! E6893 Big Data Analytics Lecture 10:! Linked Big Data Graph Computing (II)


 Beverly Peregrine Owens
 2 years ago
 Views:
Transcription
1 E6893 Big Data Analytics Lecture 10: Linked Big Data Graph Computing (II) ChingYung Lin, Ph.D. Adjunct Professor, Dept. of Electrical Engineering and Computer Science Mgr., Dept. of Network Science and Big Data Analytics, IBM Watson Research Center November 6th, 2014
2 Course Structure Class Data Number Topics Covered 09/04/14 1 Introduction to Big Data Analytics 09/11/14 2 Big Data Analytics Platforms 09/18/14 3 Big Data Storage and Processing 09/25/14 4 Big Data Analytics Algorithms  I 10/02/14 5 Big Data Analytics Algorithms  II (recommendation) 10/09/14 6 Big Data Analytics Algorithms III (clustering) 10/16/14 7 Big Data Analytics Algorithms IV (classification) 10/23/14 8 Big Data Analytics Algorithms V (classification & clustering) 10/30/14 9 Linked Big Data Graph Computing I (Graph DB) 11/06/14 10 Linked Big Data Graph Computing II (Graph Analytics) 11/13/14 11 Big Data on Hardware, Processors, and Cluster Platforms 11/20/14 12 Final Project First Presentations 11/27/14 Thanksgiving Holiday 12/04/14 13 Next Stage of Big Data Analytics 12/11/14 14 Big Data Analytics Workshop Final Project Presentations 2
3 Final Project Proposal (First) Presentation Date/Time: November 20, 7pm  9:30pm Each Team about 3 mins: 1. Team members and expected contributions of each member; 2. Motivation of your project (The problem you would like to solve); 3. Dataset, algorithm, and tools for your project; 4. Current Status of your project. Please update your team info in the Project webpage by November 11. The presentation schedule will be announced on November 13. The website will be opened to allow you upload your slides by November 20. If a project is purely by CVN students, please submit your slides without oral presentation. 3
4 ScaleGraph an Open Source version of IBM System G 4
5 ScaleGraph algorithms made Top #1 in Graph 500 benchmark 5 Source: Dr. Toyotaro Suzumura, ICPE2014 keynote E6893 Big Data Analytics Lecture 9: Linked Big Data: Graph Computing
6 Graph Definitions and Concepts A graph: G = ( V, E) V = Vertices or Nodes E = Edges or Links The number of vertices: Order The number of edges: Size Ne Nv = = E V 6
7 Subgraph A graph H is a subgraph of another graph G, if: V H V G and E H E G 7
8 Families of Graphs Complete Graph: every vertex is linked to every other vertex. Clique: a complete subgraph. 8
9 MultiGraph vs. Simple Graph Loops: MultiEdges: 9
10 Directed Graph vs. Undirected Graph Mutual arcs: Directed Edges = Arcs: { u, v} u v 10
11 Adjacency Two edges are adjacent if joined by a common endpoint in V: u and v are adjacent if joined by an edge in E: u v e 1 e 2 11
12 Decorated Graph Weighted Edges
13 Incident and Degree The degree of a vertex v, say d v, is defined as the number of edges incident on v. A vertex v V is incident on an edge if v is an endpoint of e. e E v e v d v =2 13
14 Indegrees and outdegrees For Directed graphs: Indegree = 8 Outdegree = 8 14
15 Degree Distribution Example: PowerLaw Network A. Barbasi and E. Bonabeau, Scalefree Networks, Scientific American 288: p.5059, p k k m m = e k p = C k e k / κ k τ Newman, Strogatz and Watts, 2001
16 Another example of complex network: SmallWorld Network Six Degree Separation: adding long range link, a regular graph can be transformed into a smallworld network, in which the average number of degrees between two nodes become small. from Watts and Strogatz, C: Clustering Coefficient, L: path length, (C(0), L(0) ): (C, L) as in a regular graph; (C(p), L(p)): (C,L) in a Smallworld graph with randomness p.
