On the use and abuse of Hairballs. What might graph-theory mean for cognitive neuroscience?

Olaf Sporns
Discovering the Human Connectome

Having just returned from the OHBM (Organization for Human Brain Mapping) it's not hard to be phenomenally ecstatic about the state of research and raring to apply new techniques and theories to my data.  

There is much to be hopeful for; I encountered many brilliant people who are excited about their research and how neuroscience is helping to unlock the secrets of the mind.

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That said, there did seem to be a great deal of research that was following the coat-tails of last year's big ideas.  This year the "big thing" was graph theory - the branch of mathematics and computer science that was previously applied to other fields such as social network visualization and biology  by Olaf Sporns and applied to brain networks.  

Graph theory works on the notion that you can reduce a matrix (normally a correlation matrix) to a set of nodes (which represent specific brain regions) and edges (which represent either physical or functional connections between the nodes).  Once this is done, you can then analyze the resulting graph to get metrics about each of the nodes in order to discern something about the relative significance of a location and perhaps get an idea of how much information is flowing through it relative to other nodes.  

People tend to get output that resembles this...

From Insights into multimodal imaging classification of ADHD by  Colby et al., 2012

These images are pretty cool, but also a bit difficult to parse.  Less flattering descriptors are "hairballs", "fuzzballs" etc.  The clarity of the image depends almost entirely on the researcher picking a threshold that reduces the graph to a restricted set of nodes and edges that do not overwhelm the viewer.  Needless to say, this can be a bit of an arbitrary exercise - what threshold is a "good" threshold - 0.2, 0.3?  Your guess is as good as mine.  Probably a cynic would argue that it's the threshold that makes the picture look good.  

It's easy to see how the new fascination with graph theory is a natural expression of brain connectivity - which has been the emerging topic of interest for the past few years.  For a while now, researchers have noted that the brain is generally not modular.  That is to say, when you stick someone in a scanner and ask them to perform a demanding cognitive task, their entire brain is active to a certain extent.  One can then talk about functional connectivity - which areas are connected to a critical region across time while we ask you to perform a task.   The areas that are functionally connected are said to be "networks", and as it turns out, these networks can be visualized with graphs.  

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So, how do we visualize networks without graph theory?  Generally, people would display their images on high resolution t-1 images (see below)

A now famous image of the default mode by Michael D. Fox  and Michael Greicius 

The advantage of the above mode of display is that the activity is situated in anatomy - one can look at the representation of the brain and be spatially oriented, it also meshes with the previous generation of research which would display discrete activations in a similar fashion.  

By contrast, the graphs displayed below show a representation of the brain's networks across the decades in spring-loaded graphs that are unconstrained by anatomy.  It is here that I think that graphs are truly useful - the ability to describe network dynamics and change is one of their peculiar strengths.  They become even more useful if one allows the edges to encode information such as strength of connection by either colour or line breadth   Nodes can similarly be coded by colour; a group of nodes that are highly interconnected with each other but not much else could be thought of as a "module" and uniformly coloured.  Similarly, nodes that connect to many other regions and thus are critical conveyors of information might be coded by size - larger nodes are both visually and figuratively more important.  

Fair, Damien A. et al., (2009). "Functional Brain Networks Develop from a 'Local to Distributed' Organization"\

Despite the obvious usefulness of graph-theory for comparing network dynamics, many people shy away from viewing graphs which are completely unconstrained in two-dimensions, which is why some researchers have made an effort to place nodes in approximate anatomical space.

Martijn P. van den Heuvel and Olaf Sporns Rich-Club Organization of the Human Connectome

The above representation not only respects anatomy, but also goes beyond traditional functional connectivity displays by allowing the viewer to assess relative strength and importance of areas to the emergent networks.  

"But hang on a minute", you might be thinking, "I can get all this information from a correlation matrix - no need for the pretty pictures" - and so far, you would be right...

Tal et al., Caffeine-induced global reductions in resting-state BOLD connectivity reflect widespread decreases in MEG connectivity

The above graph gives you all the information you might need to know about functional connectivity - each column and row is a node, the colour of a cell represents the strength of any two connections with each other.  So far, we've duplicated the information contained in the graph, which is hardly surprising given that graphs take matrices (including correlation matrices as input).  

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But do we gain anything beyond ease of readability and interpretation by using graphs?  What can they tell us that correlation matrices cannot?  

The answer of course is yes of course they do.  Graphs can provide us with a series of metrics that tell us more about the flow of information than a correlation matrix can on its own.  

We can calculate the degree of a node - which is to say how many connections it has.  We can also look at the distance between two nodes, and calculate the shortest path or the most efficient path between them - which is to say, we can look at the probabilistic flow of information.  Closely related is the concept of betweeness which indicates how often a node acts as a bridge along a path between other nodes etc...

There are other metrics, and this is where I feel that the literature is still catching up.  We have a marvelous new tool available to us; it makes visualizing the networks of the brain feel almost intuitive.  But the new language of graph theory (at least to many neuroscientists) is something we need to figure out.  How important is betweeness-centrality?  Should we report both that and relative degree?  How do we interpret distance?  How do any of these metrics map onto what we know about cognition?  

Graph theory is a wonderful advance for the field of behavioural neuroscience, but as with many new advances, I get the feeling that many are seduced by the pretty pictures without spending the requisite time understanding the metrics behind them, and without the metrics, graph-theory offers little beyond displaying a more aesthetic version of a correlation matrix. 

*Update*
Since writing this post I've come across this paper by Fornito et al., on Schizophrenia and connectonomics (released in 2012).  From a cursory overview, the paper looks like it starts to address not only the process of making graph theory transparent to the reader, but also makes an earnest attempt to incorporate graph metrics to help classify normal vs. schizotypal brains.  I'll be reviewing this in the near future so stay tuned...