From structure to dynamics — Networks and Complex Systems
Resource for the Networks and Complex Systems course focused on dynamical processes on graphs. It covers SIS and SIR models, epidemic thresholds, intervention strategies, and temporal networks, combining mathematical intuition with interactive simulations.
Both models define local transition rules on a network. But they answer different questions and prioritize different metrics.
| Aspect | SIS | SIR |
|---|---|---|
| Sequence | S → I → S | S → I → R |
| Central question | Does the infection persist? | How large is the outbreak? |
| Theoretical object | Stationary prevalence | Final size / percolation |
| Structural intuition | Hubs as reservoirs | Geometric reach of the outbreak |
| Classical tool | HMF, QMF, spectral radius | Transmissibility, percolation |
Watch how contagion spreads step by step. Each node changes color according to its state.
At each discrete step, every infected node attempts to infect each susceptible neighbor with probability β, and then recovers with probability μ. In SIR it moves to R; in SIS it returns to S.
The ratio λ = β/μ acts as an effective control parameter. But the threshold also depends on network structure.
Is there a critical transmissibility value such that, below it, the outbreak dies out quickly, and above it, it can be sustained? That value is the epidemic threshold, and it is a joint property of dynamics + structure.
If the network has hubs (large ⟨k²⟩), the threshold decreases: heterogeneity facilitates persistence because hubs act as dynamic reservoirs.
The spectral radius captures the structural amplification capacity of the whole network, not only the degree distribution.
Generate three networks of the same size and compare their threshold proxies.
A single run can be misleading: the initial seed and randomness dominate. That is why we repeat and average.
Vary λ = β/μ and observe where the outbreak stops being small. This shows an empirical transition zone, not an exact analytical threshold.
If I remove or immunize certain nodes, how much does the outbreak shrink? Three classical strategies:
| Strategy | Criterion | Intuition |
|---|---|---|
| Random | At random | Baseline |
| Degree | Highest degree first | Cuts many potential contacts |
| k-core | Highest coreness first | Breaks the core where diffusion is sustained |
The aggregated network answers: who was connected to whom at any time?
The temporal network answers: did a chronologically valid sequence for transmission exist?
These are not the same question. The order of contacts is part of the mechanism, not a formatting detail.
A chain of contacts with timestamps. Node 0 starts infected. Compare which nodes are reached temporally versus in the aggregated network.
"The importance of a node depends on the dynamical process taking place on the network, the time horizon, and the performance criterion."
There is no universally correct centrality. The appropriate measure depends on the propagation mechanism and the intervention objective.