Non Overlapping Communities
|
|
- Louise Hensley
- 6 years ago
- Views:
Transcription
1 Non Overlapping Communities Davide Mottin, Konstantina Lazaridou HassoPlattner Institute Graph Mining course Winter Semester 2016
2 Acknowledgements Most of this lecture is taken from: Some other material: 2
3 Lecture road Introduction to graph clustering Hierarchical approaches Spectral clustering 3
4 Networks & Communities We often think of networks being organized into modules, cluster, communities: 4
5 Goal: Find Densely Linked Clusters 5
6 Micro-Markets in Sponsored Search Find micro-markets by partitioning the query-to-advertiser graph: query advertiser Andersen, R. and Lang, K.J. Communities from seed sets. WWW,
7 Movies and Actors Clusters in Movies-to-Actors graph: Andersen, R. and Lang, K.J. Communities from seed sets. WWW,
8 Twitter & Facebook Discovering social circles, circles of trust: Mcauley, J. and Leskovec, J. Discovering social circles in ego networks. TKDD,
9 Lecture road Introduction to graph clustering Hierarchical approaches Spectral clustering 9
10 Community Detection How to find communities? We will work with undirected (unweighted) networks 10
11 Method 1: Strength of Weak Ties Edge betweenness: Number of shortest paths passing over the edge Intuition: b=16 b=7.5 Edge strengths (call volume) in a real network Edge betweenness in a real network 11
12 [Girvan-Newman 02] Method 1: Girvan-Newman Divisive hierarchical clustering based on the notion of edge betweenness: Number of shortest paths passing through the edge Girvan-Newman Algorithm: Undirected unweighted networks Repeat until no edges are left: Calculate betweenness of edges Remove edges with highest betweenness Recompute betweenness Connected components are communities Gives a hierarchical decomposition of the network 12
13 Girvan-Newman: Example Need to re-compute betweenness at every step 13
14 Girvan-Newman: Example Step 1: Step 2: Step 3: Hierarchical network decomposition: 14
15 Girvan-Newman: Results Communities in physics collaborations 15
16 Girvan-Newman: Results Zachary s Karate club: Hierarchical decomposition 16
17 We need to resolve 2 questions 1. How to compute betweenness? 2. How to select the number of clusters? 17
18 How to Compute Betweenness? Want to compute betweenness of paths starting at node A Breath first search starting from A:
19 Betweenness centrality Betweennes is a measure of the brokerage power of a node: the higher the betweenness the higher the capacity of a node of acting as a broker and spreading a particular information Gives also the idea of the capacity of a graph to be easily disconnected Shortest paths from s to t including v betweenness v = * σ,- v σ,-,/0/- Shortest paths from s to t Takes O n 2 Require all the shortest paths! in the worst case: all nodes shortest path computation Defined for nodes 19
20 How to Compute Betweenness? Count the number of shortest paths from A to all other nodes of the network : Variant of Dijkstra or Floyd-Warshall algorithms 20
21 How to Compute Betweenness? * σ,- v σ,-,/0/- Compute betweenness by working up the tree: If there are multiple paths count them fractionally The algorithm: Add edge flows: -- node flow = 1+ child edges -- split the flow up based on the parent value Repeat the BFS procedure for each starting node U 1 path to K. Split evenly paths to J Split 1:2 1+1 paths to H Split evenly 21
22 How to Compute Betweenness? Compute betweenness by working up the tree: If there are multiple paths count them fractionally The algorithm: Add edge flows: -- node flow = 1+ child edges -- split the flow up based on the parent value Repeat the BFS procedure for each starting node U 1 path to K. Split evenly paths to J Split 1:2 1+1 paths to H Split evenly 22
23 Summary of Betweenness computation 1. Build one BFS structures for each node 2. Determine flow values for each edge using the previous procedure and (3 steps) 3. Sum up the flow values of each edge in all BFS structures to get its betweenness value Notice: we are counting the flow between each pair of nodes X and Y twice (once when BFS from X and once when BFS from Y) at the end we divide everything by two 4. Use these betweenness values to identify the edges of highest betweenness for purposes of removing them in the Girvan- Newman method 23
24 We need to resolve 2 questions 1. How to compute betweenness? 2. How to select the number of clusters? 24
25 Network Communities Communities: sets of tightly connected nodes Define: Modularity Q A measure of how well a network is partitioned into communities Given a partitioning of the network into groups s Î S: Q sî S [ (# edges within group s) (expected # edges within group s) ] Need a null model! 25
