1. The Shape of the Web So, the Web is a directed graph, but what does it look like?

2. What is the shape of the web? Broder et.al: Graph structure of the web (2000)

3. The Bow-Tie Shape of the Web CORE: A giant strongly connected component (SCC) IN: The part of pages leading to CORE OUT: Pages reachable from CORE TENDRILS: Not part of CORE, leading out of IN, into OUT, or bypassing CORE (TUBES). ISLANDS: Disconnected pages.

4. 8 1 2 5 3 4 6 7 9 Within a SCC, a path can be found from any node to any other node: 1  2, 2  3, 3  1. How do you compute a SCC?

5. Strongly Connected Component SCC is defined on a digraph G = (V,E) only! (Why?) SCC is a subset C of V with the following properties: 1.  u,v  C, u is reachable from v in G and v is reachable from u in G 2. If C is a proper subset of another subset D of V, then D does not satisfy property 1 Translation: C is a maximal subset of Vertices with mutual reachability Which graph traversal algorithm(s) produce reachability information? How many SCCs does G have?

6. Transpose of a Digraph If G = (V,E) is a digraph, then its transpose G T is the digraph G = (V, E T ) where E T = { (v,u) | (u,v)  E } Which graph has more SCCs? G or G T ?

7. A fact of life… A directed graph and its transpose have exactly the same strongly connected components Why?

8. Detecting the STRONGLY CONNECTED COMPONENT requires 2 DFS traversals 1. Run DFS to compute the finishing times ƒ[u]. 2. Computer the graph’s transpose. 3. Run a 2 nd DFS, while considering vertices in the order of decreasing ƒ[u]. 4. Output the vertices in each tree in the forest as a separate SCC CRLS Algorithms textbook page 553 9 – 8 – 4 – 7 – 6 – 1 – 2 – 5 – 3 8 1 2 5 3 4 6 7 9 1/8 3/4 2/7 12/13 10/11 9/14 5/6 15/16 17/18 8 1 2 5 3 4 6 7 9 11/16 12/15 13/14 7/8 9/10 5/6 17/18 3/4 1/2

9. 8 1 2 5 3 4 6 7 9 IN: OUT: TENDRILS: ISLANDS: SCC identified. How about the rest?

10. Visualizing a small Web

11. But Why is it a Bowtie? Maybe is a teapot, a daisy? A bugle? A cauliflower? It is a collection of Bowties, because. (it could not be anything else) Proof by construction M: Why the Shape of the Web is a Bowtie? (2010)

12. Bowtie Web: Proof by Construction Start by considering one link per page Pseudo-trees appear The cycles of pseudotrees are like budding COREs INs created, no OUTs

13. The Second link creates a Bowtie Consider the 2 nd link It will reduce the number of components, enlarge the CORE, create IN and TENDRILS OUTs may appear as smaller cycles (than CORE)

14. Nodes w/out links and Third links Now include nodes w/out links (as possible targets for Bowtie nodes). They start off as ISLANDS Consider the effect of the 3 rd link. What happens when you link: IN node to IN, CORE, OUT, ISLAND CORE node to IN, CORE, OUT, ISLAND OUT node to IN, CORE, OUT, ISLAND ISLAND node to IN, CORE, OUT, ISLAND

15. Third links enlarge Bowties! INCOREOUTISLAND INuninteresting TUBETENDRIL COREEnlarged CORE uninteresting Enlarged OUT OUT ISLAND

16. Web is many Bowties!

17. So, why is the Web shaped as a Bowtie? Because. That’s the only thing it could be. A randomly generated digraph is a bowtie (or a daisy, or a teapot, or a bugle, or a ____)

19. How do you explore the Web Graph? Depth-first search? Breadth-first search? Best-first search? Random walk? How do you even know you have explored the whole graph? Where do you store it?

20. Our Random Selections: 1 st Link CORE: 5 IN: 12 OUT: 0 ISLAND: 1 TENDRIL: 0

21. Our Random Selections: 2 nd Link CORE: 14 IN: 4 OUT: 0 ISLAND: 0 TENDRIL: 0