Tuesday, August 26, 2008

What is Hiearchy?

In my primate cognition class last semester, we were having a discussion about baboon social structure, I think, and we were talking about it being hierarchical. The professor asked me what "hierarchical" meant. I was taken aback for a few moments, and couldn't come up with a response. It's good to question the meaning of words you tend to take for granted.

After reading On Intelligence in 2005, Hawkins convinced me that time and hierarchy are crucial to understanding cognition. Whether his company's particular implementation is the right technological path is an open question, but I think in terms of identifying the important theoretical concepts, he's dead on.

So what is a hierarchy?

In The Sciences of the Artificial, Herbert Simon says:

By a hierarchical system, or hierarchy, I mean a system that is composed of interrelated subsystems, each of the latter being, in turn, hierarchic in structure until we reach some lowest level of elementary subsystem. (emphasis mine)

I didn't find this definition very satisfying, though it's all right.

Wikipedia does a nice job, I think:

A hierarchy is an arrangement of objects, people, elements, values, grades, orders, classes, etc., in a ranked or graduated series.

...

Items in a hierarchy are typically thought of as being "above," "below," or "at the same level as" one another.

Not bad, though I'd truncate it down to simply:

A system of ranked elements.

Rank, it seems to me, is the most important aspect of hierarchy, and this simple working definition seems to capture nicely the essentials of hierarchy.

So why is hierarchy important to understanding cognition?

Well for starters, because the world is hierarchical. The universe is composed of galaxies, which are composed of subsystems, including solar systems. All matter is composed of atoms and molecules, which are composed of subatomic particles. And multicellular organisms, including ourselves, are composed of subsystems (nervous, circulatory, etc.) that are composed of organs composed of cells composed of molecules and so on.

And that's just spatially. Temporally, our lives are composed of stages (early childhood, adolescence, the college years, etc.), which are hierarchically divided down to individual moments.

Our neocortex is hierarchically-arranged, and I believe its architecture has evolved specifically to exploit and encode the hierarchical nature of the world around us. In The Quest for Consciousness, Christof Koch refers to the neocortex as "quasi-hierarchical," by which he means that it is not a strict hierarchy. Some cortical areas don't report directly to the area just above them. Many connections skip levels, like a private reporting not only to his sargeant, but to the general as well.

I don't really like the term "quasi-hierarchy" much, though. The neocortex is no less hierarchical because of this arrangement. If a system is composed of elements which are all the same rank, it's not a hierarchy. If there is any difference in the ranking of elements in a system, then it's a hierarchy. I'd prefer qualifiers like "strict" and "flexible" to describe the extent to which the relationship between elements in a hierarchy is between adjacent levels.

And I'll leave you to ponder this adapted diagram of the organization of the macaque monkey's visual from Felleman and Van Essen (1991). The information flows from bottom to top. The bottom part is information coming from the eyes, and the hippocampus sits on top. Our visual processing system is very probably arranged in the same way.

8 comments:

Laurie said...

"By a hierarchical system, or hierarchy, I mean a system that is composed of interrelated subsystems, each of the latter being, in turn, hierarchic in structure until we reach some lowest level of elementary subsystem."

I didn't find this definition very satisfying, though it's all right.


This description is not simply less than satisfying. It outright sucks because it's tautological.

Kenny Wyland said...

A system of ranked elements.

Hm. I think this is missing an essential quality of a hierarchy. Hierarchies have fewer nodes at the higher ranks and more nodes at the lower ranks. Even if this tree-style structure isn't what you want to include with the idea of hierarchy, the missing idea that I'm moving toward is a disconnection between elements of the same rank.

When you describe a system of ranked elements, it implies that elements of "Rank 6" are related. However, their connections to the top of the hierarchy could be completely unrelated.

When it comes down to it though, I really believe that the idea of hierarchy has to include the concept of few at the top and many at the bottom. Otherwise you're really just describing an ordered queue.

Derek James said...

Laurie...I think the part in Simon's definition that I italicized is the basic definition, and I don't think it's tautological. It's definitely confusing, though, and not a very good definition.

Kenny...So you wouldn't consider a social unit with only three members, each with more social status than the next, to be hierarchical? I'd agree that hierarchies tend to have fewer elements at the top than the bottom, but I'd still classify systems with more elements on the top than the bottom, or the same number, as hierarchies.

Wouldn't you say a tree has hierarchical structure, even if it has more branches and leaves than roots?

Philip said...

Wouldn't you say a tree has hierarchical structure, even if it has more branches and leaves than roots?

That's a misleading analogy. The branches and roots are 2 separate hierarchies, and there's a clear gradation from large and few (near the trunk) to small and many (near the leaves or root tips).

I don't really like the term "quasi-hierarchy" much, though. The neocortex is no less hierarchical because of this arrangement.

Sure it is. It's still hierarchical, but less so than if it were a proper tree structure.

In computer science the difference between data structures often comes down to cardinality - ie, how many incoming and outgoing (or up and down) connections does each element have.

1) list - each element has 0 or 1 predecessor and successor
2) tree - each element has 0 or 1 parents and 0...many children.
3) network - each element has multiple peer elements

They aren't all or nothing - the brain seems mostly to be a tree structure with a few backward connections making it a network. Most human organizations are like this - we usually have 1 primary supervisor and multiple subordinates, but we might report to other supervisors for a minority of our duties. And it's certainly true that some organizations are more hierarchical (eg, the military) than others.

Kenny Wyland said...

Kenny...So you wouldn't consider a social unit with only three members, each with more social status than the next, to be hierarchical?

Yes, I would, but that's because the system you describe is still really a tree. In social hierarchies, you never really get just 1 to 1 to 1. A person who is socially elevated will have more than just 1 person below them.... but even if they don't, they still have the potential. It's a tree that just doesn't have that particular element filled in yet.

If you have one person at Rank 1, two people at Rank 2, four people at Rank 3.... those four people all have the same Rank, but they aren't all subordinate to all of those in Rank 2. They are only subordinate to their parent. So, simply describing it as a system of ranked elements is missing the essential quality of a hierarchy. An ordered list is a system of ranked elements, but a 1 dimensional list is not a hierarchy.

Derek James said...

Philip, you say the tree analogy is misleading, but you'd consider a tree a cohesive system, right? Viewed as a single entity, it has many elements at the bottom, few in the middle, and many at the top. Why is this a bad analogy for a hierarchy with many elements at both the top and bottom of the system?

Philip said...

I suppose it can be viewed as a single hierarchy, but "top" and "bottom" are just consequences of gravity, sunlight, and soil. Topologically, everything is still smaller/more toward the edges and larger/fewer toward the trunk. You could still lay it out like this, as a single hierarchy.

trunk
/ / \ \
big roots big branches
/ / \ \ / / \ \
medium roots medium branches
etc.

Philip said...

Gah, blogger squashes my spaces in that previous post - it was much prettier when I wrote it. I think you can get the idea though.