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Computational Complexity pp 490–512Cite as

Community Structure in Graphs

  • Reference work entry

Article Outline

Glossary

Definition of the Subject

Introduction

Elements of Community Detection

Computer Science: Graph Partitioning

Social Science: Hierarchical and k‑Means Clustering

New Methods

Testing Methods

The Mesoscopic Description of a Graph

Future Directions

Bibliography

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Abbreviations

Graph :

A graph is a set of elements, called vertices or nodes, where pairs of vertices are connected by relational links, or edges.A graph can be considered as the simplest representation of a complex system, where the vertices are the elementary units of the system and the edges represent their mutual interactions.

Community :

A community is a group of graph vertices that “belong together” according to some precisely defined criteria which can be measured. Many definitions have been proposed. A common approach is to define a community as a group of vertices such that the density of edges between vertices of the group is higher than the average edge density in the graph. In the text also the terms module or cluster are used when referring to a community.

Partition :

A partition is a split of a graph in subsets with each vertex assigned to only one of them. This last condition may be relaxed to include the case of overlapping communities, imposing that each vertex is assigned to at least one subset.

Dendrogram :

A dendrogram, or hierarchical tree, is a branching diagram representing successive divisions of a graph into communities.Dendrograms are frequently used in social network analysis and computational biology, especially in biological taxonomy.

Scalability :

Scalability expresses the computational complexity of an algorithm. If the running time of a community detection algorithm, working on a graph with n vertices and m edges, is proportional to the product \( { n^\alpha m^\beta } \), one says that the algorithm scales as \( { O(n^\alpha m^\beta) } \).Knowing the scalability allows to estimate the range of applicability of an algorithm.

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Author information

Authors and Affiliations

  1. Complex Networks Lagrange Laboratory (CNLL), ISI Foundation, Torino, Italy

    Santo Fortunato

  2. SMC, INFM-CNR and Dipartimento di Fisica, “Sapienza” Università di Roma, Roma, Italy

    Claudio Castellano

Authors
  1. Santo Fortunato
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  2. Claudio Castellano
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Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Fortunato, S., Castellano, C. (2012). Community Structure in Graphs. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_33

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