# American Institute of Mathematical Sciences

December  2019, 14(4): 771-788. doi: 10.3934/nhm.2019031

## A discrete districting plan

 1 Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze 53/A, I-43124 Parma, Italy 2 Dipartimento di Matematica, Università di Pavia, via Ferrata 5, I-27100 Pavia, Italy

* Corresponding author: G. Saracco

Received  January 2019 Revised  June 2019 Published  October 2019

Fund Project: A. Saracco was partially supported by INdAM-GNSAGA. G. Saracco was partially supported by the INdAM-GNAMPA 2019 project "Problemi isoperimetrici in spazi Euclidei e non".

The outcome of elections is strongly dependent on the districting choices, making thus possible (and frequent) the gerrymandering phenomenon, i.e. politicians suitably changing the shape of electoral districts in order to win the forthcoming elections. While so far the problem has been treated using continuous analysis tools, it has been recently pointed out that a more reality-adherent model would use the discrete geometry of graphs or networks. Here we propose a parameter-dependent discrete model for choosing an "optimal" districting plan. We analyze several properties of the model and lay foundations for further analysis on the subject.

Citation: Alberto Saracco, Giorgio Saracco. A discrete districting plan. Networks & Heterogeneous Media, 2019, 14 (4) : 771-788. doi: 10.3934/nhm.2019031
##### References:

show all references

##### References:
Removing or adding a vertex; numbers correspond to the weight of the vertexes; all edges are supposed to have weight $1$
Removing or adding an edge; numbers correspond to the weight of the related vertexes and edges
Splitting two vertexes is not always possible by simply modifying the weight of their common edge
A graph where the optimal $4$-partition is not a $2$-refining of the optimal $2$-partition
A graph where the optimal $4$-partition is a $2$-refining of the optimal $2$-partition, but does not induce the optimal $2$-partitions on its components
A graph where the optimal $4$-partition is not a $2$-refining of the optimal $2$-partition for suitable choices of $\alpha = \alpha(p)$
A graph where the optimal $4$-partition is not a $2$-refining of the optimal $2$-partition
 [1] Yong Lin, Gábor Lippner, Dan Mangoubi, Shing-Tung Yau. Nodal geometry of graphs on surfaces. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1291-1298. doi: 10.3934/dcds.2010.28.1291 [2] Annalisa Cesaroni, Matteo Novaga. The isoperimetric problem for nonlocal perimeters. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 425-440. doi: 10.3934/dcdss.2018023 [3] Len G. Margolin, Roy S. Baty. Conservation laws in discrete geometry. Journal of Geometric Mechanics, 2019, 11 (2) : 187-203. doi: 10.3934/jgm.2019010 [4] Tatiana Odzijewicz. Generalized fractional isoperimetric problem of several variables. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2617-2629. doi: 10.3934/dcdsb.2014.19.2617 [5] Stan Alama, Lia Bronsard, Rustum Choksi, Ihsan Topaloglu. Droplet phase in a nonlocal isoperimetric problem under confinement. Communications on Pure & Applied Analysis, 2020, 19 (1) : 175-202. doi: 10.3934/cpaa.2020010 [6] Ihsan Topaloglu. On a nonlocal isoperimetric problem on the two-sphere. Communications on Pure & Applied Analysis, 2013, 12 (1) : 597-620. doi: 10.3934/cpaa.2013.12.597 [7] Moon Duchin, Tom Needham, Thomas Weighill. The (homological) persistence of gerrymandering. Foundations of Data Science, 2021  doi: 10.3934/fods.2021007 [8] Gyula Csató. On the isoperimetric problem with perimeter density $r^p$. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2729-2749. doi: 10.3934/cpaa.2018129 [9] Andreas Kreuml. The anisotropic fractional isoperimetric problem with respect to unconditional unit balls. Communications on Pure & Applied Analysis, 2021, 20 (2) : 783-799. doi: 10.3934/cpaa.2020290 [10] Joachim von Below, José A. Lubary. Isospectral infinite graphs and networks and infinite eigenvalue multiplicities. Networks & Heterogeneous Media, 2009, 4 (3) : 453-468. doi: 10.3934/nhm.2009.4.453 [11] Fabio Camilli, Adriano Festa, Silvia Tozza. A discrete Hughes model for pedestrian flow on graphs. Networks & Heterogeneous Media, 2017, 12 (1) : 93-112. doi: 10.3934/nhm.2017004 [12] Joshua E.S. Socolar. Discrete models of force chain networks. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 601-618. doi: 10.3934/dcdsb.2003.3.601 [13] Seunghee Lee, Ganguk Hwang. A new analytical model for optimized cognitive radio networks based on stochastic geometry. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1883-1899. doi: 10.3934/jimo.2017023 [14] Christina Knox, Amir Moradifam. Electrical networks with prescribed current and applications to random walks on graphs. Inverse Problems & Imaging, 2019, 13 (2) : 353-375. doi: 10.3934/ipi.2019018 [15] Regino Criado, Julio Flores, Alejandro J. García del Amo, Miguel Romance. Structural properties of the line-graphs associated to directed networks. Networks & Heterogeneous Media, 2012, 7 (3) : 373-384. doi: 10.3934/nhm.2012.7.373 [16] Annalisa Iuorio, Christian Kuehn, Peter Szmolyan. Geometry and numerical continuation of multiscale orbits in a nonconvex variational problem. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1269-1290. doi: 10.3934/dcdss.2020073 [17] Jorge A. Becerril, Javier F. Rosenblueth. Necessity for isoperimetric inequality constraints. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1129-1158. doi: 10.3934/dcds.2017047 [18] Sungwoo Ahn, Winfried Just. Digraphs vs. dynamics in discrete models of neuronal networks. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1365-1381. doi: 10.3934/dcdsb.2012.17.1365 [19] Meiyu Sui, Yejuan Wang, Peter E. Kloeden. Pullback attractors for stochastic recurrent neural networks with discrete and distributed delays. Electronic Research Archive, 2021, 29 (2) : 2187-2221. doi: 10.3934/era.2020112 [20] Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A model problem for Mean Field Games on networks. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4173-4192. doi: 10.3934/dcds.2015.35.4173

2020 Impact Factor: 1.213