NETZCOPE - a tool to analyze and display complex R&D collaboration networks

Size: px
Start display at page:

Download "NETZCOPE - a tool to analyze and display complex R&D collaboration networks"

Transcription

1 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots NETZCOPE - a tool to analyze and display complex R&D collaboration networks L. Streit & O. Strogan BiBoS, Univ. Bielefeld and CCM, Univ. da Madeira ZiF Conference Complexity, Mathematics and Socio-Economic Problems August 31 - September 12, 2009 L. Streit & O. Strogan Complex Networks

2 Outline The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots 1 The Task 2 Concepts from Spectral Graph Theory 3 EU R&D Network Analysis 4 Netzcope Screenshots L. Streit Complex Networks

3 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots Collaborative R&D in the EU Framework Programmes The empirical infrastructure sysres EUPRO Framework Programme (FP) Period Projects Projects > 1 partner Organisations Subentities FP FP FP FP FP FP Total Based on information from CORDIS, with major standardisation and cleaning process applied Current release January 2008 Covering FP 1 to FP 6 systems research L. Streit Complex Networks

4 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots The consortium NEMO partners and website CCM, Universidade da Madeira systems research L. Streit Complex Networks

5 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots NEMO Tasks L. Streit Complex Networks

6 The Introduction Task Concepts Concepts from Spectral from Spectral Graph Graph Theory Theory EU R&D EU R&D Network Network Analysis Analysis Netzcope Screenshots The State Sizeof of the the art Problem Weighted Graphs Spectral Gaps Comparison of Centralities The Effect of Fiedler Ordering Visualizing the EU Research Networks L. Streit with T. Krueger Spectral Properties of a Real World Network l L. Streit Complex Networks

7 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots Adjacency Matrix A simple graph G = (V,E) is descibed by an adjacency matrix indicating whether verices i and k are connected by an edge: { 1 i k a ik = 0 otherwise The degree of edge k is and we shall set d k = a ik i D = diag {d 1,...,d n } L. Streit Complex Networks

8 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots Bipartite Graphs Two types of vertices V = {V 1,V 2 } No edges between vertices of the same type. L. Streit Complex Networks

9 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots The Laplacian(s) The Laplacian and the normalized Laplacian L = D A L = 1 D 1/2 AD 1/2 play a central role, in particular L is (up to a similarity transformation) the generator of a continuous time random walk on the graph, with equal probability 1/d k along each edge. L. Streit Complex Networks

10 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots Eigenvalues of L Have 0 = λ 0 λ 1...λ n 2 The multiplicity of λ 0 equals the number of disconnected subgraphs in G. Fully connected graphs with n vertices are characterized by λ 1 =... = λ n = Bipartite graphs are characterized by λ n = 2 n n 1 and have a symmetric spectrum: with λ k, 2 λ n is also an eigenvalue. L. Streit Complex Networks

11 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots A Variational Formula Consider the map on R n x ˆx = D 1/2 x Have (x,l x) = 1 2 a ik (ˆx i ˆx k ) 2 Hence one eigenvector, corresponding to the minimal eigenvalue λ 0 = 0, is given by ê k 1. L. Streit & O. Strogan Complex Networks

12 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots I.e. and hence Theorem e k = d 1/2 k The invariant distribution for the random walk is p k = d k d k L. Streit Complex Networks

13 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots Keeping the Little Guy Waiting (Previously r B A = 1, r A B = 1/3) ( 1 Now : r A B = r B A = 1/3 = min, 1 ) d A d B L. Streit Complex Networks

14 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots The Laplacian For this random walk with waiting, the normalized Laplacian will be ) min( 1 d i L ik =, 1 d k i k L ik i = k k:k i L. Streit Complex Networks

15 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots Spectral Gap For a connected graph the spectral gap λ 1 measures the rate of knowledge diffusion through the network. Note that λ 1 = min (x,l x) Definition x e, x =1 The minimizing vector f is called the Fiedler vector (1). L. Streit Complex Networks

16 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots The Fiedler Vector (2, normalized) Recall (f,l f ) = 1 2 a ik (ˆfi ˆf k ) 2 Imagine having arranged the vertices of the graph at points y k on the real line, attracting each other by an elastic force acting between them along each edge. Then the total energy of the configuration is E = 1 2 a ik (y i y k ) 2 We recognize the positions y k = ˆf k as the configuration of minimal energy, subject to the constraints y = D 1/2 ŷ = 1, y e which fix the extension of the configuration and its localization L. Streit Complex Networks

17 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots Let us say that the centrality c k of vertex k is proportional to the centrality of all its neighbors in the network: c k = 1 λ a kl c l. (A web page, e.g., is important if it is linked to other important pages.) In other words, Ac = λc By Perron-Frobenius, there is a solution with non negative c k, corresponding to the largest eigenvalue of the adjacency matrix A. Definition The resulting c k are called eigenvalue centralities. L. Streit Complex Networks

