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Part of the Search Engine course given in the Technion (2011)

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- 1. Evolutionary Models of the Web Graph Kira Radinsky Web Size estimation models are based on the Standford slides by Christopher Manning and Prabhakar Raghavan
- 2. 7 December 2010 2 Stochastic Models for the Web’s Graph So what can explain the observed Power Law in/out degree distributions of Web pages? • Standard G(n,p) Erdös-Rényi random graphs: – A graph contains n nodes, and every two nodes are connected with probability p – Degrees are distributed B(n-1,p), and since on the Web np<<n, they can be viewed as distributed Poisson(np-p) – Such distributions have light-weight, exponentially decreasing tails - nodes with very large in-degrees are practically impossible – yet, they abound on the Web Erdös-Rényi random graphs do not model the Web graph
- 3. 7 December 2010 3 Evolutionary Models – First Attempt • The Web wasn’t built in a day; in fact, it is constantly growing and evolving • Models should (somewhat) reflect the authoring process of Web pages • Observation: older, well-established nodes should be better connected as they’ve been around longer and are better known • A corresponding model: – Start at time 0 with a single node. – At step t, add a new node with a single new edge that connects to one of the t pre-exiting nodes chosen uniformly at random – The expected in-degree at time T of the node added at time t: j=t+1,…,T 1/j log T – log t – Doesn’t result in a power law – P(2x)/P(x) is not a constant
- 4. 7 December 2010 236620 Search Engine Technology 4 Preferential Attachment • Observation: while older, well-established nodes are better known, it is not strictly because of their age but rather because of them having more in-links • The preferential attachment model: – Start at time 0 with a single node. – At step t, add a new node with a single new edge that connects to one of the t pre-exiting nodes • The probability of linking to node v: (1+in-degree(v)) / (2t-1) • A variant involves a parameter α: – Start at time 0 with a single node. – At step t, add a new node with a single new edge that connects to node v with probability α/t+(1- α)*in-degree(v)/(t-1) • Both variants indeed result in a Power-Law distribution of in-degrees (different exponents)
- 5. 7 December 2010 5 Preferential Attachment (cont.) • Another observation: if search engine rankings are influenced by PageRank, then new pages will link to high- PageRank pages more than to low PageRank pages • The model uses two positive parameters d, p such that d+p<1 • The evolution: – Start at time 0 with a single node. – At step t, add a new node with a single new edge as follows: • With probability d, connect the edge to one of the existing nodes in proportion to the in-degree (or 1+in-degree) of that node • With probability p, connect the edge to a node chosen at random according to the PageRank distribution at time t • With probability 1-p-d, connect the edge to an existing node chosen uniformly at random • With properly chosen parameters, this model can fit both the in-degree and PageRank Power-Law distributions Raghavan et al., “Using PageRank to characterize Web Structure”, 2002
- 6. 7 December 2010 6 The Copy Model The “Copy Model” assumes the following authoring model: • Each page is on a topic of interest to its author. – Some of its links will be copied from a previous page on the same topic, that the author found useful – Some links will be “original”, i.e. chosen independently by the author of the page • The stochastic process creates nodes with an out-degree of d (parallel edges are allowed) – Start at time 0 with a single node and d self-loops – At step t, add a new node with d out-links as follows • Choose an intermediate node v chosen u.a.r. from the t existing nodes • For j=1,…,d: – With probability α, connect link j to a node chosen u.a.r. from the t existing nodes – With probability 1-α, copy the j’th link of v • The copy model results in Power-Law in-degree distributions
- 7. 7 December 2010 7 Evolutionary Models - Summary • Overall, models exist that can simultaneously fit the observed Power- Law distributions of in-degrees, out-degrees and PageRank – Many other properties of the graph are still unexplained by theoretical evolutionary models • The accepted models mix-and-match the principles of preferential attachment (degrees/PageRank), copying, and random connectivity • These models have the “rich get richer” property, and favor seniority (i.e. nodes from earlier rounds tend to have higher degrees) – One can add some random “fitness” to nodes, with preferential attachment considering fitness as well, to give new nodes better chances of competing with existing nodes • Note that there’s a difference between “rich get richer” and “winner takes all” – the Web’s graph doesn’t exhibit the dominance of a single winner
- 8. 7 December 2010 236620 Search Engine Technology 8 Related Research Area: The Science of Networks • Power-law and scale-free networks • “Small World” networks and the importance of weak ties – Kleinberg’s small-world grid • Social/collaboration networks – Milgram’s “six degrees of separation” – The six degrees of Kevin Bacon – Erdös numbers הסמג את קיבל שלי השכן של ודוד"ד אחותי של בן של אישתו סיפרהברכה שלומי,משינה
- 9. What is the size of the web ? • Issues – The web is really infinite • Dynamic content, e.g., calendar • Soft 404: www.yahoo.com/<anything> is a valid page – Static web contains syntactic duplication, mostly due to mirroring (~30%) – Some servers are seldom connected • Who cares? – Media, and consequently the user – Engine design – Engine crawl policy. Impact on recall.
- 10. What can we attempt to measure? (IQ is whatever the IQ tests measure.) – The statically indexable web is whatever search engines index. • Different engines have different preferences – max url depth, max count/host, anti-spam rules, priority rules, etc. • Different engines index different things under the same URL: – frames, meta-keywords, document restrictions, document extensions, ...
- 11. A B = (1/2) * Size A A B = (1/6) * Size B (1/2)*Size A = (1/6)*Size B Size A / Size B = (1/6)/(1/2) = 1/3 Sample URLs randomly from A Check if contained in B and vice versa A B Each test involves: (i) Sampling (ii) Checking Relative Size from Overlap Given two engines A and B
- 12. Sampling URLs • Ideal strategy: Generate a random URL and check for containment in each index. • Problem: Random URLs are hard to find! Enough to generate a random URL contained in a given Engine. • Approach 1: Generate a random URL contained in a given engine – Random queries – Random searches • Approach 2: Give us a true estimate of the size of the web (as opposed to just relative sizes of indexes) – Random IP addresses – Random walks
- 13. Random URLs from random queries • Generate random query: how? – Lexicon: 400,000+ words from a web crawl – Conjunctive Queries: w1 and w2 e.g., vocalists AND rsi • Get 100 result URLs from engine A • Choose a random URL as the candidate to check for presence in engine B • This distribution induces a probability weight W(p) for each page. • Conjecture: W(SEA) / W(SEB) ~ |SEA| / |SEB| Not an English dictionary
- 14. Random searches • Choose random searches extracted from a local log [Lawrence & Giles 97] or build “random searches” [Notess] – Use only queries with small result sets. – Count normalized URLs in result sets. – Use ratio statistics
- 15. Random IP addresses • Generate random IP addresses • Find a web server at the given address – If there’s one • Collect all pages from server – From this, choose a page at random
- 16. Random walks • View the Web as a directed graph • Build a random walk on this graph – Includes various “jump” rules back to visited sites • Does not get stuck in spider traps! • Can follow all links! – Converges to a stationary distribution • Must assume graph is finite and independent of the walk. • Conditions are not satisfied (cookie crumbs, flooding) • Time to convergence not really known – Sample from stationary distribution of walk – Use the “strong query” method to check coverage by SE

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