Complex Networks Analysis: Clustering Methods


 Lisa Mathews
 2 years ago
 Views:
Transcription
1 Complex Networks Analysis: Clustering Methods Nikolai Nefedov Spring 2013 ISI ETH Zurich 1
2 Outline Purpose to give an overview of modern graphclustering methods and their applications for analysis of complex dynamic networks. Planned topics short introduction to complex networks discrete vector calculus, graph Laplacian, graph spectral analysis methods of community detection based on modularity maximization random walk on graphs, Laplacian dynamics, stability of community detection multilayer graphs: clustering and regularization topology detection via system dynamics dynamic network analysis and missing links prediction applications for realworld datasets (multidimensional time series and network analysis) 2
3 Complex Systems Complex vs Complicated Complex systems (no unique definition): a (large) number of interacting elements stochastic interactions no centralized authority, selforganized Emerging properties system behavior arises from interaction structure: detailed understanding of elements in isolation is not enough even if elements follow simple rules (chaotic behavior) evolving structures, system adaptation hierarchies, heavytails,... Complex Systems => Statistical physics large scale regularities microscopic origins of marcoscopic behavior multiple (hierarchical) scales 3
4 Complex Systems Complex Systems => Complex Networks Stat. Physics approach a fixed level of abstraction vertices => interacting elements edges => interactions (statistical) analysis of network structure dynamical processes taking place on a network dynamics of a network Graph theory approach (mostly static graphs) simple graphs => cuts, structure, factorization, spanning trees,... multigraphs => multiple edges and selfloops hypergraphs => hyperedge as a set of vertices multilayer graphs => a set of graphs on the same vertices => tensors multiplexing graphs 4
5 Graph Theory Origin: Leonhard Euler (1736) Königsberg L. Euler, Solutio problematis ad geometriam situs pertinentis, Comment. Academiae Sci. J. Petropolitanae 8, (1736) (Euler theorem: when we can draw a graph with a single line) 5
6 6
7 Complex Networks Stat. Physics approach network analysis statistical analysis (random networks, smallworld, scalefree networks) network structure analysis clustering network partition classification (taxonomy => hierarchical classification) clustering => unsupervised classification (problem dependent) relates data to knowledge (basic human activity) dynamical processes taking place on a network random walk, opinion (voting) dynamics, synchronization gamestrategies... convergence, stability... distributed computations/control dynamics of a network evolving networks interplay between network topology and dynamics on a network adaptive /learning networks 7
8 Outline Purpose to give an overview of modern graphclustering methods and their applications for analysis of complex dynamic networks. Planned topics short introduction to complex networks discrete vector calculus, graph Laplacian, graph spectral analysis methods of community detection based on modularity maximization random walk on graphs, Laplacian dynamics, stability of community detection multilayer graphs: clustering and regularization topology detection via system dynamics dynamic network analysis and missing links prediction applications for realworld datasets (multidimensional time series and network analysis) 8
9 Outline Purpose to give an overview of modern graphclustering methods and their applications for analysis of complex dynamic networks. Planned topics short introduction to complex networks complex networks, definitions, basics Graph partition mincut, normalizedcut, minratiocut Brief overview of vector calculus: differential operators (gradient, divergence, Laplace operator) Graph Laplacian as a discrete version of LaplaceBeltrami operator Spectral analysis based on graph Laplacian Limits of spectral analysis 9
