# Networks in Archaeology: Phenomena, Abstraction, Representation

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## Abstract

The application of method and theory from network science to archaeology has dramatically increased over the last decade. In this article, we document this growth over time, discuss several of the important concepts that are used in the application of network approaches to archaeology, and introduce the other articles in this special issue on networks in archaeology. We argue that the suitability and contribution of network science techniques within particular archaeological research contexts can be usefully explored by scrutinizing the past phenomena under study, how these are abstracted into concepts, and how these in turn are represented as network data. For this reason, each of the articles in this special issue is discussed in terms of the phenomena that they seek to address, the abstraction in terms of concepts that they use to study connectivity, and the representations of network data that they employ in their analyses. The approaches currently being used are diverse and interdisciplinary, which we think are evidence of a healthy exploratory stage in the application of network science in archaeology. To facilitate further innovation, application, and collaboration, we also provide a glossary of terms that are currently being used in network science and especially those in the applications to archaeological case studies.

## Keywords

Archaeology Network science Social network analysis Relational archaeology## Notes

### Acknowledgments

The authors are grateful to the contributors to this special issue and to all speakers at the session of the Society of American Anthropology in Hawaii in 2013 from which this special issue is derived, including the discussant Ian Hodder. We would also like to thank Catherine Cameron and James Skibo for their help in compiling and editing the contributions.

## Glossary

*e.g.*, 1–2 indicates that node one is connected to node 2). A key reference work for most of the concepts described here is that by Wasserman and Faust (1994), in which more elaborate descriptions, mathematical formulations, and additional bibliographic resources can be found. A limited number of additional primary sources are given in this glossary and included in a separate bibliography below.

The glossary presented here benefited greatly from discussions with members of the algorithmics group at the Department of Computer and Information Science of the University of Konstanz, John M. Roberts Jr., and the contributors to this special issue. It is coauthored with Habiba. The authors of this paper are solely responsible for any remaining mistakes in this glossary.

See node.

Defined as a directed network with no cycles.

For example, the directed network in Fig. 5b is not acyclic because it includes the cycle 2-3-4-2. Examples of acyclic networks include citation networks and dendrograms.

Defined as a way of representing a network where there is a row and a column for each node, and the values in the cells indicate whether an edge exists between a pair of nodes.

See two-mode network.

See directed edge.

See degree.

See geodesic.

A node’s betweenness centrality is defined as the fraction of the number of geodesics passing through this node over the number of geodesics between all pairs of nodes in the network.

Nodes with a high betweenness centrality are often considered to be important intermediaries for controlling the flow of resources between other nodes, because they are located on paths between many other node pairs. The concept of brokerage is often mentioned in relation to betweenness centrality. Nodes which are incident to the *only* edge connecting two subsets of nodes in the network are in a position to broker the relationship between these nodes. These nodes will typically have a high betweenness centrality but not necessarily a high degree or closeness centrality. For example, in Fig. 4 if the only route between towns A and C travels through B, town B may be in a position to tax any goods traveling between them, to control what information travels between towns, or to isolate one or both. Betweenness centrality was first quantitatively expressed by Anthonisse (1971) and Freeman (1977).

See two-mode networks.

Defined as a partitioning into blocks of structurally equivalent nodes, where the blocks are connected by hypothesized edges.

In blockmodeling, the rows and columns of adjacency matrices are arranged so that structurally equivalent nodes are in adjacent positions in the matrix, and the edges between different blocks can be studied. First introduced by White, Boorman, and Breiger (1976).

See betweenness centrality.

Defined as a family of measures of the node’s position within the network, which represent a ranking of nodes (see betweenness, closeness, degree, and eigenvector centralities).

Centrality measures are used to identify the most important or prominent nodes in the network, depending on the different definitions of importance or prominence implemented in the network measure used.

A clique is a subset of nodes in a network, where every pair of nodes is connected by an edge.

In the social sciences, only cliques of three nodes or more are usually considered. For example, in Fig. 5a, nodes 2, 3, and 4 form a clique of size 3. The definition of a clique is independent of whether it applies to the whole network or not. For example, a network can consist of multiple cliques, or an entire network can be one clique. The latter can also be called a complete network.

The closeness centrality of a node is defined as the inverse of the sum of the geodesics of that node to all other nodes divided by the number of nodes in the network.

The closeness centrality of a node gives an indication of how close this node is to all other nodes in the network, represented as the number of steps in the network that are necessary on average to reach another node. Nodes with a high closeness centrality score could be considered important or prominent, since they can share and obtain resources in less steps than other nodes. Early quantitative implementations of closeness centrality are reviewed by Freeman (1979).

