Local vs regional road connections

[muon2]

So if I can arrange to do a little mini chop into Yates in the map below, two highway cuts disappear. Is that right?

Yes it could.

Moving part of Yates into 23(?) wound make it protrude less. It makes sense to me that it would reduce erosity.

[muon2]

And here I thought I had been so consistent with this use of chops to reduce erosity for the better part of three years. I know it came up in our recent discussion of NC maps. Perhaps it was a case of non liquet rather than a lacuna? Either way it should be clear now, I hope.

[muon2]

Here's the analysis I get of our two currently best plans for NY. I'll look at upstate only and treat the boundary in the Catskills with the NYC UCC CDs as a state line.



I'll neglect the bridge chop for this exercise. There are 7 chops with macrochops in Erie and Saratoga. No subunits are chopped and there are no UCC penalties. CHOP = 7.

Not counting the effect of subunits in the macrochops, the erosity is 37. The effect of subunits raises the erosity of the Buffalo CD from 3 to 16 (internal Erie is 11 instead of 1, Erie to Niagara goes from 1 to 3, Erie to Chatauqua stays at 1, and Erie to Cattaragus goes from 0 to 1). The effect of the Saratoga macrochop raises the Albany CD erosity from 7 to 15 (internal Saratoga is 7 instead of 1, Saratoga to Washington adds 1 at NY-29, and Saratoga to Schenectady adds 1 at NY-147. EROSITY = 58.



This version has 4 chops including the macrochop of Erie. There is a UCC pack penalty for Albany. CHOP = 5.

Not counting the effect of the Erie macrochop the plan has an erosity of 36. The macrochop increases the Buffalo-Niagara erosity from 3 to 11. EROSITY = 44.

Note that even without the effect of the macrochops the plan with the UCC penalty gets lower erosity, and at best they are even if my arrangement of the chop in Allegany is used. The Albany pack plan would be eliminated on the chop score.

One thing we did in the MI exercise at train's suggestion was sum the CHOP and INEQUALITY scores, instead of using INEQUALITY as a tie breaker. I suspect the Albany pack plan has lower INEQUALITY by virtue of the chops, I know I pushed 3 of my CDs right to the 0.5% limit. That might be a method to keep the Albany pack plan in the hunt.

[muon2]

The Albany pack plan would be eliminated on the chop score.

Vis a vis which other map?


My plan in that post.

One thing we did in the MI exercise at train's suggestion was sum the CHOP and INEQUALITY scores, instead of using INEQUALITY as a tie breaker. I suspect the Albany pack plan has lower INEQUALITY by virtue of the chops, I know I pushed 3 of my CDs right to the 0.5% limit. That might be a method to keep the Albany pack plan in the hunt.

That is just an accident is it not, as opposed to something systemic favoring pack plans? Accidents should not drive policy.

Not entirely an accident. More chops, including macrochops, tend to reduce inequality. A plan that chops one county and one subunit for every CD above the first should be able to get perfect equality.

[muon2]

A regions is a group of whole counties with a whole number districts within the required variance form the quota. A district of whole counties is a region with 1 district. Two districts that are whole counties except that they share a chopped county are effectively a region with 2 districts. Extending that to a whole state, each chop reduces the number of regions in a plan by one.

It turns out that there is a strong correlation between the average number of counties per region in a plan and the inequality. A few years ago we drew up inequality-minimizing plans with no chops and without regard to erosity. I summarized those results for the Forum Redistricting Commission thread in this post.

Let's add to the discussion the I in SPICE: Inequality.

Definition: Quota. The quota is the total population of a state divided by the number of districts rounded to the nearest whole number.
Definition: Deviation. The deviation is the difference between the population of a district and the quota. Negative numbers indicate a district that has a population that is smaller than the quota.
Definition: Range. The range is the difference in population between the largest and smallest district in a plan.
Definition: Average Deviation. The average deviation is the average of the absolute values of the deviations for all districts in a plan.

Background: SCOTUS has set two different standards for districts. Legislative and local districts must be substantially equal and that has been interpreted to be a range not exceeding 10% of the quota. Congressional districts must be as equal as practicable, and for some time that was assumed to mean that only exact equality would do. However, the recent WV case makes it clear that a range of up to 1% of the quota is acceptable when driven by other neutral redistricting factors. Greater than 1% might also be acceptable, but 10% would presumably not be because that is set by a different standard. It's an evolving area in the law.

