Local vs regional road connections

[Torie]

Before responding to the AL example, I want to make sure I understand your suggestion about East Sparta. Am I correct that this only applies to paths involving a regional connection across a county line? That is, I want to make sure we aren't talking about a highway entering a fragmented subunit from an adjacent subunit (eg Plain twp along OH 687 from Jackson twp to the west), then following a highway path through an intervening subunit (eg Canton city) before reaching the node.  I think we agreed that Jackson and Plain aren't connected since the only local road path has to go through either Canton city or North Canton city.


It does not apply to fragments, precisely because we want to encourage them to be all in one CD, so we need special rules for that, so as not to penalize such unification with more highway cuts. However, it would otherwise apply, and need not involve crossing a county line. Assume in the drawing below, that we have subunits within a  macro-chopped county. I think we have two highway cuts here, where each of the red lines are. The intervening subunit generating the chops does not preclude a state highway connection (relevant if the subunit to the left is in a different county), even if the only state highway connection is through the intervening subunit, but it does generate highway cuts. Do you disagree? By the way, I like these little graphics. They make things easier to understand than referring to real maps, with a lot of names, and then you have to look up the names, and so forth. The intervening subunit is named Torieville by the way, and the township it intrudes into is named Muon2.

[muon2]

I'm not sure I agree. Suppose there is no road through Torieville as you show, but instead there is a corner cut in the NW corner of Muon2 into the rectangular subunit in the north and from there I can get to the Muon 2 node. Does that still count? I doesn't really make sense to me that I can swing through the northern subunit to get my path. Does it only make sense because Torieville is an intrusion in Muon2?

BTW, the diagram for your proposal would create a connection between Jackson and Plain twps regardless of the fragment. They are contiguous at the NW corner where a small subdivision in Plain exists. However there is no road that connects it to the rest of Plain without going through North Canton. I'm not sure that makes sense either.

Basically I am advocating that when a subunit has bits of population at the edges that don't directly connect to the node due to geography or intervening subunits, I don't want that those bits of population to create erosity through connections. To me such disconnected populations are effectively fragments but for a bridge of land with no roads.

[Torie]

I'm not sure I agree. Suppose there is no road through Torieville as you show, but instead there is a corner cut in the NW corner of Muon2 into the rectangular subunit in the north and from there I can get to the Muon 2 node. Does that still count? I doesn't really make sense to me that I can swing through the northern subunit to get my path. Does it only make sense because Torieville is an intrusion in Muon2?

That's not the most direct route, so it's irrelevant. If you are telling me the most direct route is what you describe, from node to node, I will live with that. It won't happen very often that a road nicks a subunit, and then wanders off here and there, before getting back to the node. Don't make the perfect the enemy of the good.

BTW, the diagram for your proposal would create a connection between Jackson and Plain twps regardless of the fragment. They are contiguous at the NW corner where a small subdivision in Plain exists. However there is no road that connects it to the rest of Plain without going through North Canton. I'm not sure that makes sense either.

That is not the most direct route from node to node, so it's irrelevant.

Basically I am advocating that when a subunit has bits of population at the edges that don't directly connect to the node due to geography or intervening subunits, I don't want that those bits of population to create erosity through connections. To me such disconnected populations are effectively fragments but for a bridge of land with no roads.

I don't think that makes sense. And where anything intervened, it would cleanse the cut. I would really prefer we use drawings, rather than toss these names around. Is there any highway cut that you would count in the map below?

[muon2]

After some time off to concentrate on some other work, it's time to bring back the discussion. At Torie's suggestion I've made some generic maps to show the connection rules as I understand them. Let me start with the basic rules, and for this purpose we can consider these to be local units so I won't worry about state highways.

Here's the basic rule: Two units are locally connected if there is a continuous path of public roads that allow one to travel between the two nodes without entering any other unit.

An important definition: A road that is along a border between two units is considered to be in each of those units.

Another definition: If there are two or more connections between two units then the connection is considered to be on the shortest path (by time as determined by generally available mapping software).



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.

Based on 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.

To be used to compute erosity the map with connections is converted to an equivalent graph. The equivalent graph reduces each connection to a single link between nodes.



If we are ok with this rendering of connections, I'll go on to the next example that gets at Torie's issue in the preceding post.

[Torie]

This will have to wait until I finish writing my referendum language. There is no template out there, that precisely fits (nothing seems to have passed by petition), and the language in the 2003 referendum put on the ballot by the Common Council is a total mess, and a disaster, and incomplete, and thus I found it necessary to reorganize it, and write my own language.