18 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots Newman s Modularity "A good division of a network into communities is not merely one in which there are few edges between communities; it is one in which there are fewer than expected edges between communities". Newman, Modularity is - up to a normalization constant - the number of edges within communities c minus those for a null model: Q 1 2 E c (A ij P ij ). i,j c L. Streit Complex Networks

19 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots The Null Model The standard choice for the null model will be dictated by the degrees of the vertices i and j: P ij d id j 2 E corresponding to a random graph models with a fixed set of vertices and the constraint that on average they should reproduce a given degree distribution (Chung,Lu 2002). The goal now is to find a division of the vertices into communities such that the modularity Q is maximal. An exhaustive search for a decomposition is out of the question: even for moderately big graphs there are far too many ways to decompose them into communities; fast approximate algorithms do exist. L. Streit Complex Networks

20 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots Barber s BRIM For bipartite graphs the null model should be modified, to reproduce the characteristic form of bipartite adjacency matrices ( ) 0 B A = B T 0 also for the null model. Recently M. Barber (2007) proposed an appropriate algorithm ("BRIM": bipartite, recursively induced modules) to find communities for bipartite networks. Starting from a (more or less) ad hoc partition of the vertices of type 1 it is straightforward to optimize a corresponding decomposition of the verices of type 2. From there, optimize the decomposition of vertices of type 1, and iterate. In this fashion, modularity will increase until a (local) maximum is reached. L. Streit Complex Networks

21

22

23

24

25

26

27

28

29

30 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots Organization Graph Spectra FP1, FP2, FP3 L. Streit Complex Networks

31 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots A Simulated Network L. Streit Complex Networks

32 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots Weighted vs. Unweighted, FP 2 L. Streit Complex Networks

33 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots Spectrum with the Little Guy waiting L. Streit Complex Networks

34

35

36

37

38

39

40

41

42

43

44 Zooming in

45

46

47

48

49

50

51

52 A Small Community

53 Geographic Profile of a Community

54 Organization Profile

55 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots Summary of Netzcope Features Netzcope extracts the connected components produces 2D plot of the networks displays the adjacency matrix same after re-ordering the nodes w.r.t. their strength of interaction finds communities within the network following Newman uses the BRIM algorithm to enhance the modularity computes mutual information of different decompositions produces a scatter plot to identify central players and gate keepers displays the composition of communities exhibits degree and eigenvalue centralities within communities displays the graph portrait following Bagrow et al. permits iterative decomposition and analysis ( zooming into communities ) L. Streit Complex Networks

56 The Task Concepts from Spectral Graph Theory EU R&D Network Analysis Netzcope Screenshots M. J. Barber: Modularity and community detection in bipartite networks. Phys. Rev. E 76, (2007). F. Chung and L. Lu: Connected components in random graphs with given degree sequences. Annals of Combinatorics 6, (2002). M. E. J. Newman and M. Girvan: Finding and evaluating community structure in networks. Phys. Rev. E 69, (2004). M. E. J. Newman: Finding community structure in networks using the eigenvectors of matrices. arxiv:physics/ v3 23 Jul 2006 M. E. J. Newman: Modularity and community structure in networks. URL L. Streit Complex Networks

Netzcope - A Tool to Display and Analyze Complex Networks

Netzcope - A Tool to Display and Analyze Complex Networks NEMO Working Paper #16 Netzcope - A Tool to Display and Analyze Complex Networks Oleg Strogan and Ludwig Streit (CCM, University of Madeira) Supported by the EU FP6-NEST-Adventure Programme Contract n

More information

DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II. Matrix Algorithms DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

More information

Part 2: Community Detection

Part 2: Community Detection Chapter 8: Graph Data Part 2: Community Detection Based on Leskovec, Rajaraman, Ullman 2014: Mining of Massive Datasets Big Data Management and Analytics Outline Community Detection - Social networks -

More information

USING SPECTRAL RADIUS RATIO FOR NODE DEGREE TO ANALYZE THE EVOLUTION OF SCALE- FREE NETWORKS AND SMALL-WORLD NETWORKS

USING SPECTRAL RADIUS RATIO FOR NODE DEGREE TO ANALYZE THE EVOLUTION OF SCALE- FREE NETWORKS AND SMALL-WORLD NETWORKS USING SPECTRAL RADIUS RATIO FOR NODE DEGREE TO ANALYZE THE EVOLUTION OF SCALE- FREE NETWORKS AND SMALL-WORLD NETWORKS Natarajan Meghanathan Jackson State University, 1400 Lynch St, Jackson, MS, USA natarajan.meghanathan@jsums.edu

More information

A scalable multilevel algorithm for graph clustering and community structure detection

A scalable multilevel algorithm for graph clustering and community structure detection A scalable multilevel algorithm for graph clustering and community structure detection Hristo N. Djidjev 1 Los Alamos National Laboratory, Los Alamos, NM 87545 Abstract. One of the most useful measures

