Link Structure Analysis Kira Radinsky All of the following slides are courtesy of Ronny Lempel (Yahoo!)

236620 Search Engine Technology Link Analysis In the Lecture HITS: topic-specific algorithm Assigns each page two scores – a hub score and an authority score – with respect to a topic PageRank: query independent algorithm Assigns each page a single, global importance score Both algorithms reduced to the computation of principal eigenvectors of certain matrices Today’s Tutorial: Graph modifications in link analysis algorithms SALSA – HITS with a random-walk twist Topic-Sensitive PageRank 9 December 2018 236620 Search Engine Technology

Graph Modifications in Link-Analysis Algorithms Delete irrelevant elements (pages, links) from the collection. Non-informative links Pages that are deemed irrelevant (mostly by similarity of content to the query), and their incident links [Bharat and Henzinger, 1998] Assign varying (positive) link weights to the non-deleted links. Similarity of anchor text to the query [CLEVER] Links incident to pre-defined relevant pages [CLEVER] Multiple links from pages of site A to pages of site B [Bharat and Henzinger, 1998] Note that some of the above modifications are only applicable to topic distillation algorithms Several works suggested performing modifications, that intuitively should improve precision, to the link structure of the analyzed collections: Pages that are deemed irrelevant - for HITS, pruning the initial HITS graph by removing irrelevant pages. “Links incident to pre-defined relevant pages “ - If you know some authorities/hubs and want to find additional ones, you can assign high link weights to links incident to your known good pages. These weights will propagate to the immediate vicinity and will increase the scores of the neighborhood. In general, links from/to known trusted sources can get a weight bonus.   “Multiple links from pages of site A to pages of site B “ – if site A links (from many of its pages) 100 times to site B, it is unreasonable to count all those links with a weight of 1, as they are all under the control of a single entity/site administrator. Therefore, what that paper did was allow an overall weight of 1 to all links between any two sites (in the earlier example this would mean decreasing the weight of each link from A to B to the value of 0.01) CLEVER – IBM project implementing HITS and various variations thereof, started during Kleinberg’s sabbatical at IBM Almaden. Many publications came out the CLEVER project, seehttp://www.almaden.ibm.com/projects/clever.shtml 9 December 2018 236620 Search Engine Technology

SALSA – Stochastic Approach to Link Structure Analysis SALSA, like HITS, is a topic-distillation algorithm that aims to assign pages both hub and authority scores SALSA analyzes the same topic-centric graph as HITS, but splits each node into two – a “hub personality” without in-links and an “authority personality” without out-links Examines the resulting bipartite graph Innovation: incorporate stochastic analysis with the authority-hub paradigm Examine two separate random walk Markov chains: an authority chain A, and a hub chain H. A single step in each chain is composed of two link traversals on the Web - one link forward, and one link backwards. The principal community of each type: the most frequently visited pages in the corresponding Markov Chain 9 December 2018 236620 Search Engine Technology

Forming bi-pirate graph in Salsa Hub walk Follow a Web link from a page uh to a page wa (a forward link) and then Immediately traverse a backlink going from wa to vh, where (u,w) Є E and (v,w) Є E Authority Walk Follow a Web link from a page w(a) to a page u(h) (a backward link) and then Immediately traverse a forward link going back from vh to wa where (u,w) Є E and (v,w) Є E

SALSA – Authority Chain Example Pr (23) = 2/5*1/3 Formally, The transition probability matrix: Hub weight computed from the sum of the product of the inverse degree of the in-links and the out-links A_uv = sum_(w,u),(w,v) [1/ deg(v_a) * 1/deg(w_h)] H_uv = sum_(u,w),(v,w) [1/ deg(u_h) * 1/deg(w_a)] [PA]i,j =  {k| ki, kj} (iin)-1(kout)-1 9 December 2018 236620 Search Engine Technology

236620 Search Engine Technology SALSA: Analysis The transition probabilities induce a probability distribution on the authorities (hubs) in the authority (hub) Markov chain The principal community of authorities (hubs) is defined as the k most probable pages in the authority (hub) chain While one can compute the scores by calculating the principal eigenvector of the stochastic transition matrices, a more efficient way exists 9 December 2018 236620 Search Engine Technology

SALSA: Analysis (cont.) Mathematical Analysis of SALSA leads to the following theorem: SALSA’s authority weights reflect The normalized in-degree of each page, multiplied by the relative size of the page’s component in the authority side of the graph x 3 4 a(x) = ----- x ----- = 0.25 3 +5 4 +2 9 December 2018 236620 Search Engine Technology

SALSA: Proof for Irreducible Authority Chains The proof assumes a weighted graph, in which the link kj has weight w(kj) The examples shown so far assumed that all links have a weight of 1 Define W as the sum of all links weights Define a distribution vector π by πj = din(j)/W, where din(j) is the sum of weights of j’s incoming links Similarly, dout(k) is the sum of weights of k’s outgoing links It is enough to prove that πPA=π, since PA has a single stationary eigenvector (Ergodic Theorem) Recall that PA is the transition matrix of the authority chain PA is always aperiodic 9 December 2018 236620 Search Engine Technology

SALSA: Proof for Irreducible Authority Chains 9 December 2018 236620 Search Engine Technology

Topic Sensitive PageRank [T. Haveliwala, 2002] A topic T is defined by a set of on-topic pages ST. A T-biased PageRank is PageRank where the random jumps (teleportations) land u.a.r. on ST rather than on any arbitrary Web page Recall the alternative interpretation of PageRank, as walking random paths of geometrically distributed lengths between resets Here, a reset returns to some on-topic page If we assume that pages tend to link to pages with topical affinity, short paths starting at ST will not stray too far away from on-topic pages, hence the PageRanks will be T-biased Note that pages unreachable from ST will receive a T-biased PageRank of 0 Where would be a good place to find sets ST for certain topics? The pages classified under the 16 top-level topics of the Open Directory Project (see next slide) 9 December 2018 236620 Search Engine Technology

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Topic-Sensitive PageRank (cont.) 16 PageRank vectors are computed, PR1,…,PR16 Given a query q, its affinity to the 16 topics T1,…,T16 is computed Based on the probability of generating the query by the language model induced by the set of pages ST A distribution vector [α1,…,α16] is computed, where αj ~ Prob(q | language model of STj) The PageRank vector that will be used to serve q is PRq = αjPRj The idea of biasing PageRank’s random jump destinations is also used for personalized PageRank flavors [e.g. Jeh and Widom 2003] 9 December 2018 236620 Search Engine Technology

Link Analysis Algorithms - Summary Many variants and refinements of both HITS and PageRank have been proposed. Other approaches include: Max-flow techniques [Flake et al., SIGKDD 2000] Machine learning and Bayesian techniques Examples of applications: Ranking pages (topic specific/global importance/ personalized rankings) Categorization, clustering, finding related pages Identifying virtual communities Computational issues: Distributed computations of eigenvectors of massive, sparse matrices Convergence acceleration, approximations A wealth of literature 9 December 2018 236620 Search Engine Technology