SLIDE 21 21
Bounding the Spectral Bounding the Spectral-
gap
- Neighbor data ratio is important
Neighbor data ratio is important
For a network of size n n, lower bound of spectral gap , lower bound of spectral gap
For each node, if ρ ρi
i ≥
≥ ρ ρT
T , a universal threshold
, a universal threshold value: value:
If ρ ρi
i =
= O O (n) for all nodes, then (n) for all nodes, then
Hence, L Lwalk
walk =
= O O (log ( (log (Datasize Datasize) ) ) )
i j j i
n n
i
∑
Γ ∈
=
) (
ρ
∑ =
+ − ≥ −
n i i 1 2
1 1 2 1 ρ λ
T
n ρ λ + − ≤ − 1 2 1 1 1
2
) 1 ( 1 1
2
O = − λ
Effect on Communication Topology Effect on Communication Topology
For all nodes Ni
i in the network,
in the network, ρ ρi
i =
= O O (n) (n) implies : implies :
Total Data Contained by Neighbors ( Total Data Contained by Neighbors (n nj
j in
in Γ Γi
i) ≥
) ≥ O O (n) times local (n) times local data data
- Real world network data distribution often follows
Real world network data distribution often follows power power-
law (
(“Measuring and Analyzing the Characteristics of “Measuring and Analyzing the Characteristics of Napster and Gnutella hosts” by Stefan Napster and Gnutella hosts” by Stefan Saroiu Saroiu et. al. , 2003)
- et. al. , 2003)
- Majority of the data content by few peers forming a ‘data
Majority of the data content by few peers forming a ‘data-
hub’
Peers with small amount of local data connecting to
with small amount of local data connecting to ‘data ‘data-
hub’ achieves O O (n) neighbor data ratio (n) neighbor data ratio
- Communication topology: A central hub consisting of few
Communication topology: A central hub consisting of few peers sharing most of the data, and rest of the peers peers sharing most of the data, and rest of the peers sharing few data are directly connected to this hub. sharing few data are directly connected to this hub.
