1. 6CCS3WSN--7CCSMWAL Algorithms for WWW and Social Networks Algorithmic Issues in the WWW
Lecture 1

2. Lecturer’s Contact Lecturer: Colin Cooper
Office: S6.23
Office hours: Thursday 1:15-3:15pm

3. Course Information No coursework, simply a final exam Course Web Site
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Lecture slides (Also on KEATS)
Exercises
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Papers (references)

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5. Brief Syllabus Structure of large networks such as WWW
Algorithmic issues related to information retrieval. Main topics
Webpage Ranking
Text processing
Information retrieval
Clustering data
Recommender systems

6. Internet and WWW Internet ≠ World Wide Web (WWW)
Internet is a network of computers, linked by undirected (two-way) data connections
WWW is a network of web pages, linked by directed hyperlinks

7. Characteristics of WWW
DYNAMIC: Cho & Garcia-Molina)
40% of all web pages changed weekly
23% of the .com pages changed daily
SELF-ORGANISED
Anyone can post a webpage
No standard, structure, and format
Huge. Google indexes at least 4.5 billion pages
Look unstructured/random from global perspective

8. Graph Models of Networks
Definition of graph G = (V, E),
V is a set of vertices,
E is a set of pairs (v1, v2) for some vertices v1 and v2 in V
Directed graphs (or digraphs)
Pairs (v1, v2) and (v2, v1) are not equivalent
Undirected graphs
Pairs (v1, v2) and (v2, v1) are equivalent
(v,w) also written {v,w} or vw

9. Directed Graph
A set of hyperlinked web pages can be represented by a directed graph
G1 = (V1, E1)
V1 = { P1, P2, P3, P4 }
E1 = { (P1, P2), (P1, P4), (P3, P1), (P3, P2), (P4, P1) }
Web page P1
Webpage P4
Web page P2
Web page P3

10. Undirected Graph
A computer network can be represented by an undirected graph
G2 = (V2, E2)
V2 = { S1, S2, S3, S4 }
E2 = { (S1, S2), (S1, S3), (S1, S4), (S2, S3) }
Computer S1
(S1, S2) and (S2, S1) are equivalent, only one is shown in E2
Computer S4
Computer S2
Computer S3

11. The Internet graph

12. The graph of main subdomains of www.kcl.ac.uk
(Note: hyperlinks are shown as undirected edges)

13. Example: Wikipedia page links (wikimedia.org/wiki/User:Chris73)

14. Graph Representation List of (vertices and) edges Adjacency list
Adjacency matrix
EXAMPLES……….

15. Graph Representation Adjacency-list representation
For i = 1 to the number of vertices, Adj[i] = list of vertices adjacent to vertex i
Undirected graph

16. Graph Representation Adjacency-list representation
For i = 1 to the number of vertices, Adj[i] = list of vertices adjacent to vertex i
Directed graph

17. Graph Representation Adjacency-matrix representation
Let n be the number of vertices
A is a n x n matrix where A(i, j) = 1 if (i, j) exists, A(i, j) = 0 otherwise
Undirected graph

18. Graph Representation Adjacency-matrix representation
Let n be the number of vertices
A is a n x n matrix where A(i, j) = 1 if (i, j) exists, A(i, j) = 0 otherwise
Directed graph

19. Terminology in Graphs Degree of a vertex v, d(v) Undirected graphs
Number of edges incident with v
Undirected graphs
E.g. d(1) = 2, d(5) = 3

20. Terminology in Graphs Directed graphs
In-degree d– (v) = number of edges (x, v) for x in the vertex set
E.g. d–(1) = 0, d–(4) = 2
Out-degree d+(v) = number of edges (v, x) for x in the vertex set
E.g., d+(1) = 2, d+(4) = 1
d(v) = d– (v) + d+(v)

21. Terminology in Graphs
A path of length k from vertices u to v is a sequence of k arcs/edges (u, v1), (v1, v2), ..., (vk-1, v)
.... u v1 v2
vk-1 v
E.g.,
A path from vertex 1 to vertex 4
with length 2 (Note: a path can take either direction of an edge in an undirected graphs)
A path from vertex 1 to vertex 5
with length 3

