23 Jan 2020

* Review last class; k-connectivity, directed graphs, strong/weak connectivity

- Homework (for Slota): fix sentence noting directed/undirected

k-connectivity
- related to network resilience
- how many vertices must we remove to disconnect the graph?

directed graphs
- assume some directivity to the graphs

strong and weak connectivity
- strong relates to connectivity in that there exists a u,v-path for all u,v
- weak is equivalent to connectivity if we remove directivity from the edges

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* Degree distributions

Degree distribution: how many vertices are there of each unique degree in G?

We consider the degree distribution in many different ways:
- Quantifying the degree skew, irregularity (inherent property of the dataset)
  * Meshes ==> very regular
  * Real-world graphs ==> not so much
- Use in generating random graphs for null model graphs

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* Power law exponent

Often, degrees within a graph exhibit a power law distribution

If we want to explicitly quantify and compare between graphs
- Fit a power law exponent
 *  Many different ways to do this
- We'll consider the maximum likelihood estimator
  * alpha = 1 + n * sum ( ln(d(v)) )^-1, for all v in V(G)
  * alpha = power law exponent
  * n = |V|
  * d(v) = degree of v
- Most real-world graph have: 1 <= alpha <= 3
- Generally, most real-world graphs exhibit skew not just in degree distribution
  * Connectivity structure, cluster sizes, etc.

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* Shortest paths and small-worldness

How small-world a graph is:
- What's the average shortest paths length?
  * Considering all shortest u,v-paths G
  * The averaged path length
- O(n^2) paths need calculated ==> infeasible for |V(G)| > ~1 million
  * So ==> do sampling
  * Calculate shortest paths for a subset of u,v-paths

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* Coding
  - Loading real datasets
  - Directed graphs
  - Connectivity stuff
  - Plotting degree distributions