17 Indication of Small A graph is small which usually indicates the average distance between distinct vertices is small l = 1 ( 1) / 2 u N N + v V v v dist( u, v) For instance, a protein interaction network would be considered to have the smallworld property, as there is an average distance of 3.68 among the 5,128 vertices in its giant component. 17
18 Some examples of Degree Distribution (a) scientist collaboration: biologists (circle) physicists (square), (b) collaboration of move actors, (d) network of directors of Fortune 1000 companies 18
19 Degree Distribution Kolaczyk, Statistical Analysis of Network Data: Methods and Models, Springer
20 ScaleGraph Analytics Algorithms 20
21 Centrality There is certainly no unanimity on exactly what centrality is or its conceptual foundations, and there is little agreement on the procedure of its measurement. Freeman Degree (centrality) Closeness (centrality) Betweeness (centrality) Eigenvector (centrality) 21
22 Conceptual Descriptions of Three Centrality Measurements Kolaczyk, Statistical Analysis of Network Data: Methods and Models, Springer
23 Distance Distance of two vertices: The length of the shortest path between the vertices. Geodesic: another name for shortest path. Diameter: the value of the longest distance in a graph 23
24 Closeness Closeness: A vertex is close to the other vertices c CI ( v) = u V 1 dist( v, u) where dist(v,u) is the geodesic distance between vertices v and u. 24
25 Betweenness Betweenness measures are aimed at summarizing the extent to which a vertex is located between other pairs of vertices. Freeman s definition: c B ( v) = s t v V σ σ ( s, t v) ( s, t) Calculation of all betweenness centralities requires calculating the lengths of shortest paths among all pairs of vertices Computing the summation in the above definition for each vertex 25
26 Betweeness ==> Bridges Example: Healthcare experts in the world Connections between different divisions Example: Healthcare experts in the U.S. Key social bridges 46 E6893 Big Data Analytics Lecture 1: Overview
27 Network Value Analysis First LargeScale Economical Social Network Study Productivity effect from network variables An additional person in network size ~ $986 revenue per year Each person that can be reached in 3 steps ~ $0.163 in revenue per month A link to manager ~ $1074 in revenue per month 1 standard deviation of network diversity (1  constraint) ~ $758 1 standard deviation of btw ~ $300K 1 strong link ~ $7.9 per month Structural Diverse networks with abundance of structural holes are associated with higher performance. Having diverse friends helps. Betweenness is negatively correlated to people but highly positive correlated to projects. Being a bridge between a lot of people is bottleneck. Being a bridge of a lot of projects is good. Network reach are highly corrected. The number of people reachable in 3 steps is positively correlated with higher performance. Having too many strong links the same set of people one communicates frequently is negatively correlated with performance. Perhaps frequent communication to the same person may imply redundant information exchange. 49 E6893 Big Data Analytics Lecture 1: Overview
28 Eigenvector Centrality Try to capture the status, prestige, or rank. More central the neighbors of a vertex are, the more central the vertex itself is. cei ( v) = α cei ( u) { u, v} E The vector c = ( c (1),..., c ( N )) T Ei Ei Ei v is the solution of the eigenvalue problem: A c = α Ei 1 c Ei 28
29 PageRank Algorithm (Simplified) 29
30 PageRank Steps Example: Simplified Initial State: R(A) = R(B) = R(C) = R(D) = 0.25 Iterative Procedure: R(A) = R(B) / 2 + R(C) / 1 + R(D) / 3 where R( u) R( u) = d + e N v B v v A B F u The set of pages u points to B u The set of pages point to u C D N u = d F u Number of links from u Normalization / damping factor e 1 d N = In general, d=
31 Solution of PageRank The PageRank values are the entries of the dominant eigenvector of the modified adjacency matrix. R R( p1 ) " # R( p2) $ = # $ # : $ # $ % R( pn )& where R is the solution of the equation 31 where R is the adjacency function l( pi, p j ) = 0 if page pj does not link to pi, and normalized such that for each j, N i= 1 l( p, p ) = 1 i j
32 Walk A walk on a graph G, from v 0 to v l, is an alternating sequence: The length of this walk is l. { v0, e1, v1, e2,..., vl 1, el, vl} A walk may be: Trail  no repeated edges Path  trails without repeated vertices. 32
33 Connectivity of Graph A measure related to the flow of information in the graph Connected every vertex is reachable from every other A connected component of a graph is a maximally connected subgraph. A graph usually has one dominating the others in magnitude giant component. 33
34 Reachable, Connected, Component Reachable: A vertex v in a graph G is said to be reachable from another vertex u if there exists a walk from u to v. Connected: A graph is said to be connected if every vertex is reachable from every other. Component: A component of a graph is a maximally connected subgraph. 34
35 Local Density A coherent subset of nodes should be locally dense. Cliques: 3cliques A sufficient condition for a clique of size n to exist in G is: N e 2 N " v n 2 " > $ %$ % & 2 '& n 1 ' 35
