Local vs regional road connections

[muon2]

Non-isolated fragments only require local connections within a county. That was defined earlier and is consistent with subunits inside a macrochopped county. It doesn't come into play for non-isolated fragments when there is only one chop (since by definition there is a state highway to the county seat interrupted by the fragment), but when there is more than one chop it may. Think about three fragments in a county where two have only a county road connecting them. They are connected in my initial implementation of fragments.

The only place I think my rule precludes one of your locally connected chops is in the following circumstance. There is a large county that is macrochopped between two or more districts. Another district almost entirely outside of the macrochopped county takes a small bite in the form of a town not regionally connected to the rest of the district, but the town does have a state highway connecting it to one of the dominant districts in the macrochopped county. Essentially it would have to have a state highway running parallel to the county border through the town in question, but no fork in the town to cross the border. I think this would be a very rare case, and I'm not sure it's worth a special rule.

[muon2]

Non-isolated fragments only require local connections within a county. That was defined earlier and is consistent with subunits inside a macrochopped county. It doesn't come into play for non-isolated fragments when there is only one chop (since by definition there is a state highway to the county seat interrupted by the fragment), but when there is more than one chop it may. Think about three fragments in a county where two have only a county road connecting them. They are connected in my initial implementation of fragments.

I don't see how the above is responsive to my comments.

I thought I was restating your comment about fragments that did or did not have state highway connections to the node. In my following example I thought echoed the type of precluded chop you said was OK, and I just extended it to a macrochop.

The only place I think my rule precludes one of your locally connected chops is in the following circumstance. There is a large county that is macrochopped between two or more districts. Another district almost entirely outside of the macrochopped county takes a small bite in the form of a town not regionally connected to the rest of the district, but the town does have a state highway connecting it to one of the dominant districts in the macrochopped county. Essentially it would have to have a state highway running parallel to the county border through the town in question, but no fork in the town to cross the border. I think this would be a very rare case, and I'm not sure it's worth a special rule.

I don't follow the "special rule" bit. Any chop creating a fragment without a state highway connection should lose under the preference regime to a map that chops creating a fragment that does have a state highway connection.

I can't using scoring preference on connections. All parts of each district must be connected to be a valid plan. Scoring takes place after a plan is deemed valid. Established connections are the basis for measuring erosity, and they are the only basis for measuring erosity. I won't say a connection is OK for a validity check and then vanishes for erosity scoring.

[muon2]

I guess a "special rule" means a preference in your mind. Yes, maybe it will be rare as to your example. But if the rule is that a chop using a highway connection is preferred to one that is not, it has a pretty broad application, as opposed to being so limited that it is "special."

Your second comment to me seems to be using a lot of words that basically say that you don't like preferences, as sort of a conclusory statement. It certainly is easy enough to apply a preference regime. You just don't like it based on a rather compelling desire for elegance. On that one, we just disagree. Your approach does not hew to the common sense regime in my judgment.

I intend my second comment to say something quite a bit more than a disdain for preferences that are not part of a scoring rubric. It goes to the fundamental linkage between connections and erosity in my model. The connections used to test for a valid plan are one and the same as the connections used to measure erosity. That has always been the case. A plan with internally disconnected districts is invalid on its face, just as is a plan with discontiguous districts. In crafting a rule to allow for certain local connections across county lines, the rule must function both for connections that are intact and those that are cut.

[muon2]

"In crafting a rule to allow for certain local connections across county lines, the rule must function both for connections that are intact and those that are cut."

Why is that? As I say, for example, I want to allow for chops using local highways, in a way that would not be allowed for determining if whole counties are connected. You yourself carve out exceptions.

The model is predicated on the ability to transform a map of connected areas to a graph of nodes and links. I included many of those equivalent graphs with my artificial maps. From the graph one can test that a plan has fully connected districts. The same graph then provides the cut set of links to measure erosity. Thus my obsession with the definition of connections.

Graph theory is a well-developed and still active area in discrete mathematics. It has found its way in research in both hard sciences and social sciences, and is critical to computer networks and algorithm development. I take advantage of some of its results in this model.

[muon2]

"In crafting a rule to allow for certain local connections across county lines, the rule must function both for connections that are intact and those that are cut."

Why is that? As I say, for example, I want to allow for chops using local highways, in a way that would not be allowed for determining if whole counties are connected. You yourself carve out exceptions.

The model is predicated on the ability to transform a map of connected areas to a graph of nodes and links. I included many of those equivalent graphs with my artificial maps. From the graph one can test that a plan has fully connected districts. The same graph then provides the cut set of links to measure erosity. Thus my obsession with the definition of connections.

Graph theory is a well-developed and still active area in discrete mathematics. It has found its way in research in both hard sciences and social sciences, and is critical to computer networks and algorithm development. I take advantage of some of its results in this model.

Just write the program, so that it has forks in the road. I'm not buying into any of this. Sorry!

We may need another phone call soon. When "techies" tell lawyers it can't be done, it raises the blood pressure of the latter.

