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Random Geometric Graphs

Random Geometric Graphs. Discussion of Markov Lecture of Francois Baccelli Devavrat Shah Laboratory for Information & Decision System Massachusetts Institute of Technology. Random geometric graph. G( n,r ) Place n node uniformly at random in unit square

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Random Geometric Graphs

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  1. Random Geometric Graphs Discussion of Markov Lecture of Francois Baccelli Devavrat Shah Laboratory for Information & Decision System Massachusetts Institute of Technology

  2. Random geometric graph G(n,r) • Place n node uniformly at random in unit square • Connect two nodes that are within distance r r Unit length Unit length

  3. Random geometric graph Quantity of interest: discrepancy • How “far” is G(n,r) from “expected” node placement At what r does G(n,r) become connected? near ?

  4. Random geometric graph Quantity of interest: discrepancy • How “far” is G(n,r) from “expected” node placement At what r does G(n,r) become connected? near ? Connectivity threshold (Penrose ‘97, Gupta-Kumar ‘00) • Let , then • “Connectivity discrepancy” Additional

  5. Random geometric graph Quantity of interest: discrepancy • How “far” is G(n,r) from “expected” node placement Minimum of total edge-lengths over all perfect matchings L1 grid-matching threshold: (Ajtai-Komlos-Tusnady ‘80) • With high probability, the minimal total length of matching is Similar to (and implies) connectivity threshold • Additional discrepancy

  6. Random geometric graph Quantity of interest: discrepancy • How “far” is G(n,r) from “expected” node placement Minimum of maximum edge-length over all perfect matchings L grid-matching threshold: (Leighton-Shor ‘86) • With high probability, minimal max length over all matchings is Further, additional discrepancy of max’l length

  7. Why worry about discrepancy ? For r scaling as L grid-matching threshold (say L) • G(n,r) contains “expected” grid as it’s sub-graph w.h.p Implications: • “edge conductance” of G(n,r) for r = (L) scales as (1/n) (ignoring log n term) • Hence Capacity scales as (1/n) (Gupta-Kumar ’00) Hierarchical schemes for info. th. scaling (Ozgur et al ‘06, Niesen et al ‘08) Monotone graph properties have sharp threshold (Goel-Rai-Krish ‘06) Mixing time of RW scales (n) (Boyd-Ghosh-Prabhakar-Shah ‘06) Information diffuses in time (n) (MoskAoyama-Shah ‘08) (L)  2L + 1/n  L  L 1/n

  8. Why worry about discrepancy ? For r scaling as L grid-matching threshold (say L) • G(n,r) contains “expected” grid as it’s sub-graph w.h.p Implication: • n RED, n BLUE points thrown at random in unit square • Match a RED point to a BLUE point that is UP-RIGHT • Number of unmatched points scale as (n L) ~ (n) Online bin-packing analysis (Talagrand-Rhee ‘88) Mean Glivanko-Cantelli convergence (Shor-Yukich ‘91) Bin-packing with queues (Shah-Tsitsiklis ‘08)

  9. The ultimate matching conjecture Talagrand ‘01 proposed the following conjecture • Unifies L1 and L threshold results (and more) Throw n RED, n BLUE points at random in unit square • (X1i,Y1i) : position of ithRED pt • (X2i,Y2i) : position of ithBLUE pt • For any 1/1+ 1/2=2, and some constant C, there exists a matching  such that for j =1, 2:

  10. Point process view Poisson process and stochastic geometry • Useful, for example understanding Structure of radial spanning trees (cf. Baccelli-Bordenave ‘07) Behavior of wireless protocols (cf. Baccelli-Blaszczyszyn ‘10) • Hope: resolution of The ultimate matching conjecture (or a variant of it)

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