A note to readers. This presentation was delivered on March 29th, - PowerPoint PPT Presentation

A note to readers. This presentation was delivered on March 29th, 2019 during the Texas Section MAA Conference at Tarleton State University in Stephenville, Texas. The footnotes for this presentation are (basically) what was said to the audience, and the slides served primarily as a guide for me. The University of Iowa Department of Mathematics � 1

seeds of democracy a k-medoids approach to exploring districting plans Anthony Pizzimenti • Texas MAA Section Conference @ Tarleton State • March 29th, 2019 The University of Iowa Department of Mathematics � 2

in this talk The University of Iowa Department of Mathematics � 3

in this talk 1. brief districting overview The University of Iowa Department of Mathematics � 3

in this talk 1. brief districting overview 2. what is gerrymandering? The University of Iowa Department of Mathematics � 3

in this talk 1. brief districting overview 2. what is gerrymandering? 3. maps, MCMC random walks, and seeding The University of Iowa Department of Mathematics � 3

in this talk 1. brief districting overview 2. what is gerrymandering? 3. maps, MCMC random walks, and seeding 4. k-medoids and its results The University of Iowa Department of Mathematics � 3

1. brief districting overview The University of Iowa Department of Mathematics � 4

1. census and apportionment The University of Iowa Department of Mathematics � 5

2. subdividing The University of Iowa Department of Mathematics � 6

3. districting! The University of Iowa Department of Mathematics � 7

2. what is gerrymandering? The University of Iowa Department of Mathematics � 8

partisan gerrymandering The University of Iowa Department of Mathematics � 9

partisan gerrymandering • a process wherein politicians are able to draw district boundaries that benefit their party The University of Iowa Department of Mathematics � 9

partisan gerrymandering • a process wherein politicians are able to draw district boundaries that benefit their party • a tactic for political parties to retain power The University of Iowa Department of Mathematics � 9

partisan gerrymandering • a process wherein politicians are able to draw district boundaries that benefit their party • a tactic for political parties to retain power • a means for politicians to pick their voters The University of Iowa Department of Mathematics � 9

partisan gerrymandering • a process wherein politicians are able to draw district boundaries that benefit their party • a tactic for political parties to retain power • a means for politicians to pick their voters • a threat to any representative democracy The University of Iowa Department of Mathematics � 9

q: can we tell when a districting plan is gerrymandered? The University of Iowa Department of Mathematics � 10

q: can we tell when a districting plan is gerrymandered? a: no, but we can make a really good guess. The University of Iowa Department of Mathematics � 10

3. maps, MCMC random walks, and seeding The University of Iowa Department of Mathematics � 11

gerrymandering measurement heuristic The University of Iowa Department of Mathematics � 12

gerrymandering measurement heuristic 1. get a map of the state’s subdivisions The University of Iowa Department of Mathematics � 12

gerrymandering measurement heuristic 1. get a map of the state’s subdivisions 2. create the dual graph of the map The University of Iowa Department of Mathematics � 12

gerrymandering measurement heuristic 1. get a map of the state’s subdivisions 2. create the dual graph of the map 3. impose a districting plan on the dual graph (i.e. assign each vertex to a district) The University of Iowa Department of Mathematics � 12

gerrymandering measurement heuristic 1. get a map of the state’s subdivisions 2. create the dual graph of the map 3. impose a districting plan on the dual graph (i.e. assign each vertex to a district) 4. run an MCMC random walk on the dual graph The University of Iowa Department of Mathematics � 12

gerrymandering measurement heuristic 1. get a map of the state’s subdivisions 2. create the dual graph of the map 3. impose a districting plan on the dual graph (i.e. assign each vertex to a district) 4. run an MCMC random walk on the dual graph 5. compare real-life plans to the distribution found by the MCMC random walk The University of Iowa Department of Mathematics � 12

q: how much of the sample space do we explore? The University of Iowa Department of Mathematics � 13

q: how much of the sample space do we explore? a: we usually don’t know. The University of Iowa Department of Mathematics � 13

q: how can we explore more of the sample space? The University of Iowa Department of Mathematics � 14

q: how can we explore more of the sample space? a: we can pick good starting points. The University of Iowa Department of Mathematics � 14