More information

USE OF EIGENVALUES AND EIGENVECTORS TO ANALYZE BIPARTIVITY OF NETWORK GRAPHS

USE OF EIGENVALUES AND EIGENVECTORS TO ANALYZE BIPARTIVITY OF NETWORK GRAPHS USE OF EIGENVALUES AND EIGENVECTORS TO ANALYZE BIPARTIVITY OF NETWORK GRAPHS Natarajan Meghanathan Jackson State University, 1400 Lynch St, Jackson, MS, USA natarajan.meghanathan@jsums.edu ABSTRACT This

More information

Small Maximal Independent Sets and Faster Exact Graph Coloring

Small Maximal Independent Sets and Faster Exact Graph Coloring Small Maximal Independent Sets and Faster Exact Graph Coloring David Eppstein Univ. of California, Irvine Dept. of Information and Computer Science The Exact Graph Coloring Problem: Given an undirected

More information

Francesco Sorrentino Department of Mechanical Engineering

Francesco Sorrentino Department of Mechanical Engineering Master stability function approaches to analyze stability of the synchronous evolution for hypernetworks and of synchronized clusters for networks with symmetries Francesco Sorrentino Department of Mechanical

More information

Bindel, Fall 2012 Matrix Computations (CS 6210) Week 8: Friday, Oct 12

Bindel, Fall 2012 Matrix Computations (CS 6210) Week 8: Friday, Oct 12 Why eigenvalues? Week 8: Friday, Oct 12 I spend a lot of time thinking about eigenvalue problems. In part, this is because I look for problems that can be solved via eigenvalues. But I might have fewer

More information

SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH 31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

More information

An Empirical Study of Two MIS Algorithms

An Empirical Study of Two MIS Algorithms An Empirical Study of Two MIS Algorithms Email: Tushar Bisht and Kishore Kothapalli International Institute of Information Technology, Hyderabad Hyderabad, Andhra Pradesh, India 32. tushar.bisht@research.iiit.ac.in,

More information

Why graph clustering is useful?

Why graph clustering is useful? Graph Clustering Why graph clustering is useful? Distance matrices are graphs as useful as any other clustering Identification of communities in social networks Webpage clustering for better data management

More information

Counting spanning trees

Counting spanning trees Counting spanning trees Question Given a graph G, howmanyspanningtreesdoesg have? (G) =numberofdistinctspanningtreesofg Definition If G =(V,E) isamultigraphwithe 2 E, theng e (said G contract e ) is the

More information

Walk-Based Centrality and Communicability Measures for Network Analysis

Walk-Based Centrality and Communicability Measures for Network Analysis Walk-Based Centrality and Communicability Measures for Network Analysis Michele Benzi Department of Mathematics and Computer Science Emory University Atlanta, Georgia, USA Workshop on Innovative Clustering

More information

Social Media Mining. Graph Essentials

Social Media Mining. Graph Essentials Graph Essentials Graph Basics Measures Graph and Essentials Metrics 2 2 Nodes and Edges A network is a graph nodes, actors, or vertices (plural of vertex) Connections, edges or ties Edge Node Measures

More information

Lesson 3. Algebraic graph theory. Sergio Barbarossa. Rome - February 2010

Lesson 3. Algebraic graph theory. Sergio Barbarossa. Rome - February 2010 Lesson 3 Algebraic graph theory Sergio Barbarossa Basic notions Definition: A directed graph (or digraph) composed by a set of vertices and a set of edges We adopt the convention that the information flows

More information

A Spectral Clustering Approach to Validating Sensors via Their Peers in Distributed Sensor Networks

A Spectral Clustering Approach to Validating Sensors via Their Peers in Distributed Sensor Networks A Spectral Clustering Approach to Validating Sensors via Their Peers in Distributed Sensor Networks H. T. Kung Dario Vlah {htk, dario}@eecs.harvard.edu Harvard School of Engineering and Applied Sciences

More information

Split Nonthreshold Laplacian Integral Graphs

Split Nonthreshold Laplacian Integral Graphs Split Nonthreshold Laplacian Integral Graphs Stephen Kirkland University of Regina, Canada kirkland@math.uregina.ca Maria Aguieiras Alvarez de Freitas Federal University of Rio de Janeiro, Brazil maguieiras@im.ufrj.br

More information

COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS

COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS Alexander Burstein Department of Mathematics Howard University Washington, DC 259, USA aburstein@howard.edu Sergey Kitaev Mathematics

More information

Social Media Mining. Network Measures

Social Media Mining. Network Measures Klout Measures and Metrics 22 Why Do We Need Measures? Who are the central figures (influential individuals) in the network? What interaction patterns are common in friends? Who are the like-minded users

More information

Solutions to Exercises 8

Solutions to Exercises 8 Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 8 (1) Suppose that G is a graph in which every vertex has degree at least k, where k 1, and in which every cycle contains at least 4 vertices.

