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Community detection incomplex networks
Vinh Loc DAO
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Summary
1 Introduction
2 Datasets and Benchmarks
3 Community detection in network
4 Evaluating partition quality
5 Objectives and Perspectives
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Summary
1 Introduction
What is a network or a graph ?
Some notions
Structure properties of real networks
2 Datasets and Benchmarks
3 Community detection in network
4 Evaluating partition quality
5 Objectives and Perspectives
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What is a network or a graph ?
Example : Internet, transport network, power grid, food web, social network
Figure – A very simple network G(V ,E) with |V | Vertices and |E | Edges
Node : Entity in real life
Edge : Relation between two entities to which it connects
A natural language to describe complex systems.
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A local mesh network - very small network
Node : Computer
Edge : Connection between two computer
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Student network in a university - relatively small network
Node : Student
Edge : Relation (Could be any kind of relation)
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The French highway network - large network
Node : City
Edge : Highway
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The Internet - international network
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Some notions
Complex network : Graph with non-trivial topology features
Network analysis : Studies of graph to extract non-trivial features
Community detection algorithm : Divide nodes into groups calledcommunities whose members are connected densely.
Figure – Uncover graph modules without specifying clusters’ size
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Structure properties of real networks
- Graph representing real systems are normaly neither regular nor random.- Degree (nb of connections of a node) distribution often follows a power law,as connections often follow preferential patterns.- Nodes are often found to cluster into high density groups.
Figure – Regular lattice graph Figure – Graph with modular structure
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Why community detection ?
- Comprehend network global organization- Reveal modular structures- Reveal hidden properties between nodes- Understand information diffusion process throughout network
APPLICATIONS :- Detect web clients with similar interests- Prevent epidemic transmission- Managing collaboration network- etc,- You name it
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Summary
1 Introduction
2 Datasets and Benchmarks
Datasets - Real networks
Datasets - Artificial networks
3 Community detection in network
4 Evaluating partition quality
5 Objectives and Perspectives
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Datasets - Real networks
- Social network of friendships between 34 members of a karate club at a USuniversity in the 1970s. (left)- An identity graph with 25 vertices and 31 edges. An identity graph has asingle graph auto-morphism, the trivial one. (right)
Figure – Zachary karate network Figure – Walther network
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Datasets - Artificial networks
Parameters : Graph size, node distribution, link distribution, densitydistribution, etc.
Figure – GN benchmark network Figure – LFR benchmark network
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Summary
1 Introduction
2 Datasets and Benchmarks
3 Community detection in network
Dense structure in modular network
From dense structure to community
4 Evaluating partition quality
5 Objectives and Perspectives
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An example of graph partitioning by K-means
Apply K-means graph partitioning on karate club network (K = 2)
Figure – The Zachary karate networkFigure – K-means partition on theZachary karate club
- Good solution ?- How do we know the club has 2 communities ?- Wait, tell me again what is a community ?
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Did we say a ”community” ? Isn’t it a ”cluster” ?
- What is a community ? - Answer : ”I know it when I see it”- No universal accepted definition.- More edges inside the community than edges linking its vertices with the restof the network.- Many detection methods : overlapping/non-overlapping, fast/slow,single-scale/multi-scale.
Figure – A graph division
Figure – The karate club is separateddue to a conflict coach/president
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Summary
1 Introduction
2 Datasets and Benchmarks
3 Community detection in network
4 Evaluating partition quality
5 Objectives and Perspectives
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Partition quality
- What is a good partition of a network into modules ?
- Quality function : assigns score to each partition of a graph- The most popular quality function is modularity
Q = 12|V |Σij(Aij − Pij)δ(Ci ,Cj)
Aij : Adjacency matrix, Pij : Expected connection adjacency matrixQ = Fraction of edges within communities - expected fraction of such edgesin a randomModularity favors inter community links.
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Community detection on the karate club
Edge betweenness O(|V ||E |2)
Fast greedy O(|V ||E |log(|V |))
Label propagation O(|V |+ |E |)Louvain method O(|V |log(|V |)
Leading eigenvector O(|V |2 + |E |)Modular optimization (NP-complete)
Infomap O (|V |(|V |+ |E |))
Walktrap O(|E ||V |2)
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Community detection on the Walther network
Many possible meaning divisions on a less modular network.
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Quality is relative ? Goodness is subjective
- Community characteristics :
Community density
Community connectiveness
Robustness to perturbation, etc
Question : How to choose appropriate method to satisfy certaincharacteristics and utilizing as much as possible available information ?
Ex : Which method to chose to :
- Divide students into the most cohesive groups.
- Establish geographic sites to minimize remote works of collaborators.
- Compromise between community density and calculation time.
- Maximize range of ages in a dancing community.
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Summary
1 Introduction
2 Datasets and Benchmarks
3 Community detection in network
4 Evaluating partition quality
5 Objectives and Perspectives
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Objectives and Perspectives
Guide to choose best methods base on expected quality indicators, graphcharacteristics and available resources
Create generative model to summarize community characteristics
Create a benchmark base on generative model
Construct criteria for evaluating partition quality base on end user pointof view
Propose methods to improve detection quality
Tools :- R for network analyzing, data analyzing, visualization- Gephi : Visualization
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