10 Basics: Network Structure Network or graph G = (V,E) => set of vertices joined by edges, V = {vi } set of vertices i =1,, N, E = {e (i, j ) } set of links/edges => (ordered) pair elements from V, max E = N (N 1) /2 ; vi is a neighbor of vj if there is e ( i, j ) in E number of neighbors k of a vertex vi is called its degree in directed networks: in and out degrees k in, k out edge density of the graph: ρ= E / N N 1 /2 ρ = 1 => fully connected, ρ << 1 => sparse graph Cycle/loop = closed path (distinct vertices/edges) Graph types: regular, tree, forest Bipartite network: 2 types of nodes, links only between nodes of different types. 10
11 Basics: Network Structure Shortest path between i and j => a path with min number of edges Distance d(i,j) => measure associated with the shortest path between i and j 2d i,j / N N 1 Average shortest distance l = Diameter of the graph d = max d i,j Connected graph: there is a path between any pair of nodes Min connected graph => no loops => tree, E = N  1 edges Forest => collection of trees Fully connected (complete) graph: d (i,j) = 1 for all i,j E = N(N 1) /2 Adjacency matrix A (i,j) = 1 if e {i,j } in E, 0 otherwise Clique: a fully connected subgraph kclique: clique with k vertices Motifs: subgraphs which often occur in a network (wrt to a null model) 11
12 Basics: Network Structure Centrality measures: node degree = number of neighbors Closeness centrality: d c i =1/ Σ j i d i,j measures how far (on the average) a vertex is from all other vertices Betweenness centrality = number of shortest paths going through vertex/edge, measures the amount of flow through a vertex/edge,computationally demanding. b i = d i l,m /d l,m l,m d(l,m) shortest paths between l and m; di(l,m) shortest paths going through node i Clustering coefficient of a node N 1 C i = e e e k i k i 1 j k ij jk ki 2 Ei C i = k i k i 1 Average clustering coefficient of a graph 12 C G = C i / N = triangles connected triples
13 Network: Statistical characterization Degree distribution p(k) => probability that a randomly chosen vertex has degree k P(k k ): => cond. prob. that a vertex of degree k is connected to a vertex of degree k Average degree <k > = 2 E /N Sparse graphs: <k> << N Average degree of nearest neighbors of node i : Average degree fluctuations: <k2> Clustering spectrum (of vertices which have the same degree) Topological heterogeneity: homogeneous networks: light tails heterogeneous networks: skewed, heavy tails 13
14 Stochastic Networks Stochastic network > not s single graph, but a statistical ensemble Erdős Rényi (random) networks: G (N,p)  connect N vertices randomly, each pair is connected with probability p  ensemble of possible realizations: network properties => averages over the ensemble  average number of edges E = pn N 1 /2  average degree k = 2 E / N = p N 1 pn triangles Clustering coefficient C G = ER networks C ER = p = connected triples k N practically there is no clustering large random networks are treelike networks 14
15 Erdős Rényi Networks Example N = 3, p = 1/3
16 Erdős Rényi Networks Probability pi k that vertex i has a degree k connected to k vertices, not connected to the other N k 1 k pi k = C N 1 pk 1 p N Degree distribution for the whole network P k = pi k / N i= 1 For ER networks average degree: k = 2 E / N=p N 1 pn N s. t. k = const => Poisson distribution k k k k pn P k = e = exp pn k! k! 16 N 1 k
17 Erdős Rényi Networks Connected component sizes N s. t. k = const mean component size giant component relative giant component size 17 small subgraphs k k 1 : many small subgraphs k >>1 : giant component + small subgraphs k =1 : phase transition (percolation)
18 Erdős Rényi Networks Degree distribution: Poisson (degrees of all nodes close to average) No correlations, all edges exist independently of each other Path lengths grow logarithmically with system size, <l> ~ ln (N) Connectivity depends on average degree <k> small <k> => several disjoint components, high <k> => giant connected component there is a percolation transition phase (from a fragmented to a connected) Very homogeneous networks 18
