Network Exposure Mappings by Neighbor Treatment

Representing Peer Treatment in a Network

When treatment can spread through social, clinical, or organizational connections, a person’s outcome may depend on more than their own treatment. It may also depend on who around them is treated, how many of those people are treated, or how strongly they are connected to them. An exposure mapping compresses the peer-treatment assignment into a summary that can be used to describe that dependence.

The choice of summary matters. A count, a proportion, and a weighted average can assign different exposure labels to the same treatment pattern. Even when two different patterns receive the same label, treating them as equivalent imposes a substantive restriction on potential outcomes. The mapping should therefore reflect the causal question and the network information, not just computational convenience.

This notation expresses a substantive assumption, not merely a relabeling convention. The exposure-mapping assumption is that for any two peer-treatment assignments \(z_{-i}\) and \(z'_{-i}\),

In words, once own treatment and mapped exposure are fixed, other details of the peer-treatment assignment do not change the outcome. This does not assert that peers have no effects. It asserts that their effects are fully represented by the chosen summary. The mapping and its adequacy must be distinguished: defining a count is straightforward, but assuming that the count captures all relevant peer influence requires substantive justification.

Counts and Proportions

Let \(Z_j=1\) indicate that peer \(j\) is treated and \(Z_j=0\) otherwise. The treated-neighbor count and treated-neighbor proportion for \(i\) are

where \(d_i=|N_i|\) is the number of peers in the specified set. The count preserves the absolute number treated; the proportion preserves the fraction treated. Each mapping treats peer identities as irrelevant when they yield the same summary, but count and proportion do not generally treat the same assignments as equivalent.

A proportion also loses the absolute count: if one person has one of two neighbors treated and another has five of ten treated, both have exposure \(0.5\). A proportion mapping assumes those situations have the same peer-exposure meaning, conditional on own treatment, even though the number of treated peers differs. Conversely, a count mapping assumes that equal counts have the same meaning even across different neighborhood sizes. Neither choice is universally preferable.

Threshold Categories

A threshold mapping groups counts or proportions into categories. It can be useful when the scientific question concerns crossing a meaningful exposure level, or when detailed exposure values are not needed. But categorization makes all values within a category equivalent under the mapping assumption.

Thresholds may be defined on proportions instead. For example, a category could indicate whether at least half of a person’s neighbors are treated. That rule classifies different counts differently depending on neighborhood size. A threshold should be stated precisely, including how equality at a cut point is handled, so that every assignment has an unambiguous exposure.

Weighted Ties

If ties differ in strength or relevance, a count gives each specified neighbor equal influence. A weighted mapping instead uses fixed, nonnegative tie weights \(w_{ij}\), where \(w_{ij}\) represents the prespecified importance of peer \(j\)’s treatment for individual \(i\). One common summary is

This normalized weighted exposure ranges from 0 to 1. The weights and the neighbor set should be defined independently of the treatment assignments being summarized; otherwise the exposure definition itself can change with treatment. The formula also requires a positive total weight. For an individual with no specified ties, it is necessary to define a separate exposure category rather than divide by zero.

What Each Mapping Assumes

All of these choices create groups of peer-treatment assignments that receive the same exposure label. The mapping assumption says that assignments in the same group have the same potential outcome at each fixed own-treatment level. The choice determines which distinctions are kept and which are discarded.

A weighted sum does not, by itself, assert that each peer has an additive effect on the outcome. It asserts that the relevant peer-treatment pattern can be summarized by the weighted value. The outcome may vary nonlinearly with that value unless a further outcome model says otherwise. Likewise, using a count does not mean that the outcome must increase linearly with the count; it means only that assignments with the same count are treated as equivalent.

Why the Mapping Choice Matters

A mapping makes a complex assignment vector manageable, but it can erase meaningful features. If a close contact matters more than a distant contact, an unweighted count may obscure that distinction. If absolute numbers matter, a proportion may equate situations with very different numbers of treated peers. If effects change gradually, a coarse threshold can hide relevant variation. These are scientific concerns about how treatment can affect outcomes, not just questions of notation.

At minimum, a defensible mapping should specify the network or tie information used, the function that converts peer assignments to exposure, and the equivalences the mapping entails. It should also distinguish the exposure from own treatment: \(Y_i(1,e)\) and \(Y_i(0,e)\) refer to different own-treatment conditions even when peer exposure \(e\) is identical. Defining exposure does not alone establish that outcomes or causal effects can be identified from available data; it describes which treatment contrasts the exposure representation regards as meaningful.

A Short Justification: Why Coarsening Restricts Potential Outcomes

Consider two full peer-treatment assignments \(a\) and \(b\) that receive the same exposure value under a chosen mapping. The mapping assumption gives \(Y_i(z,a)=Y_i(z,b)\) for either fixed own treatment \(z\). Thus, if a count mapping assigns both patterns the value 2, the outcomes for those patterns must be equal at \(z=0\), and they must also be equal at \(z=1\). This follows directly from the assumption; it is not a consequence of the arithmetic calculation of the count.

More generally, the mapping partitions peer assignments into sets with common exposure labels. The assumption requires potential outcomes to be constant within each such set, separately for each own-treatment level. A more detailed mapping usually creates smaller sets and asserts fewer equalities; a coarser mapping creates larger sets and asserts more. This explains why adding thresholds or collapsing categories can simplify analysis while strengthening the substantive restriction.

Common Mistake

A common mistake is to treat an exposure mapping as a harmless data-processing step: “These two assignments both have two treated neighbors, so they have the same exposure.” The calculation is correct, but the causal conclusion does not follow from arithmetic alone. Calling both assignments exposure 2 adopts the assumption that peer identities and other features of those assignments do not matter once the count is fixed. The correction is to report the calculation and explicitly assess whether that equivalence is scientifically credible. If identities or tie strengths plausibly matter, a count-only mapping may be too coarse.