Community detection in complex networks Vinh Loc DAO Summary 1 - - PowerPoint PPT Presentation

Community detection in complex networks

Vinh Loc DAO

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 called communities 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 US

university in the 1970s. (left)

• An identity graph with 25 vertices and 31 edges. An identity graph has a

single 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, density distribution, 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 network Figure – K-means partition on the Zachary 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 rest
• f the network.
• Many detection methods : overlapping/non-overlapping, fast/slow,

single-scale/multi-scale.

Figure – A graph division Figure – The karate club is separated due 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 =

1 2|V |Σij(Aij − Pij)δ(Ci, Cj)

Aij : Adjacency matrix, Pij : Expected connection adjacency matrix Q = Fraction of edges within communities - expected fraction of such edges in a random Modularity 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 certain characteristics 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, graph characteristics 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 point

• f view

Propose methods to improve detection quality Tools :

• R for network analyzing, data analyzing, visualization
• Gephi : Visualization

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