PageRank is well known as a highly successful link analysis method for deducing the importance of a web page. While the context in which this method is used has been limited to web page analysis, the algorithm does not make much assumptions about the nature of the nodes. This raises an interesting question. Can we utilize PageRank in a different kind of network graph? For example, if we run PageRank on a social network, what do the converged PageRank values mean, in the context of this social network? Or, if we run PageRank on a pipe network flow, does it tell us anything about the flow itself? Does it even make any sense?

While I do not have the answers, it’s worthwhile to investigate these questions, just to see where it leads us. We know PageRank revolves around the concept of Hub and Authority, so we can ask what do Hub and Authority mean in the context of, say, a social network. Before we move on, it should be noted that PageRank runs on a directed graph. A social network is not a directed graph; however, it is trivia to convert it into one, depending on what the directed edges between nodes mean. For simplicity’s sake, we can just convert each undirected edge (u, v) into two directed eges (u, v) and (v, u), so that every node has as many incoming edges as outgoing edges. In a social network, popular nodes have many incoming edges as well as outgoing edges. It seems obvious that these nodes have very high authority scores. Then what about unpopular nodes? Do they necessarily have low authority scores? If we think about it, these nodes, on their own, are not able to generate very high authority score due to their lack of incoming edges. However, as soon as an unpopular node makes a connection with a popular node, the popular node’s high authority score serves as a strong endorsement on that unpopular node. Thus, even if an unpopular node has very few incoming edges, if all the incoming edges are connected to popular nodes, this unpopular node would have very high authority score as well (though not as high as a popular node’s).

But what could authority scores possibly mean in the context of a social network? A node’s popularity? power? influence? helpful connections? authority (seems apt)? Knowledge? All these seem equally valid, but are they? As mentioned earlier, even unpopular nodes can have high authority scores, so at least we know authority scores do not indicate popularity. What about power? When we talk about power, ability to manipulate/control other nodes comes to mind. Using this definition, can we infer that from authority scores? If we think in terms of network exchange, nodes with many incoming edges usually (but not always) have bargaining power. Their bargaining power is greatest when at least one of their neighbors is a lonely node (a node whose incoming edges are very few), more so when most of that lonely node’s edges connect to non-lonely nodes. Do all this correlate well with authority scores? Not really. Again, unpopular, or lonely nodes, can have high authority scores, so having high authority scores don’t really mean one can control others. How about influence? We commonly talk about peer influence, so influence could well mean the ability to affect other nodes. It is one of the mechanisms by which a fad, a trend, or a meme spreads out. Strong/weak influence usually means how much we are able to affect a particular person, so such measure is very much the same as power. However, having wide influence would mean the ability to affect nodes very far out. Does this correlate well with authority scores? Seems likely (even an unpopular node can spread his/her idea if the node can affect a popular node), however, this depends on the edge’s strength, i.e. how close a person is to another. Because we have not taken this into account, we’ll leave it as inconclusive for the moment.

What about helpful connections? Does a node with high authority score get help easily compared to a node with low authority score? Consider an unpopular node with high authority score. The node has a higher chance of getting help from his/her popular neighbors. But if we think about it, a popular node, more often than not, has to interact with many nodes. His/her attention span to any particular node is often very short. It seems reasonable to expect that popular node not to pay much attention to this unpopular node (unless we take in account of edge strength). So this unpopular node, even with high authority score, is pretty much hopeless, so to speak. However, it’s also conceivable that helping the unpopular node doesn’t require much effort, as the popular node can often relegate the request to one of his/her many neighbors. Then that unpopular node is effectively gaining help through a friend’s friend. As the unpopular node does so, he/she gains a new friend. Now we have made quite a lot of assumptions, but this seems to point to a totally different and new direction. The ability to gain new friends. Does high authority score mean one can easily gain new friends? Seems very likely. To see this, read ‘The Tipping Point’ chapter two. It talks of connectors and non-connectors. A non-connector gets most of his/her friends through connectors, who are well connected by its very definition. The non-connector’s high authority score correspond well to this. If he/she has low authority score, it also seems reasonable that he/she is not connected to that many connectors, so that he/she can’t make new friends easily.

Much of these conclusions are purely thought experiments using common sense and a lot of assumptions, without much buttress from real-life experiments or examples. However, they do serve to illuminate the point that utilizing PageRank on different networks can let us gain new insights.