More information

Math 4707: Introduction to Combinatorics and Graph Theory

Math 4707: Introduction to Combinatorics and Graph Theory Math 4707: Introduction to Combinatorics and Graph Theory Lecture Addendum, November 3rd and 8th, 200 Counting Closed Walks and Spanning Trees in Graphs via Linear Algebra and Matrices Adjacency Matrices

More information

AN UPPER BOUND ON THE LAPLACIAN SPECTRAL RADIUS OF THE SIGNED GRAPHS

AN UPPER BOUND ON THE LAPLACIAN SPECTRAL RADIUS OF THE SIGNED GRAPHS Discussiones Mathematicae Graph Theory 28 (2008 ) 345 359 AN UPPER BOUND ON THE LAPLACIAN SPECTRAL RADIUS OF THE SIGNED GRAPHS Hong-Hai Li College of Mathematic and Information Science Jiangxi Normal University

More information

A CHARACTERIZATION OF TREE TYPE

A CHARACTERIZATION OF TREE TYPE A CHARACTERIZATION OF TREE TYPE LON H MITCHELL Abstract Let L(G) be the Laplacian matrix of a simple graph G The characteristic valuation associated with the algebraic connectivity a(g) is used in classifying

More information

Lecture 9. 1 Introduction. 2 Random Walks in Graphs. 1.1 How To Explore a Graph? CS-621 Theory Gems October 17, 2012

Lecture 9. 1 Introduction. 2 Random Walks in Graphs. 1.1 How To Explore a Graph? CS-621 Theory Gems October 17, 2012 CS-62 Theory Gems October 7, 202 Lecture 9 Lecturer: Aleksander Mądry Scribes: Dorina Thanou, Xiaowen Dong Introduction Over the next couple of lectures, our focus will be on graphs. Graphs are one of

More information

Complex Networks Analysis: Clustering Methods

Complex Networks Analysis: Clustering Methods Complex Networks Analysis: Clustering Methods Nikolai Nefedov Spring 2013 ISI ETH Zurich nefedov@isi.ee.ethz.ch 1 Outline Purpose to give an overview of modern graph-clustering methods and their applications

More information

Electrical Resistances in Products of Graphs

Electrical Resistances in Products of Graphs Electrical Resistances in Products of Graphs By Shelley Welke Under the direction of Dr. John S. Caughman In partial fulfillment of the requirements for the degree of: Masters of Science in Teaching Mathematics

More information

The spectra of random graphs with given expected degrees

The spectra of random graphs with given expected degrees Classification: Physical Sciences, Mathematics The spectra of random graphs with given expected degrees by Fan Chung Linyuan Lu Van Vu Department of Mathematics University of California at San Diego La

More information

Performance of Dynamic Load Balancing Algorithms for Unstructured Mesh Calculations

Performance of Dynamic Load Balancing Algorithms for Unstructured Mesh Calculations Performance of Dynamic Load Balancing Algorithms for Unstructured Mesh Calculations Roy D. Williams, 1990 Presented by Chris Eldred Outline Summary Finite Element Solver Load Balancing Results Types Conclusions

More information

LABEL PROPAGATION ON GRAPHS. SEMI-SUPERVISED LEARNING. ----Changsheng Liu 10-30-2014

LABEL PROPAGATION ON GRAPHS. SEMI-SUPERVISED LEARNING. ----Changsheng Liu 10-30-2014 LABEL PROPAGATION ON GRAPHS. SEMI-SUPERVISED LEARNING ----Changsheng Liu 10-30-2014 Agenda Semi Supervised Learning Topics in Semi Supervised Learning Label Propagation Local and global consistency Graph

More information

Laplacian spectrum of weakly quasi-threshold graphs

Laplacian spectrum of weakly quasi-threshold graphs Laplacian spectrum of weakly quasi-threshold graphs R. B. Bapat 1, A. K. Lal 2, Sukanta Pati 3 1 Stat-Math Unit, Indian Statistical Institute Delhi, 7-SJSS Marg, New Delhi - 110 016, India; e-mail: rbb@isid.ac.in

More information

A Tutorial on Spectral Clustering

A Tutorial on Spectral Clustering A Tutorial on Spectral Clustering Ulrike von Luxburg Max Planck Institute for Biological Cybernetics Spemannstr. 38, 7276 Tübingen, Germany ulrike.luxburg@tuebingen.mpg.de This article appears in Statistics

More information

ON THE DEGREES OF FREEDOM OF SIGNALS ON GRAPHS. Mikhail Tsitsvero and Sergio Barbarossa

ON THE DEGREES OF FREEDOM OF SIGNALS ON GRAPHS. Mikhail Tsitsvero and Sergio Barbarossa ON THE DEGREES OF FREEDOM OF SIGNALS ON GRAPHS Mikhail Tsitsvero and Sergio Barbarossa Sapienza Univ. of Rome, DIET Dept., Via Eudossiana 18, 00184 Rome, Italy E-mail: tsitsvero@gmail.com, sergio.barbarossa@uniroma1.it

More information

Practical Graph Mining with R. 5. Link Analysis

Practical Graph Mining with R. 5. Link Analysis Practical Graph Mining with R 5. Link Analysis Outline Link Analysis Concepts Metrics for Analyzing Networks PageRank HITS Link Prediction 2 Link Analysis Concepts Link A relationship between two entities