36 Weakened Versions of Cliques  Plexes A subgraph H consisting of m vertices is called nplex, for m > n, if no vertex has degree less than m n. 1plex 1plex No vertex is missing more than 1 of its possible m1 edges. 36
37 Another Weakened Versions of Cliques  Cores A kcore of a graph G is a subgraph H in which all vertices have degree at least k. 3core Batagelj et. al., A maximal kcore subgraph may be computed in as little as O( Nv + Ne) time. Computes the shell indices for every vertex in the graph Shell index of v = the largest value, say c, such that v belongs to the ccore of G but not its (c+1)core. For a given vertex, those neighbors with lesser degree lead to a decrease in the potential shell index of that vertex. 37
38 Density measurement The density of a subgraph H = ( VH, EH ) is: den( H ) = V H E H ( V 1) / 2 H Range of density and 0 den( H ) 1 den( H ) = ( V 1) d ( H ) H average degree of H 38
39 Use of the density measure Density of a graph: let H=G Clustering of edges local to v: let H=Hv, which is the set of neighbors of a vertex v, and the edges between them Clustering Coefficient of a graph: The average of den(hv) over all vertices 39
40 An insight of clustering coefficient A triangle is a complete subgraph of order three. A connected triple is a subgraph of three vertices connected by two edges (regardless how the other two nodes connect). The local clustering coefficient can be expressed as: den H The clustering coefficient of G is then: τ ( v ) = cl( v) = τ Δ 3 ( v) ( v) # of triangles # of connected triples for which 2 edges are both incident to v. 1 cl( G) = cl( v) V v V Where V V is the set of vertices v with dv 2. 40
41 An example 41
42 Transitivity of a graph A variation of the clustering coefficient takes weighted average where τ Δ τ cl T ( G) 1 ( G) = τ Δ ( v) 3 v V ( G) = τ ( v) 3 3 v V τ 3( v) cl( v) v V " τ Δ = = τ 3 v τ 3 v V " 3 ( G) ( ) ( G) The friend of your friend is also a friend of yours is the number of triangles in the graph is the number of connected triples Clustering coefficients have become a standard quantity for network structure analysis. But, it is important on reporting which clustering coefficients are used. 42
43 Vertex / Edge Connectivity If an arbitrary subset of k vertices or edges is removed from a graph, is the remaining subgraph connected? A graph G is called kvertexconnected, if (1) Nv>k, and (2) the removal of any subset of vertices X in V of cardinality X smaller than k leaves a subgraph G X that is connected. The vertex connectivity of G is the largest integer such that G is k vertexconnected. Similar measurement for edge connectivity 43
44 Vertex / Edge Cut If the removal of a particular set of vertices in G disconnects the graph, that set is called a vertex cut. For a given pair of vertices (u,v), a uvcut is a partition of V into two disjoint nonempty subsets, S and S, where u is in S and v is in S. Minimum uvcut: the sum of the weights on edges connecting vertices in S to vertices in S is a minimum. 44
45 Minimum cut and flow Find a minimum uvcut is an equivalent problem of maximizing a measure of flow on the edges of a derived directed graph. Ford and Fulkerson, MaxFlow MinCut theorem. 45
46 Graph Partitioning Many uses of graph partitioning: E.g., community structure in social networks A cohesive subset of vertices generally is taken to refer to a subset of vertices that (1) are well connected among themselves, and (2) are relatively well separated from the remaining vertices Graph partitioning algorithms typically seek a partition of the vertex set of a graph in such a manner that the sets E( Ck, Ck ) of edges connecting vertices in Ck to vertices in Ck are relatively small in size compared to the sets E(Ck) = E( Ck, Ck ) of edges connecting vertices within Ck. 46
47 Classify the nodes 47
48 Example: Karate Club Network 48
49 Hierarchical Clustering Agglomerative Divisive In agglomerative algorithms, given two sets of vertices C1 and C2, two standard approaches to assigning a similarity value to this pair of sets is to use the maximum (called singlelinkage) or the minimum (called complete linkage) of the similarity xij over all pairs. x ij = v i d( N ) + d( N 1) v N ΔN v j v The normalized number of neighbors of vi and vj that are not shared. 49
50 Hierarchical Clustering Algorithms Types Primarily differ in [Jain et. al. 1999]: (1) how they evaluate the quality of proposed clusters, and (2) the algorithms by which they seek to optimze that quality. Agglomerative: successive coarsening of parittions through the process of merging. Divisive: successive refinement of partitions through the process of splitting. At each stage, the current candidate partition is modified in a way that minizes a specific measure of cost. In agglomerative methods, the least costly merge of two previously existing partition elements is executed In divisive methods, it is the least costly split of a single existing partition element into two that is executed. 50