This isn't even about the program, but goes well beyond that. There are theoretical papers about using graph theory to attack redistricting. I saw some old ones that got me thinking about this during the last cycle. Now, like me, the mathematicians have figured out that maps can transform into graphs, and have some nice theorems, too. However, they haven't yet figured out how to get erosity from the transformation or balance erosity against chops. I hope that's where we come in.

It's the transformation that's a mathematical certainty, not a techie limitation.

[muon2]

Let me add a higher level of statements above definitions and items. These are principles that guide the rules. Then I'll go back to illustrations of planar graphs and open questions about their definitions and applications.

Principle: Values for parameters that describe a plan are based on simple integer measures.

Principle: Each plan can be represented by a planar graph of nodes and links, where each node corresponds to a discrete area in the plan (unit, subunit, or fragment), and each link corresponds to a connection between nodes.

Principle: Each node must have a link to at least one other node in the graph, a pair of nodes may not have more than one link between them, and links may not cross in the plane (ie the nodes and links form a simple connected planar graph).

Principle: Internal connectivity and the shape of districts (erosity) are determined from the graph that represents the plan.

Principle: The effects of scaling in a map from low density to high density areas are represented by the division of nodes due to district lines (chops) and the creation of new links.

Principle: There are qualitative differences in the division of nodes and their grouping in districts, such as differences that reflect quantitative measures of communities of interest, population equality, and political responsiveness.

Principle: An optimal plan is on the Pareto frontier that balances the shape of the districts against the number and quality of node divisions and their grouping.

[muon2]

The items you added on nodes I treat as definitions, not principles. In principle I could define every census tract as a unit and each census block in the tract as a subunit. In principle I could use contiguity to define connections. The rules would work mathematically, but would not be satisfying since it would ignore the preference to follow existing political subdivisions and highways between them. That's why the principle uses nodes and links only in their broadest sense - a node represents a discrete area on a map and a link represents a connection between two areas in the map that are represented by nodes.

In principle any binary preference can be represented by a 1 or 0. So the principle of small integer values supports preferences. I think my resistance is that your application of preferences in sequence tends to run against the use of the Pareto frontier since it overly winnows the set of choices. I'd rather use the preferences in sum total (once reduced to binary or other values) as it preserves a larger space for a Pareto test

Thinking about it today, there is a modified version of one of your suggested principles worth stating.

Principle: Each complete map is divided into geographic units based on governmental jurisdictions such that every area in the map falls in a unit. Geographic units of large population may be divided into subunits based on governmental jurisdictions such that every area in the geographic unit falls in a subunit. Units and subunits should be identifiable from Census geography.

[muon2]

Here's a situation involving two chops into a county creating two isolated fragments. As before consider that all contiguous counties are locally connected, the black lines represent state highways, and the connecting path is based on the shortest distance.



The East Agnew fragment is clearly isolated and has no state highways at all. The clear interpretation is that it uses its local connections, and is linked to Dawes, Elbridge and central Agnew.

The West Agnew fragment is more complicated. The state highway from Dawes to West Agnew is not the connecting path and the path from Calhoun to Agnew enters in central Agnew. It qualifies as isolated.

Option A: If state highway connections have priority in determining links to West Agnew, then there are connections to Dawes and Central Agnew, but none to Calhoun. That makes the Calhoun district disconnected from West Agnew and would be disallowed.

Option B: If as an isolated fragment it goes straight to local connections, then the connection to Calhoun would exist as well. That would make the district internally connected and allowable.

As a side note, if the node for Dawes were in the western part of that county, then the Dawes-West Agnew path would be the connecting path and the fragment would not be isolated. It also would not be connected to Calhoun in that case and the district would be disallowed as in option A.

Here's the equivalent graph showing the link in question as a dashed line.

[muon2]

Here's a situation involving two chops into a county creating two isolated fragments. As before consider that all contiguous counties are locally connected, the black lines represent state highways, and the connecting path is based on the shortest distance.



As a side note, if the node for Dawes were in the western part of that county, then the Dawes-West Agnew path would be the connecting path and the fragment would not be isolated. It also would not be connected to Calhoun in that case and the district would be disallowed as in option A.

I don't follow this at all. Given where you put the state highways, and the chops, I don't see the relevance in this example of where the D node is.

In the map above there are two paths between Dawes and Agnew. Given the location of the Dawes node, the eastern path is the connecting path since it's shorter. The chops shown put that path going into the central Agnew fragment. That leaves west Agnew with no connecting paths so it is isolated.

Suppose the Dawes node were relocated to a point near its western border with Calhoun. In that case the western path between Dawes and Agnew would be the connecting path. It enters Agnew in the west Agnew fragment, so west Agnew would not be isolated. It would have a regional connection to Dawes. Thus I could not consider the local connection to Calhoun and the west district would be disallowed.

The situation as interpreted by option A is the same as if the Dawes node were in the west rather than at the shown location. That is why I brought it up.

Does this clarify or muddy my initial comment?

[muon2]

Yes, it does clarify. We are now focusing on the connections between D and the west fragment of A. I don't consider that fragment "isolated" from D. It has a state highway link, albeit not direct. To me the shortest route metric deals with erosity measuring highway cuts, not isolation. So where the D node is would affect whether or not there is a highway cut.