26 Null Model: Configuration Model Given real G on n nodes and m edges, construct rewired network G Same degree distribution but random connections Consider G as a multigraph (=multiple edges for the same pair of nodes) Question: what is the expected number of edges between two nodes? Imagine to split the edges in nodes + half-edges i The expected number of edges between nodes i and j of degrees k i and k j equals to: k i k j The expected number of edges in (multigraph) G : = 1 k i k j 2 i N j N = 1 1 k 2m 2 2m i j N k j = 1 2m 2m = m 4m i N = = k ik j 2m 2m Note: * k ` = 2m j 26
27 Modularity Modularity of partitioning S of graph G: Q sî S [ (# edges within group s) (expected # edges within group s) ] Q G, S = 1 A 2m ij k ik j s S i s j s 2m Normalizing cost.: -1<Q<1 Adjacency matrix Modularity values take range [ 1,1] It is positive if the number of edges within groups exceeds the expected number 0. 3 < Q < 0. 7 means significant community structure 27
28 Reasoning about modularity Can modularity = 1? Answer: NO!!! Range: [ x y, 1) Whatis the modularity of a two completed connected components of n/2 nodes? Can you find an example of a network partitioning with modularity < 0? 28
29 Modularity: Number of clusters Modularity is useful for selecting the number of clusters: Q Why not optimize Modularity directly? 29
30 Lecture road Introduction to graph clustering Hierarchical approaches Spectral clustering 30
31 Graph Partitioning Undirected graph G(V, E): Bi-partitioning task: Divide vertices into two disjoint groups A, B A Questions: How can we define a good partition of G? How can we efficiently identify such a partition? B 31
32 Graph Partitioning What makes a good partition? Maximize the number of within-group connections Minimize the number of between-group connections A B 32
33 Graph Cuts Express partitioning objectives as a function of the edge cut of the partition Cut: Set of edges with only one vertex in a group: A B cut(a,b) = 2 33
34 Graph Cut Criterion Criterion: Minimum-cut Minimize weight of connections between groups Degenerate case: argmin A,B cut(a, B) Optimal cut Minimum cut Problem: Only considers external cluster connections Does not consider internal cluster connectivity 34
35 Graph Cut Criteria NP-hard Criterion: Normalized-cut Connectivity between groups relative to the density of each group vol(a): total weight of the edges with at least one endpoint in A: vol A = n Why use this criterion? n Produces more balanced partitions How do we efficiently find a good partition? Problem: Computing optimal cut is NP-hard i A k i Shi, J., & Malik, J. (2000). Normalized cuts and image segmentation. TPAMI, 22(8),
36 Spectral Graph Partitioning A: adjacency matrix of undirected G A ij = 1 if (i, j) is an edge, else 0 x is a vector in Rˆ with components (x 1,, x n ) Think of it as a label/value of each node of G What is the meaning of A x? ˆ ŒŽx y = A Œ x Œ = x Œ,Œ Entry y i is a sum of labels x j of neighbors of i 36
37 What is the meaning of Ax? j th coordinate of A x : Sum of the x-values of neighbors of j Make this a new value at node j A x = λ x Spectral Graph Theory: Analyze the spectrum of matrix representing G Spectrum: Eigenvectors x i of a graph, ordered by the magnitude (strength) of their corresponding eigenvalues λ i : 37
38 Questions? 38
39 References Andersen, R. and Lang, K.J. Communities from seed sets. WWW, Mcauley, J. and Leskovec, J. Discovering social circles in ego networks. TKDD, Shi, J., & Malik, J. (2000). Normalized cuts and image segmentation. TPAMI, 22(8), Fiedler, M., Algebraic connectivity of graphs. Czechoslovak mathematical journal, 23(2), pp
Mining Social Network Graphs
Mining Social Network Graphs Analysis of Large Graphs: Community Detection Rafael Ferreira da Silva rafsilva@isi.edu http://rafaelsilva.com Note to other teachers and users of these slides: We would be
More informationCS 5614: (Big) Data Management Systems. B. Aditya Prakash Lecture #21: Graph Mining 2
CS 5614: (Big) Data Management Systems B. Aditya Prakash Lecture #21: Graph Mining 2 Networks & Communi>es We o@en think of networks being organized into modules, cluster, communi>es: VT CS 5614 2 Goal:
More informationCS224W: Analysis of Networks Jure Leskovec, Stanford University
CS224W: Analysis of Networks Jure Leskovec, Stanford University http://cs224w.stanford.edu 11/13/17 Jure Leskovec, Stanford CS224W: Analysis of Networks, http://cs224w.stanford.edu 2 Observations Models
More informationCS246: Mining Massive Datasets Jure Leskovec, Stanford University
CS246: Mining Massive Datasets Jure Leskovec, Stanford University http://cs246.stanford.edu SPAM FARMING 2/11/2013 Jure Leskovec, Stanford C246: Mining Massive Datasets 2 2/11/2013 Jure Leskovec, Stanford