19 RealWorld Networks Shortest path Clustering Random networks Short Low Real networks Short High Regulartopology networks * Long High * * [Watts & Strogatz 1998] 19
20 Random vs RealWorld Networks Degree distributions Poison distribution k k k P k = e k! [Barabási & Albert, 1999] 20 Heavy tail distributions (often power law in log axes)
21 Network Models: SmallWorld D.J. Watts and S. Strogatz, Collective dynamics of 'smallworld' networks", Nature 393, , 1998 WS model: Take a regular clustered network Rewire the endpoint of each link to a random node with probability p SWN => a simple model for interpolating between regular and random networks clustering coefficient WS model, k>2 [Barrat & Weight, 2000] Degree distribution 21 Randomness controlled by a single tuning parameter N >> k >> ln(n) >> 1 <= independent of system size
22 Network Models: SmallWorld Networks SmallWorld Network short paths, high clustering regular network Clustering Path Length 22 [Watts & Strogatz] N = 1000 k = 10 average over 20 realizations at each p random network
23 Network Models: SmallWorld Networks Epidemics: number of infected Epidemic size Density of shortcuts Network structure strongly affects processes taking place on networks [Watts & Strogatz] 23 Dynamics of sync, virus spreading : small number of shortcuts greatly speeds up the process: 3% shortcuts => 50% epidemic
24 Network Models: ScaleFree Networks A.L. Barabási & R. Albert, Emergence of Scaling in Random Networks, Science 286, 509 (1999) Degree distributions PowerLaw Distribution logarithmic axes 24
25 Power Law Distributions P k = Powerlaw tails, k>k min γ 1 γ 1 k min k γ k = k n P k dk n Fluctuations k min n n k = k min γ 1 γ 1 n for γ n 1 Level of heterogeneity: 1<γ< 2 k kc= cutoff due to finitesize diverging degree fluctuations for γ< 3 2 <γ<3 k 2 only k γ 1 Scale invariance: F k F αk =α D F k <=> shift on log scale γ Powerlaw: P k = Ak γ P αk = A αk 25 for most of real world networks 2 <γ<3 = α γ P k
26 Power Law Distributions Networks with Power Law Distributions => ScaleFree Networks logarithmic axes powerlaw no characteristic scale (node degree) in the distribution 26 P k = k k exp k /k!
27 BarabásiAlbert Model ScaleFree Networks Where networks come from? Networks are not static => growth networks BA model of network growth based on the principle of preferential attachment  the rich get richer 2 results in networks with a powerlaw degree distribution P k =2m /k (average degree <k> = 2m ) 1. Take a small seed network, e.g. a few connected nodes 2. Let a new node of degree m enter the network 3. Connect the new node to existing nodes such that the probability πi Degree distribution 27 2m2 P k = 3 k of connecting to node i of degree ki is π i= Average shortest path lengths 3 ki ki Clustering coefficient:
28 Network Models Random p = Small world p = 0.1 Scale free <k> = 2
29 Network Models: Summary ErdösRenyi model short path lengths Poisson distribution (no hubs) no clustering WattsStrogatz Small World model short path lengths high clustering (N independent) almost constant degrees 29 BarabásiAlbert scalefree model short path lengths powerlaw distribution for degrees robustness no clustering (may be ﬁxed) Realworld networks short path lengths high clustering broad degree distributions, often power laws
30 Similarity Graphs Graphs embedded in space Euclidean distance (L2 norm) Manhattan distance (L1 norm) Cosine similarity Graphs built from data: Data points from Euclidean space, sampling of some underlying distribution,... Connectivity parameter: k (KNN), ε  neighborhood graph,... Similarity measure => fully connected (weighted ) matrix Graphs not embedded in space Neighborhood measures  structural equivalence: share the same neighbors => Jaccard coeﬃcient  regular equivalence: if neighbors of a node are similar Pearson correlation coeﬃcient Path dependent measures Measures based on random walk:  commutetime: average number of steps for a random to hit a target and return  escape probability: probability to hit a target before coming back 30