It is kind of fun to draw a map that has a pretty good chance of actually becoming law. Perhaps it will be the first of many, if we get our act together with this project, and perhaps get a court interested in the approach to prime the legislative pumps.

[muon2]

I'll go ahead and post a second example. Like the first example, this one does not involve any chops. If this makes sense, I can move to the impact of chops.



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.



Now suppose that Dewey annexes all of the river in Adlai. Adlai becomes discontiguous, but none of the connections change since there re no river crossings in Adlai. Therefore the equivalent graph would remain the same. Furthermore, one can say that there is an equivalency between the version with the river in Adlai and the river in Dewey in terms of their connections despite the fact that Adlai is discontiguous in the latter version.

[Torie]

After some time off to concentrate on some other work, it's time to bring back the discussion. At Torie's suggestion I've made some generic maps to show the connection rules as I understand them. Let me start with the basic rules, and for this purpose we can consider these to be local units so I won't worry about state highways.

Here's the basic rule: Two units are locally connected if there is a continuous path of public roads that allow one to travel between the two nodes without entering any other unit.

An important definition: A road that is along a border between two units is considered to be in each of those units.

Another definition: If there are two or more connections between two units then the connection is considered to be on the shortest path (by time as determined by generally available mapping software).



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.

Based on 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.

Yes,  because there is no uninterrupted public pavement connection. But bear in mind the nick rule for state highway inter-county connections. That is not relevant here however.

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.

To be used to compute erosity the map with connections is converted to an equivalent graph. The equivalent graph reduces each connection to a single link between nodes.



If we are ok with this rendering of connections, I'll go on to the next example that gets at Torie's issue in the preceding post.

I agree with everything you said in both posts, but have some comments in red, one of which is above in this post.

[Torie]

I'll go ahead and post a second example. Like the first example, this one does not involve any chops. If this makes sense, I can move to the impact of chops.



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.

Agreed, unless there is a public pavement connection between C and A that is a more circuitous path not shown on the map to its north. If there were such a pavement connection, then C and A are connected, but for erosity purposes in a macro chopped county, there would be two highway cuts counted since the most direct public pavement route between the nodes of C and A must go into D and then out of D again to get to the node of A.

Here's the equivalent graph.



Now suppose that Dewey annexes all of the river in Adlai. Adlai becomes discontiguous, but none of the connections change since there re no river crossings in Adlai. Therefore the equivalent graph would remain the same. Furthermore, one can say that there is an equivalency between the version with the river in Adlai and the river in Dewey in terms of their connections despite the fact that Adlai is discontiguous in the latter version.

Yes, and just for record here, a CD taking in C and the disconnected portion of A on the west side of the river, with another CD taking in the portion of A on the east side of the river, does not count as a chop.

[muon2]

So, let's try this one (now with edits based on discussion that followed the original post). I've taken the 5 unit example and overlaid a district that chops Agnew into two fragments. I am assuming a simple chop that involves no defined subunits. I'll start with my working definition of fragment.

A fragment is a contiguous geographic unit created by a chop of a larger whole geographic unit. A fragment is like other geographic units that are determined by fixed political boundaries in that fragments have all the same data and can be built from individual census blocks. However, fragments are artifacts of a specific redistricting plan and have no special legal status other than to facilitate mapping.

Like all other geographic units a fragment has a node and links. A fragment that contains the node of the whole geographic unit has that same node as the node of the whole unit. A fragment that does not contain the node of the whole geographic unit has as its node the node of the subunit in the fragment with the largest population.

Connections for fragments follow from the connections that exist in the whole geographic unit. Paths used as connections that enter the whole unit within a fragment are links to the node of that fragment. There is a link between two fragments in the same whole unit if the fragments are locally connected.



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.



My comments from the original post:
Graphs require nodes and links. Each geographic unit or relevant part gets a node, and here the relevant parts are the fragments created by the chop. The node for the east fragment of Agnew is defined as the original node. At this point I'm going to defer a discussion of the exact location of the node in the west fragment as it doesn't affect this example, but I have included a node with an outline only to complete the graph. I've colored the nodes to reflect which districts they are in.

The links inherited from the unchopped graph are shown as solid lines, based on the shortest path between nodes. An additional solid line shows the link between the fragments, which exists since the fragments have a path between them. The red lines indicate links that go between districts. On this part we have been consistent.