More informationCS246: Mining Massive Datasets Jure Leskovec, Stanford University
CS246: Mining Massive Datasets Jure Leskovec, Stanford University http://cs246.stanford.edu HITS (Hypertext Induced Topic Selection) Is a measure of importance of pages or documents, similar to PageRank
More informationSocial-Network Graphs
Social-Network Graphs Mining Social Networks Facebook, Google+, Twitter Email Networks, Collaboration Networks Identify communities Similar to clustering Communities usually overlap Identify similarities
More informationOnline Social Networks and Media. Community detection
Online Social Networks and Media Community detection 1 Notes on Homework 1 1. You should write your own code for generating the graphs. You may use SNAP graph primitives (e.g., add node/edge) 2. For the
More informationCSE 258 Lecture 6. Web Mining and Recommender Systems. Community Detection
CSE 258 Lecture 6 Web Mining and Recommender Systems Community Detection Dimensionality reduction Goal: take high-dimensional data, and describe it compactly using a small number of dimensions Assumption:
More informationCSE 158 Lecture 6. Web Mining and Recommender Systems. Community Detection
CSE 158 Lecture 6 Web Mining and Recommender Systems Community Detection Dimensionality reduction Goal: take high-dimensional data, and describe it compactly using a small number of dimensions Assumption:
More informationMy favorite application using eigenvalues: partitioning and community detection in social networks
My favorite application using eigenvalues: partitioning and community detection in social networks Will Hobbs February 17, 2013 Abstract Social networks are often organized into families, friendship groups,
More informationCommunity Detection. Community
Community Detection Community In social sciences: Community is formed by individuals such that those within a group interact with each other more frequently than with those outside the group a.k.a. group,
More informationJure Leskovec, Cornell/Stanford University. Joint work with Kevin Lang, Anirban Dasgupta and Michael Mahoney, Yahoo! Research
Jure Leskovec, Cornell/Stanford University Joint work with Kevin Lang, Anirban Dasgupta and Michael Mahoney, Yahoo! Research Network: an interaction graph: Nodes represent entities Edges represent interaction
More informationModularity CMSC 858L
Modularity CMSC 858L Module-detection for Function Prediction Biological networks generally modular (Hartwell+, 1999) We can try to find the modules within a network. Once we find modules, we can look
More informationBig Data Analytics. Special Topics for Computer Science CSE CSE Feb 11
Big Data Analytics Special Topics for Computer Science CSE 4095-001 CSE 5095-005 Feb 11 Fei Wang Associate Professor Department of Computer Science and Engineering fei_wang@uconn.edu Clustering II Spectral
More informationClustering Algorithms on Graphs Community Detection 6CCS3WSN-7CCSMWAL
Clustering Algorithms on Graphs Community Detection 6CCS3WSN-7CCSMWAL Contents Zachary s famous example Community structure Modularity The Girvan-Newman edge betweenness algorithm In the beginning: Zachary
More informationWeb Structure Mining Community Detection and Evaluation
Web Structure Mining Community Detection and Evaluation 1 Community Community. It is formed by individuals such that those within a group interact with each other more frequently than with those outside
More informationBetweenness Measures and Graph Partitioning
Betweenness Measures and Graph Partitioning Objectives Define densely connected regions of a network Graph partitioning Algorithm to identify densely connected regions breaking a network into a set of
More informationCSE 255 Lecture 6. Data Mining and Predictive Analytics. Community Detection
CSE 255 Lecture 6 Data Mining and Predictive Analytics Community Detection Dimensionality reduction Goal: take high-dimensional data, and describe it compactly using a small number of dimensions Assumption:
More informationSocial Data Management Communities
Social Data Management Communities Antoine Amarilli 1, Silviu Maniu 2 January 9th, 2018 1 Télécom ParisTech 2 Université Paris-Sud 1/20 Table of contents Communities in Graphs 2/20 Graph Communities Communities
More informationClusters and Communities
Clusters and Communities Lecture 7 CSCI 4974/6971 22 Sep 2016 1 / 14 Today s Biz 1. Reminders 2. Review 3. Communities 4. Betweenness and Graph Partitioning 5. Label Propagation 2 / 14 Today s Biz 1. Reminders
More informationExtracting Information from Complex Networks
Extracting Information from Complex Networks 1 Complex Networks Networks that arise from modeling complex systems: relationships Social networks Biological networks Distinguish from random networks uniform
