## Social Network Analysis

August 30, 2011 § Leave a comment

Social network analysis is a methodological approach using theoretical graph concepts to understand and explain social structure. Purposes can be divided into identifying; Centrality, Cohesive Subgroups, Structural Roles and Network statistics.

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**Centrality** can be locally measured by degree, or via global distance measures. Though more expensive, global means are often more valuable, with example methods being the measure of **Betweeness** (‘gatekeepers’), **Closeness**, (sum of shortest paths to all vertices), and **Eccentricity **(length of the longest shortest path). The use of any of these methods are, of course, dependent on purpose. Feedback measures such as Status/Hub/Authority (useful for page length)/eigenvector can also assess centrality.

Centrality can be displayed by node graphs, radial drawings, hierarchical drawings(,dendrograms), etc.

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**Cohesive Subgroups** help identify meaningful social groups. Their components can be strong/weak, cycles/cyclic, connected (k-connectivity)/isolated, or, cut vertex/separation-pair.

Cliques (complete subgraphs, where all nodes are connected), can be found via **n-clique**, with ‘n’ dictating the maximum path length of members of a clique, allowing the relaxation of the definition with it’s increase. **N-Clan** usefully extends n-clique, requiring the diameter of the clique to be no greater than n. Dense areas can also be found with **k-core**, in which every vertex is adjacent to at least k other vertices. **K-plex** finds a set of vertices in which every vertex is adjacent to All except k of the other vertices (connected to n-k vertices).** ****n-clique **and **n-clan** are about **reachability** (path length). **k-core** and **k-plex** are about **degree**.

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**Structural Equivalence and Network Positions** can greatly reduce complex networks via methods such as the Block Model (or image matrix). This is done by clustering via cliques, distance, or similarity. Similarity of social positions can be conceptualised by; structural equivalence, automorphic equivalence, regular equivalence, outdegree and indegree equivalence, blockmodelling and generalised blockmodelling.

Nodes are structurally equivalence if they hold identical positions in the network. Blockmodel essentially combines similar nodes into one. Nodes that are structurally equivalent are also automorphically equivalent. (common – vast majority)

Automorphic nodes do not have to be connected to exactly the same nodes, but to nodes that play analogous roles in the network.

Nodes are regularly equivalent if they have ties to the same role, revealing social structures. Regular equivalences relaxes the definition further, no longer requiring degree, but only that you know one person in a class. (common – niche applications)

All of these forms of equivalence and several others [Pattison 1993], have the property that: • There is a path at the block level if and only if there is at least one path at the node level.

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**Network Measures/Statistics** can be used to analyse and compare networks. These statistics consist of;

– Degree Distribution

– Clustering Coefficient

– Diameter

– Average Path Length

– Connected Component

– Density

Such analysis’ can be performed using pajek by Vladimir Batagelj.

Blockmodelling can aid with matrix rearrangement views and clustering. The identification of ‘Triads’ / Pattern Searching has also been useful in social network analysis, along with the normalization of data.

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Sources:

- Social Network Analysis: Methods and Applications [Wasserman and Faust 94]
- Network Analysis: Methodological Foundations LNCS 3418 Tutorial [Brandes and Erlebach eds. 04]

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