The Muon Rules
muon2:
That brings me to the basic score for erosity.
Principle: The quality of the shape of a district is based on how well its areas are connected to each other and to the extent that connected areas are split between different districts. Irregular shapes not due to natural barriers and political boundaries at the borders suggest a poor quality district shape. Unrelated areas connected through unusual means suggest a poor quality district shape. A small district in a densely populated area should be measured in a way that is comparable to a large district in a sparsely populated area.
Definition: Cut link. A cut link is a connection between nodes in different districts.
Definition: Erosity. The erosity of a district is the number of cut links to nodes in that district. (nb. Cut links indicate that areas that share transporation interests are in different districts. This reflects on the external quality of a district and correlates to the geographic shape of the district.)
Item E7: The erosity of a plan is the number of cut links in the plan. Mathematically the number of cut links in a plan will equal one half of the sum of the number of cut links for all the districts. This is because each cut link for a district shows up in the count of two districts.
Example 1:
In this example there is a state with nine counties labeled A through I with regional roads shown as heavy black lines. Counties A through D form one district and the rest form a second district.
The equivalent graph replaces roads with links between nodes. The nodes are colored to show which district they are in. Blue links are internal to a district. Red links are cut links between districts.
There are seven cut links in the plan so the erosity is 7. Each district also has an erosity of 7 which is the number of cut links associated with nodes in the district.
Example 2:
This example uses the same underlying map as in the previous example. One district now consists of whole counties B through D plus a fragment of E, shown with an outlined node. The remaining counties and fragment form the other district.
The rules for links to fragments changes the graph in county E. The new graph has only five cut links for an erosity of 5. The chop of E created a more compact shape that was reflected in the lower score. The trade off between chops and erosity is the key element in scoring.
muon2:
This special rule deals with the issue of nick paths and district that grab a fragment without a local connection.
It is worthwhile reiterating a couple of previous principles with emphasis added.
Principle: Road connections between areas are an indication of a community of interest between those areas. State and federal highways are a stronger indication of a connection than local roads. Large areas at the regional scale of counties should be connected by better roads than small areas at the scale of smaller governmental units.
Principle: The quality of the shape of a district is based on how well its areas are connected to each other and to the extent that connected areas are split between different districts. Irregular shapes not due to natural barriers and political boundaries at the borders suggest a poor quality district shape. Unrelated areas connected through unusual means suggest a poor quality district shape. A small district in a densely populated area should be measured in a way that is comparable to a large district in a sparsely populated area.
Those lead me to this rule.
Definition: Component. A component of a district is a subset of all nodes in the district such that any two nodes of the set are connected to each other by a sequence of links in the district. Each district will consist of one or more disconnected components.
Item E8: Each component in a district in excess of one increases the erosity of the plan by one. (nb. Disconnected components indicate that the district includes areas that are of a regional scale but rely on local not regional connections. This reflects on the internal quality of the district. This item allows districts to be connected by a regional path that nicks another county as long as there is a separate local connection. It is not preferred.)
Example:
This example was used previously to show links and other local paths with isolated fragments. The 5 geographic units are now counties Agnew, Burr, Calhoun, Dawes and Elbridge, labeled A through E. There are are three districts creating three fragments from Agnew, called West Agnew, Central Agnew, and East Agnew. The nodes for the counties are indicated with solid stars and the nodes for the fragments are shown with hollow stars.
Roads that count for regional connections are shown with heavy brown lines. They are in the same configuration as the roads in the example for connections. The fine brown lines are other local roads. There are sufficient local roads so that all contiguous counties and fragments are locally connected.
This is the equivalent graph that was created from this example. The nodes of the different districts are in different colors. The solid lines count as links. The dashed lines represent local connections between nodes that do not count as links. The red lines indicate links and local connections between nodes in different districts.
The cut links (solid red lines) give the basic erosity of the plan. Here there are eight cut links so the basic plan erosity is 8. Note that the orange node western district has an erosity of 4, the green node eastern district has an erosity of 6, and the black node central district has an erosity of 6. Those sum to 16 which is two times the plan erosity of 8.
The eastern district has two components since Calhoun is only locally connected to West Agnew. One component consists of Calhoun and the other consists of West Agnew. The extra component increases the plan erosity by 1.
The western district also has two components. Dawes and Elbridge are regionally connected and have a link in the district, but East Agnew is only locally connected to each of those counties. The extra component adds another 1 point to the plan erosity.
The central district has only one component consisting of Burr and Central Agnew.
With the two excess components the total plan erosity is 10.
muon2:
Definition: A nick path is a regional road that primarily links two counties, but goes slightly into a third county along its path. The simple definition would ban the use of a nick path for lack of a regional connection. An exact definition to provide an exception for nick paths proved elusive.
