I suspect some of may have found the use of matrix arithmetic and graph theory in this week's paper a bit overwhelming. I still think its a paper worth reading. First, its good to know that there's more to Matrixes than Morpheus. Second, its interesting to see how people are trying to fit formal models to intuitions.
So lets focus on the intuitions. As a first aid measure, try applying masking tape (preferably purple) to every paragraph that has squigly brackets, and see how it's already become more friendly.
Now lets talk about hypergraphs. This here picture shows Actors (AKA users or plain people), instances (AKA resources, documents or web pages) and concepts (AKA words, tags, categories).
When a person tags a page, she creates a triparty relationship between person, tag and page. Mika calls this a hyperedge. I draw a triangle.
When many people do the same to many pages with many tags, you get what Mika calls a hypergraph, and would probably be refered to in plain English as a 'pretty big mess' (not to be confused with a mesh).
So what do mathematicians do with big messy problems? They find a way to break them down to many small less messy problems. In the case at hand, we do so by looking only at one type of connection at a time, say the dotted lines (tag – page) and measuring it along another (say number of persons). So, we get nice and simple graphs that connect only tags (concepts) by counting the number of things they have in common. For example, if a lot of pages are tagged 'sex' and 'table' we induce there's a semantic connection between the two.
As Mika explains:
In words, the bipartite graph AC links the persons to the concepts that they have used for tagging at least one object. Each link is weighted by the number oftimes the person has used that concept as a tag. This kind of graph is known in the social network analysis literature as an afiliation network [7], linking people to affliations with weights corresponding to the strength of the afiliation. An afiliation network can be used to generate two simple, weighted graphs (onemodenetworks) showing the similarities between actors and events, respectively. (At this point it is recommended to dichotomize the graph by applying somethreshold.)
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In summary, the AC graph, the a±liation network of people and concepts can be folded into two graphs: a social network of users based on overlappingsets of objects and a lightweight ontology of concepts based on overlapping setsof communities. Thus in this simple model, social networks and semantics are just flip-sides of the same coin: the original bipartite graph contains all theinformation to generate these networks, while it it not possible to re-generatethe original graph from them.
Mika compares two ontologies (= networks of concepts) derived in this process. One links concepts by the number of people who use them together, the other by the number of pages they are mutualy associated with. It turns out that looking at peopls (communities) is a better way to discover semantic relationships than looking at objects. In other words, the same resource means different things to different people, but people in the same community share a common set of meanings.
Well, I think Wenger and Lave would be pleased to hear this.
But there are other networks that can be derived. By fixing the concepts, we can plot the potential social relationships, common interests or implicit communities. Which brings us back to Lyndsay's point. If a tagging system wants to survive its own growth, perhaps it should make these networks explict. For example, identify clusters, show them to me, and show prioretize resources tagged by people who share my social cluster.
