Local vs regional road connections

[Torie]

Again, I am not entirely sure what your are talking about (I really like our artificial maps), but I think I know what you are saying, and yes, if a sensitive highway between nodes within a county, or a state highway between counties to the respective county seats, is cut, it is one cut per crossing of a CD line. So yes, there should be two cuts counted.

Of course, if there are policy reasons to do otherwise, following the common sense rule, I am all ears, but I don't see any. Just minimize those sensitive highway cuts.

[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.

[Torie]

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.

Draw some artificial maps. I'm lost. I don't understand the public policy issue. If there is a real one, well then that is very important for me to understand.

[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.

[Torie]

Yes, I agree.

[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.

[Torie]

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?

[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).

[Torie]

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?

[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.

[Torie]

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

[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.

[Torie]

Somehow, I knew that is what you would say. This lacunae of yours never entered my synapses before somehow. We shall see how it goes. It's barely possible that a gratuitous chop could cause a map to enter the pareto optimal frontier, by shedding two cuts in exchange for one chop. Hopefully, in real life, that situation will not arise.

[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.

[Torie]

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.

You taught me something. I had to look that up. A gap in the law versus no law. I am not quite sure what the distinction is. Maybe the connotation of non liquet is that it's more novel, and applicable law is farther away, as opposed to where there are precedents, but precisely where the line is drawn at the margins has not been legally resolved.

[Torie]

Going back to the topic at hand, how do you thing gutting the Albany metro area that way comports with the common sense rule? How do you think the commentators would react to such a gutting being the cat's meow map? Isn't the common sense approach, that you start with the core metro county, and branch out to its burbs first?

I am not in the mood right now to go to the mat with you over this, but I find the map disturbing. It is something for us to seriously think about, as to whether there is some workable fix for this. Now please don't go all stubborn on my here! Just think about it, in some dark room and muse about it for awhile. Thanks my friend.

[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.

[Torie]

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

Vis a vis which other map?


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.

[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.

[Torie]

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.

I see. If based on experimentation, that actually proves to be the case, I will be more open to the suggestion. It is a rather important policy consideration in my view. But I don't have that much problem with 3 out of the 4 counties being in one CD, avoiding a chop, with the core county one of the three (which is your map). That is a reasonable tradeoff. It's the shredding of a group of counties in a metro area (as I defined that term), that concerns me.

[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).

[Torie]

Yes, I see. You are feeling quite proud of yourself this morning aren't you. 

We shall see whether we need to raise the salience of inequality to get the kind of maps of which I approve.   We seem to have dodge the bullet in NY, after temporary panic, and maybe we have enough to so dodge elsewhere.

[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.

[Torie]

My edits are in red. None of this should come as a surprise to you. I am quite firm in my opinion on the bridge chop issue. I am unaware of any argument to the contrary that I find of much merit thus far. I thought we did away with pseudo nodes for fragments, except for computer mapping purposes.
Where is the language that a fragment can create but one highway cut that we worked through?

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. 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.

Definition: There is a bridge chop if two counties wholly or partially within one district are connected sole by an intervening county that contains a fragment.

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.

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). is disfavored.

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. [I don't understand the one half the sum business.]

[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.