Two-mode networks

In a two-mode (bipartite, affiliation) network there are two kinds of nodes, such as people and the events they attend, and ties run only between the kinds. This page tells you how the program knows that a network is two-mode, what every command does with it, what the results mean and where they differ from R. The tutorial on two-mode networks shows the commands on the Southern women and the Swedish directors.

1. How a network becomes two-mode
2. Commands that make a new network keep the two modes
3. The rule: a two-mode network gets its two-mode form
4. What each command does
5. Measures with a two-mode form in R
6. Edge lists and files
7. Not done yet
8. References

How a network becomes two-mode

The program keeps a flag, Two-mode, with every network, together with the mode (1 or 2) of every node and a label for each mode. The flag is set when the network is created, by declaration, and nowhere else:

You create the network with	Two-mode?
nwset ..., twomode (edge list of the ties between the two modes) or bipartite (a rectangular matrix)	yes
nw2fromedge (two id columns), nwfromedge ..., twomode, nw2set	yes
nwimport of a Pajek file with *Vertices n n1, a rectangular UCINET matrix, a five-column edge list or a GML file with the node attribute nwmode (all written by nwexport)	yes
nwuse of a saved two-mode network	yes, with its modes and labels
any other command	no

No command works the flag out from the ties, because the adjacency matrix of a two-mode network looks like that of a one-mode network. The declaration is checked: both modes must hold nodes and no tie may join two nodes of the same mode (nwname net, new2mode(true) on a one-mode network, or an import of a file with such a tie, stops with error 459; nwreplacemat refuses a matrix with a tie inside a mode).

A two-mode network is stored as one square matrix over the nodes of both modes. The cells between nodes of the same mode hold missing values, not 0. The nodes are ordered by name, not by mode; _nwmode (1 or 2) says which mode a row belongs to.

Commands that make a new network keep the two modes

nwdropnodes, nwkeepnodes, nwsubset, nwduplicate, nwsym, nwtranspose, nwrecode, nwdichotomize, nwaddnodes (the option mode() is required) and nwpermute return a two-mode network with the modes of the remaining nodes and the level labels. (Before this version, nwdropnodes and nwkeepnodes returned a one-mode network, and the other commands lost the level labels.) nwcollapse refuses a two-mode network. nw2project and nwproject return an ordinary one-mode network of one mode.

The rule: a two-mode network gets its two-mode form

A network declares itself two-mode when it is created. A command that has a two-mode form of the same concept uses that form automatically on a two-mode network, and the option onemode forces the one-mode computation on the bipartite graph, which is what igraph does with a bipartite graph. The commands are nwdegree, nwclustering, nwmotifs, nwcloseness, nwbetween, nwcommunity, nwmodularity, nwsimilar and nwdissimilar. The older names nw2degree, nw2clustering and nw2motifs stay as aliases of nwdegree, nwclustering and nwmotifs (same options, same results) and still stop on a one-mode network. nw2project and nw2nested have no one-mode counterpart and keep their own names.

What each command does

The help file of every command states in its section “Supported network types” one of four fixed wordings:

Wording	Meaning	Commands
Two-mode: yes - what changes	The command uses the two modes the way R does.	nw2project, nw2nested, nwmatching, nwpermute, nwdyads, nwsummarize, nwexport, nwimport, nwtoedge, nw2toedge, nwplot, nwplotmatrix, nwturnover and the commands that keep the modes above
Two-mode: two-mode form by default …	The command has a two-mode form of the same concept and uses it on a two-mode network; onemode forces the one-mode computation.	nwdegree, nwclustering, nwmotifs, nwcloseness, nwbetween, nwcommunity, nwmodularity, nwsimilar, nwdissimilar (nw2degree, nw2clustering, nw2motifs are aliases)
Two-mode: same as for the bipartite graph read as a one-mode network	The measure is defined on any graph and the number is right for the bipartite graph; nothing is printed.	nwcomponents, nwkcore, nwgeodesic, nwevcent, nwpagerank, nwburt, nwconstraint, nwbridges, nwcohesion, nwtriads, … (see the table)
Two-mode: not supported - error 6088	The command would read the blocks between nodes of the same mode, which hold no ties. It stops with one message.	nwspectral, nwrandomwalk, nwcorrelate, nwhierarchy, nwconcor, nwfactions, nwcoreperiphery, nwqap, nwinduced, nwcollapse

The message is the same for all of them: “command does not support two-mode networks (net): it reads the blocks between nodes of the same mode, which hold no ties. Make a one-mode network of one mode first (nwproject).” nwspectral and nwrandomwalk add that the option onemode reads the bipartite graph as a one-mode graph, which is what igraph does (the Laplacian spectrum and the hitting times of a random walk exist for a bipartite graph).