More informationBasics of Network Analysis
Basics of Network Analysis Hiroki Sayama sayama@binghamton.edu Graph = Network G(V, E): graph (network) V: vertices (nodes), E: edges (links) 1 Nodes = 1, 2, 3, 4, 5 2 3 Links = 12, 13, 15, 23,
More informationCS 534: Computer Vision Segmentation II Graph Cuts and Image Segmentation
CS 534: Computer Vision Segmentation II Graph Cuts and Image Segmentation Spring 2005 Ahmed Elgammal Dept of Computer Science CS 534 Segmentation II - 1 Outlines What is Graph cuts Graph-based clustering
More informationLesson 2 7 Graph Partitioning
Lesson 2 7 Graph Partitioning The Graph Partitioning Problem Look at the problem from a different angle: Let s multiply a sparse matrix A by a vector X. Recall the duality between matrices and graphs:
More informationSpectral Methods for Network Community Detection and Graph Partitioning
Spectral Methods for Network Community Detection and Graph Partitioning M. E. J. Newman Department of Physics, University of Michigan Presenters: Yunqi Guo Xueyin Yu Yuanqi Li 1 Outline: Community Detection
More informationSpectral Clustering on Handwritten Digits Database
October 6, 2015 Spectral Clustering on Handwritten Digits Database Danielle dmiddle1@math.umd.edu Advisor: Kasso Okoudjou kasso@umd.edu Department of Mathematics University of Maryland- College Park Advance
More informationMCL. (and other clustering algorithms) 858L
MCL (and other clustering algorithms) 858L Comparing Clustering Algorithms Brohee and van Helden (2006) compared 4 graph clustering algorithms for the task of finding protein complexes: MCODE RNSC Restricted
More informationGraph Mining: Overview of different graph models
Graph Mining: Overview of different graph models Davide Mottin, Konstantina Lazaridou Hasso Plattner Institute Graph Mining course Winter Semester 2016 Lecture road Anomaly detection (previous lecture)
More informationV4 Matrix algorithms and graph partitioning
V4 Matrix algorithms and graph partitioning - Community detection - Simple modularity maximization - Spectral modularity maximization - Division into more than two groups - Other algorithms for community
More informationTHE KNOWLEDGE MANAGEMENT STRATEGY IN ORGANIZATIONS. Summer semester, 2016/2017
THE KNOWLEDGE MANAGEMENT STRATEGY IN ORGANIZATIONS Summer semester, 2016/2017 SOCIAL NETWORK ANALYSIS: THEORY AND APPLICATIONS 1. A FEW THINGS ABOUT NETWORKS NETWORKS IN THE REAL WORLD There are four categories
More informationIntroduction to Machine Learning
Introduction to Machine Learning Clustering Varun Chandola Computer Science & Engineering State University of New York at Buffalo Buffalo, NY, USA chandola@buffalo.edu Chandola@UB CSE 474/574 1 / 19 Outline
More informationCAIM: Cerca i Anàlisi d Informació Massiva
1 / 72 CAIM: Cerca i Anàlisi d Informació Massiva FIB, Grau en Enginyeria Informàtica Slides by Marta Arias, José Balcázar, Ricard Gavaldá Department of Computer Science, UPC Fall 2016 http://www.cs.upc.edu/~caim
More informationGraph Theory Review. January 30, Network Science Analytics Graph Theory Review 1
Graph Theory Review Gonzalo Mateos Dept. of ECE and Goergen Institute for Data Science University of Rochester gmateosb@ece.rochester.edu http://www.ece.rochester.edu/~gmateosb/ January 30, 2018 Network
More informationOverlapping Communities
Yangyang Hou, Mu Wang, Yongyang Yu Purdue Univiersity Department of Computer Science April 25, 2013 Overview Datasets Algorithm I Algorithm II Algorithm III Evaluation Overview Graph models of many real
More informationCommunity Structure and Beyond
Community Structure and Beyond Elizabeth A. Leicht MAE: 298 April 9, 2009 Why do we care about community structure? Large Networks Discussion Outline Overview of past work on community structure. How to
More informationVisual Representations for Machine Learning
Visual Representations for Machine Learning Spectral Clustering and Channel Representations Lecture 1 Spectral Clustering: introduction and confusion Michael Felsberg Klas Nordberg The Spectral Clustering
More informationCS 534: Computer Vision Segmentation and Perceptual Grouping
CS 534: Computer Vision Segmentation and Perceptual Grouping Ahmed Elgammal Dept of Computer Science CS 534 Segmentation - 1 Outlines Mid-level vision What is segmentation Perceptual Grouping Segmentation
More informationAlgorithms and Applications in Social Networks. 2017/2018, Semester B Slava Novgorodov
Algorithms and Applications in Social Networks 2017/2018, Semester B Slava Novgorodov 1 Lesson #1 Administrative questions Course overview Introduction to Social Networks Basic definitions Network properties
