Overview Analysis of Social Networks

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Overview Analysis of Social Networks

• phanquan_at_labri.fr

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Plan

• Introduction
• Analysis of Social Network
• Properties of network
• The small-world effect
• Clustering coefficient
• Scale-free network
• Betweenness centrality
• Community structure
• Finding community structure
• Conclusion

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Introduction

• Social Network
• Collaboration networks
• Film Actors
• Telephone call graph
• Webograph (WebOfPeople)
• Technological Network
• Internet, the World-Wide Web
• Sofware packages
• Biological Network
• Metabolic network
• Protein interactions
• Neural network

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Analysis of Social Network

• Social Networks
• Social Entities persons, organizations, things,
  cities (Actor/Node/Point/Agent)
• Binary Relations social relations, dependencies,
  exchange (Tie,Link,Edge,Line,Arc)
• directed or undirected, weighted or unweighted
• Weight increasing or decreasing the tie between
  the two entities
• A labeled directed graph G (or Matrices)
• Vertex or edge attribute a partial function
  assiging nominal or numerical values to vertices
  or edges

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Analysis of Social Network

• Purpose
• Identify important vertices, crucial
  relationships,subgroups, roles,
• Answer questions about structures
• Interest
• Element Properties(absolute and relative)
• Single actors, links, incidences
• Group classifying the elements of networks and
  properties of subnetworks
• Actor equivalence classes, cluster identification
• Network Connectivity or balance

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Analysis of Social Network

• What makes a vertex importance or central ?
• Centrality of a vertex
• Network tend to build clusters ?
• Clustering coefficient
• Network evolve ?
• Degree distribution
• Overall structure ?
• Small-world phenomenon

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Properties of networks

• The small-world effect
• Clustering coefficient
• Degree distribution
• Scale-free network
• Betweenness Centrality
• Community structure

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The small world effect

• Milgram's experiment (1967)
• The participants could only pass the letters (by
  hand) to personal acquaintances who they thought
  might be able to reach the target whether
  directly or via a "friend of a friend"
• Letters passed person to person were able to
  reach a designated target individual step in only
  a small numbers of steps

Small world phenomeon It is said that all
strangers can be linked through six degrees of
separation
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Online experiment
Home page of http//smallworld.columbia.edu/
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Clustering coefficient

• Determine whether or not a graph is a small-world
  network (Watts and Strogatz)
• ?G(v) number of subgraphs 3 edges 3 vertices,
  one of which is v
• tG(v) number of subgraphs (not necessarily
  induced) with 2 edges 3 vertices, one of which is
  v and such that v is incident to both edges
• This average higher then random graph with same
  vertex set small-world

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Scale Free Network

• Power law distribution
• in number of connections between nodes
• Some few nodes
• Extremely high connectivity
• Essentially scale-free
• Vast majority
• Relatively poorly connected

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Scale Free Network

• Some scale-free network graphs
• Protein interaction networks
• Protein binding relation
• Metabolic pathway
• Enzyme, Substrate links by chemical interactions
• WWW ou Weblog links
• Web pages links pointing from one page to
  another
• Actor collaborations
• Actors in the same movies
• Airline traffic routes

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Scale Free Network

• Main properties
• have scaling (power law) degree distribution
• have growth and preferential attachment
• have hightly connected hubs which hold the
  network together
• self-similar
• universal in the sense of not depending on
  domain-specific details

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Comparing Random and Scale-Free Distribution

• Random Scale-Free
• Source the journal Nature

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Scale Free Network

• Cumulative degree distributions for six different
  networks 1

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Scale Free Network

• Strength and weakness
• extremely tolerant of random failures
• inhomogeneity of the nodes on the network
• extremely vulnerable to intentional attacks on
  their hubs
• Hub is important
• extremely vulnerable to epidemics
• Critical threshold (number of nodes infected)

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Betweenness centrality

• Vertex betweenness
• Determine the role of each actor (node) in a
  social network
• Based on shortest path
• Edge betweenness

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Modularity

• A specific proposed division of that network into
  communities
• Division is good many edges within communities,
  only a few between them
• NC total number of clusters in a given set C
• dc number of edges between nodes of a given
  cluster c
• lc total degree of nodes in cluster c L
  total number of edges in whole network
• For detecting community structure in social
  network
• Optimal set of clusters the one with highest
  modularity during cluster building

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Communities in network

• Social network display a community structure
• Families, groups of close friends, etc.
• Community subgraph V with the internal
  connections denser than the external ones
• Detect the presence of communities ?
• Find members of possible communities ?