More information

Protein Protein Interaction Networks

Protein Protein Interaction Networks Functional Pattern Mining from Genome Scale Protein Protein Interaction Networks Young-Rae Cho, Ph.D. Assistant Professor Department of Computer Science Baylor University it My Definition of Bioinformatics

More information

Rank one SVD: un algorithm pour la visualisation d une matrice non négative

Rank one SVD: un algorithm pour la visualisation d une matrice non négative Rank one SVD: un algorithm pour la visualisation d une matrice non négative L. Labiod and M. Nadif LIPADE - Universite ParisDescartes, France ECAIS 2013 November 7, 2013 Outline Outline 1 Data visualization

More information

Network (Tree) Topology Inference Based on Prüfer Sequence

Network (Tree) Topology Inference Based on Prüfer Sequence Network (Tree) Topology Inference Based on Prüfer Sequence C. Vanniarajan and Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology Madras Chennai 600036 vanniarajanc@hcl.in,

More information

Chapter 4. Trees. 4.1 Basics

Chapter 4. Trees. 4.1 Basics Chapter 4 Trees 4.1 Basics A tree is a connected graph with no cycles. A forest is a collection of trees. A vertex of degree one, particularly in a tree, is called a leaf. Trees arise in a variety of applications.

More information

Chapter 6. Orthogonality

Chapter 6. Orthogonality 6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be

More information

Exponential time algorithms for graph coloring

Exponential time algorithms for graph coloring Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A k-labeling of vertices of a graph G(V, E) is a function V [k].

More information

Lecture 12: Partitioning and Load Balancing

Lecture 12: Partitioning and Load Balancing Lecture 12: Partitioning and Load Balancing G63.2011.002/G22.2945.001 November 16, 2010 thanks to Schloegel,Karypis and Kumar survey paper and Zoltan website for many of today s slides and pictures Partitioning

More information

Nonlinear Iterative Partial Least Squares Method

Nonlinear Iterative Partial Least Squares Method Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., Richard-Plouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for

More information

Factor analysis. Angela Montanari

Factor analysis. Angela Montanari Factor analysis Angela Montanari 1 Introduction Factor analysis is a statistical model that allows to explain the correlations between a large number of observed correlated variables through a small number

More information

(67902) Topics in Theory and Complexity Nov 2, 2006. Lecture 7

(67902) Topics in Theory and Complexity Nov 2, 2006. Lecture 7 (67902) Topics in Theory and Complexity Nov 2, 2006 Lecturer: Irit Dinur Lecture 7 Scribe: Rani Lekach 1 Lecture overview This Lecture consists of two parts In the first part we will refresh the definition

More information

Performance Metrics for Graph Mining Tasks

Performance Metrics for Graph Mining Tasks Performance Metrics for Graph Mining Tasks 1 Outline Introduction to Performance Metrics Supervised Learning Performance Metrics Unsupervised Learning Performance Metrics Optimizing Metrics Statistical

More information

SCAN: A Structural Clustering Algorithm for Networks

SCAN: A Structural Clustering Algorithm for Networks SCAN: A Structural Clustering Algorithm for Networks Xiaowei Xu, Nurcan Yuruk, Zhidan Feng (University of Arkansas at Little Rock) Thomas A. J. Schweiger (Acxiom Corporation) Networks scaling: #edges connected

More information

Nimble Algorithms for Cloud Computing. Ravi Kannan, Santosh Vempala and David Woodruff

Nimble Algorithms for Cloud Computing. Ravi Kannan, Santosh Vempala and David Woodruff Nimble Algorithms for Cloud Computing Ravi Kannan, Santosh Vempala and David Woodruff Cloud computing Data is distributed arbitrarily on many servers Parallel algorithms: time Streaming algorithms: sublinear

More information

Spectral Analysis of Signed Graphs for Clustering, Prediction and Visualization

Spectral Analysis of Signed Graphs for Clustering, Prediction and Visualization Spectral Analysis of Signed Graphs for Clustering, Prediction and Visualization Jérôme Kunegis Stephan Schmidt Andreas Lommatzsch Jürgen Lerner Ernesto W. De Luca Sahin Albayrak Abstract We study the application

More information

Geometrical Segmentation of Point Cloud Data using Spectral Clustering

Geometrical Segmentation of Point Cloud Data using Spectral Clustering Geometrical Segmentation of Point Cloud Data using Spectral Clustering Sergey Alexandrov and Rainer Herpers University of Applied Sciences Bonn-Rhein-Sieg 15th July 2014 1/29 Addressed Problem Given: a

More information

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph

More information

Network Algorithms for Homeland Security

Network Algorithms for Homeland Security Network Algorithms for Homeland Security Mark Goldberg and Malik Magdon-Ismail Rensselaer Polytechnic Institute September 27, 2004. Collaborators J. Baumes, M. Krishmamoorthy, N. Preston, W. Wallace. Partially

