The Muon Rules
muon2:
Here's the basic rule for connections. I have three special rules that will follow.
Principle: It should be possible to travel between all areas in a district without leaving the district. There may be parts of discrete areas that are not accessible as long as the node for the area can be reached.
Definition: Local connection. There is a local connection between two nodes if there is a continuous path of public roads and ferries that allow one to travel between the two nodes without entering any other unit. Roads along the border of two units are considered to be in either or both of the units as needed to form a connection.
Item E1: Each node in a district must be able to trace a path of local connections to every other node in the district by way only of nodes in that district.
Here were the examples that were fine for the basic connection rule.
Example 1:
In this example I'll call the 5 geographic units Agnew, Burr, Calhoun, Dawes and Elbridge, labeled A through E. The nodes are indicated with stars and the roads are shown with heavy lines.
From the basic rule the following connections exist:
Dawes is connected to both Calhoun and Elbridge each by a single path.
Agnew is connected to both Burr and Dawes, each by two separate paths. The shortest one by time would count as the connecting path.
Burr is not connected to Calhoun. The obvious shortest path cuts a corner of Agnew and no other path stays only within those two units.
Agnew is connected to Calhoun by a single path. The shortest path cuts a corner of Dawes, so it's not a connection. There is a valid connection that dips south towards Burr first but stays only in Agnew and Calhoun.
Elbridge is connected to both Agnew and Burr by virtue of a road that runs along the boundary of Agnew and Burr.
The equivalent graph reduces each connection to a single link between nodes.
Example 2:
In this example there are 4 geographic units: Adlai, Bryan, Clay, and Dewey. As before the nodes are indicated with stars and the roads are shown with heavy lines. The thick shaded area running roughly vertical represents a natural barrier such as a river. Think of Dewey as an independent city that has grown along the river annexing land in Adlai.
Adlai, Bryan and Dewey are all mutually connected to each other.
Clay is connected to Bryan, but not to Adlai or Dewey. The path from Clay to Dewey must go through either Adlai or Bryan. One path from Clay to Adlai initially goes into Adlai but then goes through Dewey before reaching the node of Adlai.
Here's the equivalent graph. Note that on the graph the nodes need not be in their actual location. It's the relative connection of nodes that matters.
muon2:
The first special rule for connections is for fragments. This follows what was worked out earlier in the thread.
Definition: Fragment. A fragment is the contiguous part of a unit entirely within a district formed by a chop of that unit. Fragments are artifacts of a specific redistricting plan and need not correspond to a recognized political unit. The node of the fragment containing the node of the chopped unit is that same node. For a fragment that does not contain the node of the whole political unit, the node is that of the most populous subunit in the fragment.
Item E2: Fragments trace paths to their nodes as if they were part of the original unit. A connection to the node of a fragment exists if the connecting path to the unit with the fragment enters the unit in that fragment. Fragments within the same unit are locally connected if their nodes are locally connected.
This example is based on the the previous unchopped one. The 5 geographic units are Agnew, Burr, Calhoun, Dawes and Elbridge, labeled A through E. The nodes are indicated with stars and the roads are shown with heavy lines.
The shaded area represents a district that chops unit Agnew.
The East Agnew fragment has a node from the original whole Agnew. The West Agnew fragment has a newly created node shown as a hollow star that will be used as a placeholder for mapping.
The path from Agnew (before the chop) to Calhoun without a chop enters Agnew in the West Agnew fragment, so there is a link from West Agnew to Calhoun.
The path from Agnew to Elbridge enters Agnew in the East Agnew fragment, so there is a link from East Agnew to Elbridge.
The primary path from Agnew to Dawes enters in East Agnew, so there is a link from East Agnew to Dawes. A secondary path from Agnew to Dawes enters in West Agnew, but does not form a link.
The primary path from Agnew to Burr enters in East Agnew, so there is a link from East Agnew to Burr. A secondary path from Agnew to Burr enters in West Agnew, but does not form a link.
There is a path between West Agnew and East Agnew that forms a local connection, so there is a link between those fragments.
