1. CIKM’ 09 November 3rd, 2009, Hong Kong
P-Rank: A Comprehensive Structural Similarity Measure over Information Networks
Peixiang Zhao, Jiawei Han, Yizhou Sun
University of Illinois at Urbana-Champaign
Presented by Prof. Hong Cheng, CUHK
CIKM’ 09 November 3rd, 2009, Hong Kong

2. Outline Introduction & Motivation P-Rank Experimental Studies
Formula
Derivatives
Computation
Experimental Studies
Future direction & Conclusion
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3. Introduction Information Networks (INs)
Physical, conceptual, and human/societal entities
Interconnected relationships among different entities
INs are ubiquitous and form a critical component of modern information infrastructure
The Web
highway or urban transportation networks
research collaboration and publication networks
Biological networks
social networks
Information networks have graph as their underlying data model, which consist of the vertex set V and the edge set E.
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4. Problem Similarity computation on entities of INs
How similar is webpage A with webpage B in the Web ?
How similar is researcher A with researcher B in DBLP co-authorship network ?
First of all, how to define “similarity” within a massive IN?
Textual proximity of entity labels/contents
Structural proximity conveyed through links!
A good structural similarity measure in INs: SimRank (KDD’02)
1.Similiarty means how similar two nodes in the graph are.
2. Previous studies show that the structural similarity can generate more meaningful similarity results, in comparison with text based similarity measures, for example, string based edit distance, cosine measure, etc. The main reason is that, the structural similarity is more homogeneous and language independent. By making use of the link information in the graph, the structural similarity between two nodes in the graph can make use of the neighborhood structure to help reinforce the similarity belief.
3. SimRank is a de facto standard for structural similarity on large graphs. There are a lot of work done to extend SimRank in different scenarios.
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5. Why SimRank is not Enough?
Philosophy
two entities are similar if they are referenced by similar entities
Potential problems
Semantic incomplete
Only partial structural information from in-link direction is considered during similarity computation
Biased similarity results
May fail in different IN settings !
Inefficient in computation
Worst-case O(n4), can be improved to O(n3), where n is the number of vertices in the information network
Note Simrank is defined in a recursive way.
Because Simrank only consider those node who “cite” the vertex-pair (in-link information), while neglecting those vertices “cited” by the vertex-pair (out-link information), so the measure is somehow biased and the scenario when the vertex pair doesn’t have common in-link neighbors, their Simrank score is UNDEFINED.
SimRank is EXTREMELY inefficient in large real information networks.
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6. Why SimRank is not Enough?
(a) A Heterogeneous IN and Structural Similarity Scores
These two diagrams show the infeasibility of SimRank, compared with P-Rank.
The first diagram is a heterogeneous IN. It contains nodes belonging to different categories: conference (C), committee member (m1, m2, m3) and paper (p1, p2, p3, p4). The edges have different meanings, like Conference “invite” PC member and PC member “bid” for paper and paper is “reviewed” by PC member and paper is “accepted” by the conference. Note for vertex pair (m1, m2), (m1, m3) and (m2, m3), SimRank gives the same similarity score. It can not distinguish these three pairs. However, by making use of the structural information from both in-link and out-link sides, P-Rank can differentiate them successfully. Similar things occur for (p1, p3), (p3, p4)
The second diagram is a homogeneous IN. It contains nodes belonging to the same category, for example, all nodes are papers and edges are citations. Similarly, SimRank cannot differentiate (p2, p3), (p3, p4). Even worse, for node pair (p4, p5), (P2, p5), the SimRank is UNAVAILABLE. Because the vertex pair do not have common in-link neighbors which can pass similarity scores to them. However, P-Rank can successfully measure the similarity for all vertex pairs.
The main philosophy is that, P-Rank can jointly take into account of similarity flow from both directions and if one direction is unavailable (SimRank may fail), P-Rank can still work successfully.
(b) A Homogeneous IN and Structural Similarity Scores
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7. P(enetrating)-Rank Philosophy: Two entities are similar, if Advantages
they are referenced by similar entities
they reference similar entities
Advantages
Semantic complete
Structural information from both in-link and out-link directions are considered during similarity computation
Robust in different IN settings
A unified structural similarity framework
SimRank is just a special case
P-rank extends the definition by adding the similarity computation for out-link direction
For robustness, Simrank may fail in some IN, like the previous example, however, P-Rank can be employed into different IN settings
P-Rank is a general similarity framework and many structural similarity measures, including Simrank, are its special cases. (Will be explained in details later)
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8. P-Rank Formula
The structural similarity between vertex a and vertex b (a ≠ b), s(a, b):
Recursive form
Approximate iterative form
In-link similarity
Out-link similarity
For the recursive form, it is a linear combination of similarity computation from both sides. Lambda is a parameter controlling the relative weight between in-link side and out-link side. C is a dampen factor. I(a) is all the nodes with a link to a and O(a) are all nodes pointed by a
For the iterative form, k is the iterative number. It means that the similarity score of iteration k+1 can be computation based on the similarity score in iteration k.