51 Hierarchical Clustering The resulting hierarchy typically is represented in the form of a tree, called a dendrogram. The measure of cost incorporated into a hierarchical clustering method used in graph partitioning should reflect our sense of what defines a cohesive subset of vertices. In agglomerative algorithms, given two sets of vertices C1 and C2, two standard approaches to assigning a similarity value to this pair of sets is to use the maximum (called singlelinkage) or the minimum (called complete linkage) of the dissimilarity xij over all pairs. Dissimlarities for subsets of vertices were calculated from the xij using the extension of Ward (1963) s method and the lengths of the branches in the dendrogram are in relative proportion to the changes in dissimilarity. x ij = v i d( N ) + d( N 1) v N ΔN v j v Nv is the set of neighbors of a vertex. Δ is the symmetric difference of two sets which is the set of elements that are in one or the other but not both. xij is the normalized number of neighbors of vi and vj that are not shared. 51
52 Other dissimilarity measures There are various other common choices of dissimilarity measures, such as: x = ( A A ) ij ik jk k i, j 2 Hierarchical clustering algorithms based on dissimilarities of this sort are reasonably efficient, running in time. 2 O( N log N ) v v 52
53 Hierarchical Clustering Example 53
54 Several Graph Open Source on Tools Titan is a native Blueprints enabled graph database 54
55 Graph Language 55
56 Performance Comparison of Titan and others Dataset: 12.2 million edges, 2.2 million vertices Goal: Find paths in a property graph. One of the vertex property is call TYPE. In this scenario, the user provides either a particular vertex, or a set of particular vertices of the same TYPE (say, "DRUG"). In addition, the user also provides another TYPE (say, "TARGET"). Then, we need find all the paths from the starting vertex to a vertex of TYPE TARGET. Therefore, we need to 1) find the paths using graph traversal; 2) keep trace of the paths, so that we can list them after the traversal. Even for the shortest paths, it can be multiple between two nodes, such as: drug>assay>target, drug>moa>target First test (coldstart) Avg time (100 tests) Requested depth 5 traversal Requested full depth traversal IBM System G (NativeStore C++) 39 sec 3.0 sec 4.2 sec IBM System G (NativeStore JNI) 57 sec 4.0 sec 6.2 sec Java Neo4j (Blueprints 2.4) 105 sec 5.9 sec 8.3 sec Titan (Berkeley DB) 3861 sec 641 sec 794 sec Titan (HBase) 3046 sec 1597 sec 2682 sec 56 First full test  full depth 23. All data pulled from disk. Nothing initially cached. Modes  All tests in default modes of each graph implementation. Titan can only be run in transactional mode. Other implementations do not default to transactional mode.
57 ScaleGraph DB System G DB s open source version Prereqs Linux Intel 64 OpenJDK 6 or higher Maven  maveninfiveminutes.html 57
58 ScaleGraph DB (a.k.a. PropelGraph) Installation 1a) git clone https://github.com/scalegraph/propelgraph.git or 1b) wget https://github.com/scalegraph/propelgraph/archive/master.zip ; unzip master.zip 2) cd propelgraph/propelgraphgremlin 3)./makepackage.sh 58
59 ScaleGraph DB Trying It Out 3) cd propelgraphgremlin ) bin/gremlin.sh 5) optional: read a gremlin tutorial 6) g = CreateGraph.openGraph("nativemem_authors","awesome") 7) new LoadCSV().populateFromVertexFile(g, "data/movies.movies.v.csv", "movies", ) 8) new LoadCSV().populateFromVertexFile(g, "data/movies.appearances.e.csv", "appearances", ) 9) g.v(20).both.bothv 10) Analytics.collaborativeFilter(g, 20, "appearance", Direction.OUT, "appearance", Direction.IN) 59
60 ScaleGraph DB Help https://github.com/scalegraph/scalegraph/propelgraph 60
61 Questions? 61
! E6893 Big Data Analytics Lecture 9:! Linked Big Data Graph Computing (I)
! E6893 Big Data Analytics Lecture 9:! Linked Big Data Graph Computing (I) ChingYung Lin, Ph.D. Adjunct Professor, Dept. of Electrical Engineering and Computer Science Mgr., Dept. of Network Science and
More informationChingYung Lin, Ph.D. Adjunct Professor, Dept. of Electrical Engineering and Computer Science IBM Chief Scientist, Graph Computing. October 29th, 2015
E6893 Big Data Analytics Lecture 8: Spark Streams and Graph Computing (I) ChingYung Lin, Ph.D. Adjunct Professor, Dept. of Electrical Engineering and Computer Science IBM Chief Scientist, Graph Computing
More informationSocial 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 informationComplex 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 graphclustering methods and their applications
More informationSocial 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 likeminded users
More informationCourse on Social Network Analysis Graphs and Networks
Course on Social Network Analysis Graphs and Networks Vladimir Batagelj University of Ljubljana Slovenia V. Batagelj: Social Network Analysis / Graphs and Networks 1 Outline 1 Graph...............................
More informationAsking Hard Graph Questions. Paul Burkhardt. February 3, 2014
Beyond Watson: Predictive Analytics and Big Data U.S. National Security Agency Research Directorate  R6 Technical Report February 3, 2014 300 years before Watson there was Euler! The first (Jeopardy!)