More information

A modularity-based spectral graph analysis

A modularity-based spectral graph analysis A modularity-based spectral graph analysis Dario Fasino (Udine), Francesco Tudisco (Roma TV) Cagliari, VDM60 D. Fasino, F. Tudisco Modularity-based spectral graph analysis 1/ 18 Introduction Graphs and

More information

YILUN SHANG. e λi. i=1

YILUN SHANG. e λi. i=1 LOWER BOUNDS FOR THE ESTRADA INDEX OF GRAPHS YILUN SHANG Abstract. Let G be a graph with n vertices and λ 1,λ,...,λ n be its eigenvalues. The Estrada index of G is defined as EE(G = n eλ i. In this paper,

More information

Mining Social-Network Graphs

Mining 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 information

GRAPH THEORY and APPLICATIONS. Trees

GRAPH THEORY and APPLICATIONS. Trees GRAPH THEORY and APPLICATIONS Trees Properties Tree: a connected graph with no cycle (acyclic) Forest: a graph with no cycle Paths are trees. Star: A tree consisting of one vertex adjacent to all the others.

More information

Studying E-mail Graphs for Intelligence Monitoring and Analysis in the Absence of Semantic Information

Studying E-mail Graphs for Intelligence Monitoring and Analysis in the Absence of Semantic Information Studying E-mail Graphs for Intelligence Monitoring and Analysis in the Absence of Semantic Information Petros Drineas, Mukkai S. Krishnamoorthy, Michael D. Sofka Bülent Yener Department of Computer Science,

More information

6.042/18.062J Mathematics for Computer Science October 3, 2006 Tom Leighton and Ronitt Rubinfeld. Graph Theory III

6.042/18.062J Mathematics for Computer Science October 3, 2006 Tom Leighton and Ronitt Rubinfeld. Graph Theory III 6.04/8.06J Mathematics for Computer Science October 3, 006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Graph Theory III Draft: please check back in a couple of days for a modified version of these

More information

Random graphs with a given degree sequence

Random graphs with a given degree sequence Sourav Chatterjee (NYU) Persi Diaconis (Stanford) Allan Sly (Microsoft) Let G be an undirected simple graph on n vertices. Let d 1,..., d n be the degrees of the vertices of G arranged in descending order.

More information

Definition. A graph is a collection of vertices, and edges between them. They are often represented by a drawing:

Definition. A graph is a collection of vertices, and edges between them. They are often represented by a drawing: 1. GRAPHS AND COLORINGS Definition. A graph is a collection of vertices, and edges between them. They are often represented by a drawing: 3 vertices 3 edges 4 vertices 4 edges 4 vertices 6 edges A graph

More information

The Characteristic Polynomial

The Characteristic Polynomial Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

More information

Linear Programming I

Linear Programming I Linear Programming I November 30, 2003 1 Introduction In the VCR/guns/nuclear bombs/napkins/star wars/professors/butter/mice problem, the benevolent dictator, Bigus Piguinus, of south Antarctica penguins

More information

Relations Graphical View

Relations Graphical View Relations Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Introduction Recall that a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). A binary

More information

Math 550 Notes. Chapter 7. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010

Math 550 Notes. Chapter 7. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010 Math 550 Notes Chapter 7 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 7 Fall 2010 1 / 34 Outline 1 Self-Adjoint and Normal Operators

More information

Efficient Load Balancing by Adaptive Bypasses for the Migration on the Internet

Efficient Load Balancing by Adaptive Bypasses for the Migration on the Internet Efficient Load Balancing by Adaptive Bypasses for the Migration on the Internet Yukio Hayashi yhayashi@jaist.ac.jp Japan Advanced Institute of Science and Technology ICCS 03 Workshop on Grid Computing,

More information

Block designs/1. 1 Background

Block designs/1. 1 Background Block designs 1 Background In a typical experiment, we have a set Ω of experimental units or plots, and (after some preparation) we make a measurement on each plot (for example, the yield of the plot).

More information

Combinatorics: The Fine Art of Counting

Combinatorics: The Fine Art of Counting Combinatorics: The Fine Art of Counting Week 9 Lecture Notes Graph Theory For completeness I have included the definitions from last week s lecture which we will be using in today s lecture along with

More information

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

More information

Lecture Notes on Spanning Trees

Lecture Notes on Spanning Trees Lecture Notes on Spanning Trees 15-122: Principles of Imperative Computation Frank Pfenning Lecture 26 April 26, 2011 1 Introduction In this lecture we introduce graphs. Graphs provide a uniform model

More information

THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE. Alexander Barvinok

THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE. Alexander Barvinok THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE Alexer Barvinok Papers are available at http://www.math.lsa.umich.edu/ barvinok/papers.html This is a joint work with J.A. Hartigan

More information

Notes on Matrix Multiplication and the Transitive Closure

Notes on Matrix Multiplication and the Transitive Closure ICS 6D Due: Wednesday, February 25, 2015 Instructor: Sandy Irani Notes on Matrix Multiplication and the Transitive Closure An n m matrix over a set S is an array of elements from S with n rows and m columns.