Defined as the number of closed triplets over the total number of triplets in a network, where a triplet is a set of three nodes with two (open triplet) or three (closed triplet) undirected edges between them.

The clustering coefficient represents the average probability that two nodes connected to a third node are themselves connected, and it is commonly used for this purpose since the publication of the “small-world” network model (Watts and Strogatz 1998). The clustering coefficient of a network is closely related to the concept of transitivity in the social sciences, which captures the notion that “a friend of a friend is a friend.” Transitivity refers to the tendency of an open triplet to become a closed triplet.

See density.

See clique.

Defined as a subset of an undirected network in which any pair of nodes can be connected to each other *via* at least one path, and where there can be no paths to any nodes outside this subset.

For example, in the undirected network in Fig. 5a, there are two connected components: node 5 and nodes 1, 2, 3, and 4.

See edge.

Defined as a path in a directed network in which the starting node and ending node are the same. It is also called a closed path.

For example, in Fig. 5b, the path 2-3-4-2 is a cycle. See also acyclic network.

The degree of a node is defined as the number of edges incident to this node.

The average degree of a network is the sum of the degrees of all nodes in this network divided by the number of nodes. In a directed network, the indegree of a node refers to the number of incoming incident edges of a node. In a directed network, the outdegree of a node refers to the number of outgoing incident edges of a node. For example, node 2 in Fig. 5a has a degree of 3, while the same node can be said in Fig. 5b to have an indegree of 2 and an outdegree of 1.

Defined as the centrality of a node based on the number of edges incident to this node.

According to the degree centrality measure, a node is important or prominent if it has edges to a high number of other nodes.

Defined as the probability distribution of all degrees over the whole network.

The measure is commonly used to compare the structure of networks since the publication of the “scale-free” network structure (Albert and Barabási 2002, p. 49; Barabási and Albert 1999; Newman 2010, p. 243–247). In scale-free networks, the degree distribution follows a power-law.

Defined as the fraction of the number of edges that are present to the maximum possible number of edges in the network.

Cohesion is a commonly used concept which is often operationalized using the density measure. For other cohesion measures, see Wasserman and Faust (1994, 249–290).

Defined as the length of the longest geodesic in the network.

Defined as an ordered pair of nodes, which is often graphically represented as an arrow drawn from a starting node to an end node.

A directed edge is asymmetric. It connects a starting node with an ending node and cannot be traversed in the other direction. For example, all edges in Fig. 5b are directed edges.

Defined as a set of nodes and a set of directed edges.

A path through a directed network will need to follow the direction of the directed edges. For example, the network in Fig. 5b is a directed network.

See path length.

Defined as any pair of nodes in a network that may or may not have an edge between them.

For undirected edges, there are two possible dyadic relationships: connected, or not connected. For directed edges, there are four: unconnected; connected in one direction; connected in the other direction; and connected in both directions.

Defined as a line between a pair of nodes, representing some kind of relationship between them.

Many synonyms exist to refer to an edge, including tie, arc, relationship, link, connection, and line. An edge can be directed or undirected, and weighted or unweighted. The concept “arc” is often used to refer to a directed edge.

Defined as a network consisting of a node (called ego), the nodes it is directly connected to, and the edges between these nodes.

The eigenvector centrality of a node is defined in terms of the eigenvector centrality of nodes incident on it.

More descriptively, instead of assigning a single centrality score to a node, a node’s eigenvector centrality is defined in terms proportional to the nodes incident on it. A node with a high eigenvector centrality is a node that is connected to other nodes with a high eigenvector centrality. See Newman (Newman 2010, 169–172) for the procedure to calculate eigenvector centrality.

A polyvalent concept that comprises two variants. The first is the structural integration of a node or any group of nodes within the network. Different measures for structural integration exist. The E/I index is one example of this, which is calculated as a ratio of the number of edges within a group of nodes (internal) and between groups of nodes (external). The second variant, as popularized in economic theory through the work of Polanyi (1944) and Granovetter (1985), relates to the intertwined nature of social, economic, political, religious, and cultural interactions. See Borck *et al.* (2015) and Hess (2004) for overviews of this concept.

See structural equivalence.

Defined as the path between a pair of nodes with the shortest path length. Sometimes referred to as the shortest path length between a pair of nodes. For example, the geodesic between nodes 1 and 3 in Fig. 5b is the path 1–3 with a length of 1. The average shortest path length is the average of all geodesics in a network.

See network.

Defined as a tendency of nodes to become connected to other nodes that are dissimilar under a certain definition of dissimilarity.

Defined as a tendency of nodes to become connected to other nodes that are similar under a certain definition of similarity.

For example, in Fig. 6b, a node with an attribute value represented in gray will have a tendency of being connected to a node with the same attribute value.

