The promise of “blissful ignorance” appears to have some theoretical merit after coming to the current lemma plaguing my thesis research.
Consider the problem of following a reliable path through WKU’s campus during the summer, during which construction projects have sidewalks momentarily removed for utility work. Let’s say, for simplicity, that each section of sidewalk—that is, the smooth space between cracks mothers are so afraid of—has the same probability of being removed, and let’s call this probability α. It does not particularly matter what the actual value of α is, at least not at this point along the train of thought. We will leave α be much like a “fill in the blank” to be answered by data in the future.
This is where much of the research carried out by others studying Network Reliability begins to differ. Some, like us, consider α to represent “failure,” while others let it represent “success.” Some shape their models such that they function properly on any network shape, while we and others restrict our interests to only networks exhibiting certain qualities. Such qualities could be “3-regular,” describing those where every intersection connects exactly three roads, or “planar,” describing those which can be drawn on a sheet of paper without lines crossing. Most city roadmaps fit this latter description until bridges show up.
We limit our model to only “directed acyclic graphs,” describing something like a city where every road is a one-way street and once an intersection is left, it is impossible to return—without driving into oncoming traffic at least. Although this is far from reality, it reduces the complexity of the math drastically.
Now consider the task at hand again—finding a path on campus. Imagine that the campus is symmetrical and that we are standing at an intersection where going left is in no way, now or further along the path, different. Only there is one difference.
To the left, a marching band has recently held practice, and a “band tower” has been left out. These are scaffolding-like structures designed for the band director to gain a bird’s eye view of his band so that he or she might catch errors in the marchers’ pattern formation.
Obviously there is some advantage to taking the route on the left over the route on the right. By climbing the band tower, we might be able to see where construction crews had obstructed sidewalks on that particular day. Once with this knowledge, finding a path to our destination is as simple as just taking it.
This example, trivial as it is, shows that the promise of knowledge is attracting, and the certainty provided by a path is just as important when it comes to “reliability” as the path’s connectedness to back-up paths—which may be likewise connected to back-up-back-up paths, and so on and so forth. Without this knowledge, the choice would have simply been to just pick one. But in a more exotic case—say, one involving a phone call from a friend and the Biblical prophet Moses—it’s hard to say which path we’d expect to take until we know more. ∎