Item 6. All plans for congressional districts shall have a range not exceeding 1% of the quota. All other plans shall have a range not exceeding 10% of the quota except when otherwise limited by state law.

Background: Some time ago there were some threads that tried to optimize the population equality of districts with no county splits. The result of that exercise was the following graph.



Each square represents a state. New England states used towns instead of counties, and states with counties too large for a district assumed that a whole number of counties would nest inside the large county. The more counties available per district, the closer to equality one could achieve, and the relation is logarithmic in population. The green line represents the best fit to the data. Data for average deviation can be fit as well, but the result is not substantially different other than the scale factor that has the average deviation equal to about 1/4 the range.

The average state has about 72 counties and if one divides that number into 2, 3, 4, etc. districts then one can use the fit from the data in the graph to predict a likely range. That in turn can be built into a table.

Item 7. The INEQUALITY score for a plan is found by taking the range for a plan and comparing it to the table below.

Range	Inequality
0-1	0
2-10	1
11-100	2
101-400	3
401-900	4
901-1600	5
1601-2400	6
2401-3200	7
3201-4000	8
4001-4800	9
4801-5600	10
5601-6300	11
6301-7000	12
7001-7700	13

This score reflects the expected improvement one should get by adding chops to a plan. So adding it to the chop score creates an automatic balance between chops and inequality. It also provides for a Pareto choice vs erosity where plans with no chops are likely (eg IA).

[muon2]

So here is where I think we are so far on definitions and their applicability to connections and erosity.

Definition: Node. A node is a reference point for a geographic unit. For a political unit the node is the primary place of government for that unit.

Definition: Local Connection. There is a local connection between two subunits within a county 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 both units on either side of the border.

Definition: Regional Connection. There is regional connection between two counties or subunits in different counties 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 two nodes without entering any other county. If the node is not on a numbered highway, then the connection is measured from the point of the nearest numbered highway in the county to the node.

Definition: Connecting Path. There is often more than one possible path to connect two nodes. For both local and regional connections the connection between two units is considered to be the path that takes the shortest time as determined by generally available mapping software.

Definition: Fragment. A fragment is a contiguous unit entirely within a district formed by the chop of a political unit. 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. Fragments are connected to other fragments in the same political unit if their nodes are locally connected. A fragment is connected to another county or fragment in another county if the connecting path between the counties enters the county in that fragment.

Item: Each unit or fragment in a district must be connected to every other unit or fragment in the district. The connection may either be direct or by way of other units in the district.

Item: Two whole units in a district cannot be connected solely by way of a fragment (bridge chop).

Definition: Cut link. A cut link is a connection between nodes in different districts.

Definition: Erosity. The erosity of a district is the set of cut links to nodes in that district.

Item: The EROSITY score for a districting plan is the set of all cut links in that plan. EROSITY is also equal to one half the sum of the erosity for the districts in the plan.

[muon2]

Thanks for the comments. I think a number of them are based on discussions that predate my artificial maps. I tried to make these definition reflect the state of things based just on those maps. That's why nick cuts aren't there, I want to address those with artificial maps, too. The definitions I wrote were what I needed to assess NY, because I didn't think there were any qualifying nick cuts in NY.

Even if it's just to make the mapping happy, I want to place a fragment node somewhere. Fragments have to have links so that there can be cut links for measuring erosity. Links have to be between nodes, so fragments have to have nodes, even if they aren't used for anything on the physical map.

Bridge chops were the only thing that weren't part of the artificial maps, and I mentioned them only because they came up in the NY plans. They are really about the definition of chops, not erosity. I'm open to something other than a ban on bridge chops, but we will have to be much more precise for scoring. "Disfavored" won't do. I suggest that this may also be a subject for artificial maps.

Here's a longer exposition on the erosity score definition. I hope it helps with the math.

The EROSITY of a plan is equal to all the cut links in the plan.
The erosity of a district is equal to all the cut links where one node of a cut link is in the district. That is to say it is the total of all cut links associated with that district.
Each cut link has exactly two nodes associated with it, and those nodes are in different districts.
Each cut link between two districts will count twice for district erosity, once in the score of each of those districts with nodes associated with the cut link.
The total of the erosity for all districts in a plan is equal to two times the number of cut links.
Therefore the EROSITY of a plan is equal to half the sum total of the erosity for the districts in the plan.