More informationExtracting Communities from Networks
Extracting Communities from Networks Ji Zhu Department of Statistics, University of Michigan Joint work with Yunpeng Zhao and Elizaveta Levina Outline Review of community detection Community extraction
More informationOh Pott, Oh Pott! or how to detect community structure in complex networks
Oh Pott, Oh Pott! or how to detect community structure in complex networks Jörg Reichardt Interdisciplinary Centre for Bioinformatics, Leipzig, Germany (Host of the 2012 Olympics) Questions to start from
More informationCommunity Structure Detection. Amar Chandole Ameya Kabre Atishay Aggarwal
Community Structure Detection Amar Chandole Ameya Kabre Atishay Aggarwal What is a network? Group or system of interconnected people or things Ways to represent a network: Matrices Sets Sequences Time
More informationSegmentation and Grouping
Segmentation and Grouping How and what do we see? Fundamental Problems ' Focus of attention, or grouping ' What subsets of pixels do we consider as possible objects? ' All connected subsets? ' Representation
More informationTie strength, social capital, betweenness and homophily. Rik Sarkar
Tie strength, social capital, betweenness and homophily Rik Sarkar Networks Position of a node in a network determines its role/importance Structure of a network determines its properties 2 Today Notion
More informationGraph Theory for Network Science
Graph Theory for Network Science Dr. Natarajan Meghanathan Professor Department of Computer Science Jackson State University, Jackson, MS E-mail: natarajan.meghanathan@jsums.edu Networks or Graphs We typically
More informationCommunity Analysis. Chapter 6
This chapter is from Social Media Mining: An Introduction. By Reza Zafarani, Mohammad Ali Abbasi, and Huan Liu. Cambridge University Press, 2014. Draft version: April 20, 2014. Complete Draft and Slides
More informationCentralities (4) By: Ralucca Gera, NPS. Excellence Through Knowledge
Centralities (4) By: Ralucca Gera, NPS Excellence Through Knowledge Some slide from last week that we didn t talk about in class: 2 PageRank algorithm Eigenvector centrality: i s Rank score is the sum
More informationk-means demo Administrative Machine learning: Unsupervised learning" Assignment 5 out
Machine learning: Unsupervised learning" David Kauchak cs Spring 0 adapted from: http://www.stanford.edu/class/cs76/handouts/lecture7-clustering.ppt http://www.youtube.com/watch?v=or_-y-eilqo Administrative
More informationImage Segmentation continued Graph Based Methods
Image Segmentation continued Graph Based Methods Previously Images as graphs Fully-connected graph node (vertex) for every pixel link between every pair of pixels, p,q affinity weight w pq for each link
More informationNetwork Community Detection
Network Community Detection Gonzalo Mateos Dept. of ECE and Goergen Institute for Data Science University of Rochester gmateosb@ece.rochester.edu http://www.ece.rochester.edu/~gmateosb/ March 20, 2018
More informationAn introduction to graph analysis and modeling Descriptive Analysis of Network Data
An introduction to graph analysis and modeling Descriptive Analysis of Network Data MSc in Statistics for Smart Data ENSAI Autumn semester 2017 http://julien.cremeriefamily.info 1 / 68 Outline 1 Basic
More informationClustering. SC4/SM4 Data Mining and Machine Learning, Hilary Term 2017 Dino Sejdinovic
Clustering SC4/SM4 Data Mining and Machine Learning, Hilary Term 2017 Dino Sejdinovic Clustering is one of the fundamental and ubiquitous tasks in exploratory data analysis a first intuition about the
More informationCS 5220: Parallel Graph Algorithms. David Bindel
CS 5220: Parallel Graph Algorithms David Bindel 2017-11-14 1 Graphs Mathematically: G = (V, E) where E V V Convention: V = n and E = m May be directed or undirected May have weights w V : V R or w E :
More informationAn Exploratory Journey Into Network Analysis A Gentle Introduction to Network Science and Graph Visualization
An Exploratory Journey Into Network Analysis A Gentle Introduction to Network Science and Graph Visualization Pedro Ribeiro (DCC/FCUP & CRACS/INESC-TEC) Part 1 Motivation and emergence of Network Science
More informationhttp://www.xkcd.com/233/ Text Clustering David Kauchak cs160 Fall 2009 adapted from: http://www.stanford.edu/class/cs276/handouts/lecture17-clustering.ppt Administrative 2 nd status reports Paper review
More informationScalable Clustering of Signed Networks Using Balance Normalized Cut
Scalable Clustering of Signed Networks Using Balance Normalized Cut Kai-Yang Chiang,, Inderjit S. Dhillon The 21st ACM International Conference on Information and Knowledge Management (CIKM 2012) Oct.
