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Butterfly Topological Model: Understanding Behavioral Mechanisms in Network Dynamics

The Butterfly Topological Model is designed to elucidate the behavioral mechanisms that lead to observed network properties and enhance forecasting capabilities. Key features include constant and oscillating Non-Local Clustering Coefficients (NLCCs), network densification, and a shrinking diameter after the 'gelling point.' The model employs three main parameters: 'phost' for multiple host selection, 'pstep' for local network exploration via random walks, and 'plink' for probabilistic linking. The emergent behaviors observed support heavy-tailed degree distributions and unique weight properties, contributing to a deeper understanding of network dynamics.

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Butterfly Topological Model: Understanding Behavioral Mechanisms in Network Dynamics

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  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

  3. Topological Model: “Butterfly” • Main idea: 3 parameters • phost: Chooses several hosts (“social butterfly”) • pstep: Explores local networks in random walk • plink: Links probabilistically

  4. Topological Model: “Butterfly” • Main idea: 3 parameters • phost: Chooses several hosts (“social butterfly”) • pstep: Explores local networks in random walk • plink: Links probabilistically

  5. Topological Model: “Butterfly” • Main idea: 3 parameters • phost: Chooses several hosts (“social butterfly”) • pstep: Explores local networks in random walk • plink: Links probabilistically

  6. Topological Model: “Butterfly” • Main idea: 3 parameters • phost: Chooses several hosts (“social butterfly”) • pstep: Explores local networks in random walk • plink: Links probabilistically

  7. Topological Model: “Butterfly” • Main idea: 3 parameters • phost: Chooses several hosts (“social butterfly”) • pstep: Explores local networks in random walk • plink: Links probabilistically

  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.

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

  10. Topological Model: “Butterfly” • Power-law degree distribution • Oscillating NLCCs Model(synthetic) NLCC size Log(count) slope=-2 Postnet (real) Nodes Log(degree)

  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

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