[muon2]

Definition: Node. A node is a reference point for a geographic unit. For a political unit the node is the primary place of government for that unit.

Definition: Local Connection. There is a local connection between two subunits within a county 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 both units on either side of the border.

Definition: Regional Connection. There is regional connection between two counties or subunits in different counties 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 two nodes without entering any other county, unless such highway in such other county is a qualifying nick cut path. If the node is not on a numbered highway, then the connection is measured from the point of the nearest numbered highway in the county to the node.

Definition: Connecting Path. There is often more than one possible path to connect two nodes. For both local and regional connections the connection between two units is considered to be the path that takes the shortest time as determined by generally available mapping software.

Definition: There is a  nick cut if a highway between the nodes of two counties enters another county without traveling through the node of such other county and its distance in such other county is one third of less of the distance of the length of the county measured from the two on its boundary that are farthest apart. The nick cut is a qualifying nick cut if there is also a local connection between the two counties connected by the highway with a nick cut.

The reliance on a particular fraction of the connecting path gives me pause. It isn't really consistent with the notion of paths. It's particularly at odds with the notion that the scale can float to suit the scale of the counties or townships (urban vs rural) in question. I have another thought that is more consistent and I think gets to much the same result.

Definition: Regional Path. A regional path between two units is a continuous path of all-season numbered state or federal highways or regularly scheduled ferries that allow one to travel between the two nodes. If the node is not on a numbered highway, then the path is measured from the point of the nearest numbered highway in the county to the node.

Definition: Nick Path. A nick path is a regional path between two counties or subunits in different counties that are otherwise locally connected. The nick path may traverse counties other than the two counties to be connected. The nick path cannot be part of any valid path between the counties to be connected and any of the other counties traversed by the path.

Definition: Regional Connection. There is regional connection between two counties or subunits in different counties if there is a regional path between the two nodes that enter no other unit than those connected by the path, or if there is a nick path between the two nodes.

Examples. Consider these maps to show only numbered state highways. Assume that any contiguous counties are locally connected.

Let's start with this map. We agreed that the connection from Burr to Calhoun was not a link for local connections.



With the definition above Burr remains regionally unconnected to Calhoun. The path that passes through Agnew provides the path from Agnew to Calhoun, so it's not valid as a nick path. It also provides a path from Agnew to Burr, and though that's not the connecting path (it's not the shortest), it's still a valid path from Agnew to Burr.

Next consider this map. We agreed that the connection from Clay to Dewey was not a link for local connections. As before we'll now consider these to be counties, and there's a local road with a bridge that connects Clay and Dewey.



Here the highway that passes through Adlai on the way between Clay and Dewey is not a path to the node in Adlai. Since we've assumed that Clay and Dewey are otherwise locally connected, the regional path from Clay to Dewey through Adlai qualifies as a nick path. Thus as counties Clay would be regionally connected to Dewey.

[muon2]

To show an application of my nick path definition, I'll look at NY. This is the map based on direct regional connections.



Here are the contiguous county pairs in western NY not directly linked with a regional path.

Allegany-Livingston; locally connected, no nick path.
Steuben-Ontario; locally connected, no nick path. NY-21 crosses through a bit of Yates south of Canandaigua Lake, but that's also on a path from Ontario to Yates.
Steuben-Schuyler; locally connected, and there is a nick path. NY-414 cuts across a corner of Chemung and that stretch is not part of a direct regional path to Chemung from either county. Note that NY-226 does not qualify, since it goes into Yates and overlaps the path from Yates to Schuyler on NY-14A.
Yates-Seneca; not locally connected.
Chemung-Tompkins; locally connected, no nick path. The potential nick paths all are part of other direct paths.

[muon2]

The underlined sentence I can answer directly. I'll give thought to the other parts.

Definition: Node. A node is a reference point for a geographic unit. For a political unit the node is the primary place of government for that unit.

The node is a specific point not a jurisdiction. For a county it is the place where the county government meets. A couple of years ago I tried to use the whole county seat, but I ran into problems when chops took a bite out of the seat and I tried to score plans with those chops. From a software perspective a single point is preferred to a polygonal shape as well.

Government buildings will be on roads so there is always a clear definition of a local path between two nodes. Regional paths can only use state and federal highways. Government buildings may not directly be on a state highway, so I need to place the seat of government on some state or federal highway to create a path. If there are two regional paths between a pair of county nodes then I need to know the reference point to determine which regional path is the shortest connecting path in case there are chops of that county. I doubt the ambiguity of lengths comes up much if at all, but the software still needs a reference point for measurement.