More informationUSING SPECTRAL RADIUS RATIO FOR NODE DEGREE TO ANALYZE THE EVOLUTION OF SCALE FREE NETWORKS AND SMALLWORLD NETWORKS
USING SPECTRAL RADIUS RATIO FOR NODE DEGREE TO ANALYZE THE EVOLUTION OF SCALE FREE NETWORKS AND SMALLWORLD NETWORKS Natarajan Meghanathan Jackson State University, 1400 Lynch St, Jackson, MS, USA natarajan.meghanathan@jsums.edu
More informationPart 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! E6893 Big Data Analytics Lecture 5:! Big Data Analytics Algorithms  II
! E6893 Big Data Analytics Lecture 5:! Big Data Analytics Algorithms  II ChingYung Lin, Ph.D. Adjunct Professor, Dept. of Electrical Engineering and Computer Science Mgr., Dept. of Network Science and
More informationBig Data Analytics of MultiRelationship Online Social Network Based on MultiSubnet Composited Complex Network
, pp.273284 http://dx.doi.org/10.14257/ijdta.2015.8.5.24 Big Data Analytics of MultiRelationship Online Social Network Based on MultiSubnet Composited Complex Network Gengxin Sun 1, Sheng Bin 2 and
More informationV. Adamchik 1. Graph Theory. Victor Adamchik. Fall of 2005
V. Adamchik 1 Graph Theory Victor Adamchik Fall of 2005 Plan 1. Basic Vocabulary 2. Regular graph 3. Connectivity 4. Representing Graphs Introduction A.Aho and J.Ulman acknowledge that Fundamentally, computer
More informationGeneral Network Analysis: Graphtheoretic. COMP572 Fall 2009
General Network Analysis: Graphtheoretic Techniques COMP572 Fall 2009 Networks (aka Graphs) A network is a set of vertices, or nodes, and edges that connect pairs of vertices Example: a network with 5
More informationDATA 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 informationGraphs 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 informationPractical 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 informationGRAPH 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 informationStrong and Weak Ties
Strong and Weak Ties Web Science (VU) (707.000) Elisabeth Lex KTI, TU Graz April 11, 2016 Elisabeth Lex (KTI, TU Graz) Networks April 11, 2016 1 / 66 Outline 1 Repetition 2 Strong and Weak Ties 3 General
More informationWhy 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 informationGraph models for the Web and the Internet. Elias Koutsoupias University of Athens and UCLA. Crete, July 2003
Graph models for the Web and the Internet Elias Koutsoupias University of Athens and UCLA Crete, July 2003 Outline of the lecture Small world phenomenon The shape of the Web graph Searching and navigation
More informationDiscrete 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 informationA 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 informationSCAN: 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 informationLesson 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 informationGraph Theory and Complex Networks: An Introduction. Chapter 06: Network analysis
Graph Theory and Complex Networks: An Introduction Maarten van Steen VU Amsterdam, Dept. Computer Science Room R4.0, steen@cs.vu.nl Chapter 06: Network analysis Version: April 8, 04 / 3 Contents Chapter
More informationNetwork/Graph Theory. What is a Network? What is network theory? Graphbased representations. Friendship Network. What makes a problem graphlike?
What is a Network? Network/Graph Theory Network = graph Informally a graph is a set of nodes joined by a set of lines or arrows. 1 1 2 3 2 3 4 5 6 4 5 6 Graphbased representations Representing a problem
More informationProtein Protein Interaction Networks
Functional Pattern Mining from Genome Scale Protein Protein Interaction Networks YoungRae Cho, Ph.D. Assistant Professor Department of Computer Science Baylor University it My Definition of Bioinformatics
More informationGraph definition Degree, in, out degree, oriented graph. Complete, regular, bipartite graph. Graph representation, connectivity, adjacency.
Mária Markošová Graph definition Degree, in, out degree, oriented graph. Complete, regular, bipartite graph. Graph representation, connectivity, adjacency. Isomorphism of graphs. Paths, cycles, trials.
More informationWalkBased Centrality and Communicability Measures for Network Analysis
WalkBased 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 informationA discussion of Statistical Mechanics of Complex Networks P. Part I
A discussion of Statistical Mechanics of Complex Networks Part I Review of Modern Physics, Vol. 74, 2002 Small Word Networks Clustering Coefficient ScaleFree Networks ErdösRényi model cover only parts
More informationHomework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS
Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS 1.1.25 Prove that the Petersen graph has no cycle of length 7. Solution: There are 10 vertices in the Petersen graph G. Assume there is a cycle C
More informationSGL: Stata graph library for network analysis
SGL: Stata graph library for network analysis Hirotaka Miura Federal Reserve Bank of San Francisco Stata Conference Chicago 2011 The views presented here are my own and do not necessarily represent the
More informationHandout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs
MCS236: Graph Theory Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010 Chapter 7: Digraphs Strong Digraphs Definitions. A digraph is an ordered pair (V, E), where V is the set
More information2.3 Scheduling jobs on identical parallel machines
2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed
More information1 Basic Definitions and Concepts in Graph Theory
CME 305: Discrete Mathematics and Algorithms 1 Basic Definitions and Concepts in Graph Theory A graph G(V, E) is a set V of vertices and a set E of edges. In an undirected graph, an edge is an unordered
More informationChapter 2. Basic Terminology and Preliminaries
Chapter 2 Basic Terminology and Preliminaries 6 Chapter 2. Basic Terminology and Preliminaries 7 2.1 Introduction This chapter is intended to provide all the fundamental terminology and notations which
More informationGraphs over Time Densification Laws, Shrinking Diameters and Possible Explanations
Graphs over Time Densification Laws, Shrinking Diameters and Possible Explanations Jurij Leskovec, CMU Jon Kleinberg, Cornell Christos Faloutsos, CMU 1 Introduction What can we do with graphs? What patterns
More informationDistributed Computing over Communication Networks: Maximal Independent Set
Distributed Computing over Communication Networks: Maximal Independent Set What is a MIS? MIS An independent set (IS) of an undirected graph is a subset U of nodes such that no two nodes in U are adjacent.
