Centrality and Layout

Le3 - Mon Oct 6, 2025 Layered networs; Paths, cycles, diameter, and small world networks; Connectedness and the condensation grapg; Degree centrality; Power law versus Poisson distribution; Estimation methods for the degree exponent: ML approach; Explaining the power law; Erdos-Renyi random model: binomial and Poisson descriptions of the model; Small world; Power law versus Poisson distribution; Growth and preferential attachment; the Barabasi-Albert model.

Le4 - Fri Oct 10, 2025 Mathematics of the Barabasi-Albert model; Attractiveness and the Bianconi-Barabasi model; Analysis of the Bianconi-Barabasi model; Examples with equal and uniform fitness; Measuring fitness: the www; Other ideas for modelling the power-law.

Le5 - Mon Oct 13, 2025 Properties of the power-law: largest hub, moments and scale-free networks, distances; PageRank: random walk, dead ends and spider traps, teleportation; PageRank with restart; PageRank versus degree centrality; convergence properties of PageRank; PageRank matrix structure via the condensation graph.

Le6 - Fri Oct 17, 2025 Action of the teleport vector: one leaf-only, one eigenvalue set to 1 and the rest to (at most) c; Interpretation; Tuning the teleport vector: Local PageRank; Examples of application; Approximate PageRank and the push operation: proof of the precision guarantee; Application to link prediction.

Le7 - Wed Oct 20, 2025 Topic specific PageRank; Signed PageRank; Preventing spamming; Row-normalized PageRank; HITs centrality; Eigenvector and Katz centralities and their relation to PageRank and Degree centrality; Closeness and harmonic centrality; Betweenness centrality as a measure of brokerage.

Le8 - Fri Oct 24, 2025 The clustering coefficient; Wrap-up on centrality measures; The layout problem; Aesthetic criteria; Spring-embedder algorithm; Repulsive and attractive forces; Fruchterman & Reingold; Force atlas 2; Gravity; Approximate repulsion; UMAP as a force-layout algorithm; Software tools.