Improving the Resilience of Mutualistic Networks Varun Rao December 3, 2018 1 / 34
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 β eff Degree Distributions Bifurcations Overall graphs Future Work 2 / 34
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 3 / 34
Network Dataset How we will be modeling the network! 4 / 34
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. 4 / 34
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. 5 / 34
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 ) 5 / 34
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: d x dt = f ( A ij , x ) 5 / 34
Modeling Multi-Dimensionality is hard... 6 / 34
Modeling Mutli-Dimensional Dynamics N dx i � dt = F ( x i ) + A ij G ( x i , x j ) j =1 x i → time dependent activities of all N nodes F ( x i ) , G ( x i , x j ) → dynamics of systems interactions A ij → rate at which j impacts i 7 / 34
Modeling Mutli-Dimensional Dynamics N dx i � dt = F ( x i ) + A ij G ( x i , x j ) j =1 x i → time dependent activities of all N nodes F ( x i ) , G ( x i , x j ) → dynamics of systems interactions A ij → rate at which j impacts i But...can we reduce this equation even further? 7 / 34
Reduced Equation dx i dt = F ( x i ) + β eff G ( x i , x j ) 8 / 34
Reduced Equation dx i dt = F ( x i ) + β eff G ( x i , x j ) β eff = < s out >< s in > s With this, we can get rid of the adjacency matrix! 8 / 34
Reduced Equation dx i dt = F ( x i ) + β eff G ( x i , x j ) β eff = < s out >< s in > s With this, we can get rid of the adjacency matrix! Also defining new variable x eff x eff = < s out > x < s > 8 / 34
Mutualistic Network Dynamics Mutualistic networks have specific dynamical equations N dx i dt = B i + x i (1 − x i )( x i x i x j � − 1) + A ij D i + E i x i + H j x j K i C i j =1 Term on left replaces F , while term on right G deals with the dynamics 9 / 34
Mutualistic Network Dynamics Mutualistic networks have specific dynamical equations N dx i dt = B i + x i (1 − x i )( x i x i x j � − 1) + A ij D i + E i x i + H j x j K i C i j =1 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! 9 / 34
Mutualistic Network Dynamics Mutualistic networks have specific dynamical equations N dx i dt = B i + x i (1 − x i )( x i x i x j � − 1) + A ij D i + E i x i + H j x j K i C i j =1 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! x 2 dx eff = B + x eff (1 − x eff K )( x eff eff − 1) + β eff dt C D + ( E + H ) x eff 9 / 34
Model Used β eff is what we care about! x eff relates to overall low/high state of system Can adapt above equation to any dynamic; β eff will change depending on system Want to have β eff be greater than 7! 10 / 34
Modeling in β eff space 11 / 34
Quick Summary! Taking M bi-partite matrix matrix and separating it into projection networks A and B. 12 / 34
Quick Summary! Taking M bi-partite matrix matrix and separating it into projection networks A and B. Then find β eff of each projection Can then understand resilience! 12 / 34
Okay....we have this cool framework, what does this have to do with your research? 13 / 34
Main research question! How will modifying the A ij matrix affect the bipartite networks? 13 / 34
Now, have to understand how A ij is formulated to keep going! 13 / 34
Modifying A ij m M ik M jk � A ij = � N s =1 M sk k =1 m σ ( M ik M jk )( M ik + M jk ) � A ij = � N s =1 M sk k =1 � 0 M ik � = M jk σ = 1 M ik + M jk = 2 Now, what occurs when adding species? M ∗ = � � M λ � M � M ∗ = λ 14 / 34
Modifying A ij Remember, A ij is the projection matrix! M → A , B A ∗ = A 0 + f ( λ ) B ∗ = B 0 + f ( λ ) When adding pollinator � � f ( λ ) = a b c d When adding flower . . f ( λ ) = . . 15 / 34
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 17 / 34
Locations and Species 18 / 34
Intial β eff Distributions Figure 1: Projection Network Distributions 19 / 34
β eff Distribution-Flower β eff = D + H 20 / 34
β eff Distribution-Flower β eff = D + H 20 / 34
Adding Species-Small Figure 2: Top, Adding Flower. Bottom, Adding Pollinator 21 / 34
Adding Species-Medium Figure 3: Top, Adding Flower. Bottom, Adding Pollinator 22 / 34
Adding Species-Large Figure 4: Top, Adding Flower. Bottom, Adding Pollinator 23 / 34
Overall Changes Figure 5: Top, Overall Changes for Small Locations. Botthom, Overall Changes for Medium Locations 24 / 34
Current Summary of Results Adding species benefits opposite projection network. Improvement of β eff varies depending on location size Analyze how degree affects ∆ beta eff 25 / 34
Bifurcations-Small Locations Figure 6: Top, Bifurcating Flower. Bottom, Bifurcating Pollinator 26 / 34
Bifurcations-Medium Locations Figure 7: Top, Bifurcating Flower. Bottom, Bifurcating Pollinator 27 / 34
Why do these bifurcations occur? 28 / 34
Why do these bifurcations occur? Maybe nearest neighbor degree has something to do with it? 28 / 34
Nearest Neighbor Degree From bifurcation graph, we saw that a bifurcation of some type was occuring Graphed three degrees k NN → degree of all nearest neighbors k Proj → degree in projection network k deg → normal degree For now, only have graphed these degrees after adding a flower 29 / 34
Nearest Neighbor Degrees - Small Locations Figure 8: Small Degrees 30 / 34
Nearest Neighbor Degrees - Medium Locations Figure 9: Medium Degrees 31 / 34
Future Work 1 Fixing input errors in the code 2 Analyzing why bifurcations occur 3 Graphing how changing β eff affects H , D 4 Finding optimal species, optimal k 5 Generating theoretical framework 32 / 34
Questions? Thank you! 33 / 34
References 1. Gao,J.,Barzel,B.and Barbasi,A.”Universal resilience patterns in complex networks”. Nature , 530 ,307-312 (2016). 34 / 34
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