Yesterday, the International Conference on Complex Systems wrapped up with five talks on networks. For me, the most interesting was that by Dan Braha, who spoke about what happens when you analyze a system as a network which changes over time, instead of the aggregate network formed by lumping all the timesteps together. Imagine a system made out of a whole pile of parts. At time [tex]t[/tex], part number [tex]i[/tex] might or might not be interacting with part number [tex]j[/tex], which we could represent as a time-varying matrix [tex]C_{ij}(t)[/tex]. Many studies of network-related phenomena obscure the time-dependence part. For example, in a living cell, genes are switching on and off, concentrations of enzymes are going up and down, and all sorts of stuff is changing over time. You can mix proteins A, B and C in a test tube; perhaps A bonds both to B and to C. You’d then draw an interaction network with links connecting A to B and to C — but what if B and C are never present in the cell at the same time?
Braha and company looked at a collection of e-mails sent over 113 days, exchanged among 57,138 users. (The data comes from arXiv:cond-mat/0201476v2, published five years ago in Phys. Rev. E, and were gathered at Kiel University.) A node is an individual e-mail address, and a link is established when a message is sent from one address to another. They found, among other things, that whether or not a particular node is a “hub” changes over time: popular today, an outcast tomorrow. Moreover, a node which is in the top 1000 most connected on one day may or may not be in the top 1000 for the aggregate network. Furthermoreover, when the window of aggregation is gradually increased — from one day to two days, to a week, up to the entire time period — the similarity to the total aggregate network increases, as you’d expect, but without any threshold.
In the last few minutes of his talk, Braha did a brief overview of a related investigation, in which they studied a “social network” derived from Bluetooth devices. If my Bluetooth gizmo is within two meters of yours, we’ll call that a link. The network of Bluetooth devices will naturally change over time, so we can do the same comparison between the graphs observed at short timesteps to the graph formed by aggregating all connections. During the Q&A session afterwards — before I had to, ironically enough, run off to find my cell phone — I pointed out something which it appears Braha hadn’t fully grasped.
Continue reading ICCS: Time-Dependent Networks