Here is a quick section-by-section summary of chapter 7, "the rich get richer" in Linked
7.0 - An epiphany occured to me while I was in Portugal that networks may follow the rich get richer phenomenom
7.1 - In considering networks that are abstracted with graph theory, one must add in the notion of a changing node count over time, and that the nodes have differences among them
7.2 - The examples of the web and a graph of hollywood actors with their connections to other actors both demonstrate small beginnings, with one or a handful of nodes, and then continuous growth. The author mentions that most real networks share this feature, growth.
7.3 - Model A - start with 2 nodes, and then start adding new nodes, with each one randomly connecting with 2 of the existing nodes. What results is in a power law (y = 1/x) distribution where the original are tremendously rich with connections and the vast majority are extremely weak. However, this did not model the web nor hollywood in that it didn't explain later occuring hubs. Hence growth alone is not enough to look at.
7.4 - preferential attachment - when ppl have a bias for a specific node, they'll be more likely to link it. As that node gets more links, it becomes more visible and more likely to be linked even more. The web and other networks are not democratic (or rather, not meritocratic). This is the rich get richer concept.
Given that this system is natural does that mean it's moral. Or the odd question is, since it is natural, why should it be wrong? Or maybe, I want to suggest that we don't start attaching sinister objectives to the popular or the rich when we notice this power-law occur in society, but instead reconigze that it's natural. Although, just because it's a natural phenomena, doesn't mean we can't correct it.
7.5 - scale-free model is a synthesis of the growth and preferential treatment nature of these networks. You make the probability of connecting to a node proportional to its existing popularity, and as a result, various later nodes who got lucky with more links, could continue to get lucky, so hubs would form not just in the early nodes, but with later ones as well.
By scale-free, I think that means the "average" doesn't matter. As a mnemonic, I like to think that the majority of nodes are below average on link count, and therefore these are situations where the median differs too widely from the mean, therefore making them irrelevent, or making the network free from scale (I think I'm not explaining this perfectly, sorry).
7.6-7.7 - when we have both growth and preferential treatment, we get the hub and power laws. There are other complex factors that can shape these models, and there's been a lot of recent research these past couple of years. Other ideas: internal links, rewiring, removal of nodes and links, aging, nonlinear effects.
7.8 - the big picture - wherever there is growth and preferential attachment, there will be hubs and power laws (like in Hollywood, the metabolic network within the cell, citation networks, economic webs, and the network behind language.) The internet gave good, large data to aid this process. Next up, how do late-comers come into the picture.
I really like the way this book is formatted. Each chapter is a catchy term or phenomenom that encaprsulates an illuminating model of something originally cumbersome. Plus, it lends itself to bite-sized pieces, with sections that are a couple of pages long, and follow the same basic flow: a personal introduction, a technical introduction, the guts, and a neat summary. Also, the author is repetitive, but in a good way. He emphasizes the same concept from different angles. The benefit of this is that you really get a sense for how this knowledge is useful or should be applied. Without the "big picture" I think these theories would be rendered to the trash bin of boring math equations.
Wednesday, Sep 24, 2003
