Local vs regional road connections

[muon2]

If there are no subunit chops then the nodes are where the stars are in this map. The fragments are all considered connected since there is no chop. There is no connection between the Meyers Lake node and either township regardless of the the existence of adjacent fragments. That's the sufficient condition to trigger the rule I wrote.



The existence of the fragments becomes important in the remedy outlined by the rule. The township fragments are contiguous to Meyers Lake and local connections exist, so both connections become defined by the rule. Once defined the connections can be cut by a district line adding to erosity.

It's important to define the connections before considering where the district lines are. Each chop generates a new look at connections, and this is repeated at each level - county chops to subunits, subunit chops to fragments, etc.

Suppose that neither Plain nor Canton twp are chopped, but are in different districts. Meyers Lake will have a cut link to whichever township is in the other district. Now suppose that Plain twp is chopped and Meyers Lake is in the same district as the adjacent Plain twp fragment but not in the same district as Canton twp (like your example below on the right). If we waited to define connections after the twp chop, then the Canton fragment would be assigned a pseudonode, Meyers Lake would be connected to it, and the special rule would not be invoked. This would lead to no cut connection between Meyers Lake and Plain (the question mark would vanish). In my view the question mark is a cut link whether or not Plain is chopped.

Below is the graphic. I put up the question mark, because there is not Canton township node between Meyers and Canton township node. So if no chop of Canton township, I guess no highway cuts at all around Meyers. But if it is chopped, does that generate a highway cut not only from Canton Township to Canton City, but also from Meyers to Canton township to the south? Having two highway cuts generated by the chop of Canton township there, seems a bit much. But maybe that is necessary to make the rule work in general, even if a bit much in this instance, since the rule is that when a fragmented subunit is chopped, each fragment generates its own node. So I guess maybe there is no escape.

I added emphasis to your quote since the critical point is that the creation of nodes by chopped subunit fragments not happen until connections are defined at the whole subunit level.

[muon2]

How about the below? I see what you are saying I think, but all of this special rule making only obtains when a unit is fragmented, and not otherwise.

"If there is no connecting path from the node of a geographic unit to the node of another contiguous unit which is fragmented, then a local connection exists from that geographic unit to such contiguous fragmented unit."

The above generates a highway cut where one would not otherwise exist due to fragmentation of a contiguous subunit (even if the fragmented subunit is not chopped). That gets where we want to go, right?

There are examples where this happens at the county level and fragmented units don't come into play. For example Pitkin county CO has no regional connection from its node at Aspen to any contiguous county. There are roads that aren't all-season state highways that can be used to make a path, and Pitkin has to be linked to at least one other county. I suspect there are other such examples at both the county and subunit level, so I didn't want to restrict the special rule to cases like Meyers Lake that do involve fragmented units.

[muon2]

I brought up Pitkin because there are two or three possible ways to connect it to other counties. If I treat it like Meyers Lake and allow all local paths to become connections, then that establishes where cuts will occur. I think it should have those connections and it would then affect the erosity measurement, just like Meyers Lake.

It's also possible that there is no connection due to geography even when there is no discontiguous fragment in play. We had a similar case for the cross county connection from Pike twp to Tuscawaras county. The highway connection is blocked by East Sparta, and there is no local road in the unincorporated area that connects to OH-800 south of East Sparta. Even if the township had no local roads west of East Sparta we would still define that as a connection to be cut based on OH-800. That causes it to affect erosity.



I don't want three separate rules if one rule captures the same basic idea. The idea is that in some cases a link that should exist doesn't because of specific geography - fragment from other units, mountains, rivers or any other geographic isolation.

[muon2]

In both Meyers Lake and Pitkin the standard rule provides for no connections to their respective nodes. The aim of the special rule is to insure that all units have at least one connection that could be cut. My intent is to say that if geography leaves no connection to a unit then one must look at local connections that don't provide a path under the standard rule. Those connections created under the special rule can be cut for erosity just as one would cut the connections formed by the standard rule.

My interpretation of the right hand image is that the horizontal red line is not a link. Plain twp is kept whole so there is only one link between it and Canton city based on the standard rule. The special rule (in the second quote below) is never invoked since the fragments are never viewed separately (per the first quoted rule) and there is a connecting path between the whole subunits.

