Mincuts

1. SI/EECS 767 Yang Liu Apr 2, 2010 Mincuts

2. A minimum cut is the smallest cut that will disconnect a graph into two disjoint subsets. • Application: • Graph partitioning • Data clustering • Graph-based machine learning Introduction

3. Cut • A cutC = (S,T) is a partition of V of a graph G = (V, E). • An s-t cutC = (S,T) of a network N = (V, E) is a cut of N such that s∈S and t∈T, where s and t are the source and the sink of N respectively. • The cut-set of a cut C = (S,T) is the set {(u,v)∈E | u∈S, v∈T}. • The size of a cut C = (S,T) is the number of edges in the cut-set. If the edges are weighted, the value of the cut is the sum of the weights. (http://en.wikipedia.org/wiki/Cut_(graph_theory)) Background knowledge

4. Minimum cut • A cut is minimum if the size of the cut is not larger than the size of any other cut. • Max-flow-min-cut theorem • The maximum flow between two vertices is always equal to the size of the minimum cut times the capacity of a single pipe. • Also applies to weighted networks in which individual pipes can have different capacities. Background knowledge

5. Max-flow min-cut theorem is very useful because there are simple computer algorithms that can calculate maximum flows quite quickly (in polynomial time) for any given networks. We can use these same algorithms to quickly calculate the size of a cut set. Background knowledge

6. Basic idea: • First find a path from source s to sink t using the breadth-first search; • Then find another path from s to t among the remaining edges and repeat this procedure until no more paths can be found. The augmenting path algorithm s t

7. Allow fluid to flow simultaneously both ways down an edge in the network. A simple fix Mark Newman’s text book (preprint version)

8. Graph Clustering and Minimum Cut Tress (Flake et al 2004)

9. Clustering data into disjoint groups • Data sets can be represented as weighted graphs • Nodes = entities to be clustered • Edges = a similarity measure between entities • Present a new clustering algorithm based on maximum flow. (in particular minimum cut tree) Introduction

10. Also known as Gomory–Hu tree A weighted tree that consists of edges representing all pairs minimum s-t cut in the graph For every undirected graph, there always exists a min-cut tree. See [Gomory and Hu 61] for detail and the algorithm for calculating min-cut trees. Minimum cut tree

11. Cut clustering algorithm

12. α→0, the trivial cut ({t}, V) α→∞, n trivial clusters, all singletons The exact value of α depends on the structure of G and the distribution of the weights over the edges. The algorithm finds all clusters either in increasing or decreasing order, we can stop the algorithm as soon as a desired cluster has been found. Choosing α

13. Once a clustering is produced, contract the clusters into single nodes and apply the same algorithm to the resulting graph. When contracting a set of nodes, they get replaced by a single new node; possible loops get deleted and parallel edges are combined into a single edge with weight equal to the sum of their weights. break if ((clusters returned are of desired number and size) or (clustering failed to create nontrivial clusters)) Hierarchical cut-clustering algorithm

15. CiteSeer • Citation network (documents as nodes, citations as edges) Low level high level Experimental results

16. Minimum cut trees, based on expanded graphs, provide a means for producing quality clusterings and for extracting heavily connected components. A single parameter, α, can be used as a strict bound on the expansion of the clustering while simultaneously serving to bound the intercluster weight as well. Conclusion

17. Bipartite Graph Partitioning and Data Clustering (Zha et al 2001)

18. Bipartite graph • Two kinds of vertices • One representing the original vertices and the other representing the groups to which they belong • Examples: terms and documents, authors and authors of an article • Adapt undirected graphs criteria for bipartite graph partitioning and therefore solve the bi-clustering problem. introduction

19. Bipartite graph G(X, Y, W) • In the context of document clustering • X represents the set of terms • Y represents the set of documents • W = (wij) represents term frequency of i in document j. Bipartite graph partitioning

20. Tends to produce unbalanced clusters The problem becomes following optimization problem

21. Computational complexity: general linear in the number of documents to be clustered

22. 20 news groups experiments

23. Learning from Labeled and Unlabeled Data using Graph Mincuts (Blum & Chawla 2001)

24. Many application domains suffer from not having enough labeled training data for learning. Large amounts of unlabeled examples How unlabeled data can be used to aid classification Introduction

25. A set L of labeled examples • A set U of unlabeled examples • Binary classification • L+ to denote the set of positive examples • L- to denote the set of negtive examples The Graph mincut learning algorithm

