Connected Streets, Connected Neighbors?

A common concept in city planning is that interconnected streets promote inclusion and increase the likelihood of social interaction between neighbors.  An interconnected street network, also often referred to as a grid street network, also increases safety and accessibility.
The graphic below shows two different street networks.  The street network on the top half of the graphic is based on a grid system and is interconnected.  The street network on the bottom half of the graphic is disconnected, with cul-de-sacs and streets that do not connect through to adjacent streets.

The area shown in blue is the area that is accessible within a certain amount of time (say 10 minutes) from the red dot in the center.  As you can see, there is much more blue area in the interconnected street network.

Think of this street interconnected-ness in another way.  If your child’s friend lives one street over, but that street is not directly connected to your street, it could take much longer for your child to walk to their friend’s house, even though the houses and streets are very close together.  This would mean that your child has to walk further to visit, perhaps even having to walk out to a larger, busier street to get to their friend’s house.  In this way, your neighborhood’s design and street network could start to influence your behavior and sense of safety – if it is further and less safe to walk, maybe you don’t feel comfortable letting your child walk to their friend’s house.

Should we strive for interconnected streets in our neighborhoods?  When street connections aren’t possible (due to topography or other natural features), should there still be pedestrian connections between streets?