But we already concluded that it could. Here's a simple chop of Agnew that we've discussed before.



We agreed that the only highway that counts between counties is the one with the shortest path between nodes that doesn't cross into another county (setting aside nick paths). That shortest path highway is the connecting path. It is cut and counts for erosity if it crosses the county line between districts. In the above map the shortest path from Agnew to Dawes is the eastern path and it goes between the two districts. It is cut and counts for erosity.

Now consider if the Dawes node was in the west end of Dawes. That would make the western path the shortest path between Agnew and Dawes. It enters Agnew (the chopped county) in the same district as Dawes so there is no cut link between Agnew and Dawes.

Policywise this makes some sense. The shape of the chop is well-fitted to the notch corner of the western district and really doesn't change the geometric shape much. If the Dawes node is in the west the chop fits better with the population/government centers (with the nodes as proxies) than it does if the node is in the east of Dawes. There is a more obvious cut between centers with the Dawes node in the east.

[muon2]

The agreement was for whole county connections, not chops. I see the policy issue for erosity evaluation, but not for a total ban based on lack of connectivity.

We agreed that when there is a chopped county, each connection from the whole county to its neighbors would be assigned to only one fragment of the chopped county. The assignment is based on which fragment the connecting path first enters. Your only expressed concern with my definition was on the subject of pseudonodes, which we have been hashing out separately.

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.

[muon2]

Well, irrespective to what I agreed to (perhaps not understanding the implications at the time), I still don't see the policy reason for a flat ban. The fragment does not purport to have anything to do with a node connection. It is connected to the other subunit by a state highway. Perhaps it is not that important an issue, particularly in as much as the map would be docked on erosity, but that is my view until I understand a policy reason to the contrary. The computer can be programed differently when evaluating highway cuts versus connectivity.

Let me pose a different mechanism that may be what you seek.

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.

Item: Each unit or fragment in a district must be locally 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.

Definition: Link. A link is a representation of a connecting path between nodes. A link is based on a local connection if the nodes are in the same county. A link is based on a regional connection if nodes are in different counties.

Definition: Component. A component of a district is a set of nodes in the district such that any two nodes are connected to each other by a sequence of links in the district, and that set is connected to no other nodes in the district by a sequence of links in the district.

Item: It is preferred to have only one component for each district. Each component in a district in excess of one increases the erosity by one.

The first underlined change allows isolated fragments to be appended onto districts as you would like. The new underlined item says that isolated fragments are in different components. The cost for extra components is a penalty in erosity reflecting the preference to have regionally-connected districts when possible. Note that this opens the door to allowing locally, but not regionally, connected counties to form a district with an erosity penalty, and obviates the need for a nick path definition.

[muon2]

Definition: Component. A component of a district is a set of nodes in the district such that any two nodes are connected to each other by a sequence of links in the district, and that set is connected to no other nodes in the district by a sequence of links in the district.

I don't understand what the bold means; in particular "that set is connected to no other nodes ..."

It just means that the set of nodes that are linked is complete and is missing no other linked node in the district. The form of the definition is intentionally mathy. Connected components are a concept from graph theory and having disjoint components (ie missing links) in a district can be interpreted as equivalent to extra cut links at the boundary.

Item: It is preferred to have only one component for each district. Each component in a district in excess of one increases the erosity by one.

The first underlined change allows isolated fragments to be appended onto districts as you would like. The new underlined item says that isolated fragments are in different components. The cost for extra components is a penalty in erosity reflecting the preference to have regionally-connected districts when possible. Note that this opens the door to allowing locally, but not regionally, connected counties to form a district with an erosity penalty, and obviates the need for a nick path definition.

Where a whole county is appended via a qualifying nick path, I don't think any penalty should obtain. I wish to distinguish between fragments and whole counties. There is a good policy reason for this, as we saw in the Charlotte metro area.

Please recall that my preference is to exclude nicks entirely. I'm willing to concede your ability to include isolated fragments, and I think giving me some preference to avoid nicks is a reasonable trade. I was not as sold on the need to give equal weight to some of the Charlotte arrangements. This formulation also greatly simplifies the rules by creating a general rule to replace a large number of exception rules (nicks, isolation for fragments and units, subunit connections, etc.), and that is a good thing as far as selling the system.

If you'd rather we can return to haggling over each of the exceptions. We haven't been very productive with that process IMO.

[muon2]

Well I don't think bargaining is really appropriate here. I opined as to what I think is best based on policy considerations. We can just leave this one on the disagreement table. Maybe over time, it will go away, or maybe not. I feel more strongly about the nick exception, than banning highway connected fragments that don't use the most direct route, which in the end will not ver very important, if at all, but I still take exception to that one too.

I'm not sure where we go from here. We seem to disagree on the fundamental nature of connections and what should be counted towards erosity. We seem to disagree on the importance of a cleaner system of general rules versus one that includes a longer list of exceptions to handle with special rules. We seem to disagree on what features the public will see as critical and which ones are acceptable artifacts of a neutral system. Hmm ...