4. k-medoids and its results The University of Iowa Department of Mathematics � 15

k-medoids like k-means , but the centroids must be points in the cluster. The University of Iowa Department of Mathematics � 16

k-medoids on graphs: an algorithm The University of Iowa Department of Mathematics � 17

k-medoids on graphs: an algorithm 1. take in a dual graph The University of Iowa Department of Mathematics � 17

k-medoids on graphs: an algorithm 1. take in a dual graph 2. find a spanning tree of the dual graph The University of Iowa Department of Mathematics � 17

k-medoids on graphs: an algorithm 1. take in a dual graph 2. find a spanning tree of the dual graph 3. pick an initial set of medoids , where each medoid is a vertex of the dual graph The University of Iowa Department of Mathematics � 17

k-medoids on graphs: an algorithm 1. take in a dual graph 2. find a spanning tree of the dual graph 3. pick an initial set of medoids , where each medoid is a vertex of the dual graph 4. based on the medoids, assign each vertex to an initial cluster , which partitions the tree into subtrees The University of Iowa Department of Mathematics � 17

k-medoids on graphs: an algorithm 1. take in a dual graph 2. find a spanning tree of the dual graph 3. pick an initial set of medoids , where each medoid is a vertex of the dual graph 4. based on the medoids, assign each vertex to an initial cluster , which partitions the tree into subtrees 5. pick new medoids The University of Iowa Department of Mathematics � 17

k-medoids on graphs: an algorithm 1. take in a dual graph 2. find a spanning tree of the dual graph 3. pick an initial set of medoids , where each medoid is a vertex of the dual graph 4. based on the medoids, assign each vertex to an initial cluster , which partitions the tree into subtrees 5. pick new medoids 6. re-cluster vertices according to new medoids The University of Iowa Department of Mathematics � 17

k-medoids on graphs: an algorithm 1. take in a dual graph 2. find a spanning tree of the dual graph 3. pick an initial set of medoids , where each medoid is a vertex of the dual graph 4. based on the medoids, assign each vertex to an initial cluster , which partitions the tree into subtrees 5. pick new medoids 6. re-cluster vertices according to new medoids 7. have we seen this clustering k times in a row ? The University of Iowa Department of Mathematics � 17

k-medoids on graphs: an algorithm 1. take in a dual graph 2. find a spanning tree of the dual graph 3. pick an initial set of medoids , where each medoid is a vertex of the dual graph 4. based on the medoids, assign each vertex to an initial cluster , which partitions the tree into subtrees 5. pick new medoids 6. re-cluster vertices according to new medoids 7. have we seen this clustering k times in a row ? 8. if yes , we’re done! The University of Iowa Department of Mathematics � 17

k-medoids on graphs: an algorithm 1. take in a dual graph 2. find a spanning tree of the dual graph 3. pick an initial set of medoids , where each medoid is a vertex of the dual graph 4. based on the medoids, assign each vertex to an initial cluster , which partitions the tree into subtrees 5. pick new medoids 6. re-cluster vertices according to new medoids 7. have we seen this clustering k times in a row ? 8. if yes , we’re done! 9. if no , go back to (5). The University of Iowa Department of Mathematics � 17

how do we pick new medoids? The University of Iowa Department of Mathematics � 18

method 1: median vertex find the vertex at the midpoint of each cluster. The University of Iowa Department of Mathematics � 19

method 2: tree balance for each cluster, find the vertex such that the distance from it to each other vertex is minimized. The University of Iowa Department of Mathematics � 20

method 2 method 1 The University of Iowa Department of Mathematics � 21

Note: in between steps, the clusters are colored di ff erently. The University of Iowa Department of Mathematics � 22

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