General 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 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 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 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 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 informationPUBLIC TRANSPORT SYSTEMS IN POLAND: FROM BIAŁYSTOK TO ZIELONA GÓRA BY BUS AND TRAM USING UNIVERSAL STATISTICS OF COMPLEX NETWORKS
Vol. 36 (2005) ACTA PHYSICA POLONICA B No 5 PUBLIC TRANSPORT SYSTEMS IN POLAND: FROM BIAŁYSTOK TO ZIELONA GÓRA BY BUS AND TRAM USING UNIVERSAL STATISTICS OF COMPLEX NETWORKS Julian Sienkiewicz and Janusz
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 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 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 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 informationGENERATING AN ASSORTATIVE NETWORK WITH A GIVEN DEGREE DISTRIBUTION
International Journal of Bifurcation and Chaos, Vol. 18, o. 11 (2008) 3495 3502 c World Scientific Publishing Company GEERATIG A ASSORTATIVE ETWORK WITH A GIVE DEGREE DISTRIBUTIO JI ZHOU, XIAOKE XU, JIE
More informationSmallWorld Characteristics of Internet Topologies and Implications on Multicast Scaling
SmallWorld Characteristics of Internet Topologies and Implications on Multicast Scaling Shudong Jin Department of Electrical Engineering and Computer Science, Case Western Reserve University Cleveland,
More informationTutorial, IEEE SERVICE 2014 Anchorage, Alaska
Tutorial, IEEE SERVICE 2014 Anchorage, Alaska Big Data Science: Fundamental, Techniques, and Challenges (Data Mining on Big Data) 2014. 6. 27. By Neil Y. Yen Presented by Incheon Paik University of Aizu
More information! E6893 Big Data Analytics Lecture 10:! Linked Big Data Graph Computing (II)
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
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 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 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 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 informationEmergence of Complexity in Financial Networks
Emergence of Complexity in Financial Networks Guido Caldarelli 1, Stefano Battiston 2, Diego Garlaschelli 3 and Michele Catanzaro 1 1 INFM UdR Roma1 Dipartimento di Fisica Università La Sapienza P.le Moro
More informationGraphs, Networks and Python: The Power of Interconnection. Lachlan Blackhall  lachlan@repositpower.com
Graphs, Networks and Python: The Power of Interconnection Lachlan Blackhall  lachlan@repositpower.com A little about me Graphs Graph, G = (V, E) V = Vertices / Nodes E = Edges NetworkX Native graph
More informationGraph Theory Approaches to Protein Interaction Data Analysis
Graph Theory Approaches to Protein Interaction Data Analysis Nataša Pržulj Technical Report 322/04 Department of Computer Science, University of Toronto Completed on September 8, 2003 Report issued on
More informationBioinformatics: Network Analysis
Bioinformatics: Network Analysis Graphtheoretic Properties of Biological Networks COMP 572 (BIOS 572 / BIOE 564)  Fall 2013 Luay Nakhleh, Rice University 1 Outline Architectural features Motifs, modules,
More information
IC05 Introduction on Networks &Visualization Nov. 2009.
IC05 Introduction on Networks &Visualization Nov. 2009 Overview 1. Networks Introduction Networks across disciplines Properties Models 2. Visualization InfoVis Data exploration
More informationHealthcare Analytics. Aryya Gangopadhyay UMBC
Healthcare Analytics Aryya Gangopadhyay UMBC Two of many projects Integrated network approach to personalized medicine Multidimensional and multimodal Dynamic Analyze interactions HealthMask Need for sharing
More informationComplex networks: Structure and dynamics
Physics Reports 424 (2006) 175 308 www.elsevier.com/locate/physrep Complex networks: Structure and dynamics S. Boccaletti a,, V. Latora b,c, Y. Moreno d,e, M. Chavez f, D.U. Hwang a a CNRIstituto dei
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 informationTemporal Dynamics of ScaleFree Networks
