Improving the Resilience of Mutualistic Networks Varun Rao December - - PowerPoint PPT Presentation

Improving the Resilience of Mutualistic Networks

Varun Rao December 3, 2018

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Overview

Introduction to Research

Motivation What is Resilience? Modeling the Problem

Theoretical Basis

Modifying existing equation Strategies to improve resilience Projected stats that will improve

Current Results

Change in βeff Degree Distributions Bifurcations Overall graphs

Future Work

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Motivation

Research focuses on improving a pollination network’s resilience via node additions To qualify and find resilience, heavily draw upon work done by Gao,Barzel, and Barabasi [1]

Current research uses much of the derived framework One aspect of the research focused on mutualistic networks; my research uses those dynamics!

Work focuses on positive side-effects of node addition

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Network Dataset

How we will be modeling the network!

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Network Dataset

How we will be modeling the network! We will be modifying the main network, which will change the resilience of the projection networks.

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What is Resilience?

Resilience Defintion Resilience is ability of a system to adjust its activty to retain its basic functionality when errors, failures, or disruptions occur. It is a dynamical property. Networks can be more or less resilient to node or link perturbations depending on their dynamics.

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What is Resilience?

Resilience Defintion Resilience is ability of a system to adjust its activty to retain its basic functionality when errors, failures, or disruptions occur. It is a dynamical property. Networks can be more or less resilient to node or link perturbations depending on their dynamics. Can capture resilience using a 1-d function: dx dt = f (β, x)

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What is Resilience?

Resilience Defintion Resilience is ability of a system to adjust its activty to retain its basic functionality when errors, failures, or disruptions occur. It is a dynamical property. Networks can be more or less resilient to node or link perturbations depending on their dynamics. Can capture resilience using a 1-d function: dx dt = f (β, x) More complex problems is a multi-dimensional system: dx dt = f (Aij, x)

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Modeling Multi-Dimensionality is hard...

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Modeling Mutli-Dimensional Dynamics

dxi dt = F(xi) +

N

• j=1

AijG(xi, xj) xi → time dependent activities of all N nodes F(xi), G(xi, xj) → dynamics of systems interactions Aij → rate at which j impacts i

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Modeling Mutli-Dimensional Dynamics

dxi dt = F(xi) +

N

• j=1

AijG(xi, xj) xi → time dependent activities of all N nodes F(xi), G(xi, xj) → dynamics of systems interactions Aij → rate at which j impacts i But...can we reduce this equation even further?

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Reduced Equation

dxi dt = F(xi) + βeff G(xi, xj)

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Reduced Equation

dxi dt = F(xi) + βeff G(xi, xj) βeff = < sout >< sin > s With this, we can get rid of the adjacency matrix!

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Reduced Equation

dxi dt = F(xi) + βeff G(xi, xj) βeff = < sout >< sin > s With this, we can get rid of the adjacency matrix! Also defining new variable xeff xeff = < sout > x < s >

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Mutualistic Network Dynamics

Mutualistic networks have specific dynamical equations dxi dt = Bi + xi(1 − xi Ki )( xi Ci − 1) +

N

• j=1

Aij xixj Di + Eixi + Hjxj Term on left replaces F, while term on right G deals with the dynamics

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Mutualistic Network Dynamics

Mutualistic networks have specific dynamical equations dxi dt = Bi + xi(1 − xi Ki )( xi Ci − 1) +

N

• j=1

Aij xixj Di + Eixi + Hjxj Term on left replaces F, while term on right G deals with the dynamics Now...let’s apply the new formalism to the above equation!

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Mutualistic Network Dynamics

Mutualistic networks have specific dynamical equations dxi dt = Bi + xi(1 − xi Ki )( xi Ci − 1) +

N

• j=1

Aij xixj Di + Eixi + Hjxj Term on left replaces F, while term on right G deals with the dynamics Now...let’s apply the new formalism to the above equation! dxeff dt = B + xeff (1 − xeff K )(xeff C − 1) + βeff x2

eff

D + (E + H)xeff

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Model Used

βeff is what we care about! xeff relates to overall low/high state of system Can adapt above equation to any dynamic; βeff will change depending on system Want to have βeff be greater than 7!

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Modeling in βeff space

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Quick Summary!

Taking M bi-partite matrix matrix and separating it into projection networks A and B.

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Quick Summary!

Taking M bi-partite matrix matrix and separating it into projection networks A and B. Then find βeff of each projection Can then understand resilience!

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Okay....we have this cool framework, what does this have to do with your research?

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Main research question! How will modifying the Aij matrix affect the bipartite networks?

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Now, have to understand how Aij is formulated to keep going!

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Modifying Aij

Aij =

m

• k=1

MikMjk N

s=1 Msk

Aij =

m

• k=1

σ(MikMjk)(Mik + Mjk) N

s=1 Msk

σ =

• Mik = Mjk

1 Mik + Mjk = 2 Now, what occurs when adding species? M∗ =

• M

λ

• M∗ =

M λ

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Modifying Aij

Remember, Aij is the projection matrix! M → A, B A∗ = A0 + f (λ) B∗ = B0 + f (λ) When adding pollinator f (λ) =

• a

b c d

• When adding flower

f (λ) =     . . . .    

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Steps for Implementation

1 Classify locations 2 Adding pollinator 3 Adding flower 4 Analyzing effects of additions 16 / 34

Location Analysis

Analyzed 143 locations About 50 unique locations Classified location by size Compiled tables of all species interactions as well as all each location’s interactions

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Locations and Species

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Intial βeff Distributions

Figure 1: Projection Network Distributions

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βeff Distribution-Flower

βeff = D + H

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βeff Distribution-Flower

βeff = D + H

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Adding Species-Small

Figure 2: Top, Adding Flower. Bottom, Adding Pollinator

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Adding Species-Medium

Figure 3: Top, Adding Flower. Bottom, Adding Pollinator

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Adding Species-Large

Figure 4: Top, Adding Flower. Bottom, Adding Pollinator

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Overall Changes

Figure 5: Top, Overall Changes for Small Locations. Botthom, Overall Changes for Medium Locations

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Current Summary of Results

Adding species benefits opposite projection network. Improvement of βeff varies depending on location size Analyze how degree affects ∆ betaeff

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Bifurcations-Small Locations

Figure 6: Top, Bifurcating Flower. Bottom, Bifurcating Pollinator

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Bifurcations-Medium Locations

Figure 7: Top, Bifurcating Flower. Bottom, Bifurcating Pollinator

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Why do these bifurcations occur?

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Why do these bifurcations occur? Maybe nearest neighbor degree has something to do with it?

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Nearest Neighbor Degree

From bifurcation graph, we saw that a bifurcation of some type was occuring Graphed three degrees

kNN → degree of all nearest neighbors kProj → degree in projection network kdeg → normal degree

For now, only have graphed these degrees after adding a flower

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Nearest Neighbor Degrees - Small Locations

Figure 8: Small Degrees

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Nearest Neighbor Degrees - Medium Locations

Figure 9: Medium Degrees

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Future Work

1 Fixing input errors in the code 2 Analyzing why bifurcations occur 3 Graphing how changing βeff affects H,D 4 Finding optimal species, optimal k 5 Generating theoretical framework 32 / 34

Questions?

Thank you!

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References

• 1. Gao,J.,Barzel,B.and Barbasi,A.”Universal resilience patterns in

complex networks”.Nature,530,307-312 (2016).

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