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[muon2]

One of my guiding principles is that any district should be internally connected so that one can travel within a district without leaving it. Some of the connections require a stronger standard, and those are defined as the regional connections that cross county lines. This connection principle only bows when a well-defined geographic unit that has internally unreachable areas is kept intact. I'm not thinking only of fragments, but also areas like the part of St John the Baptist parish south of the Mississippi where there is no bridge or ferry within the parish.

In order to capture the connection principle a geographic map can be transformed into a mathematical graph that many would describe as a network. In the transformation each unit in the map is represented by a node and connections that go beyond mere contiguity are represented by links. The connections must be defined so that the result is a connected graph: a path can be traced from a node to any other node through some series of links. This map of AL is that representation for counties which are the highest level of units in the state.



A district plan represents a partition of the graph for the state into a set of subgraphs corresponding to the set of districts. The connection principle is enforced by requiring that each district subgraph is itself a connected graph. Then the set of cut links between district subgraphs measures the erosity. That was how I transformed and measured your plan for AL with this graph. I would invalidate a plan that followed contiguity within the district, but not the connections in the graph.



The next step is defining rules for dealing with units that are chopped in a plan. In particular when a macrochop forces a unit to be split into subunits, I need the same principles that apply at the coarser level still apply at the finer level. That is a critical step that allows the erosity measurement to still make sense in dense population areas as I demonstrated with the King county CDs earlier in the thread.

[muon2]

I thought you wanted to step way back. I guess I stepped too far back.

Let me apply the principles I laid out to your specific questions.

Let's step way back for a moment. Can two counties be in one CD, if the state highway link does not go node to node (putting aside the nick issue)?

Two whole counties cannot be in the same CD if they are not connected directly or by a path through other nodes in the same CD. Otherwise the resulting CD would not follow the connection principle.

Has it always been your intent that if there is a highway cut, it counts for the ferocity (erosity?) count, even if does not go node to node?

Links represent connections between nodes and links only exist between nodes (eg the AL map and its kin). Cut links between nodes in different districts always count towards erosity. What isn't shown in the AL map representing your CD plan is that the node in each of the chopped counties is replaced by two linked nodes (or more if there is a macrochop) on either side of the district line.

This is a more general issue, that the highway interruption issue that East Sparta represents. This is why it is so important to fully understand the intended policy, before dealing with the language.

[muon2]

There's another principle at play here that may pertain to East Sparta. Connections that exist without a chop cannot disappear by using a chop. If a chop allowed connections to disappear then it would invite all sorts of mischief. A chop can be used judiciously to reduce erosity, but that's by burying the connections within the district and not exposing them to cuts.

When there is a simple chop of a county the county node is replaced by two nodes representing the two pieces. The existing links are assigned to one or the other of the nodes based on where the highway crosses the county line. I think we agreed on this one already from the example of Washington county AL. The point you raise is that you would like to have a chop that is only locally connected to the rest of the district, and that requires a different form of the rule assigning connections to a simple chop.

With the discussion of Stark I am presuming a macrochop. That means that Stark is replaced by a new network of its subunits. I used the Mecklenburg NC map as an example of the subnetwork created by a macrochop.

The blue links are state highway links between counties. The Mecklenburg network shows both the local (gold) and regional (blue) connections to the subunits (pink are contiguous only).



Your point here is that it should be permitted to connect the two northern subunits to Iredell in one district and connect the one just south of those two to Cabarrus in a different district, while the rest go to some other district which isn't the same as Cabarrus. Am I getting that right?

[muon2]

It sounds like we are on the same page in regards to disallowing local connections across county lines when a macrochop is present in either of the counties. I understand the distinction you are making between units due to simple chops and I am thinking on it.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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?

[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.

[muon2]

This would take us in a radically new direction. I think it means that judicious chops to reduce erosity would no longer exist. All primary paths must cross the boundary in a geographic unit at some point, so a plan can't take advantage of a convenient split to make that reduction. Washington county AL above was just such an example.