The special rule in the previous post provides for a way to include nick paths use as long as the counties are locally connected. However, the same rule slightly disfavors their use compared to true regional connections.
Nick path example:
This example was used previously to show links and other local paths with isolated fragments. The 5 geographic units are now counties Agnew, Burr, Calhoun, Dawes and Elbridge, labeled A through E. The nodes for the counties are indicated with solid stars.
Roads that count for regional connections are shown with heavy lines (brown and green). They are in the same configuration as the roads in the example for connections. The fine brown lines are other local roads. There are sufficient local roads so that all contiguous counties and fragments are locally connected.
These counties are divided into two districts. The shaded district in the south west consists of just Burr and Calhoun.
The heavy green line represents a major highway that goes between the county seats of Burr and Calhoun. Along the way it cuts across a corner of Agnew, so it is a nick path. It doesn't count as a regional connection between Burr and Calhoun, just like the other examples using this map.
The rule allows the Burr-Calhoun district to exist, because of the local connection. However Burr and Calhoun fall in two different components in that district.
The equivalent graph shows the two districts with different colored nodes in black and orange. It shows the 4 cut links between nodes in different districts in red. The dashed blue line represents the local connection between the two component orange nodes. With the point for the extra component in the southwest district the total erosity for the plan is 5.
muon2:
Definition: A bridge fragment is a fragment that connects two or more nodes in a component, such that if the bridge fragment is removed from a component then the component would become disconnected into multiple components. We've debated a lot about whether bridge fragments should be allowed or not, and if allowed should they be disfavored. In this component model, bridge fragments are fine if they provide regional links to otherwise disconnected components.
Example:
This example was used previously to show links and other local paths with isolated fragments. The 5 geographic units are now counties Agnew, Burr, Calhoun, Dawes and Elbridge, labeled A through E. The nodes for the counties are indicated with solid stars and the nodes for the fragments are shown with hollow stars.
Roads that count for regional connections are shown with heavy lines (brown and green). They are in the same configuration as the roads in the example for connections. The fine brown lines are other local roads. There are sufficient local roads so that all contiguous counties and fragments are locally connected.
The district line chops Agnew into north and south fragments. Using the connection rule for fragments the connecting path from Agnew to Dawes enters in North Agnew. The connecting paths from Dawes to the other three counties enter in South Agnew.
The effect of the connections is that Burr is connected to South Agnew and Calhoun is connected to South Agnew. Burr and Calhoun are not regionally connected, so without South Agnew they would be in separate components. South Agnew is a bridge fragment that links Burr and Calhoun into a single component.
The equivalent graph shows the two districts with different colored nodes in black and orange. It shows the 4 cut links between nodes in different districts in red. The dashed blue line represents the local connection between the two component orange nodes, but now there are regular links through the South Agnew node. The total erosity for the plan is 4, but there is a chop. Overall the shapes of the districts look fine. This is the type of tradeoff of chops for erosity that makes sense in the system.
muon2:
Here are two more examples illustrating nick paths and bridge fragments together. The idea is to illustrate them by example without a scoring definition.
As before the 5 geographic units are now counties Agnew, Burr, Calhoun, Dawes and Elbridge, labeled A through E. The nodes for the counties are indicated with solid stars and the nodes for the fragments are shown with hollow stars.
Roads that count for regional connections are shown with heavy lines (brown and green). They are in the same configuration as the roads in the example for connections. The fine brown lines are other local roads. There are sufficient local roads so that all contiguous counties and fragments are locally connected.
Example with a non-connecting fragment:
The south west fragment includes just the nick path in green. Unlike the bridge fragment in the previous post the South Agnew fragment here only has the path from Calhoun to Agnew enter it which includes some of the nick path. Importantly the nick path is not the connecting path from Burr to Agnew, which is the brown highway towards the east. Thus there is a link from Calhoun to South Agnew, but no link from Burr to South Agnew.
The equivalent graph shows the pattern of links as solid lines and local connections as dashed lines. Red links are cut links between nodes in different districts. Burr is in a different component than Calhoun and South Agnew. The graph has four cut links and one extra component for and erosity score of 5.
Example with a connecting fragment:
In this example the connecting path from Agnew to Calhoun has been moved to a separate road, north of the nick path. The nick path now is part of neither the connecting path from Agnew to Burr or Calhoun.
South Agnew is now an isolated fragment. Links a calculated based on the South Agnew node and it has regional connections to both Burr and Calhoun. That allows South Agnew to become a bridge fragment between Burr and Calhoun, putting them in the same component.
Though the extra component is gone, there are now five cut links. The erosity is 5.
One of the options considered during the debate about nick paths was to give some preference when they weren't the path to the node of the intervening county. When the nick path is not the connecting path to another node, it can be used in a connecting bridge fragment in a way that it couldn't when it shared its path.
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