22. Walk (non-simple path)
Terminology in Graphs
A cycle is a path with u = v
A simple path is path if it consists of no cycle, i.e., a vertex appear at most once on the path
Cycle
A simple path
Walk (non-simple path)

23. Terminology in Graphs
An undirected graph is connected if there is a path from every vertex to every other vertex 1 2 3 5 4
Connected
Not connected
(or disconnected)

24. Terminology in Graphs
A directed graph is strongly connected if there is a path from every vertex to every other vertex 1 2 3 4 5 6
Strongly connected
Not strongly connected
(e.g., no path from vertex 5 to vertex 6)

25. Terminology in Graphs
A directed graph is weakly connected if all the edges are considered undirected edges and the resulting (undirected) graph is connected. 1 2 3 4 5 6
Weakly connected
Not weakly connected

26. Graph Algorithms To determine properties of a graph Connected or not?
Shortest path from a vertex to another one
The average distance between vertices
How many edges, vertices, triangles
Are there any ‘clusters’
Which vertices are important

27. Graph Traversal
Try to “visit” all vertices of a graph in a systematic way
Two standard algorithms
Breath First Search (BFS)
Depth First Search (DFS)
Can be used to determine if an undirected graph is connected
If starting from a vertex, we can visit every other vertex
Simple idea of Web crawling …

28. Framework of Traversal
BFS and DFS share the same framework
A starting vertex to start the traversal
Keep a list (or priority queue) of vertices reached so far
Will mark all vertices reachable by the starting vertex
The two traversal differ in the way of adding new vertices to the list, hence result in a different traversal order

29. Traversal Algorithm Put the starting vertex in LIST
While LIST is not empty
Remove first vertex y held in LIST
Visit y & mark y as seen
For all (y, z) in E
If z is not marked and not in LIST
Add z to LIST
DFS adds z to the beginning of LIST
BFS adds z to the end of LIST

30. BFS
Traverse the vertices in a first-in-first-out manner. LIST is a QUEUE
Suppose vertex 1 is the starting vertex
Front of list on left [a,b,c] 2 3 4 1 5 6 7
To prevent ambiguity, if a vertex has more than one outgoing edges, follow the edges in ascending order of vertices they lead to

31. 1 5 2 3 4 6 7
LIST 1
Initial setting
LIST
Mark vertex 1 as seen 2
Insert vertex 2 to LIST
2, 6
Insert vertex 6 to LIST 1 5 3 4 6 7 1 5 2 3 4 6 7
LIST
Mark vertex 2
6, 3
Insert vertex 3 to LIST

32. 1 5 2 3 4 6 7
LIST
Mark vertex 6
3, 4
Insert vertex 4 to LIST
3, 4, 7
Insert vertex 7 to LIST 1 5 2 3 4 6 7
LIST
4, 7
Mark vertex 3
4, 7, 5
Insert vertex 5 to LIST 1 5 2 3 4 6 7
LIST
7, 5
Mark vertex 4

33. 5 4 6 7
LIST
Mark vertex 7 2 1 5 4 6 7
LIST
Mark vertex 5 2 1

34. The BFS traversal order of the vertices:
1, 2, 6, 3, 4, 7, 5
BFS tree
12, 16, 23, 64, 67, 35

35. DFS
Traverse the vertices in a last-in-first-out manner. LIST is a STACK
Suppose vertex 1 is the starting vertex 1 5 2 3 4 6 7
To prevent ambiguity, if a vertex has more than one outgoing arcs, follow the edges in ascending order of vertices they lead to

36. 1 5 2 3 4 6 7
LIST 1
Initial setting
LIST
Mark vertex 1 2
Insert vertex 2 to LIST
6, 2
Insert vertex 6 to LIST 1 5 3 4 6 7 1 5 2 3 4 6 7
LIST
Traverse (1,6) mark vertex 6
4, 2
Insert vertex 4 to LIST
7, 4, 2
Insert vertex 7 to LIST