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Communites and Clustering

• Two concept
• Very closely related but different
• Cluster
• Part of graph, internal edges more than external
  ones
• Community
• Set of vertices sharing the same topological
  properties
• Community Clusters
• Same set of edges

Two different communities (a bipartite clique)
that are not represented by clustered subgraphs
1
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Community identify
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Problem (1)

• Network
• n vertices with no prior information
• know structural information (edge link
  connectivity)
• Many domains are related
• Computer Science
• Mathematics
• Sociology
• Physics

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Problem (2)

• How to recognize communities within network ?
• No exact definition of the cluster
• A lot of methods
• Have their own advantages and drawback
• Are suitable to different data structures
• Mathematical view point
• Clusteringan optimization procedure according to
  clustering criterion
• Various clustering approches present different
  types of knowledge concerning the clustering
  criterion

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Problem (3)

• Clustering the graph
• Reduce visual complexity
• relatively highly connected nodes
• their associated edges are grouped to form a
  sub-graph, represented by one abstract node
• Finding the cliques
• A clique a maximal connected subgraphs (there
  is an edge between any pairs in subgraph)
• Computation issuse
• Use cliques to determine communities
• Connectivity requried in a clique is too strong

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Approaches to finding communities in networks (1)

• Divisive Method
• Girvan Newman Algorithm 10
• Calculate the betweenness score for each of the
  edges
• Remove the edge with the highest score
• Compute the modularity for the network
• Go back to step 1 until all edges of the networks
  are removed, resutlting in N non-connected nodes
• -The best division when the highest modularity
  value is obtained
• -Community detection is not graph partinioning
• -Effective for obtaining communities in several
  types networks
• . Computational cost of order O(n2m)

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Approaches to finding communities in networks (2)

• Agglomerative hierarchical clustering
• Modularity optimization algorithm 7
• Starting with a state
• Each vertex is the sole member of one of n
  communities
• Repeatedly join communities together in pairs
• Choosing at each step the join that results
• Greatest increase (or smallest decrease) in
  modularity
• Can be applied to very large networks

Visualization of the community structure at
maximum Modularity 7
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Approaches to finding communities in networks (3)

• Agglomerative hierarchical clustering
• Single linkage (nereast neighbor) methods
• merges clusters iteratively
• develop a measure of similarity between pairs of
  vertices
• many different such similarity measures are
  possible, par exemple vertices structural
  equivalence
• starting with an empty network of n vertices and
  no edges, one adds edges between pairs of
  vertices in order of decreasing similarity,
  starting with the pair with strongest similarity
• Weaknesses
• Not scale well time complexity at least 0(n2)
• Can not undo

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Approaches to finding communities in networks (4)

• Hierarchical Growth Method 6
• expanding neighborhoods
• First neighborhoods vertices at a distance one
  edge
• Second neighborhoods
• Use a threshold to obtain the best value of the
  modularity
• Consideration of successive neighborhoods of a
  set of seeds
• Start from vertex -gt Link of its successive
  neighborhood
• To verify if they belong to same community than
  the seed
• Inter-community edge removed -gt split network
  into communities
• Zachary Karate Club Network
• Simple benchmark for community
• finding methodologies

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Hierarchical Growth Method
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Approaches to finding communities in networks (5)

• Others
• SFClusters 9
• considers the characteristics of the SF network
  graph
• A lot of useful information in graph network
  (have large clustering coefficients or clustered
  regions of graph)
• finds local clusters based on the local density
  and vertex neighborhood
• Time complexity
• Polynominal O(nml3)
• n number of vertices
• m number of edges
• l vertex size of the average modified Gabriel
  influence region of the graph

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Conclusion

• Brief overview some basic indices for analysic of
  social networks
• One of difficile problem
• Relation in social network
• Strong / Weak
• Algorithm aspects in network analysis
• Concern the fast computation of such indices
• Particular method analysis needs
• A priori knowledge of the number of expected
  communities

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Conclusion

• Network analysis demandes
• Visualizations
• Two obvious criteria for the quality
• Is the information manifest in the network
  represented accurately ?
• Is this information conveyed efficiently ?
• Creating network visualizations
• Thought throught three aspects
• Substantive aspect the viewer is interested in
• Design (Par example mapping of data to
  graphical variables)
• Algorithm employed to realize the design

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Reference

• 1 M.E.J. Newman . The structure and function of
  complex network
• 2 Analysic and Visualization Social Networks
• 3 Guido Caldarelli- Scale-Free Networks
  Complex Webs in Natural, Technological and Social
  Sciences- Oxford University Press
• 4 U.Brandes. A faster algorithm for betweenness
  centrality
• 5Jan Rupnik. Finding community structure in
  social network analysis-Overview
• 6 F.A.Rodrigues, G. Travieso, L. da F.Costa
  Fast Community Identification by Hierarchical
  Growth
• 7 A. Clauset, M. E. J. Newman, and C. Moore
  Finding community structure in very large
  networks
• 8 M. Girvan and M. E. J. Newman -Community
  structure in social and biological networks
• 9 Xiaohua Hu , Jianchao Han Discovering
  Clusters from Large Scale-Free Network Graph
• 10M.E.J. Newman and M.Girvan Finding and
  evaluating community structure in networks
• 11M.E.J Newman Fast algorithm for detecting
  community structure in networks

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Thank you very much