## Projection

Two-mode networks are rarely analysed in their original form. Although this is preferable, few methods exist for that purpose. As such, these networks are often transformed into one-mode networks (only one type of nodes) to be analysed. This procedure is often referred to as projection. Projection 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 this diagram. For example, node E and node F would be connected as they have ties to a common red node. Projected networks often contains large fully-connected cliques as ties are formed among all nodes in the one-mode network that were connected to a single other node in the two-mode structure. These networks are often binary one-mode networks (admittedly, I am guilty of analysing binary projections, e.g., Seierstad and Opsahl, 2011); however, information from the two-mode network can be incorporated as tie weights. The usefulness of this approached has been illustrated by Padrón et al. (2011). By studying a number of networks, the found that this approach “allow us to make more realistic predictions on the potential competitive or facilitative interactions among species of one set (e.g. plants) that share species of the other (e.g. flower visitors or dispersers).”

*Binary Two-mode Networks*

Traditionally, the ties in projected one-mode networks do not have weights attached to them. However, recent empirical studies of two-mode networks has created a weighted one-mode network by defining the weights as the number of co-occurrences (e.g., the number of events two individuals have co-attended or the number of papers that two authors had collaborated on). To formally describe this method and ease the comparison among the different methods introduced in this post, this method can be formalised as: where where is the weight between node and node , and is the nodes of the other kind that node and node are connected to, their co-occurrences (e.g., the red nodes in the above diagram). In the sample network, the tie between node A and node B has a weight of 2 as these two nodes have connections to two common red nodes, whereas the tie between node A and node C has only a weight of 1 as these nodes have connections to merely one common red node.

Newman (2001) extended this procedure while working with scientific collaboration networks. He argued that the social bonds among scientist collaborating with few others on a paper were stronger than the bonds among scientists collaborating with many on a paper. He proposed to discount for the size of the collaboration by defining the weights among the nodes using the following formula: where is the number of authors on paper (e.g., the number of blue nodes connected to the red node ). In the context of scientific collaboration networks, this implies that if two scientists who only write a single paper together with no other co-authors get a weight of 1 (e.g., node B and node D). Moreover, if two scientists have written two papers together without any co-author, the weight of their tie would be 2 (e.g., node B and E). However, if the two scientists have a co-author, the weight on the tie between them is 0.5 (e.g., node A and node C). A more complicated example is the tie between node A and node B in the diagram. They have written two papers together: one without any other co-author and one with node C as a co-author. The first paper would give their tie a weight of 1, and the second tie would add 0.5 to the weight of this tie. Therefore, the weight attached to their tie is 1.5. By discounting for the number of blue nodes attached to the same red node, this methods creates a one-mode projection in which the strength of a node is equal to the number of ties originating from that node in the two-mode network (e.g. the sum of weights attached to ties originating from node A in the one-mode projection is 2, and node A is connected to two red nodes).

*Weighted Two-mode Networks*

So far, the methods has only dealt with binary two-mode networks. However, weighted two-mode networks also exist. For example, in online forums, where the node sets are users and threads (or topics), and a tie between two nodes is established if a user posts a message to a thread, it is possible to differentiate the two-mode ties based on the number of messages or characters posted by users to a specific thread. Another example is purchasing behaviour where the node set are people and goods, and the weight can be defined in terms of number of purchases. Similarily to weighted one-mode networks, the data used for analysis is richer if the tie weights are included in the two-mode network than if they are discarded. A weighted two-mode network is illustrated in the diagram to the right.

In a similar spirit as simply the number of co-occurrence for a binary two-mode network, the one-mode projection of a weighted two-mode network could be based on the weights the two nodes have directed towards common nodes (of different kind). In addition, with such a method it is possible to differentiate how the two nodes interact with the common node, and to project it onto a directed weighted one-mode network. In this type of projected network, the weight of a tie from a node to another is not necessarily the same as the weight attached to the tie from the latter node to the former node. It is not a truly directed network as two directed ties exist between any connected node pair. More specifically, all dyads are composed of either two directed ties (mutual) or no directed ties (null), and no dyads are made of a single directed tie (asymmetric). For example, if the diagram above referred to an online forum, node B has posted 4 messages in to a thread that node D participate in, thus giving the directed tie between node B and node D in the one-mode projection a weight of 4. Conversely, node D posted 6 messages to that thread, and therefore, the weight attached to the tie from node D to node B is equal to 6. This method can be formalised as follows: .In a similar spirit as the method used by Newman (2001), it is also possible to discount for the number of nodes when projecting weighted two-mode networks. For example, it could be argued that if many online users post to a thread, their ties should be weaker than if there were few people posting to the thread. A straight forward generalisation is the following function: . This formula would create a directed one-mode network in which the out-strength of a node is equal to the sum of the weights attached to the ties in the two-mode network that originated from that node. For example, node C has a tie with a weight of 5 in the two-mode network and an out-strength of 5 in the one-mode projection.

