BigBang Simulation for Embedding Network Distances in Geometric Space

1
Big-Bang Simulation for Embedding Network
Distances in Geometric Space

• Yuval Shavitt and Tomer Tankel
• Dept. of Electrical Eng. Systems

2
Problem and solutions

• The Problem
• Collect and disseminate distance information.
• Usage
• Closest mirror selection
• Construction of application level multicast trees
• peer-2-peer networks
• Solutions
• IDMaps ToN 01 triangulation
• GNP Ng and Zhang, 02
• Euclidean Embedding in Rd, down-hill-simplex
• Not accurate, high max/var symmetric distortion

3
The Closet Mirror Selection Problem

• Given a set of servers, si 1iK,
• and a client, c, find a server sj,
• s.t., ?i 1iK d(sj,c) ad(si,c)ß

4
First Solution

• The original IDMaps solution ToN01 sums segment
  distances to estimate end-to-end distance.
• No sense of geomerty

T1
T2
5
Problem statement
T1
T2
A
B
C
6
Problem statement
T1
T2
A
B
C
7
Embedding Problem Statement

• Ng and Zhang, 02 suggest to embed the graph in
  d dimensions geometric space.
• They use down hill simplex (DHS) to minimize the
  total all-pairs symmetric distortion.
• Namely global embedding

Distortion Max Real dist. /computed
dist., computed dist. / Real dist.
8
Basic Idea

• In our model
• particles network nodes (Tracers, clients)
• inter-particle force friction difference
  between real and estimated distance
• Kinetic energy drive particles out of local
  minima of the error function

9
Inter Particle Forces
10
(No Transcript)
11
(No Transcript)
12
(No Transcript)
13
(No Transcript)
14
BBS Features

• Particles with larger estimation errors move
  faster
• Equilibrium points of the potential function are
  points where the field force, Fi, is zero for all
  particles Vi
• Friction slows down particles so they can slip
  into potential wells.

15
No Friction Effect
16
No Friction Effect (cont.)
17
Friction Effect on Energy and Velocity
m
.7
Energy
m
0
Energy
60
2000
50
i
1500
0
i
40
Single Particle Energy (E)
Maximum Velocity of Particles max(v (t))
30
1000
20
500
10
m
Velocity _at_
.7
m
Velocity _at_
0
0
0
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
Single Particle Distortion max(v
D
/
)
i
j
i
j
0
0
18
Algorithm details

• Calculate each particle friction coefficient
  (proportional to the sum of its real distances to
  the other particles).
• Place all particles near the origin.
• Repeat
• calculate the forces on the particles
• apply Newtons 2nd Law for one time step
• adjust time step
• Until error does not improve OR error/velocity
  below threshold OR diverge to infinity
• Switch error function and goto step 3

19
Other Embedding Methods

• Semi-Definite Programming (SDP)
• Best known theoretical result- Linial et. al.
  95
• Multi-Dimensional Scaling (MDS)
• Simple and low complexity implementation
• Down-Hill Simplex (DHS)
• Used in GNP Ng Zhang 02
• Minimum-Spanning Tree / 2 Random Trees
• 3n edges network aggregation- Awerbuch 01

20
Performance Metrics

• Average Symmetric Distortion
• Two Sided Embedding Distortion
• Minc1c2 c1dij gt vi vj gt dij/c2

21
(No Transcript)
22
(No Transcript)
23
Our Main Idea
Embed the graph in hyperbolic space
Example Poincare disk D2
24
Embedding Example in D2
25
Curvature and Distances Ratio
26
Hyp. Curvature Scaled Metric
Gaussian Curvature Cmax2
Scaled Metric
Normalized Metric
A
B
A
B
1
Cmax
D
C
AB/AC
D
C
Cmax(AB/AC)
27
Metric Curvature Equation

• Arranging points A,B,C and D symmetrically at the
  hyperbolic origin and equating distances to the
  scaled metric we get
• cosh(Cmax) - cosh(Cmax AB/AC) 1 0
• Equation for metric without distortion
• cosh(Cmax)-2cosh(Cmax(r2)/(2r2)) 1 0
• Where rb/a is ratio between edges of the graph.

28
Embedding Methods

• All pair (AP)
• Embed n-nodes metric, n(n-1)/2 distance pairs, at
  once.
• Two phase (TP)
• Embed Small subset of t Tracers, t(t-1)/2
  distance pairs.
• For each of the other nodes, embed its distances
  to several nearest Tracers.
• Random Neighbors (RN)
• Embed with distances to
• The 1-neighborhood
• Order of log(n)peer nodes, selected uniformly at
  random.
• No fixed tracers

Not scalable
29
Rand. Neigh. vs. Two Phase

• Two Phase
• Non-symmetric
• Distributed
• Over-estimation of short distances
• Sensitive to Tracer failure

• Random Neighbors
• Symmetric
• Central calculation
• Equally accurate for all distances.
• Suitable for P2P
• How to calculate distributively?

30
All-Pairs Embedding for AS Graph 1/00
31
Two-Phase Embedding for AS Graph 1/00
32
Transit
LAN
33
(No Transcript)
34
(No Transcript)
35
(No Transcript)
36
(No Transcript)
37
3D Hyperbolic IGT
3D Hyperbolic Multicast Tree
AS 1/00 Graph 800 LANCORE members
100
Complementary Frequency
50
0
1
1.5
2
Relative Delay Penalty(RDP)
38
2D Hyperbolic IGT
2D Hyperbolic Multicast Tree
AS 1/00 800 LANCORE members
100
Complementary Frequency
50
0
1
1.5
2
2.5
Relative Delay Penalty(RDP)
39
Two-Phase Embedding, dim5 AS 1/00, 800 members,
15 tracers
40
Conclusions and Further Study

• Hyperbolic space improves embedding for all
  methods (e.g., GNP) and all curvatures.
• Closest mirror selection results.
• Scalable, symmetric, and distributed Tree
  Construction
• Future work
• Features of HYP Coordinates
• Density vs. Origin Distance
• Degree vs. Origin Distance