Geographic Gossip on Geometric Random Graphs via Affine Combinations

1
Geographic Gossip on Geometric Random Graphs via
Affine Combinations

• Hariharan Narayanan
• Department of Computer Science
• University of Chicago

2
Geometric Random Graphs
3
Geometric Random Graphs
4
Geometric Random Graphs
5
Distributed Averaging

• Each node starts with a value.

6
Distributed Averaging

• Each node starts with a value.
• Task Develop Decentralized Asynchronous
  algorithm to average values of nodes.

7
Distributed Averaging

• Each node starts with a value.
• Task Develop Decentralized Asynchronous
  algorithm to average values of nodes.
• (Value at each node must tend to the initial
  average.)

8
Result

• Averaging algorithm for Geometric Random graph
  with n nodes using Geographic information.

9
Result

• Averaging algorithm for Geometric Random graph
  with n nodes using Geographic information.
• Takes transmissions to decrease the
  variance by a multiplicative constant (previously
  )

10
Result

• Averaging algorithm for Geometric Random graph
  with n nodes using Geographic information.
• Takes transmissions to decrease the
  variance by a multiplicative constant (previously
  )
• Not quite decentralized but close.

11
Result

• Averaging algorithm for Geometric Random graph
  with n nodes using Geographic information.
• Takes transmissions to decrease the
  variance by a multiplicative constant (previously
  )
• Not quite decentralized but close.
• Asymptotically optimal exponent.

12
Subdivide into smaller squares
13
Convex averaging within squares
0
0
2
-1
2
2
1
0
2
-2
0
0
14
Convex averaging within squares
0
1
2
-1
1
2
1
0
2
-2
0
0
15
Convex averaging within squares
0
1
2
-1
1
2
1
0
2
-2
0
0
16
Convex averaging within squares
0
1.5
2
-1
1
2
1
0
1.5
-2
0
0
17
Convex averaging within squares
0
1.5
2
-1
1
2
1
0
1.5
-2
0
0
18
Convex averaging within squares
1
1.5
2
-1
1
1
1
0
1.5
-2
0
0
19
Convex averaging within squares
1
1.5
2
-1
1
1
1
0
1.5
-2
0
0
20
Convex averaging within squares
1
1.5
2
0
1
1
0
0
1.5
-2
0
0
21
Convex averaging within squares
1
1.5
2
0
1
1
0
0
1.5
-2
0
0
22
Convex averaging within squares
1
1.5
0
0
1
1
0
0
1.5
0
0
0
23
Affine Combination between squares

• Greater mass Transport

1
1
0
1
0
1
0
0
1
0
1
0
24
Affine Combination between squares

• Greater mass Transport

1
1
0
-2
0
1
3
0
1
0
1
0
25
Convex Combinations within squares
1
1
0
0
-2
1
3
0
1
0
1
0
26
Convex Combinations within squares
1
-.5
0
0
-.5
1
3
0
1
0
1
0
27
Convex Combinations within squares
1
-.5
0
0
-.5
1
3
0
1
0
1
0
28
Convex Combinations within squares
1
-.25
0
0
-.5
1
3
0
-.25
0
1
0
29
Convex Combinations within squares
1
-.25
0
0
-.5
1
3
0
-.25
0
1
0
30
Convex Combinations within squares
1
-.25
1.5
0
-.5
1
1.5
0
-.25
0
1
0
31
Convex Combinations within squares
1
-.25
1.5
0
-.5
1
1.5
0
-.25
0
1
0
32
Convex Combinations within squares
1
-.25
.75
.75
-.5
1
1.5
0
-.25
0
1
0
33
Convex Combinations within squares
1
-.25
.75
.75
-.5
1
1.5
0
-.25
0
1
0
34
Convex Combinations within squares
.25
-.25
.75
.75
.25
1
1.5
0
-.25
0
1
0
35
Convex Combinations within squares
.25
-.25
.75
.75
.25
1
1.5
0
-.25
0
1
0
36
Convex Combinations within squares
.25
-.25
.75
.75
-.25
1
.75
0
-.25
0
1
.75
37
Geometric Random Graphs
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5
.5