26. Construct a weighted graph G = (V, E), where V = L∪U∪{v+, v-}, e ∈ E is a weight w(e). v+, v-: classification vertices; other vertices: example vertices; w(v, v+) = ∞ for all v ∈L+ and w(v, v-) = ∞ for all v ∈L- The edge between example vertices are assigned weights based on some relationship (similarity/distance) between the examples

27. Determine a minimum (v+, v-) cut for the graph, i.e. the minimum total weight set of edges whose removal disconnects v+ and v-. (using a max-flow algorithm in which v+ is the source, v- is the sink) Assign a positive label to all unlabeled examples in the set V+ and a negative label to all unlabeled examples in the set V-. *edges between examples which are similar to each other should be given a high weight

28. If there are few labeled examples, it can cause mincut to assign the unlabeled examples to one class or the other If the graph is too sparse, it could have a number of disconnected components Therefore it is important to use a proper weighting function Potential problem

29. Datasets: UCI, 2000 • The mincut algorithm has many degrees of freedom in terms of how the edge weights are defined. • Mincut-3: each example is connected to its nearest labeled example and two other nearest examples overall • Mincut-δ: if too nodes are closer than δ , they are connected • Mincut-δ0: max δ which graph has a cut of value 0 • Mincut-δ1/2: the size of the largest connected component in the graph is half the number of datapoints • Mincut-δopt: the values of δ that corresponds to the least classification error in hindsight Experimental analysis

31. The basic idea of this algorithm is to build a graph on all the data with edges between examples that are sufficiently similar • then to partition the graph into a positive and a negative set in a way that • (a) agrees with the labeled data • (b) cuts as few edges as possible review

32. Semi-supervised Learning using Randomized Mincuts (Blum et al 2004)

33. The drawbacks of the graph mincut approach: • A graph may have many minimum cuts and the mincut algorithm produces just one, typically the “leftmost” one using standard network flow algorithms. • Produced based on joint labeling rather than per-node probabilities. • Can be improved by averaging over many small cuts. Introduction

34. Repeatedly add artificial random noise to the edge weights Solve for the minimum cut in the resulting graphs Output a fractional label for each example corresponding to the fraction of the time it was on one side or the other Basic idea

35. Given a graph G, produce a collection of cuts by repeatedly adding random noise to the edge weights and then solving for the minimum cut in the perturbed graph. Sanity check: remove those that are highly unbalanced (any cut with less than 5% of the vertices on one side in this paper) Predict based on a majority vote Randomized mincuts with sanity check

36. Overcome some of the limitations of the plain mincut algorithm. • Consider a graph which simply consists of a line with a positively labeled node at one end and a negatively labeled node at the other end with the rest being unlabeled. • Plain mincut: the cut will be the leftmost or right most one • Randomized mincut: end up using the middle of the line with confidence that increases linearly out to the endpoints example

37. Uniform distribution of minimum cuts

38. The graph should be either be connected or at least have the property that a small number of connected components cover nearly all the examples. Good to create a graph that at least has some small balanced cuts. Graph design criteria

39. MST: simply construct a minimum spanning tree on the entire dataset δ-MST: connect two points with an edge if they are within a radius δ. Then veiw the components produced as super nodes and connect them via an MST. Two graph construction methods

40. Handwritten digits 20 newsgroups Various UCI datasets Experimental analysis

42. Improve performance when the number of labeled examples is small Providing a confidence score for accuracy-coverage curves. Conclustion

43. A Sentimental Education: Sentiment Analysis using Subjectivity Summarization based on Minimum Cuts (Pang & Lee 2004)

44. machine-learning method that applies text-categorization techniques to determine the sentiment polarity—positive (“thumbs up”) or negative (“thumbs down”) introduction

45. Previous approaches focused on selecting indicative lexical features • Their approach: • Label the sentences as either subjective or objective • Apply a standard machine-learning classifier to the resulting extract.

46. methods

47. n items x1, . . . , xn to divide into two classes C1 and C2 • Individual scoresindj(xi): non-negative estimates of each xi’s preference for being in Cj based on just the features of xi alone; • Association scores assoc(xi, xk): non-negative estimates of how important it is that xi and xk be in the same class. Minimize the partition cost : Subjectivity detection

48. Build an undirected graph G with vertices {v1, . . . , vn, s, t}; the last two are, respectively, the source and sink. Add n edges(s, vi), each with weight ind1(xi), and n edges (vi, t), each with weight ind2(xi). Finally, add edges (vi, vk), each with weight assoc(xi, xk).

50. Classifying movie reviews as either positive and negative The correct label can be extracted automatically from rating information (number of stars) Evaluation framework