[muon2]

The underlined sentence I can answer directly. I'll give thought to the other parts.

Definition: Node. A node is a reference point for a geographic unit. For a political unit the node is the primary place of government for that unit.

The node is a specific point not a jurisdiction. For a county it is the place where the county government meets. A couple of years ago I tried to use the whole county seat, but I ran into problems when chops took a bite out of the seat and I tried to score plans with those chops. From a software perspective a single point is preferred to a polygonal shape as well.

Government buildings will be on roads so there is always a clear definition of a local path between two nodes. Regional paths can only use state and federal highways. Government buildings may not directly be on a state highway, so I need to place the seat of government on some state or federal highway to create a path. If there are two regional paths between a pair of county nodes then I need to know the reference point to determine which regional path is the shortest connecting path in case there are chops of that county. I doubt the ambiguity of lengths comes up much if at all, but the software still needs a reference point for measurement.

What if a city hall is not within the corporate limits?

That's a situation that we've encountered (VA county seats in independent cities, township halls in incorporated city limits). There will have to be either an exception or clarification rule. I think that should wait until we have some other broader rules pinned down. The general rules may suggest the best way to deal with these exceptions.

[muon2]

Is this the example from western NC?

And then, at the other extreme from the UCC urban area exception issue,  we have the Clay County "nick cut" link.  State highway 64 as it traverses through Clay County from Cherokee to Macon just misses the county seat of Clay, Hayesville, and there are no other urban areas in Clay County. Yet the "nick" traverses the length of the whole county. That is a no go. That is not a nick. I am not sure what rule to fashion to deal with that. That the length of the nick highway cannot be more than a third of the length of the county measured from the two points thereof that give the longest length? Or something like that? And by measuring the length of the highway for this ratio, the rule would tend to penalize twisty state highway nick cuts that twist and turn over rugged terrain, which is a good thing I would think. An absolute highway distance parameter not based on a ratio might not work too well, given counties vary a lot in size.

My definition handles this case. US-64 is on a path from both Hayesville (Clay) to Murphy (Cherokee) and Hayesville to Franklin (Macon). Either or both disqualify it by definition. So it can't be used as a nick path from Murphy to Franklin.

[muon2]

Is this the example from western NC?

And then, at the other extreme from the UCC urban area exception issue,  we have the Clay County "nick cut" link.  State highway 64 as it traverses through Clay County from Cherokee to Macon just misses the county seat of Clay, Hayesville, and there are no other urban areas in Clay County. Yet the "nick" traverses the length of the whole county. That is a no go. That is not a nick. I am not sure what rule to fashion to deal with that. That the length of the nick highway cannot be more than a third of the length of the county measured from the two points thereof that give the longest length? Or something like that? And by measuring the length of the highway for this ratio, the rule would tend to penalize twisty state highway nick cuts that twist and turn over rugged terrain, which is a good thing I would think. An absolute highway distance parameter not based on a ratio might not work too well, given counties vary a lot in size.

My definition handles this case. US-64 is on a path from both Hayesville (Clay) to Murphy (Cherokee) and Hayesville to Franklin (Macon). Either or both disqualify it by definition. So it can't be used as a nick path from Murphy to Franklin.

That is where "a" versus "the" is benign. But in other cases its malignant. You are using "a" as a substitute for a distance metric.

Then we need to find a case where it is malignant. The distance metric gives the wrong results in the artificial example below. The distance metric would have a link from Burr to Calhoun when we agreed that there should not be one.

Let's start with this map. We agreed that the connection from Burr to Calhoun was not a link for local connections.



With the definition above Burr remains regionally unconnected to Calhoun. The path that passes through Agnew provides the path from Agnew to Calhoun, so it's not valid as a nick path. It also provides a path from Agnew to Burr, and though that's not the connecting path (it's not the shortest), it's still a valid path from Agnew to Burr.

And the distance metric might fail to find the nick cut in this other example, especially if I move the node for Clay to the SE close to the border with Adlai. It is counter intuitive that as the nodes get physically closer the ability to use a nick path diminishes, since the fraction in the intervening county increases.

Next consider this map. We agreed that the connection from Clay to Dewey was not a link for local connections. As before we'll now consider these to be counties, and there's a local road with a bridge that connects Clay and Dewey.