Temporal Dynamics of ScaleFree Networks Erez Shmueli, Yaniv Altshuler, and Alex Sandy Pentland MIT Media Lab {shmueli,yanival,sandy}@media.mit.edu Abstract. Many social, biological, and technological
More informationEffects of node buffer and capacity on network traffic
Chin. Phys. B Vol. 21, No. 9 (212) 9892 Effects of node buffer and capacity on network traffic Ling Xiang( 凌 翔 ) a), Hu MaoBin( 胡 茂 彬 ) b), and Ding JianXun( 丁 建 勋 ) a) a) School of Transportation Engineering,
More informationMeaneld theory for scalefree random networks
Physica A 272 (1999) 173 187 www.elsevier.com/locate/physa Meaneld theory for scalefree random networks AlbertLaszlo Barabasi,Reka Albert, Hawoong Jeong Department of Physics, University of NotreDame,
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 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 informationGraph theory and network analysis. Devika Subramanian Comp 140 Fall 2008
Graph theory and network analysis Devika Subramanian Comp 140 Fall 2008 1 The bridges of Konigsburg Source: Wikipedia The city of Königsberg in Prussia was set on both sides of the Pregel River, and included
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 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 informationCollective behaviour in clustered social networks
Collective behaviour in clustered social networks Maciej Wołoszyn 1, Dietrich Stauffer 2, Krzysztof Kułakowski 1 1 Faculty of Physics and Applied Computer Science AGH University of Science and Technology
More informationComplex Network Visualization based on Voronoi Diagram and Smoothedparticle Hydrodynamics
Complex Network Visualization based on Voronoi Diagram and Smoothedparticle Hydrodynamics Zhao Wenbin 1, Zhao Zhengxu 2 1 School of Instrument Science and Engineering, Southeast University, Nanjing, Jiangsu
More informationarxiv:1402.2959v1 [cs.ne] 12 Feb 2014
Local Optima Networks: A New Model of Combinatorial Fitness Landscapes Gabriela Ochoa 1, Sébastien Verel 2, Fabio Daolio 3, and Marco Tomassini 3 arxiv:1402.2959v1 [cs.ne] 12 Feb 2014 1 Computing Science
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 informationGraph Theory and Networks in Biology
Graph Theory and Networks in Biology Oliver Mason and Mark Verwoerd March 14, 2006 Abstract In this paper, we present a survey of the use of graph theoretical techniques in Biology. In particular, we discuss
More informationLINEARALGEBRAIC GRAPH MINING
UCSF QB3 Seminar 5/28/215 Linear Algebraic Graph Mining, sanders29@llnl.gov 1/22 LLNLPRES671587 New Applications of Computer Analysis to Biomedical Data Sets QB3 Seminar, UCSF Medical School, May 28
More informationThe average distances in random graphs with given expected degrees
Classification: Physical Sciences, Mathematics The average distances in random graphs with given expected degrees by Fan Chung 1 and Linyuan Lu Department of Mathematics University of California at San
More informationGraph Mining Techniques for Social Media Analysis
Graph Mining Techniques for Social Media Analysis Mary McGlohon Christos Faloutsos 1 11 What is graph mining? Extracting useful knowledge (patterns, outliers, etc.) from structured data that can be represented
More informationFrom Random Graphs to Complex Networks:
Unterschrift des Betreuers DIPLOMARBEIT From Random Graphs to Complex Networks: A Modelling Approach Ausgeführt am Institut für Diskrete Mathematik und Geometrie der Technischen Universität Wien unter
More informationA MULTIMODEL DOCKING EXPERIMENT OF DYNAMIC SOCIAL NETWORK SIMULATIONS ABSTRACT
A MULTIMODEL DOCKING EXPERIMENT OF DYNAMIC SOCIAL NETWORK SIMULATIONS Jin Xu Yongqin Gao Jeffrey Goett Gregory Madey Dept. of Comp. Science University of Notre Dame Notre Dame, IN 46556 Email: {jxu, ygao,
More informationStatistical Mechanics of Complex Networks
Statistical Mechanics of Complex Networks Réka Albert 1,2 and AlbertLászló Barabási 2 1 School of Mathematics, 127 Vincent Hall, University of Minnesota, Minneapolis, Minnesota 55455 2 Department of Physics,
More informationThe architecture of complex weighted networks
