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Introduction to Graph Cluster Analysis

Introduction to Graph Cluster Analysis. Outline. Introduction to Cluster Analysis Types of Graph Cluster Analysis Algorithms for Graph Clustering k-Spanning Tree Shared Nearest Neighbor Betweenness Centrality Based Highly Connected Components Maximal Clique Enumeration Kernel k-means

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Introduction to Graph Cluster Analysis

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  1. Introduction to Graph Cluster Analysis

  2. Outline • Introduction to Cluster Analysis • Types of Graph Cluster Analysis • Algorithms for Graph Clustering • k-Spanning Tree • Shared Nearest Neighbor • Betweenness Centrality Based • Highly Connected Components • Maximal Clique Enumeration • Kernel k-means • Application

  3. Outline • Introduction to Clustering • Introduction to Graph Clustering • Algorithms for Graph Clustering • k-Spanning Tree • Shared Nearest Neighbor • Betweenness Centrality Based • Highly Connected Components • Maximal Clique Enumeration • Kernel k-means • Application

  4. What is Cluster Analysis? The process of dividing a set of input data into possibly overlapping, subsets, where elements in each subset are considered related by some similarity measure Clusters by Shape 2 Clusters Clusters by Color 3 Clusters

  5. Applications of Cluster Analysis • Summarization • Provides a macro-level view of the data-set Clustering precipitation in Australia From Tan, Steinbach, Kumar Introduction To Data Mining, Addison-Wesley, Edition 1

  6. Outline • Introduction to Clustering • Introduction to Graph Clustering • Algorithms for Graph Clustering • k-Spanning Tree • Shared Nearest Neighbor • Betweenness Centrality Based • Highly Connected Components • Maximal Clique Enumeration • Kernel k-means • Application

  7. What is Graph Clustering? • Types • Between-graph • Clustering a set of graphs • Within-graph • Clustering the nodes/edges of a single graph

  8. Between-graph Clustering Between-graph clustering methods divide a set of graphs into different clusters E.g., A set of graphs representing chemical compounds can be grouped into clusters based on their structural similarity

  9. Within-graph Clustering Within-graph clustering methods divides the nodes of a graph into clusters E.g., In a social networking graph, these clusters could represent people with same/similar hobbies Note: In this chapter we will look at different algorithms to perform within-graph clustering

  10. Outline • Introduction to Clustering • Introduction to Graph Clustering • Algorithms for Within Graph Clustering • k-Spanning Tree • Shared Nearest Neighbor • Betweenness Centrality Based • Highly Connected Components • Maximal Clique Enumeration • Kernel k-means • Application

  11. k-Spanning Tree kgroups of non-overlapping vertices k-Spanning Tree 4 Minimum Spanning Tree 2 3 1 k 3 2 5 • STEPS: • Obtains the Minimum Spanning Tree (MST) of input graph G • Removes k-1 edges from the MST • Results in k clusters 2 4

  12. What is a Spanning Tree? A connected subgraph with no cycles that includes all vertices in the graph 2 3 1 6 2 5 7 3 4 3 1 6 G 4 2 4 5 2 5 2 7 2 4 Weight = 17 Note: Weight can represent either distance or similarity between two vertices or similarity of the two vertices

  13. What is a Minimum Spanning Tree (MST)? The spanning tree of a graph with the minimum possible sum of edge weights, if the edge weights represent distance 2 3 1 2 3 1 6 2 5 3 G 2 5 7 3 Weight = 11 4 3 1 4 6 2 4 4 2 4 5 2 5 7 2 4 3 1 2 5 4 2 4 2 5 Note: maximum possible sum of edge weights, if the edge weights represent similarity Weight = 13 2 Weight = 17

  14. Algorithm to Obtain MSTPrim’s Algorithm 2 3 1 6 Given Input Graph G Select Vertex Randomly e.g., Vertex 5 Initialize Empty Graph T with Vertex 5 2 5 7 3 4 T 4 2 4 5 3 5 5 4 5 5 4 5 5 4 Repeat until all vertices are added to T Select a list of edges L from G such that at most ONE vertex of each edge is in T From L select the edge X with minimum weight Add X to T T 4 6 4

  15. k-Spanning Tree Remove k-1 edges with highest weight Note: k – is the number of clusters 4 Minimum Spanning Tree 2 2 3 3 1 1 E.g., k=3 3 3 E.g., k=3 2 2 5 5 2 2 4 4 3 1 5 4 2 4 3 Clusters