More informationGraph drawing in spectral layout
Graph drawing in spectral layout Maureen Gallagher Colleen Tygh John Urschel Ludmil Zikatanov Beginning: July 8, 203; Today is: October 2, 203 Introduction Our research focuses on the use of spectral graph
More informationMining Social-Network Graphs
342 Chapter 10 Mining Social-Network Graphs There is much information to be gained by analyzing the large-scale data that is derived from social networks. The best-known example of a social network is
More informationCommunity detection. Leonid E. Zhukov
Community detection Leonid E. Zhukov School of Data Analysis and Artificial Intelligence Department of Computer Science National Research University Higher School of Economics Network Science Leonid E.
More informationSpectral Clustering X I AO ZE N G + E L HA M TA BA S SI CS E CL A S S P R ESENTATION MA RCH 1 6,
Spectral Clustering XIAO ZENG + ELHAM TABASSI CSE 902 CLASS PRESENTATION MARCH 16, 2017 1 Presentation based on 1. Von Luxburg, Ulrike. "A tutorial on spectral clustering." Statistics and computing 17.4
More informationTELCOM2125: Network Science and Analysis
School of Information Sciences University of Pittsburgh TELCOM2125: Network Science and Analysis Konstantinos Pelechrinis Spring 2015 2 Part 4: Dividing Networks into Clusters The problem l Graph partitioning
More informationLocal Fiedler Vector Centrality for Detection of Deep and Overlapping Communities in Networks
Local Fiedler Vector Centrality for Detection of Deep and Overlapping Communities in Networks Pin-Yu Chen and Alfred O. Hero III, Fellow, IEEE Department of Electrical Engineering and Computer Science,
More informationPersistent Homology in Complex Network Analysis
Persistent Homology Summer School - Rabat Persistent Homology in Complex Network Analysis Ulderico Fugacci Kaiserslautern University of Technology Department of Computer Science July 7, 2017 Anything has
More informationSergei Silvestrov, Christopher Engström. January 29, 2013
Sergei Silvestrov, January 29, 2013 L2: t Todays lecture:. L2: t Todays lecture:. measurements. L2: t Todays lecture:. measurements. s. L2: t Todays lecture:. measurements. s.. First we would like to define
More informationDiscovery of Community Structure in Complex Networks Based on Resistance Distance and Center Nodes
Journal of Computational Information Systems 8: 23 (2012) 9807 9814 Available at http://www.jofcis.com Discovery of Community Structure in Complex Networks Based on Resistance Distance and Center Nodes
More informationLecture Note: Computation problems in social. network analysis
Lecture Note: Computation problems in social network analysis Bang Ye Wu CSIE, Chung Cheng University, Taiwan September 29, 2008 In this lecture note, several computational problems are listed, including
More informationDetecting Community Structure for Undirected Big Graphs Based on Random Walks
Detecting Community Structure for Undirected Big Graphs Based on Random Walks Xiaoming Liu 1, Yadong Zhou 1, Chengchen Hu 1, Xiaohong Guan 1,, Junyuan Leng 1 1 MOE KLNNIS Lab, Xi an Jiaotong University,
More informationarxiv: v1 [stat.ml] 2 Nov 2010
Community Detection in Networks: The Leader-Follower Algorithm arxiv:111.774v1 [stat.ml] 2 Nov 21 Devavrat Shah and Tauhid Zaman* devavrat@mit.edu, zlisto@mit.edu November 4, 21 Abstract Traditional spectral
More informationIntroduction to spectral clustering
Introduction to spectral clustering Vasileios Zografos zografos@isy.liu.se Klas Nordberg klas@isy.liu.se What this course is Basic introduction into the core ideas of spectral clustering Sufficient to
More informationCS 140: Sparse Matrix-Vector Multiplication and Graph Partitioning