More information

Search engines: ranking algorithms

Search engines: ranking algorithms Search engines: ranking algorithms Gianna M. Del Corso Dipartimento di Informatica, Università di Pisa, Italy ESP, 25 Marzo 2015 1 Statistics 2 Search Engines Ranking Algorithms HITS Web Analytics Estimated

More information

Linear Algebra and its Applications

Linear Algebra and its Applications Linear Algebra and its Applications 438 2013) 1393 1397 Contents lists available at SciVerse ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Note on the

More information

On the dual of the solvency cone

On the dual of the solvency cone On the dual of the solvency cone Andreas Löhne Friedrich-Schiller-Universität Jena Joint work with: Birgit Rudloff (WU Wien) Wien, April, 0 Simplest solvency cone example Exchange between: Currency : Nepalese

More information

10. Graph Matrices Incidence Matrix

10. Graph Matrices Incidence Matrix 10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

More information

Expansion Properties of Large Social Graphs

Expansion Properties of Large Social Graphs Expansion Properties of Large Social Graphs Fragkiskos D. Malliaros 1 and Vasileios Megalooikonomou 1,2 1 Computer Engineering and Informatics Department University of Patras, 26500 Rio, Greece 2 Data

More information

Influence Discovery in Semantic Networks: An Initial Approach

Influence Discovery in Semantic Networks: An Initial Approach 2014 UKSim-AMSS 16th International Conference on Computer Modelling and Simulation Influence Discovery in Semantic Networks: An Initial Approach Marcello Trovati and Ovidiu Bagdasar School of Computing

More information

A FAST AND HIGH QUALITY MULTILEVEL SCHEME FOR PARTITIONING IRREGULAR GRAPHS

A FAST AND HIGH QUALITY MULTILEVEL SCHEME FOR PARTITIONING IRREGULAR GRAPHS SIAM J. SCI. COMPUT. Vol. 20, No., pp. 359 392 c 998 Society for Industrial and Applied Mathematics A FAST AND HIGH QUALITY MULTILEVEL SCHEME FOR PARTITIONING IRREGULAR GRAPHS GEORGE KARYPIS AND VIPIN

More information

Good luck, veel succes!

Good luck, veel succes! Final exam Advanced Linear Programming, May 7, 13.00-16.00 Switch off your mobile phone, PDA and any other mobile device and put it far away. No books or other reading materials are allowed. This exam

More information

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various

More information

Graphs and Network Flows IE411 Lecture 1

Graphs and Network Flows IE411 Lecture 1 Graphs and Network Flows IE411 Lecture 1 Dr. Ted Ralphs IE411 Lecture 1 1 References for Today s Lecture Required reading Sections 17.1, 19.1 References AMO Chapter 1 and Section 2.1 and 2.2 IE411 Lecture

More information

Graph Theory. Introduction. Distance in Graphs. Trees. Isabela Drămnesc UVT. Computer Science Department, West University of Timişoara, Romania

Graph Theory. Introduction. Distance in Graphs. Trees. Isabela Drămnesc UVT. Computer Science Department, West University of Timişoara, Romania Graph Theory Introduction. Distance in Graphs. Trees Isabela Drămnesc UVT Computer Science Department, West University of Timişoara, Romania November 2016 Isabela Drămnesc UVT Graph Theory and Combinatorics

More information

Expander Graph based Key Distribution Mechanisms in Wireless Sensor Networks

Expander Graph based Key Distribution Mechanisms in Wireless Sensor Networks Expander Graph based Key Distribution Mechanisms in Wireless Sensor Networks Seyit Ahmet Çamtepe Computer Science Department Rensselaer Polytechnic Institute Troy, New York 12180 Email: camtes@cs.rpi.edu

More information

Entropy Waves, The Zig-Zag Graph Product, and New Constant-Degree Expanders

Entropy Waves, The Zig-Zag Graph Product, and New Constant-Degree Expanders Entropy Waves, The Zig-Zag Graph Product, and New Constant-Degree Expanders Omer Reingold Salil Vadhan Avi Wigderson. August 1, 2001 Abstract The main contribution of this work is a new type of graph product,

More information

OPTIMAL DESIGN OF DISTRIBUTED SENSOR NETWORKS FOR FIELD RECONSTRUCTION

OPTIMAL DESIGN OF DISTRIBUTED SENSOR NETWORKS FOR FIELD RECONSTRUCTION OPTIMAL DESIGN OF DISTRIBUTED SENSOR NETWORKS FOR FIELD RECONSTRUCTION Sérgio Pequito, Stephen Kruzick, Soummya Kar, José M. F. Moura, A. Pedro Aguiar Department of Electrical and Computer Engineering

More information

Application of Graph Theory to

Application of Graph Theory to Application of Graph Theory to Requirements Traceability A methodology for visualization of large requirements sets Sam Brown L-3 Communications This presentation consists of L-3 STRATIS general capabilities

More information

The Open University s repository of research publications and other research outputs

The Open University s repository of research publications and other research outputs Open Research Online The Open University s repository of research publications and other research outputs The degree-diameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:

More information

Degree distribution in random Apollonian networks structures

Degree distribution in random Apollonian networks structures Degree distribution in random Apollonian networks structures Alexis Darrasse joint work with Michèle Soria ALÉA 2007 Plan 1 Introduction 2 Properties of real-life graphs Distinctive properties Existing

More information

In the following we will only consider undirected networks.

