CS 6604: Data Mining Large Networks and Time-Series

Transcription

1 CS 6604: Data Mining Large Networks and Time-Series Pratik Anand Lecture 10/18: Community Detection Prof. B Aditya Prakash

2 Agenda Background Strong and weak ties, EB and G-N, cut and conductance Spectral clustering Laplacian matrix and properties Normalized graph laplacian and unnormalized spectral clustering algorithm Higher order organization of complex networks Motifs Conductance of motifs Spectral clustering of motifs Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 2

3 Background Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 3

4 Community Behavior Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 4

5 Need for community detection Source : Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 5

6 Need for community detection Provides a bigger picture of a large complex network Different communities act similarly Facebook, Twitter communities can help spot the origin of fake news Targeted advertisement to communities NSA!!! Source : Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 6

7 Strong and Weak ties Strong ties - ties between friends and family members Weak ties - ties between colleagues, acquaintances Strong Triadic closure Source : Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 7

8 Defining ties Different view of the same set of points by different criteria of putting edge between two nodes This results in different kind of strong ties Overlapping sections Removal of bridges results in clusters Source : Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 8

9 Betweenness of an edge (EB) Total unit flow flowing through an edge 7,7 Used by Girvan - Newman method to partition a graph Source : Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 9

10 Girvan - Newman method Remove edges with highest EBs Recalculate per node EB and repeat above Source : Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 10

11 Cuts and Conductance Goal : Split a graph into different clusters with different weights Min cut : Minimizing the weight between two subgraphs Balanced cuts Ratio cut for k partitions N-cut for k partitions NP - hard Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 11

12 SPECTRAL CLUSTERING Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 12

13 Spectral Clustering More efficient than traditional algorithms like k-means Provides better results, easier to implement Works on Laplacian matrix of similarity graph Similarity graphs usually created by different methods like k- means, ε-neighborhood graph Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 13

14 Laplacian matrix L = D - W Source : Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 14

15 Properties of Unnormalized Graph Laplacian Source : Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 15

16 Normalized Graph Laplacian Source : Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 16

17 The algorithm v1 Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 17

18 The algorithm v2 Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 18

19 Visual representation Source : Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 19

20 Combining it all : Higher-Order Organization of Complex Networks Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 20

21 Finding Higher-order structures Source : Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 21

22 Motifs Small subgraphs which are building blocks of all networks Earlier works based on frequency count of subgraphs Identifying results in understanding of overall structure which leads to accurate clustering Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 22

23 Identifying motifs in networks Source : Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 23

24 Applying Conductance to Motifs Source : Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 24

25 Clustering Find a set of node which minimizes motif conductance Such nodes become a cluster Problem? : It is computationally NP-hard Solution? Applying spectral clustering to motifs Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 25

26 Motif Spectral Clustering Input : Graph G and motif M Weighted graph G using the motif M to create motif adjacency matrix Spectral clustering on G Output : Clusters of original graph G with lowest motif conductance Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 26

27 Different phases of motif clustering Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 27

28 Problem Higher-order spectral analysis of a network of airports in Canada and the United States Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 28

29 Part A Motifs, anchored at blue nodes Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 29

30 Part B Top 50 most populous cities in the US Thicker lines represent weight in motif adjacency matrix Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 30

31 Part C Green - hubs Red - West coast nonhubs Purple - East coast nonhubs Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 31

32 Part D : Comparision Higher-order Laplacian Non -Higher-order Laplacian Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 32

33 Conclusion Spectral clustering is an effective technique Motifs are enablers to apply spectral clustering to unweighted graphs Clusters provide a macroscopic behavior of nodes in a network Motifs can be applied to large variety of networks like social network, transportation network Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 33

34 THANK YOU Pratik Anand 2017 CS 6604: DM Large Networks & Time-Series 34