Fluctuations in Growth Processes Angad Yuvraj and Dominic Yates - - PowerPoint PPT Presentation

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Fluctuations in Growth Processes Angad Yuvraj and Dominic Yates - - PowerPoint PPT Presentation

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Fluctuations in Growth Processes Angad Yuvraj and Dominic Yates Supervised by Alex Adamou, Ole Peters and Rosalba Garca Milln 1 Introduction Classical view holds that fluctuations are irrelevant to growth of a system. Recent

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Fluctuations in Growth Processes

Angad Yuvraj and Dominic Yates

Supervised by Alex Adamou, Ole Peters and Rosalba García Millán

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Introduction

  • Classical view holds that fluctuations are irrelevant to growth of a

system.

  • Recent advancements demonstrate that fluctuations do play a role

when considering the growth rate of the system.

  • It has been shown that individuals achieve boost in growth rate by

completely pooling and equally splitting their resources.

  • We extend this concept to consider more practical forms of sharing

than this type of “cooperation”.

  • We consider the effects of two types of resource sharing on growth

rate, and the impact of fluctuations.

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Geometric Brownian Motion

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Geometric Brownian Motion

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Cooperation

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Cooperation

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Our Model

  • The motivation behind our model came after studying a paper by

Jain and Krishna on the evolution of a complex system.

  • The system comprised of elements that catalysed the growth of
  • ther elements.
  • We noted that this represented a system in which individuals

were cooperating, except that there was no cost to the catalysis.

  • This meant that the system didn’t display the same conservation

present in the previous cooperation model, in which individuals pooled and split resources.

  • Their guiding equation helped us to generalise the cooperation

model.

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Our Model

  • We consider a population of N individuals which interact through

the giving and taking of resources, not necessarily in a symmetric manner.

  • We visualise this concept through graphs where each node

represented an individual and directed edges represented the flow of resources.

  • We do not allow a node to have a directed edge to itself, and

each node may have at most one directed edge towards each

  • ther node.
  • We allow edges to have different weights, corresponding to

different rates of resource sharing.

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  • The Weighted Adjacency Matrix

v v v v v v

0.5 1.1 1

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Our Model

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Our Model

  • Receiving

Term Giving Term

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Our Model

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Our Model

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The Deterministic Case

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Questions

  • What happens when individuals only engage in partial sharing?
  • What rate of sharing does it take to achieve maximum growth

rate?

  • To what extent does growth rate depend on the way resources

are distributed?

  • Does any of the above depend on how noisy the system is?

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Complete Graph on N nodes

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Fully Connected Graph for N = 5

Complete Graph on N nodes

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N-Cycle for N = 5

  • N-Cycle

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The Complete Graph for N = 2

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Normalised Growth, G

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Normalised Growth, G

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Normalised Growth, G

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Interpretation of Results for N = 2

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Questions

  • What happens when individuals only engage in partial sharing?
  • What rate of sharing does it take to achieve a higher growth

rate?

  • To what extent does growth rate depend on the way resources

are distributed?

  • Does any of the above depend on how noisy the system is?

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Finiteness of Time Length T

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Questions

  • What happens when individuals only engage in partial sharing?
  • What rate of sharing does it take to achieve maximum

growth rate?

  • To what extent does growth rate depend on the way resources

are distributed?

  • Does any of the above depend on how noisy the system is?

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  • Fully Connected Graph for Different N

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Fully Connected Graph for Different N

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  • Form One

Form Two

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  • Form One

Form Two

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Similar Behaviour in Our Model

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Questions

  • What happens when individuals only engage in partial sharing?
  • What rate of sharing does it take to achieve maximum

growth rate?

  • To what extent does growth rate depend on the way resources

are distributed?

  • Does any of the above depend on how noisy the system is?

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N-Cycle

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Questions

  • What happens when individuals only engage in partial sharing?
  • What rate of sharing does it take to achieve maximum growth

rate?

  • To what extent does growth rate depend on the way resources

are distributed?

  • Does any of the above depend on how noisy the system is?

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Conclusion

  • In the absence of fluctuations, there is no incentive to

cooperate.

  • Even a little sharing goes a long way.
  • The manner of distribution does have an impact on the growth

rate.

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Thank you for listening.

  • References:

1. “The Evolutionary Advantage of Cooperation”, O. Peters and A. Adamou http://arxiv.org/abs/1506.03414 2. “The Emergence and Growth of Complex Networks in Adaptive Systems”, S. Jain and S. Krishna Computer Phys. Comm. 121-122 (1999) 116-121. 3. “Far from equilibrium: Wealth reallocation in the United States”, Y. Berman, O. Peters and A. Adamou http://arxiv.org/abs/1605.05631 4. “Note on mean-field wealth models and the Random Energy Model”, Jean-Phillipe Bouchard (not published)

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