This is redundancy: many different sequences (genotypes) give the same shape (phenotype). Redundancy can be boring, such as non-significant whitespace in C code: different spacing compiles the same (and gives the same behaviour).

It is also local, in that equivalent genotypes are just one step away - add or remove a space - and you can continue with further steps. So a set of equivalent genotypes/programs are connected by being adjacent (one "step" away). [\tangent maybe not all the equivalent genotypes are connected]

But this is interesting redundancy, because some of those equivalent genotypes are just one step away from dramatically different phenotypes. This doesn't happen with C whitespace (though maybe it happens with other equivalent implementations, different names, ways of looping, but I can't think of an example]

Using the metaphor in the article, one set of equivalent connected genotypes is like a network of roads, on which you can take steps to move around the system without penalty, because they are all equivalent and each step is neutral. Extending the metaphor for the "interesting" aspect, another set of equivalent connected genotypes with a dramatically different phenotype is like a railway network. Mostly, the two networks are separate, but sometimes, they are very close, so that in one step, you switch to another network, like a railway station. [For correctness, we disallow level crossings, because there both road and rail would have the same phenotype. We could disallow any crossings, making it planar, or introduce the third dimension and have bridge crossings, where the position in 3D is the genotype.]

There would be a great many such networks, with distinct phenotypes.

There would be networks that have no adjacency; but it might still be possible to reach them by moving to intermediate networks (e.g. travel by car then rail then bicycle path then footpath etc), provided the phenotypes of those intermediate networks were neutral or advantageous.

I like both the article's hypotheses:
that all complex systems have this property;
or that evolved biological systems only have it because evolution is faster with it.

2. The second appeals to me because it helps explain accelerating evolution by the establishment of platforms: e.g. the body-plan collection of genes may have taken a long time to come up with, but once it did, body plan diversity exploded. Though the article complains about the number of body plans possible, it's dramatically fewer than all possible raw sequences. It's configuring a body-plan instead of coding it from scratch. Having many different possibilities is good as it makes it a powerful expressive platform - perhaps like an algebra or programming language, once it gets complex enough, it is very powerful. The key quality is that within this configuration language, the density of "useful" results is higher than without it [e.g. a random configuration is more likely to be useful than random raw code - the platform is somehow specialized to its purpose]

Similarly, perhaps this system of RNA with this quality was not the first to evolve, but several arose... and this is the one that took off.

1. But maybe all complex systems have it too, provided they have redundancy. Perhaps, if there are many sets of connected equivalent genotypes, and each set is very large, there are likely to be many adjacencies between networks? Note: It's not necessary for all networks to have adjacencies, just enough of them. You could imagine varying these properties of the system (number and size of networks, relative to the total space) and come up with parameters that give "enough" adjacent networks [though I'm not quite sure how to define "enough".]

My feeling is that getting those parameters good enough by chance might be pretty rare - something that could take a few billion years over the surface area of a planet to have reasonable chance at...