I've also drawn two dashed lines showing connections between West Agnew and both Burr and Dawes. I found we have generally been including them as connections for local subunits, but not for chopped counties with regional (ie state highway) connections. The West Agnew-Dawes connection is internal and wouldn't affect the erosity determination, but the West Agnew-Burr connection goes between districts so it matters.

So the question is should the dashed connections always count, never count, or only count for local connections?

The answer used above is that the dashed connections never count.

[Torie]

Please give me the policy reasons for counting the dotted lines for locality cuts. My point of view, absent seeing mischief created thereby, is to not give such fragments a node, so that at least in the example above, the fragments have no impact on erosity scores, be it inter county or local.

[Torie]

Yeah, but I fail to see the need for a pseudo node. The shortest path involves highway cuts, pseudo node or not. You have not persuaded me yet, what purpose it serves that somehow changes something.

[muon2]

Yeah, but I fail to see the need for a pseudo node. The shortest path involves highway cuts, pseudo node or not. You have not persuaded me yet, what purpose it serves that somehow changes something.

I need a representation for geographic data. Suppose you were putting together a spreadsheet to help you balance populations for districts in a state. You might put one county on each row of the spreadsheet and one district on each column. At the intersection of the county row and district column you would but the population of the county. You could then sum the column to get the district population.

Minus information about the neighbors those cells with populations in them are nodes. That is they are a specific point (cell) that has information about the population and the assigned district.

Now suppose you decide to split the population of a county between two districts. In the spreadsheet you would fill population into a new cell for that county in a different column and deduct that population from the original cell. You have created a new node for that new information about the chopped county.

The only thing I have added for my nodes that wasn't part of the spreadsheet example is information about the neighbors. In my examples I display that neighbor information graphically in the form of links. It doesn't change the underlying need to create a new node to hold the split population from the fragment. It does facilitate visualizing where the links now fall after the chop.

[Torie]

I am sorry, but you are not making any sense to me whatsoever. The population of the fragment can be calculated without creating a node. I see no use for the node whatsoever. You need to show me where it would make some difference having a node or not for the fragment without a node. A fragment does not become its own subunit.

[muon2]

I am sorry, but you are not making any sense to me whatsoever. The population of the fragment can be calculated without creating a node. I see no use for the node whatsoever. You need to show me where it would make some difference having a node or not for the fragment without a node. A fragment does not become its own subunit.

Of course I can calculate the population, and the VAP, and the BVAP, etc, but it's inefficient to do all those calculations every time I reference a place. The way GIS usually works is it creates a container for each unique geographic entity. If it's an efficient GIS then it only does those calculations once and stores them until the entity is modified. A large unit like a district is going to add all the relevant pieces assigned to it. Redistricting does not permit me to assign one geographic entity to two different districts, so I split the original unit into two.

It's really no different than building up from blocks and putting a pin in it as a reference point, which is what jimrtex was doing with the Google maps of the Hudson wards. I'm just calling those pins the nodes. However by breaking down a larger unit instead of building up from locks, I more easily retain the knowledge of the links that exist before the chop. So the only difference from jimrtex's Hudson wards and my nodes is that I carry the link information as well as population, registered voters, and everything else.

The fragment is not a subunit in the sense of a political subdivision that exists prior to a specific redistricting plan. However, a fragment has all the same information associated with it as a preexisting subunit. So the fragment functions in all ways as if it were a subunit, just that it is an artifact of a specific plan.

[Torie]

I still don't understand why you have to create a pseudo node. Populations are calculated by census block. Maybe we have to talk on the phone about this one.

[muon2]

I've edited my initial example to more clearly define fragments and walk through the individual links formed and not formed in the example.

So, let's try this one (now with edits based on discussion that followed the original post). I've taken the 5 unit example and overlaid a district that chops Agnew into two fragments. I am assuming a simple chop that involves no defined subunits. I'll start with my working definition of fragment.

A fragment is a contiguous geographic unit created by a chop of a larger whole geographic unit. A fragment is like other geographic units that are determined by fixed political boundaries in that fragments have all the same data and can be built from individual census blocks. However, fragments are artifacts of a specific redistricting plan and have no special legal status other than to facilitate mapping.

Like all other geographic units a fragment has a node and links. A fragment that contains the node of the whole geographic unit has that same node as the node of the whole unit. A fragment that does not contain the node of the whole geographic unit has as its node the node of the subunit in the fragment with the largest population.

Connections for fragments follow from the connections that exist in the whole geographic unit. Paths used as connections that enter the whole unit within a fragment are links to the node of that fragment. There is a link between two fragments in the same whole unit if the fragments are locally connected.



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.