The architecture of complex weighted networks A. Barrat*, M. Barthélemy, R. PastorSatorras, and A. Vespignani* *Laboratoire de Physique Théorique (Unité Mixte de Recherche du Centre National de la Recherche
More informationStatistical theory of Internet exploration
Statistical theory of Internet exploration Luca Dall Asta, 1 Ignacio AlvarezHamelin, 1,2 Alain Barrat, 1 Alexei Vázquez, 3 and Alessandro Vespignani 1,4 1 Laboratoire de Physique Théorique, Bâtiment 210,
More informationStatistical Inference for Networks Graduate Lectures. Hilary Term 2009 Prof. Gesine Reinert
Statistical Inference for Networks Graduate Lectures Hilary Term 2009 Prof. Gesine Reinert 1 Overview 1: Network summaries. What are networks? Some examples from social science and from biology. The need
More informationarxiv:condmat/ v1 [condmat.disnn] 21 Oct 1999
Emergence of Scaling in Random Networks arxiv:condmat/9910332v1 [condmat.disnn] 21 Oct 1999 AlbertLászló Barabási and Réka Albert Department of Physics, University of NotreDame, NotreDame, IN 46556
More informationDmitri Krioukov CAIDA/UCSD
Hyperbolic geometry of complex networks Dmitri Krioukov CAIDA/UCSD dima@caida.org F. Papadopoulos, M. Boguñá, A. Vahdat, and kc claffy Complex networks Technological Internet Transportation Power grid
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 informationOn the Synchronization of Networks with Prescribed Degree Distributions
On the Synchronization of Networks with Prescribed Degree Distributions Fatihcan M. Atay Türker Bıyıkoğlu Jürgen Jost Max Planck Institute for Mathematics in the Sciences Inselstr. 22, D04103 Leipzig,
More informationRecent Progress in Complex Network Analysis. Models of Random Intersection Graphs
Recent Progress in Complex Network Analysis. Models of Random Intersection Graphs Mindaugas Bloznelis, Erhard Godehardt, Jerzy Jaworski, Valentas Kurauskas, Katarzyna Rybarczyk Adam Mickiewicz University,
More informationResearch Article A Comparison of Online Social Networks and RealLife Social Networks: A Study of Sina Microblogging
Mathematical Problems in Engineering, Article ID 578713, 6 pages http://dx.doi.org/10.1155/2014/578713 Research Article A Comparison of Online Social Networks and RealLife Social Networks: A Study of
More informationThe LargeScale Structure of Semantic Networks: Statistical Analyses and a Model of Semantic Growth
Cognitive Science 29 (2005) 41 78 Copyright 2005 Cognitive Science Society, Inc. All rights reserved. The LargeScale Structure of Semantic Networks: Statistical Analyses and a Model of Semantic Growth
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 informationSmallWorld Internet Topologies
SmallWorld Internet Topologies Possible auses and Implications on Scalability of EndSystem Multicast Shudong Jin Azer Bestavros omputer Science Department Boston University Boston, MA 0225 jins,best@cs.bu.edu
More informationThe Structure of Growing Social Networks
The Structure of Growing Social Networks Emily M. Jin Michelle Girvan M. E. J. Newman SFI WORKING PAPER: 200106032 SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily
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 informationIn 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 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 informationAnalyzing the Facebook graph?
Logistics Big Data Algorithmic Introduction Prof. Yuval Shavitt Contact: shavitt@eng.tau.ac.il Final grade: 4 6 home assignments (will try to include programing assignments as well): 2% Exam 8% Big Data
More informationA mixture model for random graphs
A mixture model for random graphs JJ Daudin, F. Picard, S. Robin robin@inapg.inra.fr UMR INAPG / ENGREF / INRA, Paris Mathématique et Informatique Appliquées Examples of networks. Social: Biological:
More informationSocial and Economic Networks: Lecture 1, Networks?
Social and Economic Networks: Lecture 1, Networks? Alper Duman Izmir University Economics, February 26, 2013 Conventional economics assume that all agents are either completely connected or totally isolated.
