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

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 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

6. 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

7.  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)

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 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:

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)