Tore Opsahl

A system whose elements are somehow connected can be represented as a network. The elements of a system is represented as nodes (also known as actors or vertices) and the connections among interacting elements are known as ties, edges, arcs, or links. The nodes might be neurons, individuals, groups, organisations, airports, or even a countries, whereas ties can take the form of friendship, communication, collaboration, alliance, flow, or trade, to name only a few.

Not all ties in a network have the same capacity. For example, in a social network some contacts are friends, whereas others are simply acquaintances. It would be grossly unfair to treat everyone in the same manner.

By recording the strength of ties, we can create a weighted network. However, these networks are more difficult to analyse than if ties were simply present or absent. I devoted a large part my Ph.D. thesis to the analysis of these networks.

Basic network definitions

Throughout this blog I make use of some network terminology beyond nodes and ties, and I would like to clarify my definition of these terms.

Undirected / directed network

Networks can be directed or undirected. This refers to whether ties are created from one node and directed towards another, or simply tie two nodes together. For example, a network where ties are created when people ask for advice from others are generally recorded as directed (Lazega, 2001). This is because asking for advice refers to social interactions in which knowledge flows from one node to another in a specific direction. In these networks, the weight of the tie from a node to another might be different from the weight from the other node to the initial node (e.g. the two ties between nodes A and B in the first sample network). Conversely, a network where ties are formed when two people collaborate on a project tends to be undirected. This is due to the fact that collaboration usually implies a two-way interaction between the nodes. In these network, a tie is not “produced” by one node and directed towards another, but is a joint product of both the nodes.

Degree and Strength of nodes

The degree of a node is equal to the number of other nodes the node is connected to. For example, node E in the above undirected network has a degree of 2 since it is connected to nodes B and F. In a directed network, each node can have an out-degree and in-degree. Out-degree is the number of nodes that the node connect to, whereas in-degree is the number of nodes that connect to the node. In the directed network, node E would have an out-degree of 2, but only an in-degree of 1 since only node B has directed a tie towards it.

The strength of a node is equal to the sum of weights attached to the ties that connect a node to others. For node E in the undirected network, this would be equal to 3. In a similar fashion as degree, in a directed network, each node has an out-strength and an in-strength. Out-strength is the sum of weights attached to ties originating from the node, whereas in-strength is the sum of weights attached to ties directed toward the node. Therefore, node E would have an out-strength of 3, but only an in-strength of 2.

One-/two-mode networks

Most networks are analysed as one-mode networks. These are simple networks with one set of nodes and ties among these nodes. For example, an inter-organisational network where the nodes are employees and ties are formed between two nodes when one employee asks other one for advice. However, many network dataset are by definition two-mode networks (also known as affiliation or bipartite networks). These are a particular type of networks with two sets of nodes and ties are only established between nodes belonging to different sets. The first network in this diagram illustrates a binary two-mode network where the colour represent the node set to which a node belongs. One of the first two-mode datasets to be analysed was the Davis Southern Club Women dataset (Davis et al., 1941), which recorded the attendance of a group of women (node set 1) to a series of events (set 2). A woman would be linked to an event if she attended it. Another type of two-mode dataset that has become popular in recent years is scientific collaboration networks. In this type of networks, a tie is established between a scientist (node set 1) and a paper (node set 2) if the scientist authored that paper (e.g., Newman, 2001).

Although the two-mode structure contains a number of details of the network, few network measures exist for two-mode networks. Therefore, these networks are often projected onto a one-mode network (only one type of nodes). This is done by selecting one of the sets of nodes and linking two nodes from that set if they were connected to the same node (of the other kind). This process is illustrated for the blue nodes of the second part of the diagram above. For example, node E and node F are connected to the same red node, therefore, in the one-mode projection they are tied together.

References

Davis, A., Gardner, B. B., Gardner, M. R., 1941. Deep South. University of Chicago Press, Chicago, IL.

Lazega, E., 2001. The Collegial Phenomenon: The Social Mechanisms of Cooperation Among Peers in a Corporate Law Partnership. Oxford University Press, Oxford, UK.

Newman, M. E. J., 2001. Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality. Physical Review E 64, 016132.