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Butterfly Topological Model: Understanding Behavioral Mechanisms in Network Dynamics

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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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