I wanted to write this as part of the previous post on the problems of publishing, but felt it warrants an independent discussion.
The question is about assigning significance to observations that are scientific in nature in a very broad sense. Some of the most profound discoveries may seem like simple observations on hindsight. OTOH, an ingenious discovery or invention may prove inelegant or moot later on. (For example, Einstein's "biggest blunder" in introducing the Cosmological constant to account for a static universe.)
Anyway, let me try to illustrate with a few examples from graph theory how the same result can be seen in different ways.
Moore Bound: Given a fixed degree k (that is, every node has degree k) and a diameter d, Moore bound gives the maximum number of nodes N
max that can exist in the network. In other words, Moore Bound is the largest k-regular graph of diameter d. And, N
max = (k
(d + 1) - 1) / (k - 1). This has a lot of important implications, such as: given n nodes and a fixed degree k, you know the theoretical lower bound on the diameter (hence that of the maximum communication delay, say) of a network. These expressions themselves are rather easy to derive. How do you get the above expression for N
max? Simple. Imagine a k-ary tree of height d. You have 1 root node, k nodes at the first level, k
2 at the second level and so on. The resulting expression is, N
max = 1 + k + k
2 + ... + k
d = (k
(d + 1) - 1) / (k - 1).
To be sure, Moore Bound is a nice theoretical result. But as you can also see it is no more than a simple extension of a "fairly obvious" observation. Sure it is useful in graph theory; and it was discovered in the early stages of the development of graph Theory. But strictly speaking, it is not much more than high school mathematics. After all, geometric series have been known for a couple of centuries, if not more (Well, they were described by Euclid in 300 BCE
)
Handshaking Lemma: This is one of the first results that one studies. It states: in a graph, the sum of degrees of the vertices is twice the number of the edges. Duh! Isn't it?
Eulerian Circuits: A necessary and sufficient condition for the existence of an Eulerian circuit in a graph is that every vertex should have an even degree i.e. an even number of edges are incident on every vertex. This is also fairly evident if you think of the edge incidences as entry and exit doors.
The thing is, in hindsight all the early results of a theory seem obvious. But since we know that those early results have lead to profounder results later on, we don't reject them. But let's say you make some observations (rather elementary) that may (or may not) have implications on a new field. Say, early observations about something which may (or may not) develop into a new theory. How do we judge their significance?
The context of this post is a paper that we had written that had a number of simple, but fairly insightful (at least that's what we think!) results wrt regular graphs as applied to distributed index design. As part of my PhD work, we found that regular (or nearly regular) graphs have many optimal (efficiency, robustness etc.) properties. So, the idea of the paper was to show/prove some properties associated with regular graphs and to relate them to optimality objectives of networks. For example, we can easily construct bigger and bigger graphs while maintaining the regularity of a graph. This relates to joining of nodes in a p2p network. Or something interesting such as, there is a minimum number of nodes n, below which you cannot construct a non-hamiltonian regular graph. We put them in the context of a theory or principles of distributed index design, wherein we can provide certain basic theoretical guarantees through these results.
Most of the results in the paper were admittedly simple observations, and the paper got rejected as a result. Rejections are no big deal. But, however simple these observations might be, we haven't seen anyone else observing them
! That argument was kind of countered by the criticism that they are probably too simple even to need an observation. But now that we have observed them, what do we do with them? We can't probably publish them. We can't seem to unobserve them either. Also, can we or they be sure that these are not the early results of something that may develop well in due course? I don't know. It's a big problem.
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