1. Graph Algorithms
Ch. 5 Lin and Dyer

2. Graphs Are everywhere Manifest in the flow of emails
Connections on social network
Bus or flight routes
Social graphs: twitter friends and followers
Take a look at Jon Kleinberg’s page and book on Networks, Crowds and Markets Reasoning about a highly connected world.

3. Graph algorithms
Graph search and path planning:: shortest path to a node
Graph clustering:: diving the graphs into smaller related clusters
Minimum spanning tree:: graph that covers the nodes in an efficient way
Bipartite graph match:: div graph into two mapping sets: job seekers and employers
Maximum flow:: designate source and sink; determine max flow between the two: transportation
Identifying special nodes: authoritative nodes: containment of spread of diseases; Broad street water pump in London, cholera and beginnings of epidemiology

4. Graph Representations
How do you represent this
visual diagram as data?

5. Simple, Baseline Data Structure1 n1 n2 n3 n4 n5
n1 [n2, n4]
n2 [n3, n5]
n3 [n4]
n4 [n5]
n5 [n1,n2,n3]
Adjacency matrix – this
is good for linear algebra;
But most web links and social
Networks are sparse x/
Space req. is O(n2)
(ii) Adjacency lists

6. Problem definition: intuition
Input: graph adjacency list with edges and vertices, w edges distances, starting vertex
Output(goal): label the nodes/vertices with the shortest distance value from the starting node

7. single source shortest path problem
Sequential solution: Dijkstra’s algorithm 5.2
Dijkstra (G, w, s) // w edge distances list, s starting node, G graph
d[s]  0
for all other vertices d[v] ∞
Q  {V} // Q is priority queue based on distances
while Q # 0
u  min(Q) // node with min d value
for all vertex v in u.adjacencyList
if d[v] > d[u] + w[u,v]
d[v]  d[u] + w[u,v]
mark u and remove from Q
At each iteration of while loop, the algorithm expands the node with the shortest distance and updates distances to all reachable nodes

8. Sample graph : lets apply the algorithm 5.2n2 n4 ∞ 1 ∞ 10 9 3 2 6 4 7 n1 5 ∞ ∞ 2 n3 n5

9. Issues Sequential Need to keep global state: not possible with MR
Lets see how we can handle this graph problem for parallel processing with MR

10. Parallel Breadth First
Assume distance of 1 for all edges (simplifying assumption): later we will expand it to other distances

11. Issues in processing a graph in MR
Goal: start from a given node and label all the nodes in the graph so that we can determine the shortest distance
Representation of the graph (of course, generation of a synthetic graph)
Determining the <key,value> pair
Iterating through various stages of processing and intermediate data
When to terminate the execution

12. Input data format for MR
Node: nodeId, distanceLabel, adjancency list {nodeId, distance}
This is one split
Input as text and parse it to determine <key, value>
From mapper to reducer two types of <key, value> pairs
<nodeid n, Node N>
<nodeid n, distance until now label>
Need to keep the termination condition in the Node class
Terminate MR iterations when none of the labels change, or when the graph has reached a steady state or all the nodes have been labeled with min distance or other conditions using the counters can be used.
Now lets look at the algorithm given in the book

13. Mapper
Class Mapper method map (nid n, Node N) d  N.distance emit(nid n, N) // type 1 for all m in N. Adjacencylist emit(nid m, d+1) // type 2

14. Reducer
Class Reducer method Reduce(nid m, [d1, d2, d3..]) dmin = ∞; // or a large # Node M  null for all d in [d1,d2, ..] { if IsNode(d) then M  d else if d < dmin then dmin  d} M.distance  dmin // update the shortest distance in M emit (nid m, Node M)

15. Trace with sample Data
1 0 2:3: :4: :4: : :4

16. Intermediate data
1 0 2:3:
2 1 3:4:
3 1 2:4:5: :
:4:

17. Intermediate Data
1 0 2:3:
2 1 3:4:
3 1 2:4:5:
4 2 5:
5 2 1:4:

18. Final Data
1 0 2:3:
2 1 3:4:
3 1 2:4:5:
4 2 5:
5 2 1:4:

19. Sample Data
1 0 2:3:
:4:
:4:5 : :4
2 1 3:4:
3 1 2:4:5
4 2 5:
5 2 1:4

20. PageRank
25 Billion Dollar algorithm (huge matrix and Eigen vector problem.)
Larry Page and Sergei Brin (Standford Ph.D. students)
Rajeev Motwani and Terry Winograd (Standford Profs)

21. Consider this web problem
Nodes 1, 2, 3,4 with ranks
x1, x2,x3, x4 1 2 4 X3
Problem: How to calculate the
Ranks or “influence” of these
web linked nodes?
Solution: Treat it as linear as linear algebraic problem.. Write the linear equations,
Solve the equation system. Let’s do just that for this network

22. Linear Algebra problem
x1 = [ ½ x2 + ½x3] = [0x1+ ½ x2 + ½x3+0x4]
x2 = [ ½ x1+ 0 x2 + 0x3+ ½ x4]
x3 = [ ½ x1+ 0 x2 + 0x3+ ½ x4]
x4 = [0 x1+ ½ x2 + ½ x3+0 x4]
Web link problem develops into a problem of finding the Eigen vector for the square matrix.
We seek the Eigen vector X with value of 1 for the link matrix Ax = 1; lets do that

23. Solve Square link matrix for Eigen Vector
[0 + ½ + ½ + 0 ] [ ½ ½ ] X [x1 x2 x3 x4] [ ½ ½ ] [0 + ½ + ½ + 0 ] A x = 1 solve this for x1=? x2 =? x3=? x4=? Transpose the matrix, etc.. Now scale the problem to billions of nodes?!!

24. General idea Consider the world wide web with all its links.
Now imagine a random web surfer who visits a page and clicks a link on the page
Repeats this to infinity
Pagerank is a measure of how frequently will a page will be encountered.
In other words it is a probability distribution over nodes in the graph representing the likelihood that a random walk over the linked structure will arrive at a particular node.

25. PageRank Formula
P(n) = α 1 𝐺 +(1−𝛼) 𝑚∈𝐿(𝑛) 𝑃 𝑚 𝐶 𝑚 α randomness factor G is the total number of nodes in the graph L(n) is all the pages that link to n C(m) is the number of outgoing links of the page m Note that PageRank is recursively defined. It is implemented by iterative MRs.

26. Example
Figure 5.7
Lets assume alpha as zero
Lets look at the MR

27. Mapper for PageRank
Class Mapper method map (nid n, Node N) p  N.Pagerank/|N.AdajacencyList| emit(nid n, N) for all m in N. AdjacencyList emit(nid m, p) “divider”

28. Reducer for Pagerank
Class Reducer method Reduce(nid m, [p1, p2, p3..]) node M  null; s = 0; for all p in [p1,p2, ..] { if p is a Node then M  p else s  s+p } M.pagerank  s emit (nid m, node M) “aggregator”

29. Lets trace with sample data1 2 4 3

30. Discussion
How to account for dangling nodes: one that has many incoming links and no outgoing links
Simply redistributes its pagerank to all
One iteration requires pagerank computation + redistribution of “unused” pagerank
Pagerank is iterated until convergence: when is convergence reached?
Probability distribution over a large network means underflow of the value of pagerank.. Use log based computation