How we study this
Most studies of who works with whom ask about one relationship at a time. Does this school work with the police? Does that one work with a charity?
A two-mode network puts them back together. The easiest way to see what one is starts with something you already know.
Think of the clubs at a school. You could write down who is friends with whom: a line between Maya and Sam, another between Sam and Priya. That is an ordinary network. It has one kind of thing in it, students, and the lines run between students.
Now write down something different. List the students down the side of a page and the clubs across the top, and put a tick in a box whenever a student belongs to a club. That is a two-mode network. It has two kinds of thing in it, students and clubs, and a line only ever runs from a student to a club. A student is never in another student. A club is never in a club. The lines cross between the two kinds and never inside either one.
This study is that page, with schools instead of students and outside organizations instead of clubs. There are 13,167 rows, one for each school, and eight columns, one for each of the organizations a school can work with: parents' groups, civic groups, local businesses, churches, social services, mental-health providers, juvenile justice agencies and the police. A tick in a box means that school works with that organization. The whole study is that one page of ticks.
Why bother thinking of it as a network at all, rather than as a spreadsheet? Because writing it this way means you have to look at a school's whole row at once. Asking "does this school work with the police?" reads a single box. Asking "how many of the eight does it work with, and which ones?" reads the row. Those are different questions, and the second is the one this study asks.
Robert Breiger pointed out in 1974 that a table like this can be read in two directions at once. Read across the rows and it describes schools by the organizations they work with. Read down the columns and it describes organizations by the schools that work with them. He called that duality, and it is the reason a table of memberships is a network rather than a list.
What the study we build on did
Florence Metz, Philip Leifeld and Karin Ingold used this idea in the Journal of Public Policy in 2019. Their case was the Swiss argument over micropollutants in water: chemicals from medicines, cleaning products and farming that end up in rivers. They took 30 organizations involved in that argument, including government offices, industry groups and environmental groups, and 15 policy instruments those organizations could support, such as taxes, bans and subsidies.
The usual approach would ask, for each organization and each instrument, whether that organization supports it. Metz and her colleagues argued that this misses the point, because an organization does not decide about a tax without thinking about the ban next to it. Support comes in sets. So they modeled the whole table at once and asked which instruments tend to be supported together, and which organizations tend to support the same things.
They found the preferences were tangled up with each other, which is the finding their method was built to see. That line of work goes back to Hans Bressers and Laurence O'Toole, who argued in the same journal in 1998 that policy instruments get chosen in sets rather than one by one.
What we do with it
We use their approach on a field of the opposite shape.
Their 30 organizations all deal with each other. They sit in the same argument, they watch each other, and one of them can support an instrument because a rival did. That is interesting, and it also makes one question very hard to answer: when an organization ends up with a wide set of commitments, you cannot easily tell whether that came from pressure on the organization or from who it happens to sit next to.
Our 13,167 schools have no dealings with one another. A school in Ohio does not know what a school in Oregon works with, and cannot copy it directly. The two-mode table has the same shape as theirs, with schools where they had organizations and eight community bodies where they had fifteen instruments, and it is missing the part that tangles their case. That is what lets us ask the first question on its own: what pushes a school into working with more organizations?
Two other differences matter. Theirs is one moment in one country's argument; ours is the same eight questions asked six times over sixteen years, so we can watch the pattern change. And theirs is 30 organizations against our 13,167 schools, which is about four hundred times as many, so small differences show up here that would be invisible there.
What the model does
The picture below shows the table as a network for one survey year. Squares are the eight organizations, circles are schools, and a school sits nearer the middle the more organizations it works with.
We then fit a model of the whole network at once, called an exponential random graph model. It asks a simple question in a complicated way: given how many lines there are in total, are they spread across schools the way chance would spread them? They are not. Some schools hold far more than chance would give them, and that unevenness is the first result in this study.
The three articles this builds on
Breiger, R. L. (1974). The duality of persons and groups.
Social Forces 53(2): 181–190.
Bressers, H. and O'Toole, L. J. (1998). The selection of policy instruments: a network-based
perspective. Journal of Public Policy 18(3): 213–239.
Metz, F., Leifeld, P. and Ingold, K. (2019). Interdependent policy instrument preferences: a
two-mode network approach. Journal of Public Policy 39(4): 609–636.