Local vs regional road connections

[Torie]

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?

The map came from here. I found the page doing a search for "new york topographic maps".

[Torie]

"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.

[Torie]

The use of the term node without explaining what it is, is confusing. Thus my suggestions below in red.


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

Principle. Each map is divided into governmental jurisdictions, with a specific location in each jurisdiction assigned a node.

Principle. The node that is assigned to each governmental jurisdiction, is where the seat of government is located, or if one does not exist, a location within a sub-jurisdiction thereof with the greatest population.

Principle: Values for parameters that describe a plan are based on simple integer measures. Does this leave room for the preference mechanic, that I am so resistant to giving up?

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.

[Torie]

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.

Yes, unless there is a local connection to the E node, and no local connection to the A node.

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.

Yes, I would allow local connections to suffice only if the chop takes the entirety of the subunit, and even then, it is not preferred.


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.

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

[Torie]

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.

[Torie]

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.

[Torie]

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.

[Torie]

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 ..."

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.

[Torie]

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.

[Torie]

Just move on. Presumably you find some value in bouncing ideas off of me. In the end, a paper can lay out the options, and where people may disagree. In defending what you prefer, your challenge will be not to lapse into jargon, or abstract grand unified theories, but rather persuade folks that it is common sense, and where it isn't, it is not far off, and not worth the additional complexity. What I am receptive to, is public policy and common sense arguments of course. That is what will change my mind.

The issue we are arguing about as I say is of very marginal importance, and very deep in the weeds.