  16. k-Spanning Tree R-code • library(GraphClusterAnalysis) • library(RBGL) • library(igraph) • library(graph) • data(MST_Example) • G = graph.data.frame(MST_Example,directed=FALSE) • E(G)$weight=E(G)$V3 • MST_PRIM = minimum.spanning.tree(G,weights=G$weight, algorithm = "prim") • OutputList = k_clusterSpanningTree(MST_PRIM,3) • Clusters = OutputList[[1]] • outputGraph= OutputList[[2]] • Clusters

  17. Outline • Introduction to Clustering • Introduction to Graph Clustering • Algorithms for Within Graph Clustering • k-Spanning Tree • Shared Nearest Neighbor Clustering • Betweenness Centrality Based • Highly Connected Components • Maximal Clique Enumeration • Kernel k-means • Application

  18. Shared Nearest Neighbor Clustering Shared Nearest Neighbor Graph (SNN) 2 0 2 Groups of non-overlapping vertices 1 4 Shared Nearest Neighbor Clustering 2 3 2 1 3 2 1 2 τ • STEPS: • Obtains the Shared Nearest Neighbor Graph (SNN) of input graph G • Removes edges from the SNN with weight less than τ

  19. What is Shared Nearest Neighbor?(Refresher from Proximity Chapter) Shared Nearest Neighbor is a proximity measure and denotes the number of neighbor nodes common between any given pair of nodes u v

  20. Shared Nearest Neighbor (SNN) Graph Given input graph G, weight each edge (u,v) with the number of shared nearest neighbors between u and v 2 2 0 0 SNN G 4 4 2 1 1 3 3 1 2 3 2 2 1 1 Node 0 and Node 1 have 2 neighbors in common: Node 2 and Node 3

  21. Shared Nearest Neighbor ClusteringJarvis-Patrick Algorithm SNNgraph of input graph G 2 0 2 1 If u and v share more than τ neighbors Place them in the same cluster 4 2 3 2 1 3 2 1 2 E.g., τ =3 2 0 4 1 3

  22. SNN-Clustering R code • library(GraphClusterAnalysis) • library(RBGL) • library(igraph) • library(graph) • data(SNN_Example) • G = graph.data.frame(SNN_Example,directed=FALSE) • tkplot(G) • Output = SNN_Clustering(G,3) • Output

  23. Outline • Introduction to Clustering • Introduction to Graph Clustering • Algorithms for Within Graph Clustering • k-Spanning Tree • Shared Nearest Neighbor Clustering • Betweenness Centrality Based • Highly Connected Components • Maximal Clique Enumeration • Kernel k-means • Application

  24. What is Betweenness Centrality?(Refresher from Proximity Chapter) Betweenness centrality quantifies the degree to which a vertex (or edge) occurs on the shortest path between all the other pairs of nodes Two types: • Vertex Betweenness • Edge Betweenness

  25. Vertex Betweenness The number of shortest paths in the graph G that pass through a given node S G E.g., Sharon is likely a liaison between NCSU and DUKE and hence many connections between DUKE and NCSU pass through Sharon

  26. Edge Betweenness The number of shortest paths in the graph G that pass through given edge (S, B) E.g., Sharon and Bob both study at NCSU and they are the only link between NY DANCE and CISCO groups NCSU Vertices and Edges with high Betweenness form good starting points to identify clusters

  27. Vertex Betweenness Clustering Given Input graph G Betweenness for each vertex 1. Disconnect graph at selected vertex (e.g., vertex 3 ) 2. Copy vertex to both Components Repeat until highest vertex betweenness≤ μ Select vertex v with the highest betweenness E.g., Vertex 3 with value 0.67

  28. Vertex Betweenness Clustering R code • library(GraphClusterAnalysis) • library(RBGL) • library(igraph) • library(graph) • data(Betweenness_Vertex_Example) • G = graph.data.frame(Betweenness_Vertex_Example,directed=FALSE) • betweennessBasedClustering(G,mode="vertex",threshold=0.2)

  29. Edge-Betweenness ClusteringGirvan and Newman Algorithm Given Input Graph G Betweenness for each edge Disconnect graph at selected edge (E.g., (3,4 )) Repeat until highest edge betweenness≤ μ Select edge with Highest Betweenness E.g., edge (3,4) with value 0.571 29

  30. Edge Betweenness Clustering R code • library(GraphClusterAnalysis) • library(RBGL) • library(igraph) • library(graph) • data(Betweenness_Edge_Example) • G = graph.data.frame(Betweenness_Edge_Example,directed=FALSE) • betweennessBasedClustering(G,mode="edge",threshold=0.2)

  31. Outline • Introduction to Clustering • Introduction to Graph Clustering • Algorithms for Within Graph Clustering • k-Spanning Tree • Shared Nearest Neighbor Clustering • Betweenness Centrality Based • Highly Connected Components • Maximal Clique Enumeration • Kernel k-means • Application

