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RANDOM GEOMETRIC GRAPHS Discussion of Markov Lecture of Francois Baccelli Devavrat Shah Laboratory for Information & Decision System Massachusetts Institute of Technology

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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

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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 ?

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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

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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 L 1 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

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max’l length 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

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L 2L + 1/ n 1/ n (L) 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)

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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)

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The ultimate matching conjecture Talagrand ‘01 proposed the following conjecture Unifies L 1 and L threshold results (and more) Throw n RED, n BLUE points at random in unit square (X 1 i,Y 1 i ) : position of i th RED pt (X 2 i,Y 2 i ) : position of i th BLUE pt For any 1/ 1 + 1/ 2 =2, and some constant C, there exists a matching such that for j =1, 2:

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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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