37. 2 3 4
LIST
4, 2
Traverse (6,7) mark 7 1 5
LIST
5, 4, 2
Insert vertex 5 to LIST 6 7 1 5 2 3 4 6 7
LIST
4, 2
Traverse (7,5) mark 5
Backtrack over 7 to 6 1 5 2 3 4 6 7
LIST
Traverse (6,4) mark 4
3, 2
Insert vertex 3 to LIST

38. 2 3 4
LIST 2
Traverse (4,3) mark 3
Backtrack to 4 1 5 6 7 1 5 2 3 4 6
LIST
Traverse (4,2) mark 2
Backtrack over 4, 6 to 1
End. List empty 7

39. The DFS traversal order of the vertices:
1, 6, 7, 5, 4, 3, 2
DFS tree from vertex 1
16, 67, 75, 64, 43, 42
The BFS traversal order of the vertices:
1, 2, 6, 3, 4, 7, 5
BFS tree from vertex 1
12, 16, 23, 64, 67, 35

40. Demmo (BFS-DFS.r)
Lets see what happens……….

41. Demmo

42. Shortest Path
A shortest path from a vertex u to a vertex v is a path, among all possible paths from u to v, which consists of the least number of arcs/edges
Shortest path is not unique; there could be more than one shortest paths between two vertices
The shortest path from u to v is called the (graph) distance

43. Shortest Path
The shortest path from vertex 1 to vertex 5, can be either
(1, 2), (2, 3), (3, 5)
(1, 6), (6, 7), (7, 5)
The length of the shortest path is 3 1 5 2 3 4 6 7

44. BFS finds shortest paths
BFS tree from vertex 1
12, 16, 23, 64, 67, 35
See the BFS tree from vertex 1 1 5 2 3 4 6 7

45. Distance The (graph) distance (u,v) from vertex u to vertex v is
0 if u = v
the length of a shortest path from u to v in the BFS tree
 if there is no path from u to v

46. Distance
(u,v) 1 2 3 4 5

47. Distance
(u,v) v 1 2 3 4 5 6 u 

48. Diameter
The diameter D of a graph G = (V, E) is the maximum graph distance (longest shortest path) between any pair of vertices.
(u,v) 1 2 3 4 5
Diameter is 2

49. Diameter
(u,v) v 1 2 3 4 5 6 u 
Diameter is 

50. i.e., the path from the starting vertex to the last visited vertex
BFS and Shortest Paths
BFS can find the shortest paths and distance from the starting vertex to all other vertices in a connected component
(2) x2
(1) x1
distance from
the starting
vertex
Starting vertex
(0) x0
(2) x3
(3) x6
(1) x4
(2) x5
The longest (from level to level) path (depth) of the BFS: x0, x1, x2, x6
i.e., the path from the starting vertex to the last visited vertex

51. Shortest Path Algorithm
Modify the BFS algorithm to find the shortest path/distance from a vertex to every other vertex
Let x be the starting vertex
Dist[i] = (x, i)
the distance from vertex x to vertex I
The modified BFS computes Dist[i] for all vertices i

52. BFS Shortest Path Algorithm
Put the starting vertex x to QUEUE
Dist[x] = 0
While QUEUE is not empty
Extract the first vertex y in QUEUE
Mark y as seen
For all (y, z) in E
If z is not marked and not in QUEUE
Dist[z] = Dist[y] + 1
Add z to the end of QUEUE
For all unmarked vertices i
Dist[i] = 

53. Example with starting vertex 3
Dist[3] = 0 
Dist[5] = Dist[3] + 1 = 1  
Dist[6] = Dist[3] + 1 = 1  

54. 
Dist[4] = Dist[5] + 1 = 2     
Dist[2] = Dist[4] + 1 = 3     
Dist[1] =    

55. All shortest paths? BFS from 1 (red edges)
Not in any shortest path from 1
In shortest path but not BFS tree
Shortest (1,6)-paths: 1246, 1346, 1356

