Valued (weighted) networks

A network is valued when its ties carry a number, such as the number of joint arrests, the number of e-mails or the strength of a friendship, and not only present or absent. This page tells you when the program treats a network as valued, which commands then use the numbers, which ignore them, and how to ask for the binary result instead.

1. When is a network valued?
2. The rule for the commands
3. Commands that use the values by default
4. Commands that have their own measure option
5. Commands that look only at presence
6. Commands that refuse
7. Negative values
8. Data handling
9. Running a command on a cutoff version
10. References

When is a network valued?

The program keeps a flag, Valued, with every network. It is set when the network is created, by the way you create it, and it is not worked out again from the numbers later:

You create the network with	Valued?
nwfromedge with a value variable (the third variable), nw2fromedge with a value variable	yes, also when every value is 1
nwfromedge or nwset ..., edgelist without a value variable	no
nwimport of a file that gives tie values (Pajek lines with a third field, GML value or weight, a matrix or UCINET file with values other than 0 and 1)	yes
nwimport of a file without tie values	no
nwset from a matrix or from variables, with no statement of intent	yes if the matrix has a value other than 0 and 1, otherwise no (this is the only place where the numbers decide)
nwset, valued or nwset, unvalued; nwfromedge ..., valued or unvalued	as you say
nwuse of a saved network	the flag that was saved with it
nwdichotomize	no

Commands that transform a network (nwkeepnodes, nwdropnodes, nwsubset, nwtranspose, nwsym, nwcollapse, nwpermute, nwduplicate, nwrecode, nwreplace and others) keep the flag of their source. A binary network that gets a value other than 0 and 1 through such a command becomes valued.

nwsummarize shows the flag (Valued: true or false). If a saved network has the wrong flag, for example shares between 0 and 1 stored as binary by an earlier release, correct it with nwname netname, newvalued(true) (or newvalued(false)).

The rule for the commands

• A measure that has a weighted form in R uses the tie values of a valued network by default, as R does, and says so in its output. The option binary ignores the values and gives the result for presence only. binary is the same option in all of them. The older option weighted is still accepted where it existed and means the same as the default.
• A measure that has no weighted form in R is binary by definition. On a valued network it prints one line, (note: tie values are ignored; this measure is binary), once per call (not with silent or quietly). Use nwdichotomize to choose a cutoff.
• On a network that is not valued, nothing changes: every result is the one you got before.

Commands that use the values by default

Command	What the values do	R equivalent
nwdegree	degree is the strength (sum of the tie values); alpha(a) gives k (s/k)^a with k the number of ties and s the strength (Opsahl et al. 2010); binary or alpha(0) the number of ties; ties() stores the tie counts next to the strength; the output shows the mean number of ties and the mean strength	igraph::strength, tnet::degree_w (igraph::degree is the tie count)
nwdegree (nw2degree) on a two-mode network	the same for the two modes of a two-mode network (divided by the size of the other mode)	tnet::degree_tm
nwgeodesic	a step along a tie of value w costs 1/w^alpha, alpha = 1 by default, so a stronger tie is shorter; alpha(-1) reads the values as lengths; binary counts steps	igraph::distances(weights = 1/w); tnet::distance_w
nwcloseness	closeness from those distances	igraph::closeness(weights = 1/w)
nwbetween	shortest paths of least cost 1/w^alpha; tied shortest paths are recognised with a relative tolerance of 1e-10	igraph::betweenness(weights = 1/w); tnet::betweenness_w
nwevcent	eigenvector centrality of the matrix of tie values	igraph::eigen_centrality
nwpagerank	the surfer follows a tie with probability proportional to its value	igraph::page_rank
nwrandomwalk	hitting times of a walk that steps along a tie with probability proportional to its value	(the walk of igraph::random_walk)
nwpath	the path of least cost 1/w^alpha; r(path_cost) is the cost	igraph::shortest_paths(weights = 1/w)
nwmatching	the matching of largest total tie value, which can have fewer pairs than the largest matching; r(weight)	igraph::max_bipartite_match
nwassortativity	the node attribute is the strength instead of the degree	igraph::assortativity(values = strength)
nwkatz, walks	weighted walks	igraph::alpha_centrality
nwinduced	inherits the weighting of the centrality it uses (the degree measure counts ties)

The weighting convention differs between R packages. igraph treats the weights of its path functions as lengths; tnet and nwcommands treat them as strengths and use the cost 1/w^alpha. With alpha(-1) (nwgeodesic, nwpath, nwbetween) you get igraph’s convention. Closeness from tnet is the igraph value multiplied by the mean tie value; the ranking is the same.