See degree.

Defined as nodes in a network which have no incident edges.

For example, node 5 in Fig. 5a is an isolate.

See edge.

See edge.

Defined as a set of nodes and a set of edges.

In mathematics, a network is referred to as a graph, while networks often consist of social nodes and edges, and are referred to as social networks in the social sciences.

Defined as an atomic discrete entity representing a network concept.

A vertex (plural vertices) is a commonly used synonym to refer to a node. The term actor is sometimes used as a synonym for nodes in the social sciences.

See two-mode network.

See degree.

Defined as a walk between a pair of nodes in which no nodes and edges are repeated.

For example, nodes 1 and 3 in Fig. 5b are connected by the path 1-2-3.

Defined as the number of edges in a path.

For example, nodes 1 and 3 in Fig. 5b are connected by the path 1-2-3, which has a path length of 2.

Defined as a mathematical relationship between two entities where the frequency of one entity varies as a power of the second entity. More formally, the probability of a node with degree *k* is proportional to *k* ^{ a }.

Commonly used to describe the degree distribution of networks with a scale-free structure (Barabási and Albert 1999). When a network’s degree distribution follows a power-law, it implies that few nodes have a much higher degree than all other nodes in the network and most nodes have a very low degree. Nodes with a very high degree, sometimes referred to as “hubs” in the network, significantly reduce the average shortest path length of the network.

See edge.

See geodesic.

A small-world network is defined as a network in which the average shortest path length is almost as small as that of a uniformly random network with the same number of nodes and density, whereas the clustering coefficient is much higher than in a uniformly random network (a uniformly random network is defined as a network in which each edge exists with a fixed probability *p*).

The small-world network structure as described here was first published by Watts and Strogatz (1998). This structure illustrates that relatively few edges between clusters of nodes are needed to significantly reduce the average shortest path length. It implies that resources can flow between any pairs of nodes in the network relatively efficiently, while maintaining a high degree of clustering.

See network.

Defined as a connected component in a directed network.

In a directed network, a connected component is always either strongly or weakly connected. For example, in Fig. 5b, node 5 is a connected component, while the set of nodes 1, 2, 3, and 4 are not because node 1 cannot be reached by a path from the other nodes.

A number of theoretical network models used in the social sciences rely on a distinction between strong and weak ties (particularly those drawing on Granovetter 1973). The distinctions between the two, however, are rarely formally defined. In general, strong ties are used to describe frequently activated relationships (such as family/kin ties) whereas weak ties are used to describe infrequently accessed connections (acquaintances). Strong ties tend to be among actors with similar sets of overlapping relationships whereas weak ties more often connect sets of actors who would otherwise be unconnected. In weighted networks, thresholds on the distribution of weights across a network as a whole are often used to define strong *vs.* weak ties though there are no consistent rules used for this distinction.

Defined as two nodes are structurally equivalent if they have identical edges to and from all other nodes in the network (see Lorrain and White 1971).

Structural equivalence is used to identify nodes which have the same position in a network. It can be used to inform blockmodeling. In the social sciences, the structural similarities identified through structural equivalence are used to study social positions and social roles.

See edge.

See clustering coefficient.

Defined as a network in which two sets of nodes are defined as modes. In a two-mode network, nodes of one mode can only be connected to nodes of another node.

Defined as an unordered pair of nodes, which is often graphically represented as a line drawn between the pair of nodes.

An undirected edge is symmetric. Typically just called an edge. For example, all edges in Fig. 5a are undirected edges.

A set of nodes and a set of undirected edges.

An undirected network is symmetric. For example, the network in Fig. 5a is an undirected network.

Defined as an edge which is not weighted.

Typically just called an edge. See also weighted edge.

Defined as a set of nodes and a set of unweighted edges.

See weighted network.

See node.

A walk between a pair of nodes is defined as any sequence of nodes connected through edges which has that pair of nodes as endpoints.

In contrast to a path, nodes and edges can be repeated in a walk. For example, in Fig. 5b, a walk between nodes 2 and 3 could be 2-3-4-2-3.

Defined as a connected component in a directed network where the directionality of edges is ignored.

In a directed network, a connected component is always either strongly or weakly connected. For example, in Fig. 5b, there are two weakly connected components: node 5 and nodes 1, 2, 3, and 4.

See strong tie.

Defined as an edge with a value associated to it.

These values are often real numbers but they can also be any concept connecting the end nodes of the edge. The definition of an edge weight depends on the research context. Weights could be represented as an attribute of an edge. Thresholding can be applied to select a subset of edges with a given edge weight.

Defined as a set of nodes and a set of weighted edges.

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