What makes life worthwhile is who accompanies you through it, and moving really disrupts your support network. Making new friends is very difficult for me. Luckily, I have rather gregarious colleagues. They've been a great support network during the disequilibration of moving, but I really miss being able to walk to the coffee shop for a good discussion with my best friend from Nebraska. The Rocket Scientist is great for the talking part, but doesn't drink coffee (and there's that pesky chromosome mismatch).
I expected that my condensed matter physics colleagues and I would know people in common, but I've been surprised at the larger scale on which this is true. My colleague Owl and I have an acquaintance in common, which I learned only when we both ended up at the same party. I had met the host through the physics education community and Owl through their mutual research field. We’d both known our host more than ten years, but didn’t know the other knew. Our host is one of those types that bridges multiple worlds. I guess I'm also one of those types given my NASCAR life and my physics life.
My curiosity was further teaked when my co-blogger, Lee, finally got me on Facebook and Dave Moody, the host of the Sirius Speedway radio show, prodded me to get on LinkedIn. Yes, it took me this long. I never joined cliques in high school. OK, probably because I was never asked, but that’s a topic for the Cocktail Party Psychology blog. (Actually, now that I think of it, the one thing common to both worlds are cocktails. If you like cocktails that involve Everclear, check out Dave's Apple Pie Moonshine. Do pay close attention to the "allow to cool" part of the recipe.)
Facebook and LinkedIn let you scan how many people in your Outlook address book already belong. I was surprised, first because it was about 75% and second, because the 75% included a lot of people I didn't expect to see. I didn’t ask anybody to ‘friend’ me due to my stubborn refusal to use ‘friend’ as a verb. But I did look at the ‘friends’ the people I know have. (I hesitate now to say “my friends” since that apparently requires a reciprocity I’m not sure I have with all of them.) LinkedIn identifies people you're linked to through other people by degree. For example, in the real world, Owl and I were second-degree contacts before we met (the link being through the host), and now we're first-degree contacts.
Assume a spherical person. Well, at least a circular person, as that’s how you represent people in network diagrams. The circles (people) are called ‘nodes’ and interactions between people are represented by links. In a random network, like the one at right, two nodes are randomly chosen and a line drawn between them. Interesting properties emerge in random networks, which were described in 1959 by Erdos and Renyi. When there are only a small number of links (small being measured relative to the number of nodes), the network looks like a collection of clusters, but as more and more links are added, you eventually get a giant interconnected cluster. The formation of the giant cluster (a type of percolation) happens suddenly, like a phase transition. And if you want to get the attention of physicists, mention phase transitions.
Physicists started thinking about how statistical mechanics could be used to understand networks. A random network has a Poisson distribution of link probabilities (the probability that a given node has some number of links). This means that a few outliers will have very small or very large numbers of links, but that most nodes will have reasonably similar numbers of links.
Real-life networks (Co-authorship relationships, social networks, HIV/AIDS transmission, electrical grids and yes, movie actors) rarely look like random networks. Most real networks (like the one at left) have a small number of nodes with a very large number of links and a large number of nodes with very few links. This is in contrast to the more egalitarian behavior of random networks. The scale-free behavior is characterized by a power-law distribution of the links, instead of the Poisson distribution.
Both types of networks have the ‘small-world’ property, which is that there is a fairly short distance between any two nodes in the network. (Yes, we play six-degrees of Physicist X at meetings, where Physicist X is usually a theorist, someone who makes samples, or someone who has unique measurement apparatus.) The short path is because the the size of the network (the distance between any two nodes) is log N, with N being the number of nodes.
A big difference between random and scale-free networks is clustering: Scale-free networks are divided into groups or communities. This implies differences in how networks grow. Random networks are, well, random. Scale-free networks grow by preferential attachment, in which a new link is much more likely to attach to a node the more links the node has (a rich-get-richer type of thing, which actually has a name - the Matthew Effect). Authorship, scientific collaboration and funding follow this type of growth model.
Characterizing a network's properties are one thing, but physicists want to get the time dependence right. Various models produce clustering, small-world effects and dynamic evolution, but getting them all in a single model has been elusive.
Gonzalez, et al. modeled the interactions of people by representing them as particles random-walking through space, making links when the particles collided. This sounds like a random model; however, they introduced anisotropy via an adjustable parameter: a vector describing how two particles moved away from each other after a collision. Changing this parameter allowed them to reproduce the growth of a number of different types of networks. A similar approach is being used to investigate the networking of proteins that cause breast cancer.
It's interesting that one parameter can vary the nature of the network so much - that's the parameter responsible for why some people got asked to the prom and I can recite the lines from a lot of 1930's movies. I spoke with an anthropologist, who was skeptical about a single parameter being able to account for the richness of human behavior; however, she also told me that the unit of understanding was often the community, so perhaps that parameter is really more indicative of communal norms and thus more general. For example, if I write a grant with you and it's funded, we're probably going to write more grants.
The graph at right (credit Marta Gonzalez) from a physorg.com report shows a high school friendship network. Different colors represent different grades. You can't really make out anything in the middle, but see all those dots on the edges with few or no links? One of those would have been me. I couldn't help but be struck by the likely irony that it was the folks who probably were on the periphery of the giant cluster that are now trying to understand it.
Today, of course, my social network looks a little bit different than it did in high school. My Outlook contacts file directly links Nobel Laureates with NASCAR drivers. One of the issues with statistical mechanics models, though, is that they only work when the networks are large enough. There have to be enough links relative to the number of clusters. I've got essentially two giant clusters with very few interconnects between them.
There's no reason for the lack of interconnects except opportunity. So when The Rocket Scientist emailed me a racous joke that starred a couple of good 'ol boys, I couldn't help but forward it to a couple of my NASCAR friends. I even made a direct link between the two networks - The Rocket Scientist has now been introduced to an equally crazy (and I mean that in a good way) Ph.D. who is obsessed with making race car engines faster. I even know a NASCAR driver who is really anxious to meet a cosmologist. I'll have that network size approaching log N in no time.
Until I get those links, though, my network looks as if it were two parallel worlds, sort of like Superman' s normal world and the Bizzaro-world analog. I am hard pressed to figure out which world is the "normal" one.