A few details. nwgeodesic computes weighted distances for networks of up to 100 nodes; for a larger network it counts steps and says so, unless you add force. nwpagerank and nwrandomwalk on a valued network are computed in Mata and take longer than the binary version on very large networks. The directed nwevcent, nosym uses outgoing ties, igraph uses incoming ties.

Commands that have their own measure option

These use the values already, with an option of their own (measure(), stat(), …). The default for a valued network is the weighted form; measure(binary) or the option named in the table gives the binary form.

nwclustering (Opsahl and Panzarasa 2009; measure(binary)), nwconstraint, nwconcor, nwcoreperiphery, nwfactions, nwcommunity, nwmodularity, nwspectral (measure(binary)), nwdissimilar and nwhierarchy (euclidean and manhatten use the values), nwsimilar (pearson, crossproduct), nwcorrelate, nwcontext, nw2project (stat()), nw2clustering, nwqap, nwmaxflow (weighted: the values are capacities, as capacity is in igraph).

Commands that look only at presence

R has no weighted form of these measures. On a valued network they print the one-line note:

nwbridges, nwbrokerage, nwclique, nwcohesion, nwcomponents, nwdyads, nwego, nwkcomponents, nwkcore, nwkplex, nwmixing, nwmotifs, nwnclan, nwnclique, nwneighbor, nwreach, nwshared, nwsimindex, nwsimmelian, nwtabulate, nwtriads, nwturnover.

nwkatz without walks (decay by the number of steps), nwutility, nwlambda and nwaltergen have no weighted counterpart in R either; the first two use steps, nwlambda counts edge-disjoint paths (use nwmaxflow, weighted for capacities).

Commands that refuse

Command	When	Message
nwclustering	a weighted measure() on a directed network	“not defined for networks that are both weighted and directed. Either choose measure(binary) or symmetrize the network.”
nwcommunity, nwmodularity	directed network without symmetrize	“Either specify symmetrize or symmetrize the network first.”
nwmatching	one-mode network	requires a two-mode network

Negative values

Only nwbalance is built for signed networks. The other commands do not agree on what a negative tie is: some count it as a tie, some treat it as no tie, and some give values that make no sense. Split a signed network into a positive and a negative network with nwrecode or nwdichotomize, or use nwbalance.

Data handling

• Dichotomize: nwdichotomize net, threshold(k) generate(new) gives 1 to every dyad with a value of at least k and 0 to everything else; the result is binary. For gang_valued, threshold(2) reproduces the 133 ties of the binary gang network.
• Recode into a few levels: nwrecode net (1 2 = 1) (3/4 = 2), generate(new) (the result keeps the flag of the source).
• Combine the two directions when symmetrizing: nwsym net, mode(max) (or min, sum, mean).
• Save and load: values and flag come back unchanged in nwsave/nwuse, with full precision, and in Pajek and UCINET files; nwimport, type(gml) reads value or weight and the direction.

Running a command on a cutoff version

. nwwebuse gang, nwclear
. nwdichotomize gang_valued, threshold(2) generate(gang_strong)
. nwsummarize gang_strong
. nwbetween gang_strong, generate(btw_strong)

Report the cutoff and, ideally, show that the main result does not depend on it (try 1, 2 and 3).

References

Newman, M. E. J. (2004). Analysis of weighted networks. Physical Review E 70, 056131. Opsahl, T., Agneessens, F., and Skvoretz, J. (2010). Node centrality in weighted networks: Generalizing degree and shortest paths. Social Networks 32, 245-251. Opsahl, T., and Panzarasa, P. (2009). Clustering in weighted networks. Social Networks 31, 155-163. Leung, C. C., and Chau, H. F. (2007). Weighted assortative and disassortative networks model. Physica A 378, 591-602.