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Social Network Analysis, Graph Theoretical Approaches to

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

Social network analysis (SNA) focuses on the structure of ties within a set of social entities or actors, e.g., persons, groups, organizations, and nations, or the products of human activity or cognition such as semantic concepts, web sites, and so on. In a graph theoretical approach, a social network is conceptualized as a graph, that is, a set of vertices (or nodes, units, points) representing social actors and a set of lines representing one or more social relations among them.

A network, however, is more than a graph because it contains additional information on the vertices and lines. Characteristics of the social actors, for instance, a person’s sex, age, and income, are represented by discrete or continuous attributes of the vertices in the network, and the intensity, frequency, valence, and type of social relation are represented by line weight or value, line sign, or line type. Formally (see pp. 94–95, 127–128 in Doreian et al. 2005), a network Ncan...

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Abbreviations

Adjacent:

Two vertices are adjacent if they are connected by a line.

Arc:

An arc is a directed line. Formally, an arc is an ordered pair of vertices.

Attribute:

An attribute is a characteristic of a vertex measured independently of the network.

Bipartite network:

See “Two-mode network.”

Clique:

A clique is a maximal complete subnetwork containing three vertices or more.

Complete:

A (sub)network is complete if it has maximum density: All possible lines occur.

Component:

A (weak) component is a maximal (weakly) connected subnetwork.

Degree:

The degree of a vertex is the number of lines incident with it.

Density:

The density of a simple network is the number of lines, expressed as a proportion of the maximum possible number of lines.

Digraph:

A digraph or directed graph is a graph containing one or more arcs.

Distance:

The distance from vertex u to vertex v is the length of the geodesic from u to v.

Edge:

An edge is an undirected line. Formally, an edge is an unordered pair of vertices.

Ego network:

The ego network of a vertex contains this vertex, its neighbors, and all lines among the selected vertices.

Geodesic:

A geodesic is the shortest path between two vertices.

Graph:

A graph is a set of vertices and a set of lines between pairs of vertices.

Incident:

A line is defined by its two endpoints, which are the two vertices that are incident with the line.

Indegree:

The indegree of a vertex is the number of arcs it receives.

Line:

A line is a tie between two vertices in a network: either an arc or an edge.

Loop:

A loop is a line that connects a vertex to itself.

Neighbor:

A vertex that is adjacent to another vertex is its neighbor.

Network:

A network consists of a graph and additional information on the vertices or the lines of the graph.

One-mode network:

In a one-mode network, each vertex can be related to another vertex.

Outdegree:

The outdegree of a vertex is the number of arcs it sends.

Path:

A path is a semipath with the additional condition that none of its lines are an arc of which the end vertex is the arc’s tail.

Reachable:

We say that a vertex is reachable from another vertex if there is a path from the latter to the former.

Semicycle:

A semicycle is a closed semipath ending at the vertex at which it starts.

Semipath:

A path is a closed sequence of lines such that the end vertex of one line is the starting vertex of the next line and no vertex in between the first vertex and the last vertex of the sequence occurs more than once.

Signed graph:

A signed graph is a graph in which each line carries either a positive or a negative sign.

Simple graph:

A simple undirected graph contains neither multiple edges nor loops. A simple directed graph does not contain multiple arcs.

Star network:

A star-network is a network in which one vertex is connected to all other vertices, but these vertices are not connected among themselves.

Strong component:

A strong component is a maximal subnetwork in which each pair of vertices is connected by a path.

Strongly connected:

A (sub)network is strongly connected if each pair of vertices is connected by a path.

Structural property:

A structural property is a characteristic (value) of a vertex that is a result of network analysis.

Triad:

A triad is a (sub)network consisting of three vertices.

Two-mode network:

In a two-mode network, vertices are divided into two sets and vertices can only be related to vertices in the other set.

Undirected graph:

An undirected graph does not contain arcs: All of its lines are edges.

Vertex (vertices):

A vertex (singular of vertices) is the smallest unit in a network.

Weakly connected:

A (sub)network is weakly connected if each pair of vertices is connected by a semipath.

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  1. University of Amsterdam, Amsterdam, The Netherlands

    Wouter de Nooy

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  1. Wouter de Nooy
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Correspondence to Wouter de Nooy .

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  1. Ramtech Ltd., Larkspur, California, USA

    Robert A. Meyers

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de Nooy, W. (2015). Social Network Analysis, Graph Theoretical Approaches to. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27737-5_488-3

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