SUPERVISED RANKING OF VERTICES OF A DIRECTED GRAPH

• US 20080162453A1
• Filed: 12/29/2006
• Published: 07/03/2008
• Est. Priority Date: 12/29/2006
• Status: Active Grant

• Current Assignee: Microsoft Technology Licensing LLC (Microsoft Corporation)
• Original Assignee: Microsoft Corporation
• Inventors: Ma, Wei-Ying, Liu, Tie-Yan, Qin, Tao
• Granted Patent: US 7,617,194 B2
• Time in Patent Office: Days
• US Class Current: 707/5
• CPC Class Codes: G06F 16/951 Indexing; Web crawling techniques
• CPC Class Codes: Y10S 707/99933 Query processing, i.e. searching

Abstract

A method and system for ranking importance of vertices of a directed graph based on links between the vertices and some prior knowledge of importance of the vertices is provided. A ranking system inputs an indication of the vertices along with an indication of the links between the vertices as the directed graph. The ranking system generates a transition-probability matrix that represents the probability of transitioning from vertex to vertex. The ranking system then generates a ranking of the vertices based on the links between the vertices represented by the stationary distribution of the transition-probability matrix that is minimally perturbed to satisfy the prior knowledge, which may be a partial ranking of the vertices.

Claims

1. A computing system for ranking of vertices of a directed graph, the system comprising:

an indication of links between the vertices of the directed graph;

prior knowledge relating to ranking of the vertices of the directed graph; and

a component that generates a ranking of the vertices based on the links between the vertices consistent with the prior knowledge relating to ranking of the vertices.

2. The computing system of claim 1 wherein the vertices represent documents and the prior knowledge is a partial ranking of the documents.

3. The computing system of claim 1 wherein the indication of the links is an adjacency matrix.

4. The computing system of claim 1 wherein the component generates the ranking based on minimizing perturbation of a transition-probability matrix derived from the links between the vertices.

5. The computing system of claim 4 wherein the component that generates the ranking applies an interactive optimization to iteratively generate a ranking based on a fixed perturbation and generate a perturbation based on a fixed ranking until the ranking and perturbation converge on a solution.

6. The computing system of claim 5 wherein only elements of the transition-probability matrix that correspond to links are perturbed.

7. The computing system of claim 4 wherein the ranking is based on solving the following objective functions and constraints:

$\mathrm{\pi <br><br>}<br><br>\left(E\right)=\arg <br><br><br><br>\underset{\mathrm{\pi <br><br>}}{\min }<br><br>{<br><br>E<br><br>}_F^2+c<br><br><br><br>{\mathrm{\pi <br><br>}}^T<br><br>\left(I+E-P\right)<br><br>{\left(I+E-P\right)}^T<br><br>\mathrm{\pi <br><br>}$
s.t. e^Tπ

=1

π

≧

0

Aπ

>

Bπ

and

$E<br><br>\left(\mathrm{\pi <br><br>}\right)=\arg <br><br><br><br>\underset{E}{\min }<br><br>{<br><br>E<br><br>}_F^2+c<br><br><br><br>{\mathrm{\pi <br><br>}}^T<br><br>\left(I+E-P\right)<br><br>{\left(I+E-P\right)}^T<br><br>\mathrm{\pi <br><br>}$
s.t. (P−

E)e=e

(P−

E)≧

0.

8. The computing system of claim 1 wherein the vertices represent documents and including a component that displays an indication of the ranking of the documents.

9. The computing system of claim 1 wherein the vertices represent documents and including a component that determines an ordering of documents relating to a query based on the generated ranking.

10. A computer-readable medium embedded with computer-executable instructions for controlling a computer system to rank web pages, by a method comprising:

providing an adjacency matrix indicating links between the web pages;

providing a partial ranking of the web pages;

generating a transition-probability matrix based on the links of the web pages, the transition-probability matrix indicating probability of transitioning from one web page to another web page; and

generating a stationary distribution for the transition-probability matrix that attempts to satisfy the partial ranking of the web pages and minimizes perturbation of the transition probability matrix, the stationary distribution representing the ranking of the web pages.

11. The computer-readable medium of claim 10 wherein the partial ranking of the web pages indicates relative ranking of some web pages.

12. The computer-readable medium of claim 10 wherein the generating of the stationary distribution is performed based on a non-convex problem space being converted to a convex problem space.

13. The computer-readable medium of claim 10 wherein the generating of the stationary distribution applies an interior-point optimization technique.

14. The computer-readable medium of claim 10 wherein the generating of the stationary distribution applies an interactive optimization to iteratively generate a stationary distribution based on a fixed perturbation to the transition-probability matrix and generate a perturbation to the transition-probability matrix based on a fixed stationary distribution until the stationary distribution and perturbation converge on a solution.

15. The computer-readable medium of claim 14 wherein the stationary distribution is based on solving the following objective functions and constraints:

$\mathrm{\pi <br><br>}<br><br>\left(E\right)=\arg <br><br><br><br>\underset{\mathrm{\pi <br><br>}}{\min }<br><br>{<br><br>E<br><br>}_F^2+c<br><br><br><br>{\mathrm{\pi <br><br>}}^T<br><br>\left(I+E-P\right)<br><br>{\left(I+E-P\right)}^T<br><br>\mathrm{\pi <br><br>}$
s.t. e^Tπ

=1

π

≧

0π

Aπ

>

Bπ

and

$E<br><br>\left(\mathrm{\pi <br><br>}\right)=\arg <br><br><br><br>\underset{E}{\min }<br><br>{<br><br>E<br><br>}_F^2+c<br><br><br><br>{\mathrm{\pi <br><br>}}^T<br><br>\left(I+E-P\right)<br><br>{\left(I+E-P\right)}^T<br><br>\mathrm{\pi <br><br>}$
s.t. (P−

E)e=e

(P−

E)>

0.

16. The computer-readable medium of claim 14 wherein only elements of the transition-probability matrix that correspond to links are perturbed.

17. The computer-readable medium of claim 10 including displaying an indication of the ranking of the documents.

18. The computer-readable medium of claim 10 including determining an ordering of the web pages as results of a query based on the generated ranking.

19. A method in a computer system for ranking documents, the method comprising:

providing an adjacency matrix indicating links between the documents;

providing a partial ordering of the documents indicating relative ranking of some documents;

generating a transition-probability matrix from the adjacency matrix that indicates probability of transitioning from one document to another document; and

generating a stationary distribution for the transition-probability matrix that attempts to minimize perturbation to the transition-probability matrix needed to satisfy the partial ordering, the stationary distribution representing the ranking of the documents.

20. The method of claim 19 wherein the component that generates the stationary distribution applies an interactive optimization to iteratively generate a stationary distribution based on a fixed perturbation and generate a perturbation based on a fixed stationary distribution until the stationary distribution and perturbation converge on a solution.