Network Formation

2. Oh yeahhhhhhh! Network Formation Can we model it?

3. Let’s see what Neo Martinez has to say!

4. Firstly, a “trophic species” is a group of taxa (organisms) that share the same predators and prey in a food web Food webs are represented by a matrix with S rows and columns, which represents a food web with S trophic species. Visualizing the matrix, S^2 links are possible, but there are only L actual links Directed connectance (C)=L/S^2 General Information for Models

5. The Random (Erdős–Rényi) Model • Links in the random model all occur with a probability equal to C, which means that this model does not take into account any biological structuring (trophic levels or any kind of special relationships). There is no “pecking order”, and most nodes have about the same number of links (no hubs). Thus, the model isn’t particularly accurate for biological systems. • Most networks follow the “scale free” or power-law degree distribution, while this follows the Poisson degree distribution (there is a “modal hump” for degree) • We’ll talk about this more later

6. Cascade Model • Each trophic species gets a random value from 0-1 • Each species has a probability 2CS/(S-1) of eating species with values less than its own. • This works a little better because now things are a bit more ordered and we have a semblance of a food chain, but…

7. .8 .05 .9 “underestimates interspecific trophic similarity, overestimates food-chain length”

8. The Niche Model • Also assigns random numbers from 0-1, called “niche values” • Species can consume a range (r) of other species. C is the range from r/2 to n. • Allows for cannibalism and looping

9. Results

10. Other Models for Other Things • The Niche model is good for modeling networks that are already developed, but not necessarily for predicting how nodes are added and how the network grows. • For that, you can use the Barabasi-Albert model, which incorporates both growth and “preferential attachment”.

11. The Barabasi-Albert Model… • Power-law degree distribution: • Power-law allows for ‘preferential attachment’ • Interactions between nodes can be represented in a network model by direction of edges, or the number of “in” edges and “out” edges, which, in the case of a food web, would represent who eats what; the Barabasi-Albert model is NOT directed, because the fact that when a new node is introduced, its “in degree” is 0, so nothing would ever connect to it. The BA assumes that each new node is connected to m other existing nodes (has a degree of m when it enters).

12. …Sacrifices Realism for Simplicity • If p(k)=fraction of nodes with degree k, then the probability that a new edge will attach to a node with that degree k is (k*p(k))/(2m), where, if you recall, m is the number of edges for each new node. There is a 2 in the denominator to indicate that there is no longer a direction (so each edge provides two degrees).

13. Thus, • Based on the prior information, time plays a role and older nodes have more edges. The model assumes new nodes are added in discreet time intervals. • An example: this model can be used for the internet (the older a site, the more hits it gets, especially if it started out with a lot of hits).

14. Some helpful sources: -http://www.lce.hut.fi/teaching/S-114.220/k2005b/TH_14032005.pdf -The Newman reading: http://www.sccs.swarthmore.edu/users/08/bblonder/phys120/docs/newman.pdf -The Movie: http://vw.indiana.edu/07netsci/entries/submissionspg2.html#diversity