1. Butterfly model slides

2. Topological Model: “Butterfly” Objective: Develop model to help explain behavioral mechanisms that cause observed properties, and to aid in forecasting. Properties: – Constant/oscillating NLCC’s – Densification (nodes vs edges) – Shrinking diameter (after “gelling point”) – Heavy-tailed degree distribution – Weight properties – Emergent, local, intuitive behavior 2

3. Topological Model: “Butterfly” Main idea: 3 parameters – p host : Chooses several hosts (“social butterfly”) – p step : Explores local networks in random walk – p link : Links probabilistically 3

4. Topological Model: “Butterfly” Main idea: 3 parameters – p host : Chooses several hosts (“social butterfly”) – p step : Explores local networks in random walk – p link : Links probabilistically 4

5. Topological Model: “Butterfly” Main idea: 3 parameters – p host : Chooses several hosts (“social butterfly”) – p step : Explores local networks in random walk – p link : Links probabilistically 5

6. Topological Model: “Butterfly” Main idea: 3 parameters – p host : Chooses several hosts (“social butterfly”) – p step : Explores local networks in random walk – p link : Links probabilistically 6

7. Topological Model: “Butterfly” Main idea: 3 parameters – p host : Chooses several hosts (“social butterfly”) – p step : Explores local networks in random walk – p link : Links probabilistically 7

8. Topological Model: “Butterfly” Theorem: Number of visits in each local neighborhood will follow power law. – Helps lead to heavy tailed outdegree-distribution. Proof: See Ch. 4.1. Also proved that Butterfly reproduces the other properties related to components. 8

9. Topological Model: “Butterfly” 9 Time Diam- eter Shrinking diameter log(nodes) log(edge s) slope=1.17 Densification Model (synthetic) Time Diam- eter log(edges) log(node s) Postnet (real) slope=1.1 1.

10. Topological Model: “Butterfly” 10 Postnet (real) Log(degree) Log(cou nt) slope=-2 Power-law degree distribution Nodes NLCC size Oscillating NLCCs Model(synt hetic)

11. Topological model: “Butterfly” Observed properties: Densification Shrinking diameter Heavy-tailed degree distribution Oscillating NLCCs Also (in weighted version, see thesis): Eigenvalue power law Weight power laws Bursty weight additions 11