The importance of integrating networks

For a given amount of bike/ped improvements, it's most useful to integrate the pieces into a large network, as opposed to having smaller disconnected pieces. The reason is that a larger network makes a larger number of trips possible through bike/ped, and the growth curve is non-linear, with increasing benefits for every little additional piece tacked on.

A technical illustration follows:

If we take as a starting point a system that simply links destinations A and B, there is only one possible trip A->B (Ignore all reverse trips B->A, since they simply multiply the number by 2.)

Now, if there is a destination C that gets linked to A and B, there are three possible trips. They are as follows.
A->B
B->C
C->A

If a fourth destination D gets added to the network, the possible trips are
A->B
B->C
C->D
A->C
A->D
B->D

Notice how now 6 trips are possible. On the other hand, if A were linked to B, and C linked to D, but are in two separate networks, we would have only 2 possible trips. In general, if there are n destinations that are linked into one network, then there are n(n-1)/2 trips possible within that network. The relation is quadratic.

Of course, there is a point where distances become impractical for the effect to continue; nevertheless, an effective use of bike/ped funds would involve leveraging existing facilities to obtain the maximum usefulness for the funds expended. Breaks in the network are highly detrimental.