**Want to test it with your data?**

*Binary two-mode networks*

The following code requires a binary two-mode network to be listed in an edgelist format with two columns named i and p. The i column refers to the nodes you would like to keep in the one-mode projection (e.g., the blue nodes), and the p column refers to the nodes you would like to discard (e.g., the red nodes). The binary two-mode network in the diagram above can be loaded using the following function.

net <- cbind( i=c(1,1,2,2,2,2,2,3,4,5,5,5,6), p=c(1,2,1,2,3,4,5,2,3,4,5,6,6))

The one-projections highlighted above can be created using the following code:

# Load tnet library(tnet) # Binary one-mode projection projecting_tm(net, method="binary") # Simply the number of common red nodes projecting_tm(net, method="sum") # Newman's method projecting_tm(net, method="Newman")

*Weighted two-mode networks*

The following code requires a weighted two-mode network to be listed in an edgelist format with three columns named i, p, and w. The i column refers to the nodes you would like to keep in the one-mode projection (e.g., the blue nodes), the p column refers to the nodes you would like to discard (e.g., the red nodes), and the w column must be the weight of the ties. The weighted two-mode network in the diagram above can be loaded using the following function.

net.w <- cbind( i=c(1,1,2,2,2,2,2,3,4,5,5,5,6), p=c(1,2,1,2,3,4,5,2,3,4,5,6,6), w=c(4,2,2,1,4,3,2,5,6,2,4,1,1))

The one-projections highlighted above can be created using the following code:

# Load tnet library(tnet) # Simply the sum of weights towards common red nodes projecting_tm(net.w, method="sum") # Generalisation of Newman's method projecting_tm(net.w, method="Newman")

**References**

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

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

Opsahl, T., 2013. Triadic closure in two-mode networks: Redefining the global and local clustering coefficients. Social Networks 35, doi:10.1016/j.socnet.2011.07.001.

Padrón, B., Nogales, M., Traveset, A., 2011. Alternative approaches of transforming bimodal into unimodal mutualistic networks. The usefulness of preserving weighted information. Basic and Applied Ecology, doi:10.1016/j.baae.2011.09.004.

Seierstad, C., Opsahl, T., 2011. For the few, not the many? The effects of affirmative action on presence, prominence, and social capital of women directors in Norway. Scandinavian Journal of Management 27 (1), 44-54.

1.Snehal | September 13, 2012 at 11:16 pmHi Tore

Great resource particularly for weighted 2-mode networks.

I am working with 3-mode networks and figured that I’ll have to convert (project) it to 2 modes to get some measures. Can you suggest any literature that deals with this conversion? Right now I am just replicating the 2->1 mode procedures. Your comments/suggestions are greatly appreciated.

Thanks!

2.Tore Opsahl | September 17, 2012 at 6:33 pmHi Snehal,

Thanks! I have not directly worked with 3-mode networks, so I haven’t given too much thought. I think this is an exciting area of new research with many unanswered questions. Do you project from 3-mode to two-mode by connecting primary and secondary nodes connected to the same tertiary nodes? What do you do with existing connections between primary and secondary nodes?

Best,

Tore

3.Snehal | September 17, 2012 at 10:35 pmThanks Tore for your reply. Yes, at this point, I am considering that primary and secondary nodes are connected if they share a tertiary node. And, the number of common tertiary nodes is captured in the tie strength. Interestingly, given the nature of my original data, primary and secondary nodes are only connected to a tertiary node (women attending event, women having some sort of expertise). So I guess the 2nd question hasn’t been an issue with this dataset.

Best

Snehal

4.Mary | October 31, 2012 at 5:48 amHi Tore,

Thank you for your article. I was looking for something which would be illustrative and precise. And your article is just that. I am currently working on a two mode bipartite network. Do you know how I can use neighborhood measures like Common Neighbors in the bipartite setting? Thanks

5.Tore Opsahl | October 31, 2012 at 1:49 pmHi Mary,

Glad you found it useful. I am not entirely sure what you are referring to when you mention neighborhood measures. However, if you would like to calculate the number of common nodes between primary nodes in a two-mode network, you can use the projecting_tm(net, method=”sum”). This function will produce pairs of primary nodes (columns 1 and 2) with a third column with the number of common nodes.

Best,

Tore

6.David Fisher | January 17, 2013 at 2:24 pmHi Tore,

Great blog, this section in particular is my go-to guide to two-mode networks.

I am interesting in using the geometric mean, instead of the sum, to calculate the strength of an interaction between 2 nodes when projecting a 2-mode network onto a one-mode network.

The strength of the interaction between nodes A and B in the above example (leaving out Newman’s correction for now) would be 2 in this case (or 4.24 if you sum the weights directed at nodes they share before taking the geometric mean).

Do you think this is sensible, and can it be done in tnet?

Cheers

David

7.Tore Opsahl | January 21, 2013 at 8:37 pmHi David,

Great suggestion. It does sound like a good idea as differences in tie weights would discount the tie strength. I haven’t implemented this type of projection method for two-mode networks; however, if you connect with me by email, this code can easily be created.

Best,

Tore

8.Martin Grandjean | March 12, 2013 at 9:46 pmWhat a great description! I’ve just published a “state of my research” on my blog (www.martingrandjean.ch/analyse-de-reseau-cartographier-lactivite-de-la-societe-des-nations/) and as a historian (passionate about network analysis but without maths studies) I was wondering how to justify theoretically this “projection”. Thank you for these keys!

9.Tore Opsahl | March 12, 2013 at 10:43 pmHi Martin,

Great to see more open science being done! I’m not sure there is a specific theory I can point you too regarding projects as there is so many. For example, two-mode networks with people and events can be projected due to assumed temporal geographical co-location. However, I believe it really depends on your context which theories of connectedness would apply.

Best,

Tore