CS 140: Sparse Matrix-Vector Multiplication and Graph Partitioning Parallel sparse matrix-vector product Lay out matrix and vectors by rows y(i) = sum(a(i,j)*x(j)) Only compute terms with A(i,j) 0 P0 P1
More informationFinding and Visualizing Graph Clusters Using PageRank Optimization. Fan Chung and Alexander Tsiatas, UCSD WAW 2010
Finding and Visualizing Graph Clusters Using PageRank Optimization Fan Chung and Alexander Tsiatas, UCSD WAW 2010 What is graph clustering? The division of a graph into several partitions. Clusters should
More informationGraph Exploration: Taking the User into the Loop
Graph Exploration: Taking the User into the Loop Davide Mottin, Anja Jentzsch, Emmanuel Müller Hasso Plattner Institute, Potsdam, Germany 2016/10/24 CIKM2016, Indianapolis, US Where we are Background (5
More informationContent-based Image and Video Retrieval. Image Segmentation
Content-based Image and Video Retrieval Vorlesung, SS 2011 Image Segmentation 2.5.2011 / 9.5.2011 Image Segmentation One of the key problem in computer vision Identification of homogenous region in the
More informationCommunity Detection in Social Networks
San Jose State University SJSU ScholarWorks Master's Projects Master's Theses and Graduate Research Spring 5-24-2017 Community Detection in Social Networks Ketki Kulkarni San Jose State University Follow
More informationClustering Algorithms for general similarity measures
Types of general clustering methods Clustering Algorithms for general similarity measures general similarity measure: specified by object X object similarity matrix 1 constructive algorithms agglomerative
More informationSegmentation Computer Vision Spring 2018, Lecture 27
Segmentation http://www.cs.cmu.edu/~16385/ 16-385 Computer Vision Spring 218, Lecture 27 Course announcements Homework 7 is due on Sunday 6 th. - Any questions about homework 7? - How many of you have
More informationThe clustering in general is the task of grouping a set of objects in such a way that objects
Spectral Clustering: A Graph Partitioning Point of View Yangzihao Wang Computer Science Department, University of California, Davis yzhwang@ucdavis.edu Abstract This course project provide the basic theory
More informationLecture 11: E-M and MeanShift. CAP 5415 Fall 2007
Lecture 11: E-M and MeanShift CAP 5415 Fall 2007 Review on Segmentation by Clustering Each Pixel Data Vector Example (From Comanciu and Meer) Review of k-means Let's find three clusters in this data These
More informationClassic Graph Theory Problems
Classic Graph Theory Problems Hiroki Sayama sayama@binghamton.edu The Origin Königsberg bridge problem Pregel River (Solved negatively by Euler in 176) Representation in a graph Can all the seven edges
More informationCommunity Identification in International Weblogs
Technische Universität Kaiserslautern Fernstudium Software Engineering for Embedded Systems Masterarbeit Community Identification in International Weblogs Autor: Christoph Rueger Betreuer: Prof. Dr. Prof.
More informationMinimum Spanning Trees
Minimum Spanning Trees Problem A town has a set of houses and a set of roads. A road connects 2 and only 2 houses. A road connecting houses u and v has a repair cost w(u, v). Goal: Repair enough (and no
More informationClustering. Informal goal. General types of clustering. Applications: Clustering in information search and analysis. Example applications in search
Informal goal Clustering Given set of objects and measure of similarity between them, group similar objects together What mean by similar? What is good grouping? Computation time / quality tradeoff 1 2
More informationCSCI-B609: A Theorist s Toolkit, Fall 2016 Sept. 6, Firstly let s consider a real world problem: community detection.