Here the highway that passes through Adlai on the way between Clay and Dewey is not a path to the node in Adlai. Since we've assumed that Clay and Dewey are otherwise locally connected, the regional path from Clay to Dewey through Adlai qualifies as a nick path. Thus as counties Clay would be regionally connected to Dewey.

[muon2]

Now I'm confused. I thought the uniform highway number was a different standard than the maximum 1/3 of distance in the cut. At least they were two different proposals when you and jimrtex were debating this over NC.

Moving C closer to D does not change the length in A, but it does change the fraction of the path in A. If I put the nodes for C and D on the same numbered highway right at the border with A, the fraction in A would be the majority of the length. Yet if I move C and D farther apart I can eventually get the fraction in A under 33% and it becomes a link. That is counter intuitive to me.

[muon2]

Now I'm confused. I thought the uniform highway number was a different standard than the maximum 1/3 of distance in the cut. At least they were two different proposals when you and jimrtex were debating this over NC.

It must be both the same highway number and fit within the length parameter.

Moving C closer to D does not change the length in A, but it does change the fraction of the path in A. If I put the nodes for C and D on the same numbered highway right at the border with A, the fraction in A would be the majority of the length. Yet if I move C and D farther apart I can eventually get the fraction in A under 33% and it becomes a link. That is counter intuitive to me.


The fraction is the length of the nick as the numerator and the length of the nicked county as the denominator. The length of the road outside the nicked county is irrelevant. If I am missing something, please put up a graphic.

I don't know what the underlined means. I don't recognize it from a previous proposal. jimrtex had referenced the length of the entire path as a denominator IIRC.

I'd still like to see an example where you feel my proposal creates an untenable result. From my perspective, it would not bother me to have no nick paths, but I'm trying to accommodate those who see a value in them.

[muon2]

I don’t understand the sentence underlined in red. I thought a numbered highway needed to go into the node jurisdiction. If not, is the only requirement is that the numbered highway enter the county in question, and then can use local roads to get to the node thereafter?

The word “any” that I underlined, suggests that there can be more than one regional path between two counties. If that is the case, then the direct red line that I drew representing a regional highway connection node C to node B would not be a nick cut, because part of it also serves as a regional path from node A to node B, albeit a circuitous one and its use of the highway in question for a bit was not built to connect node A to node B (which is the point).  And the second red line from node C to the right does use the shortest direct regional path from A to B (assuming there is always only one qualifying regional path, to wit, the shortest one). Do we really want to exclude that?

My recollection is that I made the test the number of the state highway. I drew two red lines on your map, which represent additional numbered state highways in addition to what you drew (the local politicians are good a pork barreling, so thus the plethora of state highways). If the right red line and the vertical black line for the last bit of the path to node B are the same number, then it is a qualifying nick path. If not, it is not. In the nicked county, or the county where the regional connection is in issue, if we don't have the same number as the state highway at the point where it enters the nicked county, it is not a valid nick path. If the number is the same, then you do.  And by that test, if the path from node C to node D in your second graphic has a different highway number for the final portion in Dewey going north, then you would also not have a valid path there as well.

And how about that rural NC connection issue in the western part of the state, where we had a numbered highway that traversed the whole county as a nick cut without traveling through the node of that “nicked” county. We wanted to exclude that, so we did the distance metric. If that path did not use a highway that served in part as a, or the (depending on the resolution of the above), regional connection from the node of the nicked county to the node of some other adjacent county, then it would count as a regional connection?

So we have quite a bit here to think about.

 

If I understand this, you would have either of the red highways become a nick path. I would exclude them, since either of them becomes the connecting path (ie shortest) between A and C. To me that's true even if the highway was explicitly built for a connection from C to B.

Can I ask if any of the county pairs I listed in NY causes grief? I want to make sure we aren't arguing over something so artificial that it need not control the rules.

[muon2]

We shall have to disagree here until we can find a real world case that clearly divides us and is intolerable by one method or the other. You may find your metric to be common sense, but it is a metric wholly new to my definitions of paths and erosity. I only invoke distance when comparing two established paths between the same points for the purposes of determining fragment connections.

The strength of my definitions is that paths have no reliance on real distance which allows them to naturally scale in size and scope when they enter urban areas. Without scaling erosity breaks down as a measure as do many other tests of compactness. By using distance only to compare paths between the same points, the scale is preset by those points, not by the paths.