My comments from the original post:
Graphs require nodes and links. Each geographic unit or relevant part gets a node, and here the relevant parts are the fragments created by the chop. The node for the east fragment of Agnew is defined as the original node. At this point I'm going to defer a discussion of the exact location of the node in the west fragment as it doesn't affect this example, but I have included a node with an outline only to complete the graph. I've colored the nodes to reflect which districts they are in.

The links inherited from the unchopped graph are shown as solid lines, based on the shortest path between nodes. An additional solid line shows the link between the fragments, which exists since the fragments have a path between them. The red lines indicate links that go between districts. On this part we have been consistent.

I've also drawn two dashed lines showing connections between West Agnew and both Burr and Dawes. I found we have generally been including them as connections for local subunits, but not for chopped counties with regional (ie state highway) connections. The West Agnew-Dawes connection is internal and wouldn't affect the erosity determination, but the West Agnew-Burr connection goes between districts so it matters.

So the question is should the dashed connections always count, never count, or only count for local connections?

The answer used above is that the dashed connections never count.

[jimrtex]

Yeah, but I fail to see the need for a pseudo node. The shortest path involves highway cuts, pseudo node or not. You have not persuaded me yet, what purpose it serves that somehow changes something.

I need a representation for geographic data. Suppose you were putting together a spreadsheet to help you balance populations for districts in a state. You might put one county on each row of the spreadsheet and one district on each column. At the intersection of the county row and district column you would but the population of the county. You could then sum the column to get the district population.

Minus information about the neighbors those cells with populations in them are nodes. That is they are a specific point (cell) that has information about the population and the assigned district.

Now suppose you decide to split the population of a county between two districts. In the spreadsheet you would fill population into a new cell for that county in a different column and deduct that population from the original cell. You have created a new node for that new information about the chopped county.

The only thing I have added for my nodes that wasn't part of the spreadsheet example is information about the neighbors. In my examples I display that neighbor information graphically in the form of links. It doesn't change the underlying need to create a new node to hold the split population from the fragment. It does facilitate visualizing where the links now fall after the chop.

I use sumif() to calculate each district's population. I place each unit (county or block) in a row, and then in one column insert the district number. If a unit needs to be divided, I split it into multiple rows.

[muon2]

Yeah, but I fail to see the need for a pseudo node. The shortest path involves highway cuts, pseudo node or not. You have not persuaded me yet, what purpose it serves that somehow changes something.

I need a representation for geographic data. Suppose you were putting together a spreadsheet to help you balance populations for districts in a state. You might put one county on each row of the spreadsheet and one district on each column. At the intersection of the county row and district column you would but the population of the county. You could then sum the column to get the district population.

Minus information about the neighbors those cells with populations in them are nodes. That is they are a specific point (cell) that has information about the population and the assigned district.

Now suppose you decide to split the population of a county between two districts. In the spreadsheet you would fill population into a new cell for that county in a different column and deduct that population from the original cell. You have created a new node for that new information about the chopped county.

The only thing I have added for my nodes that wasn't part of the spreadsheet example is information about the neighbors. In my examples I display that neighbor information graphically in the form of links. It doesn't change the underlying need to create a new node to hold the split population from the fragment. It does facilitate visualizing where the links now fall after the chop.

I use sumif() to calculate each district's population. I place each unit (county or block) in a row, and then in one column insert the district number. If a unit needs to be divided, I split it into multiple rows.

For defined subunits I will add rows, but for direct chops I split the value into 2 columns or the two districts. It allows me to cross check the value of the row to reduce errors.

[muon2]

Here's an example that shows the difference between a fragment and a subunit when determining connections.



I've swapped Dawes for additional area in East Agnew so that East Agnew interrupts the path to West Agnew from Burr and Elbridge. As before the shaded area represents a district that chops whole unit Agnew.


This part stays the same:

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 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.

There is a path between West Agnew and East Agnew that forms a local connection, so there is a link between those fragments.



Here's the part that's different:

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 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.

For both these links to West Agnew the path enters West Agnew, but crosses into East Agnew. If the fragments were subunits, then the paths between nodes would cross into a different subunit and wouldn't count. But the nodes only were used for the whole unit connections and don't figure into the paths for the fragments.



Here's the equivalent graph with 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.

[Torie]

In that latter graph, however, you have two highway cuts from A to D for the erosity count.

[muon2]

In that latter graph, however, you have two highway cuts from A to D for the erosity count.

