A unifying framework of rating users and data items in peer-to-peer and social networks

Abstract

We propose a unifying family of quadratic cost functions to be used in Peer-to-Peer ratings. We show that our approach is general since it captures many of the existing algorithms in the fields of visual layout, collaborative filtering and Peer-to-Peer rating, among them Koren spectral layout algorithm, Katz method, Spatial ranking, Personalized PageRank and Information Centrality. Besides of the theoretical interest in finding common basis of algorithms that where not linked before, we allow a single efficient implementation for computing those various rating methods. We introduce a distributed solver based on the Gaussian Belief Propagation algorithm which is able to efficiently and distributively compute a solution to any single cost function drawn from our family of quadratic cost functions. By implementing our algorithm once, and choosing the computed cost function dynamically on the run we allow a high flexibility in the selection of the rating method deployed in the Peer-to-Peer network. Using simulations over real social network topologies obtained from various sources, including the MSN Messenger social network, we demonstrate the applicability of our approach. We report simulation results using networks of millions of nodes.

Appendix

Proof of Theorem 1

It is shown in Hall [15] that the optimal solution to the cost function is x = L ^− 1 where L is the graph Laplacian. Substitute β =  1, w _ii  =  1, w _ij  = 1, y _i  = 1 in the cost function (1) :

$$ \min E(x) \triangleq \sum\limits_i 1*(x_i - 1)^2 - 1* \sum\limits_{i,j \in E} (x_i - x_j)^2 $$

The same cost function in linear algebra form:

$$ \min E({\bf x}) \triangleq {\bf x}^T L {\bf x} - 2 {\bf x} \textbf{1} + n $$

Now we calculate the derivative and compare to zero and get

$$ \begin{array}{lll} &&{\kern-6pt}\nabla_XE({\bf x}) = 2 {\bf x}^T L - 2 {\bf x} \textbf{1} \\ &&{\kern-6pt}{\bf x} = (L)^{-1} \end{array} $$

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Proof of Theorem 2

Using the notations of Dell’Amico [16] the cost function of Koren’s spectral layout algorithm is:

$$ \min \sum\limits_{ij} w_{ij} (x_i - x_j)^2 $$

s.t

$$ \sum\limits_i deg(i) x_i^2 = n, \sum\limits_i deg(i) x_i = 0 $$

we create a weighted-sum objective:

$$ \begin{array}{lll} \mathcal{C}({\bf x},\beta,\gamma) = \sum\limits_{ij}&&w_{ij} (x_i - x_j)^2 \\ &&{\kern-22pt}- \beta \Bigg(\sum\limits_i deg(i) x_i^2 - n\Bigg) - \gamma \sum\limits_i deg(i) x_i \end{array} $$

Taking β = 1, γ = 1/2 we get:

$$ = \sum\limits_{ij} w_{ij} (x_i - x_j)^2 - \sum\limits_i deg(i) (x^2 - 1)^2 $$

Reverting to our cost function formulation we get:

$$ = \sum\limits_i deg(i) (x_i - 1)^2 + \sum\limits_{ij} w_{ij} (x_i - x_j)^2 $$

In other words, taking w _ii  = deg(i), y _i  = 1, β = 1 and we get Koren’s formulation.□

It is interesting to note, that the optimal solution according to Koren’s work is $x_i = \sum_{j \in N(i)}\frac{w_{ij}x_j}{deg(i)}$ which is equivalent to the thin plate model image processing and PDEs [17].

Proof of Theorem 5

We have shown that the fundamental matrix is equal to (I − R)^− 1. Assume that the edge weights are probabilities of Markov-chain transitions (which means that each row sums into one), substitute β = α, w _ii  = 1, y _i  = 1 in the cost function (1) :

$$ \min E(x) \triangleq \sum\limits_i 1*(x_i - 1)^2 - \alpha \sum\limits_{i,j \in E} w_{ij}(x_i - x_j)^2 $$

The same cost function in linear algebra form:

$$ \min E(x) \triangleq {\bf x} I {\bf x} - \alpha {\bf x}^T R {\bf x} - 2{\bf x} $$

Now we calculate the derivative and compare to zero and get

$$ {\bf x} = (I - \alpha R)^{-1}$$

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Proof of Theorem 6

The proof is very similar to the Spatial Ranking proof. There are two differences: the first is that the prior distribution x is set in y to weight the output towards the prior. Second, in the in the Personalized PageRank algorithm the result is multiplied by the constant (1 − α), which we omit in our cost function. This computation can be done locally at each node after the algorithm terminates, since α is a known fixed system parameter.□