Structure, centrality, graph models, and spreading on real datasets
In network models, the interaction pattern is part of the system itself. Who can contact whom changes the outcome as much as the local update rule.
Load a real graph, inspect its size, and understand what the nodes and edges mean.
Use centrality, clustering, components, and path lengths to characterize the network.
Compare the real graph against random, small-world, and scale-free models.
Run SIS and SIR on top of the same topology and explain what changed.
Communication graph between email addresses. Good for centrality, communities, robustness, and person-to-person spreading.
Mobility graph between airports built from OpenFlights data. Good for connectivity, hubs, and transport-style spreading.
Read an edge list or GraphML file into networkx.
Count nodes and edges, test connectivity, and look at node attributes.
Compute centrality, clustering, components, and path-based metrics.
Run SIS or SIR directly on the graph and compare outcomes across structures.
Degree, closeness, betweenness, and PageRank tell different stories about node influence.
Density, connected components, path lengths, and clustering quantify the overall shape of the graph.
Random, small-world, and scale-free graphs produce different path and hub structures.
Hubs and bottlenecks strongly affect thresholds, outbreak size, and control strategies.
| Model | Intuition | What to inspect |
|---|---|---|
| Erdős-Rényi \(G(N, p)\) | random edges | baseline connectivity and path lengths |
| Watts-Strogatz \(WS(N, k, p)\) | local links plus rewiring | clustering and small-world effect |
| Barabási-Albert \(BA(N, m)\) | preferential attachment | hubs and heavy-tailed degree distributions |
\[ \mathbb P(S \to I) = 1 - (1 - \beta)^m, \qquad \mathbb P(I \to S) = \mu \]
\(m\) is the number of infected neighbors.
Use the same infection mechanism, but replace recovery back to susceptible with a permanent recovered state.
Characterize one real network with metrics, plots, and centrality rankings.
Run both SIS and SIR on that same network and explain how the structure changes the propagation outcome.
Applied Math Lab