[muon2]

Isn't taking a ratio of distance in a nicked county, naturally scaling? Yes, to me, common sense, trumps introducing "novelty" to your system. That is never going to change.

The ratio could be truly scaling if the densities were generally uniform. One problem with many scale-independent measures of compactness is how they are tripped up by urban concentrations. For example one standard test is to compare the area of the smallest circle that encloses a district with the area of the district (Reock measure). That test scales with the size of the district. Yet, if the district has large regular rural areas combined with erose areas in an urban area the district scores well. I am similarly concerned that a nick passing through an urban area misses features that would be irrelevant when in cuts a rural area.

[muon2]

Isn't taking a ratio of distance in a nicked county, naturally scaling? Yes, to me, common sense, trumps introducing "novelty" to your system. That is never going to change.

The ratio could be truly scaling if the densities were generally uniform. One problem with many scale-independent measures of compactness is how they are tripped up by urban concentrations. For example one standard test is to compare the area of the smallest circle that encloses a district with the area of the district (Reock measure). That test scales with the size of the district. Yet, if the district has large regular rural areas combined with erose areas in an urban area the district scores well. I am similarly concerned that a nick passing through an urban area misses features that would be irrelevant when in cuts a rural area.

I don't understand how your comment is relevant to my point, and I don't understand your last sentence at all. What does "misses features that would be irrelevant" mean?

Sure an erose elongated county might allow for a longer nick. So what? That is hardly a big deal. The metric works well enough for government work as it were.

If it's about what works well enough, then I claim mine is simpler to test. The way I defined it, as long as the nick doesn't have a state highway fork in the county that takes you to that county's seat it's a valid nick. I don't need to measure anything. The example from Burr to Calhoun has a fork to Agnew. So it's not a nick for me.



As we know these nicks are relatively rare. I'd still like to see where my model gives an untenable result in the real world.

[muon2]

As described with my two red lines above, the result as to what is a valid nick, and what is not, appears to be very arbitrary. If a road happens to tie into another road going that goes between two notes, even with a highly circuitous route, with another route more direct, you lose the validity of the nick, and if not, you have one, even if highly circuitous.

Circuitous yes, arbitrary no. We have many circuitous connections defined directly between counties. Among these are ones that jimrtex has complained about, yet they remain for me (and I think you). There's nothing arbitrary about saying that a nick path can't already be on a direct path in the nicked county. It's saying that nick paths arise out of unusual situations where a highway passes through a third county in a truly incidental way.

Conversely using a 1/3 distance standard would be circuitous no, arbitrary yes. It tends to lead to more direct nick cuts so they would be presumably less circuitous. The use of 1/3 is itself arbitrary as it is not based on any comprehensive study of real geography. Look at the work that jimrtex put into UCCs to come up with a cutoff within MSAs that was not arbitrary.

[muon2]

As described with my two red lines above, the result as to what is a valid nick, and what is not, appears to be very arbitrary. If a road happens to tie into another road going that goes between two notes, even with a highly circuitous route, with another route more direct, you lose the validity of the nick, and if not, you have one, even if highly circuitous.

Circuitous yes, arbitrary no. We have many circuitous connections defined directly between counties. Among these are ones that jimrtex has complained about, yet they remain for me (and I think you). There's nothing arbitrary about saying that a nick path can't already be on a direct path in the nicked county. It's saying that nick paths arise out of unusual situations where a highway passes through a third county in a truly incidental way.

Conversely using a 1/3 distance standard would be circuitous no, arbitrary yes. It tends to lead to more direct nick cuts so they would be presumably less circuitous. The use of 1/3 is itself arbitrary as it is not based on any comprehensive study of real geography. Look at the work that jimrtex put into UCCs to come up with a cutoff within MSAs that was not arbitrary.

Well I don't agree that the third rule is arbitrary, and most counties are not that erose anyway. I guess we are at a dead end here. What's arbitrary is involving highways that happen to connect in some circuitous way between nodes, to invalidate a direct nick cut, or allowing nick cuts that avoid that that themselves are circuitous and not designed to connect nodes. We just are not going to agree on this one.

Perhaps you are right that this cannot go further. I think our operational definitions of arbitrary are too different. Maybe its a law vs science thing. For now I will step back to direct connections only for examples, and maybe at a future date we can revisit nicks.