Measures with a two-mode form in R

Command	Two-mode result	R
nwdegree (nw2degree)	degree divided by the size of the other mode (Borgatti and Everett 1997); raw gives the count	tnet::degree_tm (the count)
nwclustering (nw2clustering)	Opsahl’s (2013) coefficient, the share of four-paths closed by a fifth tie: Southern women .772, directors .0029	tnet::clustering_tm
nwclustering, ra	Robins and Alexander’s (2004) global coefficient 4 C4 / L3: Southern women .468, directors .0228	(formula; igraph has none)
nw2project	the projection onto one mode; stat(count) is the number of shared nodes, stat(newman) Newman’s weight 1/(k - 1) per shared node	igraph::bipartite_projection, tnet::projecting_tm
nw2nested	NODF and the C-score (Southern women NODF 48.6, C-score normalised .449)	vegan::nestednodf, bipartite::nested, bipartite::C.score
nwmotifs (nw2motifs)	the 17 bipartite motifs of two to five nodes (44 with sixnode) of Simmons et al. (2019) and the 46 (148) positions of every node; counts ties above zero (Southern women: 89 single ties, 4-cycles motif 6 = 341, as bmotif)	bmotif::mcount, bmotif::node_positions
nwcommunity, nwmodularity	Barber’s (2007) bipartite modularity; onemode gives Newman’s. Southern women: Q .3455 with 4 communities, as computeModules	bipartite::computeModules
nwsimilar, nwdissimilar	a node is compared with the nodes of its own mode over the nodes of the other mode; pairs of nodes from different modes are missing	sna::sedist, cor, dist on the incidence matrix
nwpermute	the labels are permuted within the modes; swap exchanges the partners of two ties and keeps both degree sequences	tnet::rg_reshuffling_tm, bipartite::swap.web
nwcloseness, nwbetween	the normalisations of Borgatti and Everett (1997): the smallest possible farness over the farness, the betweenness over the largest possible value of a node of its mode; onemode gives the raw numbers of the bipartite graph (twomode, the earlier way to ask for the normalisation, is accepted)	(formulas; igraph::closeness, igraph::betweenness for onemode)
nwmatching	maximum matching between the modes	igraph::max_bipartite_match
nwdyads	null dyads counted over the pairs between the modes (Southern women 163 of 252 pairs); reciprocity is shown as n/a

Two things are worth knowing. The density of a two-mode network is the number of ties over the pairs of one node of each mode (Southern women .353 = 89 / 252, the connectance of the bipartite package), not over all pairs of nodes. And nw2degree, nw2clustering, nw2motifs, nw2project and nw2nested need a two-mode network and stop on a one-mode network; nwdegree, nwclustering and nwmotifs work on both.

Edge lists and files

nwtoedge and nw2toedge write a two-mode network as its ties, one row per tie, with the mode of both nodes; allpairs writes the row of every pair of nodes of different modes as before. Read the list back with nwfromedge ..., twomode or nw2fromedge: the network has the same nodes (the isolated ones included), the same ties and the level labels.

nwexport and nwimport keep the modes in all four formats:

Format	Two-mode form
Pajek	*Vertices n n1, the nodes of the first mode first, each tie from a first-mode to a second-mode vertex
UCINET	a rectangular matrix (nr, nc) with row and column labels
edge list	five columns: ego, alter, value, mode of the ego, mode of the alter; an isolated node has no row
GML	the node attribute nwmode (1 or 2); in R, V(g)$type <- V(g)$nwmode == 2

Pajek, the edge list and GML do not carry the names of the two levels; they come back as mode 1 and mode 2. nwsave and nwuse keep them.

Not done yet

Fast tie swapping for networks with hundreds of thousands of ties and the further indices of bipartite::networklevel (H2’ and others). Two-mode exponential random graph terms (b1 and b2) belong to nwergm; the core only keeps the flag and the modes correct for them.

References

Almeida-Neto, M., Guimaraes, P., Guimaraes, P. R., Loyola, R. D., and Ulrich, W. (2008). A consistent metric for nestedness analysis in ecological systems. Oikos 117, 1227-1239. Barber, M. J. (2007). Modularity and community detection in bipartite networks. Physical Review E 76, 066102. Borgatti, S. P., and Everett, M. G. (1997). Network analysis of 2-mode data. Social Networks 19, 243-269. 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, 159-167. Robins, G., and Alexander, M. (2004). Small worlds among interlocking directors: Network structure and distance in bipartite graphs. Computational and Mathematical Organization Theory 10, 69-94. Stone, L., and Roberts, A. (1990). The checkerboard score and species distributions. Oecologia 85, 74-79.