A General Nash Equilibrium Semantic Cache Algorithm in a Sensor Grid Database - Data Management in Cloud, Grid and P2P Systems - page 17

The relationship of the semantic, life, and location weight vectors W i , W i ,and
W ” i are in Nash equilibrium. Each mobile device would want to choose scalar
coecients for its three weight vectors that would result in a cache replacement
strategy that would strive to minimize the device's costs in querying for data
and maintaining its cache and the time it takes to get queries answered. And, it
would be useful to be able to adjust the coecients separately thus enabling the
weighting scheme to be customized for different types of applications and device
movement patterns. It is their game and its objective, which is a useful one for
caching among mobile devices.

3.2 The Scalar Coe cients of the Weights of the Nodes Satisfied

α i + β i + γ i =1;0 <α i ,β i ,γ i < 1; α i ,β i ,γ i ∈ R. (3)

Theorem 1. Given three linearly independent vectors
W i , W i ,and W i and their linear combination
W i,j = α i W i,j + β i W

i,j + γ i W i,j , α i ,β i ,γ i are scalar coecient, α i ,β i ,γ i ∈ R ,

α i ,β i ,γ i ≥ 0 : then the following expression is satisfied:

α i + β i + γ i =1 , 0 <α i ,β i ,γ i < 1 , α i ,β i ,γ i ∈ R .
Proof: The rationality of the equation is obvious:
Let W i,j = α i W i,j + β i W i + γ i W i , α i ,β i ,γ i ∈ R , α i ,β i ,γ i ≥

0 .

W i,j

α i β i

W i,j + W

γ ∗ i

W =

i,j + i,j .

α i + β i + γ i α i + β i + γ i α i + β i + γ i α i + β i + γ i

W i,j

α i β i γ i

Then, W i,j = , α i = , β i = , γ i = .

α i + β i + γ i α i + β i + γ i α i + β i + γ i α i + β i + γ i

Then W i,j = α i W i,j + β i W

i,j + γ i W

i,j ,and α i + β i + γ i =1 , 0 <α i ,β i ,γ i < 1 ,

α i ,β i ,γ i ∈ R .

3.3 The Nash Equilibrium Point of the Semantic, Time and
Location Factors

W i · W i
W i

= W i · W = W i · W

i
W i

i
W i

. (4)

Theorem 2. Given three linearly independent vectors,
W i =( w i, 0 ,...,w i,j ,...,w i,m− 1 ) ,

W i =( w i, 0 ,...,w i,j ,...,w i,m− 1 ) ,

W i =( w i, 0 ,...,w i,j ,...,w i,m− 1 ) , their linear combination

W i =( w i, 0 ,...,w i,j ,...,w i,m− 1 ) ,and W i = α i W i + β i W + γ i W , α i ,β i ,γ i

i i

are scalar coecient, α i ,β i ,γ i ∈ r , α i ,β i ,γ i ≥ 0 . The Nash Equilibrium point

of W i , W i and W i is
W i ·W i

= W i ·W = W i ·W

.

i i

W i W i W i

Proof: Because W i =( w i, 0 ,...,w i,j ,...,w i,m− 1 ) , W i = w i, 0 ,...,
w i,j ,...,w i,m− 1 ) , W i =( w i, 0 ,...,w i,j ,...,w i,m− 1 ) , their linear com-
bination W i =( w i, 0 ,...,w i,j ,...,w i,m− 1 ) ,and W i = α i W i + β i W i + γ i W i (as
been shown in Fig. 2).