More informationExploration, testing, and prediction: the many roles of statistics in Network Science
Exploration, testing, and prediction: the many roles of statistics in Network Science Aaron Clauset inferred community inferred community 2 inferred community 3 Assistant Professor of Computer Science
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 informationExpander 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 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 informationChapter ML:XI (continued)
Chapter ML:XI (continued) XI. Cluster Analysis Data Mining Overview Cluster Analysis Basics Hierarchical Cluster Analysis Iterative Cluster Analysis DensityBased Cluster Analysis Cluster Evaluation Constrained
More informationIntroduced by Stuart Kauffman (ca. 1986) as a tunable family of fitness landscapes.
68 Part II. Combinatorial Models can require a number of spin flips that is exponential in N (A. Haken et al. ca. 1989), and that one can in fact embed arbitrary computations in the dynamics (Orponen 1995).
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 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 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 informationExpansion 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 informationStatistical mechanics of complex networks
Statistical mechanics of complex networks Réka Albert* and AlbertLászló Barabási Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556 (Published 30 January 2002) REVIEWS OF MODERN
More informationCommunity Detection Proseminar  Elementary Data Mining Techniques by Simon Grätzer
Community Detection Proseminar  Elementary Data Mining Techniques by Simon Grätzer 1 Content What is Community Detection? Motivation Defining a community Methods to find communities Overlapping communities
More informationData Mining Cluster Analysis: Advanced Concepts and Algorithms. Lecture Notes for Chapter 9. Introduction to Data Mining
Data Mining Cluster Analysis: Advanced Concepts and Algorithms Lecture Notes for Chapter 9 Introduction to Data Mining by Tan, Steinbach, Kumar Tan,Steinbach, Kumar Introduction to Data Mining 4/18/2004
More informationSocial Networks and Social Media
Social Networks and Social Media Social Media: ManytoMany Social Networking Content Sharing Social Media Blogs Microblogging Wiki Forum 2 Characteristics of Social Media Consumers become Producers Rich
More informationThe 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 informationB490 Mining the Big Data. 2 Clustering
B490 Mining the Big Data 2 Clustering Qin Zhang 11 Motivations Group together similar documents/webpages/images/people/proteins/products One of the most important problems in machine learning, pattern
More informationTimeDependent Complex Networks:
TimeDependent Complex Networks: Dynamic Centrality, Dynamic Motifs, and Cycles of Social Interaction* Dan Braha 1, 2 and Yaneer BarYam 2 1 University of Massachusetts Dartmouth, MA 02747, USA http://necsi.edu/affiliates/braha/dan_brahadescription.htm
More informationDECENTRALIZED SCALEFREE NETWORK CONSTRUCTION AND LOAD BALANCING IN MASSIVE MULTIUSER VIRTUAL ENVIRONMENTS
DECENTRALIZED SCALEFREE NETWORK CONSTRUCTION AND LOAD BALANCING IN MASSIVE MULTIUSER VIRTUAL ENVIRONMENTS Markus Esch, Eric Tobias  University of Luxembourg MOTIVATION HyperVerse project Massive Multiuser
More informationMANAGEMENT SCIENCE. Special Issue on Complex Systems Luis A. Nunes Amaral, Brian Uzzi, Editors
A JOURNAL OF THE INSTITUTE FOR OPERATIONS RESEARCH AND THE MANAGEMENT SCIENCES MANAGEMENT SCIENCE Volume 53 Number 7 July 7 Special Issue on Complex Systems Luis A. Nunes Amaral, Brian Uzzi, Editors Amaral,
More informationMedical Information Management & Mining. You Chen Jan,15, 2013 You.chen@vanderbilt.edu
Medical Information Management & Mining You Chen Jan,15, 2013 You.chen@vanderbilt.edu 1 Trees Building Materials Trees cannot be used to build a house directly. How can we transform trees to building materials?