The radical part is that it tilts the balancing effect of erosity vs chops. Part of the balance was to potentially tighten a border with a chop, and do it in a way that reduces erosity. With your interpretation that option is precluded.

Ironically it was a plan of yours in OH that beautifully (artistically ) sliced through one county along township lines to make a rectangle that helped me formulate the way of counting erosity in a chopped county. We didn't want to punish the artistic chop. I'm not sure I want to give up on that.

[muon2]

As I played with it, it became clear that the lieklihood of the effect increased with the number of units and number of districts in play. With some work I found a reduced arrangement that I hope illustrates the issue.

Here's a set of 9 "counties" each with one path linking them to their neighbors, so the links are in one-to-one correspondence with the paths. The district line creates no chops. The number of cut links/paths is 7 which is the erosity.



Here are the same 9 counties with a chop of E to make a perfect rectangle of the shaded district. The node for the fragment is indicated by a hollow star. My principle holds that each unit or fragment can have at most one link to a given neighbor, so there is only one link to cut between the two fragments. By my measure of counting cut links the following plan would have an erosity of 5. Both plans would go forward as Pareto equivalent, one with fewer chops and one with lower erosity.



Using a cut paths (instead of cut links) method the erosity of this second plan still has 7 cut paths. Since it has a chop that the first version lacks, it fails to gain Pareto equivalence and would be eliminated from consideration. I think that's a mistake, and that plan should be part of the set for consideration as exactly the the kind of trade off the system is intended to permit.

I hope you haven't completely abandoned your prior love for rectangular districts.

[muon2]

So does that mean we are back in agreement that there is just one cut link in Washington county, that is the link between the two fragments of the county?

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.

[muon2]

As opposed to a second cut with respect to the highway going to the county to the north?  Yes, I think that I agree with that, if that is the issue. On the other hand, if Washington County were not chopped, there would be two highway cuts, so by chopping the county, the map is deemed less erose? That is not good. Do we need a different rule for units in macro-chopped counties?

The rule is quite different in macrochopped counties. Erosity is always higher in macrochopped counties, because all the subunits create links. For me Washington county is like my rectangular district example, the line cuts through in a reasonably nice way and erosity drops.

Here are my thoughts on why that's ok, even if the line forming the chop is somewhat erose.

- The chopped unit is not macrochopped, so the unit is presumably of lower population. The principle is that in low density areas line wiggles are less important.

- In some areas one can apply the MI rule that a chop cannot split more than one subunit between the same two districts.

- The system shouldn't exclude plans that have used judicious chops to reduce erosity (eg the rectangle district).

- It is better to err on the side of inclusiveness for plans. The worst offenders need to be eliminated, but districts that have minor defects don't have to be hammered as much (eg look at the Washington county chop in the context of the whole districts in the plan).

- There are other metrics that can still apply and winnow the field of plans (skew, polarization, inequality).

[muon2]

This would take us in a radically new direction. I think it means that judicious chops to reduce erosity would no longer exist. All primary paths must cross the boundary in a geographic unit at some point, so a plan can't take advantage of a convenient split to make that reduction. Washington county AL above was just such an example.

The radical part is that it tilts the balancing effect of erosity vs chops. Part of the balance was to potentially tighten a border with a chop, and do it in a way that reduces erosity. With your interpretation that option is precluded.

Ironically it was a plan of yours in OH that beautifully (artistically ) sliced through one county along township lines to make a rectangle that helped me formulate the way of counting erosity in a chopped county. We didn't want to punish the artistic chop. I'm not sure I want to give up on that.

OK, I shall accede to your will - for the moment. It's troubling, but maybe it does tend to force chops to be put in benign places, and shut out placing chops in places with political motives. But it is something that will need to be defended, when the point is raised, that a chop can reduce erosity. That seems counter-intuitive.

What was the sliced county in Ohio that was an object d'art?

The rectangle district example will be my first line of defense. Thanks for pushing me to create it.

I can see your OH map in my mind's eye, but I can't recall which thread it was on. I believe the chop was in the north central part of the state. Your MI map with the Clinton chop could have been in that category, but it just crossed the macrochop threshold.