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