CSCI-B609: A Theorist s Toolkit, Fall 016 Sept. 6, 016 Lecture 03: The Sparsest Cut Problem and Cheeger s Inequality Lecturer: Yuan Zhou Scribe: Xuan Dong We will continue studying the spectral graph theory
More informationSupplementary material to Epidemic spreading on complex networks with community structures
Supplementary material to Epidemic spreading on complex networks with community structures Clara Stegehuis, Remco van der Hofstad, Johan S. H. van Leeuwaarden Supplementary otes Supplementary ote etwork
More informationSocial Network Analysis
Social Network Analysis Mathematics of Networks Manar Mohaisen Department of EEC Engineering Adjacency matrix Network types Edge list Adjacency list Graph representation 2 Adjacency matrix Adjacency matrix
More informationGraph Theory for Network Science
Graph Theory for Network Science Dr. Natarajan Meghanathan Professor Department of Computer Science Jackson State University, Jackson, MS E-mail: natarajan.meghanathan@jsums.edu Networks or Graphs We typically
More informationEE 701 ROBOT VISION. Segmentation
EE 701 ROBOT VISION Regions and Image Segmentation Histogram-based Segmentation Automatic Thresholding K-means Clustering Spatial Coherence Merging and Splitting Graph Theoretic Segmentation Region Growing
More informationCommunity Detection Methods using Eigenvectors of Matrices
Community Detection Methods using Eigenvectors of Matrices Yan Zhang Abstract In this paper we investigate the problem of detecting communities in graphs. We use the eigenvectors of the graph Laplacian
More informationIntroduction to Parallel & Distributed Computing Parallel Graph Algorithms
Introduction to Parallel & Distributed Computing Parallel Graph Algorithms Lecture 16, Spring 2014 Instructor: 罗国杰 gluo@pku.edu.cn In This Lecture Parallel formulations of some important and fundamental
More informationHow and what do we see? Segmentation and Grouping. Fundamental Problems. Polyhedral objects. Reducing the combinatorics of pose estimation
Segmentation and Grouping Fundamental Problems ' Focus of attention, or grouping ' What subsets of piels do we consider as possible objects? ' All connected subsets? ' Representation ' How do we model
More informationCSE 373 NOVEMBER 20 TH TOPOLOGICAL SORT
CSE 373 NOVEMBER 20 TH TOPOLOGICAL SORT PROJECT 3 500 Internal Error problems Hopefully all resolved (or close to) P3P1 grades are up (but muted) Leave canvas comment Emails tomorrow End of quarter GRAPHS
More informationReview of Graph Theory. Gregory Provan
Review of Graph Theory Gregory Provan Overview Need for graphical models of computation Cloud computing Data centres Telecommunications networks Graph theory Graph Models for Cloud Computing Integration
More informationMATH 567: Mathematical Techniques in Data
Supervised and unsupervised learning Supervised learning problems: MATH 567: Mathematical Techniques in Data (X, Y ) P (X, Y ). Data Science Clustering I is labelled (input/output) with joint density We
More informationClustering. So far in the course. Clustering. Clustering. Subhransu Maji. CMPSCI 689: Machine Learning. dist(x, y) = x y 2 2
So far in the course Clustering Subhransu Maji : Machine Learning 2 April 2015 7 April 2015 Supervised learning: learning with a teacher You had training data which was (feature, label) pairs and the goal
More information6.801/866. Segmentation and Line Fitting. T. Darrell
6.801/866 Segmentation and Line Fitting T. Darrell Segmentation and Line Fitting Gestalt grouping Background subtraction K-Means Graph cuts Hough transform Iterative fitting (Next time: Probabilistic segmentation)
More informationApplication of Spectral Clustering Algorithm
1/27 Application of Spectral Clustering Algorithm Danielle Middlebrooks dmiddle1@math.umd.edu Advisor: Kasso Okoudjou kasso@umd.edu Department of Mathematics University of Maryland- College Park Advance
More informationIntroduction to Engineering Systems, ESD.00. Networks. Lecturers: Professor Joseph Sussman Dr. Afreen Siddiqi TA: Regina Clewlow
Introduction to Engineering Systems, ESD.00 Lecture 7 Networks Lecturers: Professor Joseph Sussman Dr. Afreen Siddiqi TA: Regina Clewlow The Bridges of Königsberg The town of Konigsberg in 18 th century
More informationAPPROXIMATE SPECTRAL LEARNING USING NYSTROM METHOD. Aleksandar Trokicić
FACTA UNIVERSITATIS (NIŠ) Ser. Math. Inform. Vol. 31, No 2 (2016), 569 578 APPROXIMATE SPECTRAL LEARNING USING NYSTROM METHOD Aleksandar Trokicić Abstract. Constrained clustering algorithms as an input
More information16 - Networks and PageRank
- Networks and PageRank ST 9 - Fall 0 Contents Network Intro. Required R Packages................................ Example: Zachary s karate club network.................... Basic Network Concepts. Basic
More informationAlgorithms and Data Structures
Algorithms and Data Structures Graphs : Shortest Paths Marius Kloft Content of this Lecture Single-Source-Shortest-Paths: Dijkstra s Algorithm Single-Source-Single-Target All-Pairs Shortest Paths Transitive
More informationLocal higher-order graph clustering
Local higher-order graph clustering Hao Yin Stanford University yinh@stanford.edu Austin R. Benson Cornell University arb@cornell.edu Jure Leskovec Stanford University jure@cs.stanford.edu David F. Gleich
More information