For
a road or rail network on n cities, what is the trade-off between total
network length and the efficiency of the network in providing short
routes?
For a pair (i,j) of cities, write r(i,j) = (route-length i to j)/(straight-line distance i to j) -1.
To
make a single summary statistic for network (in)efficiency, neither the
maximum nor the average of the r(i,j) seems appropriate; we introduce a
third statistic.
Understanding the optimal trade-off between
network length and network inefficiency turns out to be surprisingly
challenging, both mathematically and algorithmically.
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