I'm not sure I follow what you are saying. By shifting Dawes to the eastern district the only cut links are from Calhoun to Dawes and between the two fragments, but the link from Dawes to East Agnew is not cut and there is no link cut from Dawes to West Agnew because that's the secondary path. When Dawes was was in the western district in the first example then there were three cut links - from Dawes to East Agnew, from Dawes to Elbridge, and between the two fragments.

[Torie]

In that latter graph, however, you have two highway cuts from A to D for the erosity count.

I'm not sure I follow what you are saying. By shifting Dawes to the eastern district the only cut links are from Calhoun to Dawes and between the two fragments, but the link from Dawes to East Agnew is not cut and there is no link cut from Dawes to West Agnew because that's the secondary path. When Dawes was was in the western district in the first example then there were three cut links - from Dawes to East Agnew, from Dawes to Elbridge, and between the two fragments.

I meant A to B. My bad. The most direct route between their respective nodes cuts through the other CD.

[muon2]

In that latter graph, however, you have two highway cuts from A to D for the erosity count.

I'm not sure I follow what you are saying. By shifting Dawes to the eastern district the only cut links are from Calhoun to Dawes and between the two fragments, but the link from Dawes to East Agnew is not cut and there is no link cut from Dawes to West Agnew because that's the secondary path. When Dawes was was in the western district in the first example then there were three cut links - from Dawes to East Agnew, from Dawes to Elbridge, and between the two fragments.

I meant A to B. My bad. The most direct route between their respective nodes cuts through the other CD.

Exactly. And that was why I drew this example. Since these are fragments all I do is look at which fragment the path from Burr to Agnew enters. That happens to be the west, so the link goes to West Agnew and since both Burr and West Agnew are in the same district no cut is observed.

It sounds like you prefer to change the interpretation and assign that path two cuts. That part of West Agnew is actually cutting the path from Elbridge to East Agnew, too, so should it count as four cuts? Either way I'm not sure how to diagram that in a way that I can automate the erosity counting process. Any thoughts?

[Torie]

In that latter graph, however, you have two highway cuts from A to D for the erosity count.

I'm not sure I follow what you are saying. By shifting Dawes to the eastern district the only cut links are from Calhoun to Dawes and between the two fragments, but the link from Dawes to East Agnew is not cut and there is no link cut from Dawes to West Agnew because that's the secondary path. When Dawes was was in the western district in the first example then there were three cut links - from Dawes to East Agnew, from Dawes to Elbridge, and between the two fragments.

I meant A to B. My bad. The most direct route between their respective nodes cuts through the other CD.

Exactly. And that was why I drew this example. Since these are fragments all I do is look at which fragment the path from Burr to Agnew enters. That happens to be the west, so the link goes to West Agnew and since both Burr and West Agnew are in the same district no cut is observed.

It sounds like you prefer to change the interpretation and assign that path two cuts. That part of West Agnew is actually cutting the path from Elbridge to East Agnew, too, so should it count as four cuts? Either way I'm not sure how to diagram that in a way that I can automate the erosity counting process. Any thoughts?

I don't know about programming, but yes, it's four cuts. Every time a node road crosses a CD line it counts as a cut. KISS. The map looks horrible, and it deserves to be dissed. I have worked very hard to avoid line drawing that causes a node road to go into another CD and then back out into the old CD again. It's a very bad practice.

[muon2]

At this point I need to return to a real map to see the implications. I'll use your AL plan that I brought up earlier in the thread. The focus is on the erosity measure in Washington county. Since these are counties, the roads that count for paths and erosity are the state (brownish lines) and federal (thick grey lines).

I've recreated the map and enlarged the chop in Washington county to illustrate the rule.



Before the chop Washington used Chatom, the county seat, as the node. It was connected to Mobile to the south via AL-17 and US-45.

Regional connections involve paths that cross county lines, and require a continuous path of numbered state and federal highways between nodes. Using Chatom, Washington was connected to Mobile to the south via AL-17 and US-45. It was connected to Clarke to the east via AL-56 and US-43. It was connected to Choctaw to the north via AL-17. The graph above has links representing those connections.

In the past we've agreed that there is only one cut link between the two parts in Washington, that is the chop in Washington only counts as 1 to erosity.

If I count paths that cut through the boundary - there are two now, one from the path from Choctaw to the north and one from the path from Clarke to the east. If instead I pick any point in the green fragment as the node (which shouldn't be necessary in this case) then either the path from Choctaw to the north or the path from Clarke to the east passes out of the green fragment and back again to get to the nominal node. Either way it seems like your interpretation would now make this count as two cuts, one for each state highway path passing through the boundary.