The equivalent graph colors the nodes of the two districts in different colors. The dashed lines represent secondary paths between nodes that do not count as links. The red lines indicate links and secondary paths that link nodes in different districts.
nb. As noted in the example the dashed lines represent real local connections, but not links. Links are based on the shortest connecting path. That's important in cases where a path of local connections is required, but no link is available. This situation will be addressed further in the special rule for isolated fragments and in the definition for components.
muon2:
The second of the special rules deals with a situation that arises in many circumstances from independent cities in VA to cities embedded in townships in OH. I'm calling these situations shared units. The treatment of shared units mirrors the language for fragments within a unit which should make it easier to apply.
Definition: Shared units. Shared units occur when a unit is surrounded or effectively surrounded by another unit, making both the surrounded and surrounding units shared units. A unit is effectively surrounded when it only has local connections to the surrounding unit or other units surrounded by the same unit. Shared units also exist if the node of a unit is in another unit.
Item E3: Shared units trace paths to their nodes as if they were a single unit. A connection to the node of a specific unit among the shared units exists if the connecting path from another node enters the shared units in that specific unit. Units within a set of shared units are locally connected if their nodes are locally connected. (nb. Shared units are often a city that formed a separate unit from its surrounding county or township, but the roads are designed to go to and through the city. The city should not act as a barrier for connection in those cases.)
Example 1:
Louisville OH (yellow) is an incorporated city in Nimishillen township (blue). Both are subunits of Stark county. The node for Louisville is shown with a white star, and the node for Nimishillen is shown with a blue star. Nimishillen township is bordered by four townships in each of the cardinal directions, and the city of Canton in the southwest (outlined in pink).
Under the basic rule for connections, every path from the node of Nimishillen to any neighbor would have to go through Louisville since the node is in Louisville. That would leave Nimishillen unconnected to any neighbor.
Louisville is surrounded by Nimishillen and contains the node for Nimishillen. That makes Louisville and Nimishillen shared units both ways under the special rule. Paths to both nodes are considered using the shared units together.
All paths to either node enter the shared unit in Nimishillen. So, Nimishillen is connected to all of the neighboring townships and Canton city. Louisville is only connected to Nimishillen.
nb. It is insufficient to define the shared unit only by considering that Louisville is surrounded. Suppose that Louisville annexed more land to the west and became directly connected to Canton city. It would not be surrounded, but the node of Nimishillen would still be inside of Louisville. The special rule would take care of that situation as well.
Example 2:
East Sparta OH (pink outline) is an incorporated city in Pike township, both shown in green. Both East Sparta city and Pike township are subunits of Stark county. Their nodes are shown by stars.
Tuscarawas county in shown in blue. There are no local roads connecting East Sparta to Tuscarawas, and no path from the node of Pike to Tuscarawas. With just the basic connection rule, neither East Sparta nor Pike have connections to Tuscarawas, despite the presence of OH-800 going from the township to Tuscarawas.
East Sparta is only connected to Pike, so it is effectively surrounded. That makes Pike and East Sparta shared units.
By the special rule for shared units, we look at paths for both East Sparta and Pike as they go through the combined shared unit. OH-800 is the shortest path to the nodes of both East Sparta and Pike. It enters the combined shared unit in Pike. By the special rule Pike is connected to Tuscarawas. East Sparta remains only connected to Pike.
muon2:
The third and final special rule for local connections is another one we have looked at earlier in this thread.
Definition: Isolated unit or fragment. An isolated unit has no connections from its node to the nodes of any other units based on the basic connection rule or the special rule for shared units. An isolated fragment has no connections based on the special rule for fragments.
Item E4: An isolated unit has a local connection to another unit if there is a local connection from its node to the node of any contiguous unit or subunit in that order of priority. Subunits can be further divided until at least one local connection is established. (nb. This is a fall back when the normal rules leave a unit with no local connections. There should always be a way to get to the population from outside the unit, even if it is by way of other parts of units.)
Example 1:
In this example there are 4 geographic units: Adlai, Bryan, Clay, and Dewey. As before the nodes are indicated with stars and the roads are shown with heavy lines. The thick shaded area running roughly vertical represents a natural barrier such as a river. Think of Dewey as an independent city that has grown along the river annexing land in Adlai.
