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
 Connect two nodes that are within distance r
r
Unit length
Unit length
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 1/ n ?
1/
n
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 1/ n ?
Connectivity threshold (Penrose ‘97, Gupta-Kumar ‘00)
 Let r 
2
log n
n

cn
n
, then
P[G(n, r) connected]
 0 if c n  
   
1 if c n  
 “Connectivity discrepancy”
Additional log n
n
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 

n log n

Similar to (and implies) connectivity threshold
 Additional discrepancy
log n
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
max’l length
matchings is
Ln ~
log
3 4
n
n
~ rc log
1 4
n
Further, additional discrepancy of log 1 4 n
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:
(L)
 “edge conductance” of
G(n,r)
 2L
+ 1/nfor r = (L)
scales as (1/n) (ignoring log n term)
L
L
 Hence
1/n
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)
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)
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 ith RED pt
 (X2i,Y2i) : position of ith BLUE pt
 For any 1/1+ 1/2=2, and some constant C, there exists a
matching  such that for j =1, 2:

i

exp  C


n
log n
j
X i -Y
j
αj
 (i)

2


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)