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9. P-Rank Property The iterative P-Rank has the following properties:
Symmetry: sk(a, b) = sk(b, a)
Monotonicity: 0 ≤ sk(a, b) ≤ sk+1(a, b) ≤ 1
Existence: The solution to the iterative P-Rank formula always exists and converges to a fixed point, s(∗, ∗), which is the theoretical solution to the recursive P-Rank formula
Uniqueness: the solution to the iterative P-Rank formula is unique when C ≠ 1
The theoretical solution to P-Rank can be reached by a repetitive computation via the iterative form
The property of P-Rank secures that we can use the iterative computation to approach its theoretical real value, and in practice, the convergence is pretty fast, usually the iteration number is around 5 or 6.
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10. P-Rank Derivatives
P-Rank proposes a unified structural similarity framework, upon which many structural similarity measures are just its special cases
Cocitation is a structural similarity measure, intuitively for a and b, cocitation is the number of nodes which point to both a and b. It is the one-step P-Rank considering only in-link directions.
Compling is the duality of Cocitation. It only consider the out-link direction. It can be regarded as one-step P-Rank considering out-link direction only.
Amsler considers both in-link and out-link but only direct neighbors without similarity score propagation throughout the whole network. It is still a one-step P-Rank
Simrank considers recursive computation but one direction is considered.
Rvs-Simrank is the duality of Simrank and only outlink is considered.
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11. P-Rank Computation
An iterative algorithm is executed until it reaches the fixed point
Space complexity: O(n2)
Time complexity: O(n4), can be improved to O(n3) by amortization
Approximation algorithms on different IN scenarios
Homogeneous IN
Radius based pruning: vertex-pairs beyond a radius of r are no longer considered in similarity computation
Heterogeneous IN
Category based pruning: vertex-pairs in different categories are no longer considered in similarity computation
N is the number of nodes in the graph
The amortization algorithm is published in VLDB’08
Approximation algorithms are pretty straightforward.
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12. Experimental Studies Data sets: Methods Metrics
Heterogeneous IN: DBLP (paper, author, conference, year)
Homogeneous IN: DBLP (paper with citation), Synthetic data R-MAT
Methods
P-Rank
SimRank
Metrics
Compactness of clusters
Algorithmic nature
Ground truth
1. The compactness of clusters can be intuitively defined as intra-cluster distance / inter-cluster distance, so the smaller, the better.
2. Algorithm nature is to test the Prank algorithm for different aspects: how fast it will converge, how it correlate with different parameters C and lambda.
3. Ground truth results are top-10 ranking results by making use of Prank as the underlying similarity measures in DBLP.
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13. Compactness of Clusters
P-Rank and SimRank are used as underlying similarity measures, respectively, and K-Medoids are used to cluster different vertices
Compactness: intra-cluster distance/inter-cluster distance
Here, Cp stands for “compactness for Prank” and Cs stands for “compactness for Simrank”. The smaller the compactness, the better the clustering results.
Heterogeneous IN
Homogeneous IN
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14. Algorithmic Nature Iterative P-Rank converges fast to the fixed point
These two diagram shows that the P-rank algorithms converges pretty fast.
For the first diagram, when C is set small, it converges faster. When C is set large, it will need more iterations to converge. The reason is that the larger C is, the smaller the loss of similarity belief during similarity propagation.
For the second diagram, it shows the relative importance of similarity scores for both in-link and out-link side. It is true that out-link plays more role in similarity computation.
P-Rank v.s. the damping factor C
P-Rank v.s. lambda
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15. Ground Truth Ranking Result
Top-10 ranking results for author vertices in DBLP by P-Rank
This results are pretty intuitive. The above one show the most similar PAIRS and the bottom two are top-10 rankings for different queries.
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16. Conclusion
The proliferation of information networks calls for effective structural similarity measures in
Ranking
Clustering
Top-k Query Processing ……
Compared with SimRank, P-Rank is witnessed to be a more effective structural similarity measure in large information networks
Semantic complete, general, robust, and flexible enough to be employed in different IN settings
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17. Thank you
CIKM’ 09 November 3rd, 2009, Hong Kong