝐷1 = 1 𝐷2 = 2𝑔

𝐷 βˆ’ 1

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Random Connection Assumption

Assume the connection between cooperators and defectors and random

  • 𝑄 𝐷1 𝐸1 = 𝑙𝑑

𝑗 𝐷1 𝑗 1 βˆ’ 𝐷1 π‘™π‘‘βˆ’π‘— π‘™π‘œπ‘‘ π‘˜ 𝐷2 π‘˜ 1 βˆ’ 𝐷2 π‘™π‘œπ‘‘βˆ’π‘˜

  • 𝑄 𝐷2 𝐸1 = π‘™π‘œπ‘‘

π‘˜ 𝐷2 π‘˜ 1 βˆ’ 𝐷2 π‘™π‘œπ‘‘βˆ’π‘˜ 𝑙𝑑 𝑗 𝐷1 𝑗 1 βˆ’ 𝐷1 π‘™π‘‘βˆ’π‘—

  • 𝑄 𝐷1 𝐸2 = π‘™π‘œπ‘‘

π‘˜ 𝐷1 π‘˜ 1 βˆ’ 𝐷1 π‘™π‘œπ‘‘βˆ’π‘˜ 𝑙𝑑 𝑗 𝐷2 𝑗 1 βˆ’ 𝐷2 π‘™π‘‘βˆ’π‘—

  • 𝑄 𝐷2 𝐸2 = 𝑙𝑑

𝑗 𝐷2 𝑗 1 βˆ’ 𝐷2 π‘™π‘‘βˆ’π‘— π‘™π‘œπ‘‘ π‘˜ 𝐷1 π‘˜ 1 βˆ’ 𝐷1 π‘™π‘œπ‘‘βˆ’π‘˜

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Analytical vs Numerical

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Conclusion

  • The existence of Hysteresis effect and alternate stable

state on socio-ecological network (possibility of Regime Shift)

  • High degree network is more robust (strong hysteresis)
  • Scale-free network is more robust than Erdos-Renyi

network

  • Community structure exhibits multiple hysteresis
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SLIDE 44

Thank You

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SLIDE 45

Appendix

  • Early Warning Signals
  • Empirical Data
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SLIDE 46

Early Warning Signals

  • Temporal EWSs
  • Spatial EWSs
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SLIDE 47
  • Time Series Data Collection
  • Filtering-detrending
  • Rolling Windows
  • Time Series Analysis for Early Warning indicators
  • Significance Test

Temporal Early Warning

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Test Model vs Null Model

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SLIDE 49

Temporal Early Warning

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SLIDE 50

The effect of network degree on temporal EWS

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SLIDE 51
  • Spatial autocorrelation is characterized by a correlation among

nearby nodes in network 𝐽 = 𝑂 π‘₯π‘—π‘˜

π‘˜ 𝑗

π‘₯π‘—π‘˜

π‘˜ 𝑗

π‘Œπ‘— βˆ’ π‘Œ π‘Œ

π‘˜ βˆ’ π‘Œ

π‘Œπ‘— βˆ’ π‘Œ 2

𝑗

π‘₯π‘—π‘˜ = 1 𝑗𝑔 𝑗 π‘π‘œπ‘’ π‘˜ 𝑗𝑑 π‘‘π‘π‘œπ‘œπ‘“π‘‘π‘’π‘“π‘’, 0 π‘π‘’β„Žπ‘“π‘ π‘₯𝑗𝑑𝑓

  • Spatial Standard deviation, skewness, and kurtosis are defined as

second, third, and fourth moment of spatial distribution 𝜏2 =

π‘Œπ‘—βˆ’π‘Œ 2

𝑗

𝑂

𝛿 =

π‘Œπ‘—βˆ’π‘Œ 3

𝑗

𝜏3

πœ† =

π‘Œπ‘—βˆ’π‘Œ 4

𝑗

𝜏4

Spatial Early Warnings

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SLIDE 52
  • Denote every nodes with strategy C with value 1

and every nodes with strategy D with value 0

D D D D D D D C C C C C C C

i j π‘₯π‘—π‘˜ = 0 π‘Œπ‘— = 0 π‘Œ

π‘˜ = 1

Moran I

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SLIDE 53
  • Denote every nodes with strategy C with value 1

and every nodes with strategy D with value 0

D D D D D D D C C C C C C C

i j π‘₯π‘—π‘˜ = 1 π‘Œπ‘— = 0 π‘Œ

π‘˜ = 1

Moran I

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SLIDE 54
  • Denote every nodes with strategy C with value 1

and every nodes with strategy D with value 0

D D D D D D D C C C C C C C

i j π‘₯π‘—π‘˜ = 1 π‘Œπ‘— = 0 π‘Œ

π‘˜ = 0

Moran I

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SLIDE 55

Spatial Indicators

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SLIDE 56

The effect of network degree on spatial EWS

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SLIDE 57

Empirical Data

  • Survey data collected by an anthropologist from

Arizona university, J S Lansing

  • Location: Bali, Indonesia
  • 20 Subaks (villages)
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SLIDE 58

Farmers Opinion From Survey

Name Caste Class Condition of Subak Water Shortage … Farmer 1 3 3 3 4 …

  • Are water shortages frequent in the subak during the dry season?

A never coded as 4 B seldom coded as 3 C sometimes 2 D Frequent 1

  • The condition of your subak now is:

A Excellent, still intact coded as 5 B Good enough coded as 4 C some problems have begun 3 D not good 2 E Bad 1 Example of the questions

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SLIDE 59

Name Caste Class Condition of Subak Water Shortage … Farmer 1 3 3 4 2 … Farmer 2 3 3 4 2 … Farmer 3 3 3 5 1 … … … … … … … Farmer i 3 2 3 2 … Farmer i+1 3 3 4 2 … Farmer i+2 1 2 2 1 … … … … … … … … … … … … … Subak 1 Subak 2

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SLIDE 60

Principal Component Analysis

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SLIDE 61

Biplot Projection

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SLIDE 62

Data vs Model

Very cooperative

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SLIDE 63

Data Model