More information5.1 Bipartite Matching
CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the FordFulkerson
More informationNetwork Analysis and Visualization of Staphylococcus aureus. by Russ Gibson
Network Analysis and Visualization of Staphylococcus aureus by Russ Gibson Network analysis Based on graph theory Probabilistic models (random graphs) developed by Erdős and Rényi in 1959 Theory and tools
More informationIE 680 Special Topics in Production Systems: Networks, Routing and Logistics*
IE 680 Special Topics in Production Systems: Networks, Routing and Logistics* Rakesh Nagi Department of Industrial Engineering University at Buffalo (SUNY) *Lecture notes from Network Flows by Ahuja, Magnanti
More informationApproximation Algorithms
Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NPCompleteness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms
More informationGephi Network Statistics
Gephi Network Statistics Google Summer of Code 2009 Project Proposal Patrick J. McSweeney pjmcswee@syr.edu 1 Introduction My name is Patrick J. McSweeney and I am a fourth year PhD candidate in computer
More information6. If there is no improvement of the categories after several steps, then choose new seeds using another criterion (e.g. the objects near the edge of
Clustering Clustering is an unsupervised learning method: there is no target value (class label) to be predicted, the goal is finding common patterns or grouping similar examples. Differences between models/algorithms
More informationData Mining Cluster Analysis: Advanced Concepts and Algorithms. ref. Chapter 9. Introduction to Data Mining
Data Mining Cluster Analysis: Advanced Concepts and Algorithms ref. Chapter 9 Introduction to Data Mining by Tan, Steinbach, Kumar 1 Outline Prototypebased Fuzzy cmeans Mixture Model Clustering Densitybased
More informationNETZCOPE  a tool to analyze and display complex R&D collaboration networks
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.
More informationIntroduction to Networks and Business Intelligence
Introduction to Networks and Business Intelligence Prof. Dr. Daning Hu Department of Informatics University of Zurich Sep 17th, 2015 Outline Network Science A Random History Network Analysis Network Topological
More informationTools and Techniques for Social Network Analysis
Tools and Techniques for Social Network Analysis Pajek Program for Analysis and Visualization of Large Networks Pajek: What is it Pajek is a program, for Windows and Linux (via Wine) Developers: Vladimir
More informationDATA MINING CLUSTER ANALYSIS: BASIC CONCEPTS
DATA MINING CLUSTER ANALYSIS: BASIC CONCEPTS 1 AND ALGORITHMS Chiara Renso KDDLAB ISTI CNR, Pisa, Italy WHAT IS CLUSTER ANALYSIS? Finding groups of objects such that the objects in a group will be similar
More informationMining SocialNetwork Graphs
342 Chapter 10 Mining SocialNetwork Graphs There is much information to be gained by analyzing the largescale data that is derived from social networks. The bestknown example of a social network is
More informationSome questions... Graphs
Uni Innsbruck Informatik  1 Uni Innsbruck Informatik  2 Some questions... Peerto topeer Systems Analysis of unstructured P2P systems How scalable is Gnutella? How robust is Gnutella? Why does FreeNet
More informationMath 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 informationData Mining Cluster Analysis: Basic Concepts and Algorithms. Lecture Notes for Chapter 8. Introduction to Data Mining
Data Mining Cluster Analysis: Basic Concepts and Algorithms Lecture Notes for Chapter 8 by Tan, Steinbach, Kumar 1 What is Cluster Analysis? Finding groups of objects such that the objects in a group will
More informationZachary 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 informationOutline. NPcompleteness. When is a problem easy? When is a problem hard? Today. Euler Circuits
Outline NPcompleteness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2pairs sum vs. general Subset Sum Reducing one problem to another Clique
More informationData Mining Cluster Analysis: Basic Concepts and Algorithms. Lecture Notes for Chapter 8. Introduction to Data Mining
Data Mining Cluster Analysis: Basic Concepts and Algorithms Lecture Notes for Chapter 8 Introduction to Data Mining by Tan, Steinbach, Kumar Tan,Steinbach, Kumar Introduction to Data Mining 4/8/2004 Hierarchical
More informationGraph Algorithms. Ananth Grama, Anshul Gupta, George Karypis, and Vipin Kumar
Graph Algorithms Ananth Grama, Anshul Gupta, George Karypis, and Vipin Kumar To accompany the text Introduction to Parallel Computing, Addison Wesley, 3. Topic Overview Definitions and Representation Minimum
More informationData Structures in Java. Session 16 Instructor: Bert Huang
Data Structures in Java Session 16 Instructor: Bert Huang http://www.cs.columbia.edu/~bert/courses/3134 Announcements Homework 4 due next class Remaining grades: hw4, hw5, hw6 25% Final exam 30% Midterm
