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Planting trees in random graphs (and finding them back)

This research explores the challenges and possibilities of detecting and reconstructing planted trees in random graphs, with a focus on the Erdös-Rényi model. The study analyzes phases, detection and reconstruction, and the implications for network security scenarios.

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Planting trees in random graphs (and finding them back)

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  1. Planting trees in random graphs(and finding them back) Laurent Massoulié Joint work with Ludovic Stephan & Don Towsley

  2. Inferenceproblemswithplantedstructure Classical examples community detection, planted clique detection, planted dense subgraph detection Typical Objectives -Detection (planted structure present or not) -Recovery (estimation of planted structure) Recurring phenomena  3 regimes, or phases:-Information-Theoretic impossibility -Computational hardness though IT-feasible -Computationally easy  Boundary phases the same for both detection and reconstruction

  3. ‘Network Security’ scenario Attackers may be “plotting” or not Plotting  communications among them Goals: -Infer from observed communication graph whether attack under way (detection) -If attack detected, identify attackers (reconstruction)

  4. Model Erdös-Rényi graph Under attack: augmented by tree on nodes chosen uniformly at random Focus on simplest tree i.e. path

  5. Outline • Phase diagram for paths • Other trees

  6. Sparse phase

  7. First moment method: no K-path in G under  poly-time detection and reconstruction

  8. The argument for undetectability Likelihood ratio between distributions without and with attack,

  9. The argument for undetectability Likelihood ratio between distributions without and with attack,

  10. Bounding Markov chain

  11. The argument for detectability, • Too many edges under • Counts of small connected components enable consistent estimation of Arguments hold for any connected K-graph, not just lines and trees

  12. The argument for reconstruction impossibility Optimal overlap between estimated and planted path: Maximum a Posteriori, i.e. nodes on largest number of K-paths

  13. The argument for reconstruction impossibility “Lures”: can construct confounding segment s.t.

  14. More generally: construct T symmetric paths

  15. Other trees: stars • Stars hard to hide! Threshold at Detection and reconstruction easy: inspection of degrees

  16. D-trees

  17. Conclusions Phase diagram for line detection & reconstruction -No computationally hard phase -Reconstruction impossible, by presence of too many copies of planted structure Planted subgraphs beyond cliques & dense subgraphs: -Phase diagram for D-regular trees? More general planted graphs? -What triggers hard phases?

  18. Thanks!

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