east point wrote:Network theory is complicated but lets take an oversimplified example. There are some 600+ destinations now on Amtrak. ( Ignoring thruway ) So you have 500 factorial different city pairs to travel. Add another station then you have 500+1 and another 500+2 more city pairs. Granted not all pairs are feasible. So as an average now it probably is 300 + . But as nodes ( connecting stations ) are added more difficult connections become easy.
Example Memphis - Florence is only for hard core train travelers. But add a Memphis - ATL - Savannah reasonable connector that MEM <> FLO becomes sellable.from o
Except in the trivial case of two cities, an n city network does not have n! (n factorial*) connections (ignoring direction: BOS-PHL and PHL-BOS would be two different city pairs): the number of city pairs (again ignoring direction) is n(n-1) = n^2 - n. If you'd consider BOS-PHL and PHL-BOS to be the same city-pair, then it's (n^2 - n)/2. So 500 cities is either 500*499 or 250*499 city pairs.
The quadratic (n^2) term is the important thing. Double the number of cities and you quadruple the number of city-pairs (whether or not you're ignoring direction). Under an assumption that every city-pair has equal value (they don't: among other things, the value of a city-pair tends to be inversely proportional to the square of the distance**), double the number of cities and you quadruple the value.
*: n! is the product of the positive integers less-than or equal to n: 3! = 3*2*1 = 6, 4! = 4*3*2*1 = 24, etc. In a network with non-stop trains in both directions between every pair of cities, n! is the number of possible trips which visit every city once.
**: by distance, we can further approximate that to travel time, including layovers waiting for transfers and taking headway into account. I don't really care that it's a 40 mile or so drive to Back Bay from where I live (unless, say, there's a nuclear bomb there which will explode and vaporize everything within 40 miles): what matters is that it's about an hour's drive. If there's a one-a-day each-way train to BBY that takes a half-hour to move from a station across the street from my house, the value of that mode is based on it taking 9.5 hours (since I'm going to be waiting an average of 9 hours between 6am and midnight for the train); double that to 2 trains a day (let's say, for argument, 9am and 9pm), then the value of that train is based on it being a 5 hour trip, which is about 3.5 times more valuable (for a rough doubling of direct operating cost).