Diversity in Ranking via Resistive Graph Centers

1. Diversity in Rankingvia Resistive Graph Centers Avinava Dubey IBM Research India Soumen Chakrabarti IIT Bombay Chiranjib Bhattacharyya IISc Bangalore

2. PageRank: Conventional view • Inputs • Graph with edge conductance matrix C • Personalized teleport distribution r • Walk with probability, teleport w.p. 1 • “Biased random surfer” • Output • Steady state visit distribution • “You should emulate theaggregate behavior ofmany random surfers” j i r

3. User view: Exact opposite! • Random search-guided surfer • Search engine knows relevant subgraph • But user can inspect only a few hits • Search engine outputs sparse teleport r Corpus

4. User view: Exact opposite! User diffuses out through sparse teleport Occasionally teleports back to search results Eventually explores green subgraph (Red, green “boundaries” are probabilistic) Corpus 4

5. Diffusion defined via subsumption • Original PageRank: diffusion via hyperlinks • But frequently used with other kinds of edges • Suppose surfer is on page i • And, having read i, there is no new info in j • Then let C(j|i), also written as C(ij) be large

6. Graph center diversity (GCD) • Suppose the searcher can click through at most three links returned by the search engine • If any of the pages could be potentially relevant, … • … then we cannot waste teleports on one cluster A natural definition of diversity

7. Formulation summary thus far • Search engine knows what’s best for query • Node i has relevance b(i) • User has limited patience scanning results • r must be sparse: at most K positive elements • Conductance matrix C and walk probability  predict user behavior once given r • Steady state visit probabilities given by • Inference, hard: design sparse r to minimize

8. Attention decay profile • Design a teleport r with decaying weights • So as to align weighted merged clouds with b Attentionprofile

9. Learning subsumption C(ij) • How strongly does irender j redundant? • Associate edge ij with features { f } • Each f has associated fixed conductance matrix Cf and personalized PageRanks Mf • Training: Given diverse node sets (r*), learn the convex combination defined by • Simple heuristic (convex optimization):

10. Divergence for r Divergence for r* Structured learning style formulation • More accurately, any r r* should do worse • Define a loss • Combine over query instances • Paper gives an online update algorithm to improve  iteratively (exponentiated gradient)

11. Marginal utility methods • Max marginal relevance (MMR) • Given q, already chose subset S; next choice is • SubTopic • Similar to MMR • sim1 and sim2 use probabilistic topic models • SVMdiv • Learns subtopic coverage from word coverage

12. PageRank based diversity models • Grasshopper • Edges associated with fixed similarity scores • Best node has highest PageRank • Make best node a sink, run PageRank again • Note, no meaningful steady state, Pr(sink)=1 • Next best node has largest expected number of visits before walk absorbed in sink • DivRank • With visits to node j, inbound edges get thicker • Rich gets even richer than you expected • Tiebreaking causes one cluster member to win

13. Submodular set selection • Sounds similar to MMR but on a graph • Undirected edge (i,j) has weight wij • Given node set V, select subset S so as to • Maximize coverage of V \S: • Minimize redundancy within S: • Additional size budget constraint • Hard, but provable approximations • No learning of edge weight/conductance S V \S

14. Experiments: Three diverse domains • Subtopic information retrieval (TREC) • Query under-specified or ambiguous • Balance responses across subtopics or facets • Social network search (IMDB) • List high-prestige actors without knowing country • Diversity  many countries covered • Extractive document summarization (DUC) • Choose subset of sentences • That are representative of the whole document • And do not render each other redundant

15. Subtopic information retrieval results • Ground truth has subtopics covered by each doc • Subtopic-aware precision vs. recall • GCD dominates other subtopic IR approaches

16. Effect of training  • Uniform  all f equal • Maxent = convex heuristic minimizing KL divergence between b and PageRank • EG = Exponentiated gradient • Successive improvements in subtopic-aware mean average precision

17. Ranking in social networks (IMDB) • 3452 actors, 1027 movies, 47 countries • Actor’s prestige depends on prestigeof movies where s/he has worked • Rank actors by prestige • GCD rapidly increases distinct countries • While also increasing number of movies

18. Document summarization • DUC 2004, task 2, ROUGE-1 • 30, 20 summaries to train, test • MMR, SubTopic not competitive • Associative graph diffusion (Grasshopper, DivRank) worse than GCD and Submodular • GCD comparable to Submodular even without using sentence size budget constraints

19. Conclusion A novel model for redundancy and diversity Based on an “inverted” notion of PageRank Inference amounts to finding centers in conductance graphs “GCD”, graph center diversity Bonus: learn conductance via edge features GCD shows better or similar performance in three diverse application domains Thank you

20. Bibliography J. Carbonell and J. Goldstein. The use of MMR, diversity-based reranking for reordering documents and producing summaries. In SIGIR Conference 1998. C. X. Zhai, W. W. Cohen, and J. Laerty. Beyond independent relevance: methods and evaluation metrics for subtopic retrieval. In SIGIR Conference 2003. X. Zhu, A. B. Goldberg, J. Van, and G. D. Andrzejewski. Improving diversity in ranking using absorbing random walks. In HLT-NAACL 2007. Y. Yue and T. Joachims. Predicting diverse subsets using structural SVMs. In ICML, 2008. Q. Mei, J. Guo, and D. Radev. DivRank: the interplay of prestige and diversity in information networks. In SIGKDD Conference, 2010. Hui Lin, Jeff Bilmes. Multi-document Summarization via Budgeted Maximization of Submodular Functions, NAACLHLT 2010.