In the example for the basic rule, Clay is connected to Bryan, but not to Adlai or Dewey. One path from Clay to Adlai initially goes into Adlai but then goes through Dewey before reaching the node of Adlai so it fails the basic connection rule.
In this case a chop follows the river splitting Adlai into east and west fragments. The east fragment has the node for Adlai. The west fragment (shaded) has a node based on its population shown by a hollow star.
There is no road that directly connects the east and west fragments so there is no local connection between the fragments. Since there is no connection from Clay to Adlai the road that cuts through the northern part of the West Adlai fragment also doesn't become a link to West Adlai. That makes West Adlai an isolated fragment.
Under the special rule for isolated fragments the node in West Adlai is used with its roads to determine connections. West Adlai has a path to both Clay and Dewey, so it becomes connected to both.
The equivalent graph colors the nodes of the two districts in different colors. The blue lines represent links between nodes in the same district. The red lines indicate links between nodes in different districts.
Example 2:
This is an example from the area around Canton OH. The colors represent different subunits of Stark county and the nodes are indicated by stars.
Canton city (white)
Canton township (green)
Plain township (dark blue)
North Canton city (light blue, actual location slightly north of image)
Meyers Lake village (red)
Both Canton and Plain townships are split into discontiguous parts by Canton city. As long as those townships are not chopped they are viewed each as single units. Canton city and the townships have sufficient other connections and do not meet the definition of shared units.
Meyers Lake is only bordered by the two townships, and those are both discontiguous from the parts with the nodes. Meyers Lake has no path to any other node so it is an isolated unit.
Based on the special rule Meyers Lake would consider counties and then townships for connections, but none would exist. The rule allows the subunits to be further divided, in this case to precincts. Meyers Lake has connections to precincts in both townships, so Meyers Lake has local connections to both townships by the isolated unit rule.
muon2:
The previous posts defined the rules for local connections which are required for all areas within a district. This post defines regional connections and what constitutes a link for erosity.
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.
Definition: Regional connection. There is regional connection if there is a continuous path of all season numbered state or federal highways or regularly scheduled ferries that allow one to travel between the nodes of two geographic units without entering any other geographic unit. If the node is not on a numbered highway, then the connection is measured from the point of the nearest numbered highway in the geographic unit to the node. Highways along the border of two units are considered to be in either or both of the units as needed to form a connection.
Item E5: Regional connections are treated like local connections for shared units, isolated units, fragments, and isolated fragments.
Item E6: A link exists between two nodes in the same county if there is a local connection between the nodes. A link exists between two nodes in different counties if there is a regional connection between nodes.
Example:
This example is based on the one for connection. 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.
Since the regional roads are in the same configuration and they follow the same rules as local roads, these are the regional connections:
Burr is regionally connected to Central Agnew. The regional path from Burr to Agnew enters Agnew in the Central Agnew fragment.
Burr is only locally connected to Calhoun. The obvious shortest regional path cuts a corner of Agnew and no other path stays only within those two units.
Burr is regionally connected to Elbridge. The regional path is along the border between Burr and Agnew which counts for staying in Burr.
Calhoun is regionally connected to Central Agnew. The regional path from Calhoun to Agnew enters Agnew in the Central Agnew fragment.
Calhoun is only locally connected to West Agnew. The regional path entered at Central Agnew, and though it passes through West Agnew, that does not count as a connection for a fragment.
Calhoun is regionally connected to Dawes.
Dawes is regionally connected to Central Agnew. The shortest regional path from Dawes to Agnew enters Agnew in the Central Agnew fragment.
Dawes is regionally connected to West Agnew. Neither shortest regional path from Calhoun or Dawes entered at West Agnew, so it is regionally isolated. As an isolated fragment it uses other regional links, and that includes the regional road from Dawes to West Agnew.
Dawes is only locally connected to East Agnew. East Agnew is regionally isolated, but there are no other regional paths to East Agnew.
Dawes is regionally connected to Elbridge.
Elbridge is regionally connected to Central Agnew. The shortest regional path from Elbridge to Agnew enters Agnew in the Central Agnew fragment using the border between Burr and Agnew.
Elbridge is only locally connected to East Agnew. East Agnew is regionally isolated, but there are no other regional paths to East Agnew.
The equivalent graph colors the nodes of the districts 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.
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