More informationGraph Theory and Complex Networks: An Introduction. Chapter 06: Network analysis. Contents. Introduction. Maarten van Steen. Version: April 28, 2014
Graph Theory and Complex Networks: An Introduction Maarten van Steen VU Amsterdam, Dept. Computer Science Room R.0, steen@cs.vu.nl Chapter 0: Version: April 8, 0 / Contents Chapter Description 0: Introduction
More informationElectrical 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 informationSPANNING 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 informationGRAPH MINING APPLICATIONS TO SOCIAL NETWORK ANALYSIS
Chapter 16 GRAPH MINING APPLICATIONS TO SOCIAL NETWORK ANALYSIS Lei Tang and Huan Liu Computer Science & Engineering Arizona State University L.Tang@asu.edu, Huan.Liu@asu.edu Abstract The prosperity of
More informationHadoop SNS. renren.com. Saturday, December 3, 11
Hadoop SNS renren.com Saturday, December 3, 11 2.2 190 40 Saturday, December 3, 11 Saturday, December 3, 11 Saturday, December 3, 11 Saturday, December 3, 11 Saturday, December 3, 11 Saturday, December
More informationFollow links for Class Use and other Permissions. For more information send email to: permissions@press.princeton.edu
COPYRIGHT NOTICE: Matthew O. Jackson: Social and Economic Networks is published by Princeton University Press and copyrighted, 2008, by Princeton University Press. All rights reserved. No part of this
More informationLecture Notes on Spanning Trees
Lecture Notes on Spanning Trees 15122: 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 informationCELLULAR MANUFACTURING
CELLULAR MANUFACTURING Grouping Machines logically so that material handling (move time, wait time for moves and using smaller batch sizes) and setup (part family tooling and sequencing) can be minimized.
More informationAn Introduction to APGL
An Introduction to APGL Charanpal Dhanjal February 2012 Abstract Another Python Graph Library (APGL) is a graph library written using pure Python, NumPy and SciPy. Users new to the library can gain an
More informationParallel Algorithms for Smallworld Network. David A. Bader and Kamesh Madduri
Parallel Algorithms for Smallworld Network Analysis ayssand Partitioning atto g(s (SNAP) David A. Bader and Kamesh Madduri Overview Informatics networks, smallworld topology Community Identification/Graph
More informationThe origins of graph theory are humble, even frivolous. Biggs, E. K. Lloyd, and R. J. Wilson)
Chapter 11 Graph Theory The origins of graph theory are humble, even frivolous. Biggs, E. K. Lloyd, and R. J. Wilson) (N. Let us start with a formal definition of what is a graph. Definition 72. A graph
More informationLecture 9. 1 Introduction. 2 Random Walks in Graphs. 1.1 How To Explore a Graph? CS621 Theory Gems October 17, 2012
CS62 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 informationPredicting Influentials in Online Social Networks
Predicting Influentials in Online Social Networks Rumi Ghosh Kristina Lerman USC Information Sciences Institute WHO is IMPORTANT? Characteristics Topology Dynamic Processes /Nature of flow What are the
More informationSTATISTICS AND DATA ANALYSIS IN GEOLOGY, 3rd ed. Clarificationof zonationprocedure described onpp. 238239
STATISTICS AND DATA ANALYSIS IN GEOLOGY, 3rd ed. by John C. Davis Clarificationof zonationprocedure described onpp. 3839 Because the notation used in this section (Eqs. 4.8 through 4.84) is inconsistent
More informationSection Summary. Introduction to Graphs Graph Taxonomy Graph Models
Chapter 10 Chapter Summary Graphs and Graph Models Graph Terminology and Special Types of Graphs Representing Graphs and Graph Isomorphism Connectivity Euler and Hamiltonian Graphs ShortestPath Problems
More informationChapter 29 ScaleFree Network Topologies with Clustering Similar to Online Social Networks
Chapter 29 ScaleFree Network Topologies with Clustering Similar to Online Social Networks Imre Varga Abstract In this paper I propose a novel method to model real online social networks where the growing
More informationMining Social Network Graphs
Mining Social Network Graphs Debapriyo Majumdar Data Mining Fall 2014 Indian Statistical Institute Kolkata November 13, 17, 2014 Social Network No introduc+on required Really? We s7ll need to understand
More informationData Mining Clustering (2) Sheets are based on the those provided by Tan, Steinbach, and Kumar. Introduction to Data Mining
Data Mining Clustering (2) Toon Calders Sheets are based on the those provided by Tan, Steinbach, and Kumar. Introduction to Data Mining Outline Partitional Clustering Distancebased Kmeans, Kmedoids,
More informationData Mining. Cluster Analysis: Advanced Concepts and Algorithms
Data Mining Cluster Analysis: Advanced Concepts and Algorithms Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004 1 More Clustering Methods Prototypebased clustering Densitybased clustering Graphbased
