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