In the following we will only consider undirected networks. Roles in Networks Roles in Networks Motivation for work: Let topology define network roles. Work by Kleinberg on directed graphs, used topology to define two types of roles: authorities and hubs. (Each

More information

Collecting Network Data in Surveys

Collecting Network Data in Surveys Collecting Network Data in Surveys Arun Advani and Bansi Malde September 203 We gratefully acknowledge funding from the ESRC-NCRM Node Programme Evaluation for Policy Analysis Grant reference RES-576-25-0042.

More information

CMSC 451: Graph Properties, DFS, BFS, etc.

CMSC 451: Graph Properties, DFS, BFS, etc. CMSC 451: Graph Properties, DFS, BFS, etc. Slides By: Carl Kingsford Department of Computer Science University of Maryland, College Park Based on Chapter 3 of Algorithm Design by Kleinberg & Tardos. Graphs

More information

The mathematics of networks

The mathematics of networks The mathematics of networks M. E. J. Newman Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109 1040 In much of economic theory it is assumed that economic agents interact,

More information

SPANNING CACTI FOR STRUCTURALLY CONTROLLABLE NETWORKS NGO THI TU ANH NATIONAL UNIVERSITY OF SINGAPORE

SPANNING CACTI FOR STRUCTURALLY CONTROLLABLE NETWORKS NGO THI TU ANH NATIONAL UNIVERSITY OF SINGAPORE SPANNING CACTI FOR STRUCTURALLY CONTROLLABLE NETWORKS NGO THI TU ANH NATIONAL UNIVERSITY OF SINGAPORE 2012 SPANNING CACTI FOR STRUCTURALLY CONTROLLABLE NETWORKS NGO THI TU ANH (M.Sc., SFU, Russia) A THESIS

More information

GRADUATE STUDENT NUMBER THEORY SEMINAR

GRADUATE STUDENT NUMBER THEORY SEMINAR GRADUATE STUDENT NUMBER THEORY SEMINAR MICHELLE DELCOURT Abstract. This talk is based primarily on the paper Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees by Marcus, Spielman, and Srivastava

More information

Analysis of Internet Topologies: A Historical View

Analysis of Internet Topologies: A Historical View Analysis of Internet Topologies: A Historical View Mohamadreza Najiminaini, Laxmi Subedi, and Ljiljana Trajković Communication Networks Laboratory http://www.ensc.sfu.ca/cnl Simon Fraser University Vancouver,

More information

Feature Point Selection using Structural Graph Matching for MLS based Image Registration

Feature Point Selection using Structural Graph Matching for MLS based Image Registration Feature Point Selection using Structural Graph Matching for MLS based Image Registration Hema P Menon Department of CSE Amrita Vishwa Vidyapeetham Coimbatore Tamil Nadu - 641 112, India K A Narayanankutty

More information

Key words. cluster analysis, k-means, eigen decomposition, Laplacian matrix, data visualization, Fisher s Iris data set

Key words. cluster analysis, k-means, eigen decomposition, Laplacian matrix, data visualization, Fisher s Iris data set SPECTRAL CLUSTERING AND VISUALIZATION: A NOVEL CLUSTERING OF FISHER S IRIS DATA SET DAVID BENSON-PUTNINS, MARGARET BONFARDIN, MEAGAN E. MAGNONI, AND DANIEL MARTIN Advisors: Carl D. Meyer and Charles D.

More information

Analysis of Internet Topologies

Analysis of Internet Topologies Analysis of Internet Topologies Ljiljana Trajković ljilja@cs.sfu.ca Communication Networks Laboratory http://www.ensc.sfu.ca/cnl School of Engineering Science Simon Fraser University, Vancouver, British

More information

Seminar assignment 1. 1 Part

Seminar assignment 1. 1 Part Seminar assignment 1 Each seminar assignment consists of two parts, one part consisting of 3 5 elementary problems for a maximum of 10 points from each assignment. For the second part consisting of problems

More information

Differential Privacy Preserving Spectral Graph Analysis

Differential Privacy Preserving Spectral Graph Analysis Differential Privacy Preserving Spectral Graph Analysis Yue Wang, Xintao Wu, and Leting Wu University of North Carolina at Charlotte, {ywang91, xwu, lwu8}@uncc.edu Abstract. In this paper, we focus on

More information

Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs

Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Yong Zhang 1.2, Francis Y.L. Chin 2, and Hing-Fung Ting 2 1 College of Mathematics and Computer Science, Hebei University,

More information