More informationSolutions 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 informationGraph 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 informationApplying Social Network Analysis to the Information in CVS Repositories
Applying Social Network Analysis to the Information in CVS Repositories Luis LopezFernandez, Gregorio Robles, Jesus M. GonzalezBarahona GSyC, Universidad Rey Juan Carlos {llopez,grex,jgb}@gsyc.escet.urjc.es
More informationNetwork (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 informationE6895 Advanced Big Data Analytics Lecture 3:! Spark and Data Analytics
E6895 Advanced Big Data Analytics Lecture 3:! Spark and Data Analytics ChingYung Lin, Ph.D. Adjunct Professor, Dept. of Electrical Engineering and Computer Science Mgr., Dept. of Network Science and Big
More informationRandom graphs and complex networks
Random graphs and complex networks Remco van der Hofstad Honours Class, spring 2008 Complex networks Figure 2 Ye a s t p ro te in in te ra c tio n n e tw o rk. A m a p o f p ro tein p ro tein in tera c
More informationFig. 1 A typical Knowledge Discovery process [2]
Volume 4, Issue 7, July 2014 ISSN: 2277 128X International Journal of Advanced Research in Computer Science and Software Engineering Research Paper Available online at: www.ijarcsse.com A Review on Clustering
More informationSoSe 2014: MTANI: Big Data Analytics
SoSe 2014: MTANI: Big Data Analytics Lecture 4 21/05/2014 Sead Izberovic Dr. Nikolaos Korfiatis Agenda Recap from the previous session Clustering Introduction Distance mesures Hierarchical Clustering
More informationChapter 11. 11.1 Load Balancing. Approximation Algorithms. Load Balancing. Load Balancing on 2 Machines. Load Balancing: Greedy Scheduling
Approximation Algorithms Chapter Approximation Algorithms Q. Suppose I need to solve an NPhard problem. What should I do? A. Theory says you're unlikely to find a polytime algorithm. Must sacrifice one
More information(a) (b) (c) Figure 1 : Graphs, multigraphs and digraphs. If the vertices of the leftmost figure are labelled {1, 2, 3, 4} in clockwise order from
4 Graph Theory Throughout these notes, a graph G is a pair (V, E) where V is a set and E is a set of unordered pairs of elements of V. The elements of V are called vertices and the elements of E are called
More informationData Mining Cluster Analysis: Basic Concepts and Algorithms. Lecture Notes for Chapter 8. Introduction to Data Mining
Data Mining Cluster Analsis: Basic Concepts and Algorithms Lecture Notes for Chapter 8 Introduction to Data Mining b Tan, Steinbach, Kumar Tan,Steinbach, Kumar Introduction to Data Mining /8/ What is Cluster
More information1. Relevant standard graph theory
Color identical pairs in 4chromatic graphs Asbjørn Brændeland I argue that, given a 4chromatic graph G and a pair of vertices {u, v} in G, if the color of u equals the color of v in every 4coloring
More informationStructural and functional analytics for community detection in largescale complex networks
Chopade and Zhan Journal of Big Data DOI 10.1186/s405370150019y RESEARCH Open Access Structural and functional analytics for community detection in largescale complex networks Pravin Chopade 1* and
More informationSolutions to Homework 6
Solutions to Homework 6 Debasish Das EECS Department, Northwestern University ddas@northwestern.edu 1 Problem 5.24 We want to find light spanning trees with certain special properties. Given is one example
More informationApplied Algorithm Design Lecture 5
Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design
More informationSmall 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 informationFinding and counting given length cycles
Finding and counting given length cycles Noga Alon Raphael Yuster Uri Zwick Abstract We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected
More information6.042/18.062J 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 informationWhy? A central concept in Computer Science. Algorithms are ubiquitous.
Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online
More informationGraph Mining and Social Network Analysis
Graph Mining and Social Network Analysis Data Mining and Text Mining (UIC 583 @ Politecnico di Milano) References Jiawei Han and Micheline Kamber, "Data Mining: Concepts and Techniques", The Morgan Kaufmann
More informationNimble 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 informationAnalysis of Algorithms, I
Analysis of Algorithms, I CSOR W4231.002 Eleni Drinea Computer Science Department Columbia University Thursday, February 26, 2015 Outline 1 Recap 2 Representing graphs 3 Breadthfirst search (BFS) 4 Applications
More information136 CHAPTER 4. INDUCTION, GRAPHS AND TREES
136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics
More information