56. Number of shortest paths from v to w of length k. N(v,w,k)
N(v,v,0)=1 One shortest path from v to v
For w at distance (k+1) from v
N(v,w,k+1) =
Sum over all neighbours u of w at distance k from v the value of N(v,u,k)
N(1,1,0)=1, N(1,2,1)=1,N(1,3,1)=1
N(1,4,2)=N(1,2,1)+N(1,3,1)=2 etc

57. More Shortest Paths Algorithms
Edges have different weights and the length of a path is the sum of the weights of edges in the path
Dijkstra’s Algorithm
All pairs shortest paths – find the shortest distances/paths for all pairs of vertices
Floyd-Warshall Algorithm

58. Diameter Diameter measures of how “well” a graph is connected
The smaller the “better”
A directed graph which is not strongly connected has a diameter of 
The WWW graph, is not strongly connected nor weakly connected.
Why? Some web pages are not hyperlinked with the rest of the WWW

59. Revised Diameter Measure
To measure the longest distance between any two “connected” pairs of vertices
Let G = (V, E) be the WWW graph
Let P be the set of all triples (u, v, k), where u, v in V and 0  k = (u, v) <  (reachable)
P is the set of finite distance vertex pairs
The revised diameter is
D*(G) = max(u, v, k) in P (u, v)

60. Average Distance
An estimate of the average length of the shortest path from a vertex to another one
Small world. Six degrees of separation is the theory that everyone and everything is six or fewer steps away.
There are several possible ways to estimate average distance….

61. Average Distance Traditional definition (all pairs)
Suppose a graph has n vertices.
There are n(n-1) pairs of vertices (u,v) giving n(n-1) paths
Definition useful only for connected undirected graphs and strongly connected directed graphs
 not applicable to WWW graph
Example

62. Revised Average Distance
Only count the pairs of reachable vertices.
Let P be the set of all triples (u, v, k) for
which u, v in V and 0 < k = (u, v) < 
i.e. distinct ordered pairs
|P| is the size of P which is  n

63. Example
(u,v) v 1 2 3 4 5 6 u 
|P| = 13
= ( ) / 13
= 1.538

64. Revised Average Distance
Average of the reciprocal of the distances
Infinite distance contributes zero to the sum
No need to distinguish between reachable and unreachable pairs
Don’t include distance from vertex to itself

65. Example
(u,v) v 1 2 3 4 5 6 u 
= 3.06
(1/1 + 1/ + 1/1 + 1/2 + 1/ / + 1/ + 1/2 + 1/1 + 1/
/ + 1/ + 1/ + 1/ /)/30 =
= /30

66. Exercise Digraph (a,b), (a,c), (b,c)
All pairs average distance in underlying graph
Average distance in digraph for reachable pairs
Average distance in digraph using reciprocal distances
Comment?
Answers 1, 1, 2

67. Connectivity in Digraphs
A graph is connected if there is a path between every pair of vertices
A digraph is strongly connected if there is a directed path between every pait of vertices
A digraph is weakly connected if the underlying graph is connected
Underlying graph: Ignore edge orientations

68. Find the strongly connected components (SCC) of a digraph
FAN-OUT(v): set of vertices obtained by BFS from v using edges directed out of vertices
FAN-IN(v): set of vertices obtained by BFS from v using edges directed into vertices
SCC(v) is the intersection of FAN-OUT(v) and FAN-IN(v)

69. Example
FO(1)={1,2,3,4,6}, FI(1)={3,2,4,5}
SCC(1)={1,2,3,4} SCC(6)={6}

70. How many SCC?
There are 3 SCC, {1,2,3,4}, {5} ,{6}

71. Examples Edges G=(V,E) Digraph
(3,1) (6,1) (9,2) (10,2) (6,3) (8,3) (1,4) (8,4) (5,10) (6,5) (9,5) (5,7) (1,8) (4,9) (7,9) (8,9) (9,10)
Draw G
Find the SCC
Exercise. Digraph V=1,2,…10, Edges
5->1 8-> 1 5-> 3 8-> 3 5->10 6->10 4-> 7 7->10 8->10 5-> 9 7-> 9

72. Draw G

73. SCC
SCC are

74. Demmo
connect-scc.r
Examples of G(n,p) and PA connectivity