  32. What is a Highly Connected Subgraph? • Requires the following definitions • Cut • Minimum Edge Cut (MinCut) • Edge Connectivity (EC)

  33. Cut • The set of edges whose removal disconnects a graph Cut = {(3,5),(4,2)} 7 7 6 6 8 8 0 0 5 5 1 1 7 6 8 4 4 0 2 2 3 3 5 1 Cut = {(0,1),(1,2),(1,3} 4 2 3

  34. Minimum Cut The minimum set of edges whose removal disconnects a graph 7 7 6 6 8 8 MinCut = {(3,5),(4,2)} 0 0 5 5 1 1 4 4 2 2 3 3

  35. Edge Connectivity (EC) • Minimum NUMBER of edges that will disconnect a graph 7 6 8 MinCut = {(3,5),(4,2)} Edge Connectivity 0 5 EC = |MinCut| = | {(3,5),(4,2)}| = 2 1 4 2 3

  36. Highly Connected Subgraph (HCS) A graph G =(V,E) is highly connected if EC(G)>V/2 G 7 6 8 0 EC(G) > V/2 2 > 9/2 5 1 4 2 G is NOT a highly connected subgraph 3

  37. HCS Clustering Return G Given Input graph G Find the Minimum Cut MinCut (G) YES 7 6 8 (3,5),(4,2)} Is EC(G)> V/2 0 Process Graph G1 5 NO 7 Process Graph G2 1 6 8 Divide G using MinCut 0 4 5 2 1 3 G2 G1 4 2 3

  38. HCS Clustering R code • library(GraphClusterAnalysis) • library(RBGL) • library(igraph) • library(graph) • data(HCS_Example) • G = graph.data.frame(HCS_Example,directed=FALSE) • HCSClustering(G,kappa=2)

  39. Outline • Introduction to Clustering • Introduction to Graph Clustering • Algorithms for Within Graph Clustering • k-Spanning Tree • Shared Nearest Neighbor Clustering • Betweenness Centrality Based • Highly Connected Components • Maximal Clique Enumeration • Kernel k-means • Application

  40. What is a Clique? A subgraph C of graph G with edges between all pairs of nodes C G 7 7 6 6 8 5 5 Clique 4

  41. What is a Maximal Clique? A maximal clique is a clique that is not part of a larger clique. Clique 7 6 7 6 8 5 5 7 4 6 Maximal Clique 8 5

  42. Maximal Clique EnumerationBron and Kerbosch Algorithm Input Graph G BK(C,P,N) C - vertices in current clique P – vertices that can be added to C N – vertices that cannot be added to C Condition: If both P and N are empty – output C as maximal clique

  43. Maximal Clique R code • library(GraphClusterAnalysis) • library(RBGL) • library(igraph) • library(graph) • data(CliqueData) • G = graph.data.frame(CliqueData,directed=FALSE) • tkplot(G) • maximalCliqueEnumerator (G)

  44. Outline • Introduction to Clustering • Introduction to Graph Clustering • Algorithms for Within Graph Clustering • k-Spanning Tree • Shared Nearest Neighbor Clustering • Betweenness Centrality Based • Highly Connected Components • Maximal Clique Enumeration • Kernel k-means • Application

  45. What is k-means? • k-means is a clustering algorithm applied to vector data points • k-means recap: • Select k data points from input as centroids • Assign other data points to the nearest centroid • Recomputecentroid for each cluster • Repeat Steps 1 and 2 until centroids don’t change

  46. k-means on GraphsKernel K-means • Basic algorithm is the same as k-means on Vector data • We utilize the “kernel trick” (recall Kernel Chapter) • “kernel trick” recap • We know that we can use within-graphkernel functions to calculate the inner product of a pair of vertices in a user-defined feature space. • We replace the standard distance/proximity measures used in k-means with this within-graph kernel function

  47. Outline • Introduction to Clustering • Introduction to Graph Clustering • Algorithms for Within Graph Clustering • k-Spanning Tree • Shared Nearest Neighbor Clustering • Betweenness Centrality Based • Highly Connected Components • Maximal Clique Enumeration • Kernel k-means • Application

  48. Application • Functional modules in protein-protein interaction networks • Subgraphs with pair-wise interacting nodes => Maximal cliques R-code • library(GraphClusterAnalysis) • library(RBGL) • library(igraph) • library(graph) • data(YeasPPI) • G = graph.data.frame(YeasPPI,directed=FALSE) • Potential_Protein_Complexes = maximalCliqueEnumerator (G) • Potential_Protein_Complexes

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