[muon2]

Setting aside nicks for now, I want to look at a particular type of chop that creates an isolated fragment. Here are the current definitions and rules.

Definition: Connecting Path. There is often more than one possible path to connect two nodes. For both local and regional connections the connecting path between two units is considered to be the path that takes the shortest time as determined by generally available mapping software.

Definition: Fragment. A fragment is a contiguous unit entirely within a district formed by the chop of a political unit. 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: A fragment is connected to another fragment in the same political unit if their nodes are locally connected. A fragment is connected to another unit if the connecting path between the other unit and the unit containing the fragment enters the unit in that fragment.

I italicized my language in the definition of fragment that was questioned. These definitions and associated item allowed us to interpret the following chop as one that reduced erosity, creating a tradeoff between chops and erosity.



Here's an example of what I call an isolated fragment that could use my questioned wording.

Definition: Isolated Fragment. An isolated fragment occurs when a chop creates a fragment that has no connections to other units.



This is certainly a chop we'd like to see as it follows the river in this example. We have agreed that Adlai and Clay are not connected since you have to go through Dewey on the most direct path or Bryan on the more circuitous path. So the path through west Adlai between Clay and Dewey is not a connecting path, so west Adlai has no connections by the existing rules. I think the natural connections for west Adlai are to both Clay and Dewey, and this should be the graph.



Unlike other fragments an isolated fragment cannot rely on the location of the prior connecting paths of its original unit. Here's a rule that resolves this situation to get the result I suggest in the graph.

Item: An isolated fragment has a connection with a unit when there is connecting path with that unit.

This treats the isolated fragment as if it were a new unit for the purposes of making a connection. Note that this requires knowledge of the node of the fragment since it now is treated as if it it were an independent unit. Without a node in the isolated fragment there is no way to define new paths to be used as connections to the isolated fragment. That takes us back to the question of if and how to define a node in a fragment.

[muon2]

Actually if I've written my definition correctly, then the fragment you would like to use is an isolated fragment, in that it has no connections to other units, at most a local connection to other fragments in the chopped unit. That leads to rules that favor your preferred chop.

Definition: Isolated County. An isolated county is a county or equivalent that has no regional connections. Example: Pitkin county, CO.
  
Item: An isolated county is connected to a unit if there is a local connection to the unit.

Item: An isolated county fragment that has no regional connections to adjacent units in other counties in the same district is connected to a unit or fragment across county lines if there is a local connection to the unit.

The underlined narrowly defines the connections for your preferred chops. Without the underlined it would treat all isolated fragments the same whether or not there is a regional link to the rest of the district. Without the underlined it is simpler and either way it accommodates your preference. Any thoughts?

Oh, and I do like your signature. What's the source?

[muon2]

It's true that isolated counties are very few in number (I'm only certain of one). But completeness requires that I address it. It turns out that the mechanism that makes most sense to address isolated counties also works for isolated fragments without regional connections even in ordinary counties.

So as you see there are two types of isolated fragments. One type has no possible regional connections and must use local connections. The other type can make regional connections using state highways that were not part of the connecting path to the unit containing the fragment. The tricky part is that if the fragment is adjacent to two counties, but there is only a regional connection to the county not in the district, it still isn't connected to its district. Not good.

Using regional connections where available requires the judge to recognize where the district extends into other counties. This tangles chops and connectivity more than is minimally necessary, but the tangle may be desirable. Human brains handle the tangle much better than computers.

One thing may help me understand the scope of your local-connected chop preference. It clearly is in play for simple chops. Is it necessary for macrochops, too? That is do you want to grab a town in a macrochopped county that is not regionally connected to the adjacent county? nb We excluded this in the Detroit metro with no complaints, and I don't recall it popping up anywhere else, unlike your simple chop usage.

[muon2]

Here's what I'd ideally like as a definition and implementation. Setting aside macrochops for now, would it cause a serious problem? The most I see is occasional extra erosity for certain isolated fragments that do have both a regional connection to a county in their district and are locally connected to a different county not in the district.

Definition: Isolated Unit. Counties or fragments may exist that have no regional connection, but are locally connected to other contiguous units. Such counties or fragments are isolated units.
  
Item: An isolated unit is connected to another unit if there is a local connection to the unit.

It is very simple to illustrate and judge. It avoids the need for separate county and fragment definitions. No knowledge of the layout of districts is necessary.