More informationGenerating Hierarchically Modular Networks via Link Switching
Generating Hierarchically Modular Networks via Link Switching Susan Khor ABSTRACT This paper introduces a method to generate hierarchically modular networks with prescribed node degree list by link switching.
More informationThe Network Structure of Hard Combinatorial Landscapes
The Network Structure of Hard Combinatorial Landscapes Marco Tomassini 1, Sebastien Verel 2, Gabriela Ochoa 3 1 University of Lausanne, Lausanne, Switzerland 2 University of Nice SophiaAntipolis, France
More informationPermanent City Research Online URL: http://openaccess.city.ac.uk/12819/
Baronchelli, A., FerreriCancho, R., PastorSatorras, R., Chater, N. & Christiansen, M. H. (2013). Networks in cognitive science. Trends in Cognitive Sciences, 17(7), pp. 348360. doi: 10.1016/j.tics.2013.04.010
More informationGreedy Routing on Hidden Metric Spaces as a Foundation of Scalable Routing Architectures
Greedy Routing on Hidden Metric Spaces as a Foundation of Scalable Routing Architectures Dmitri Krioukov, kc claffy, and Kevin Fall CAIDA/UCSD, and Intel Research, Berkeley Problem Highlevel Routing is
More informationTopology, Hierarchy, and Correlations in Internet Graphs
Topology, Hierarchy, and Correlations in Internet Graphs Romualdo PastorSatorras 1, Alexei Vázquez 2, and Alessandro Vespignani 3 1 Department de Física i Enginyeria Nuclear, Universitat Politècnica de
More informationPractical statistical network analysis (with R and igraph)
Practical statistical network analysis (with R and igraph) Gábor Csárdi csardi@rmki.kfki.hu Department of Biophysics, KFKI Research Institute for Nuclear and Particle Physics of the Hungarian Academy of
More informationTHE NETWORK ANALYSIS OF URBAN STREETS: A DUAL APPROACH. and INFN Sezione di Catania, Italy,
THE NETWORK ANALYSIS OF URBAN STREETS: A DUAL APPROACH SERGIO PORTA a, PAOLO CRUCITTI b, VITO LATORA c a Dipartimento di Progettazione dell Architettura, Politecnico di Milano, Italy, sergio.porta@polimi.it
More informationhttp://www.elsevier.com/copyright
This article was published in an Elsevier journal. The attached copy is furnished to the author for noncommercial research and education use, including for instruction at the author s institution, sharing
More informationDegree 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 reallife graphs Distinctive properties Existing
More informationCharacterization of Latent Social Networks Discovered through Computer Network Logs
Characterization of Latent Social Networks Discovered through Computer Network Logs Kevin M. Carter MIT Lincoln Laboratory 244 Wood St Lexington, MA 02420 kevin.carter@ll.mit.edu Rajmonda S. Caceres MIT
More informationInet3.0: Internet Topology Generator
Inet3.: Internet Topology Generator Jared Winick Sugih Jamin {jwinick,jamin}@eecs.umich.edu CSETR4562 Abstract In this report we present version 3. of Inet, an Autonomous System (AS) level Internet
More informationEntropy based Graph Clustering: Application to Biological and Social Networks
Entropy based Graph Clustering: Application to Biological and Social Networks Edward C Kenley YoungRae Cho Department of Computer Science Baylor University Complex Systems Definition Dynamically evolving
More informationEnhancing the robustness of scalefree networks
Enhancing the robustness of scalefree networks Jichang Zhao and Ke Xu State Key Laboratory of Software Development Environment Beihang University, 100191, Beijing, P.R.China Email: zhaojichang@nlsde.buaa.edu.cn
More informationThe Topology of LargeScale Engineering ProblemSolving Networks
The Topology of LargeScale Engineering ProblemSolving Networks by Dan Braha 1, 2 and Yaneer BarYam 2, 3 1 Faculty of Engineering Sciences BenGurion University, P.O.Box 653 BeerSheva 84105, Israel
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 information