The blogosphere if often described as an *echo chamber*. Ideas, comments, controversies and discussions are all reverberate in this online space. In the paper titled "Information Diffusion in the Blogspace", Daniel Gruhl et al. studied the dynamics of topics propagate in the Blogosphere. They say that resonance is a rare phenomenon that can be described as

fascinating phenomenon in which a massive response in the community is triggered

by a minute event in the real world.

While few topics reach the state where they "*resonate*" through out the community/blogosphere, the echo chamber is still active with millions of posts and discussions in the form of trackbacks, comments and twitter posts, etc. The blogosphere is a huge graph and the echo chamber phenomena can be easily modeled in terms of *random walks* in this graph. Imagine that your post is read by your immediate neighbors and their posts (in reply to yours) is read by their neighbors and so on. Now what is the probability that this sequence would terminate with a post or a comment back on your blog? This is nothing but the *commute time* or the round trip time for the random walk to start on a given node (say A), traverse to a node (say B) and return back to the node A.

Commute time embedding is a way to map the original graph (which lies in a high dimensional space) into a two (or any low) dimensional space in such a way that the euclidean distance in the low dimensional embedding preserves (i.e proportional to) the commute times in the original graph. Interestingly, this can be quite useful in describing how ideas or information might flow in the network. For more details about the exact method to compute the embedding please refer to the paper. It is a neat technique that can be described in terms of the eigenvectors of the graph Laplacian (and can also be related to heat diffusion in the graph).

Consider the classic Karate club example. The original graph and the corresponding embedding in a two dimensional space is shown below. Notice how the nodes 33 and 34 are very close to each other. Compare this to node 3 which is almost equidistant to the central nodes (33/34 and 1) of both the communities.

I think that this might be an interesting way to model the echo chamber of the blogosphere. In this representation, blogs that are closer to each other in Euclidean space are also closer in terms of *social distance* (another term used in the literature that approximately means the same thing as commute time).

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