Making connections:  Random graphs and network evolution

Social structures are not random graphs

• Evolution of a random graph
• Emergence of a large component at the critical phase transition
• Degree distribution will resemble Poisson (independent connections with replacement)
• Average path length will be small until critical value
• Is this a good model for how individuals and social structures build networks?
• Many local neighborhoods with overlaps; bias for within-neighborhood; some randomness

The alpha model and small world networks

• Similar to Rappoport's transitivity.
• Actors are more likely to connect via existing connections
• Neighborhood bias as a variable "alpha"
• Alpha low = caves (clustered graph); alpha high = solarians (random graph)
• The small world phenomenon (figure 3.4)
• Path length within component and degree of clustering as randomness increases creates three regions: fragmented, small-world, and random

The beta model -- substrates and rewiring

• A minor revision -- rewiring
• Figure 3.6:  lattice neighborhoods as clustering
• Rewiring:  replace a local with a random long distance
• Same small-world emerges:  high clustering and low path length
• Is this a better model?  The lattice sub-strate as a picture of social structure.

Applying the random biased network model

• Wildly different types of networks display small worlds
• Why might this be from an evolutionary point of view?
• Why might this be from an agency point of view?