Interference Radius and Ego-Network Exposure

Why Define a Radius?

Under interference, one unit’s outcome may depend on treatments assigned to other units. A network makes some of those relationships explicit: it records which units are connected and how many steps separate them. An interference radius sets a limit on which network locations are included when defining a unit’s exposure. The resulting ego-network gives a local description of the assignment around that unit.

The radius is not just a computational convenience. Treating it as sufficient for an outcome is a substantive assumption: it says that, once the relevant local treatments are specified, assignments beyond the radius do not further change that outcome. A second modeling decision is whether to preserve the full local treatment pattern or summarize it, for example by the number of treated neighbors. A summary is useful only if the distinctions it discards do not matter for the outcome being modeled.

Network Distance and the Ego-Network

Represent the study network by a graph whose vertices are units and whose edges denote the relationships of interest. For units \(i\) and \(j\), let \(d(i,j)\) be their shortest-path distance: the number of edges in the shortest path joining them. If no path joins them, set \(d(i,j)=\infty\). The graph and the meaning of an edge should be specified for the application; a friendship tie, for example, is not automatically interchangeable with a household or communication tie.

Definition: For a nonnegative integer radius \(r\), the radius-\(r\) neighbor set of unit \(i\) is \(N_i^{(r)}=\{j:j\ne i,\ d(i,j)\le r\}\). The radius-\(r\) ego-network of \(i\) contains \(i\), the units in \(N_i^{(r)}\), and the network edges among those units. The set of units is \(E_i^{(r)}=\{i\}\cup N_i^{(r)}\). At radius zero, the ego-network contains only \(i\); at radius one, it adds directly connected neighbors.

Let \(Z_j\in\{0,1\}\) indicate whether unit \(j\) is treated, and write the complete assignment as \(\mathbf Z=(Z_1,\ldots,Z_n)\). The treatment assignment within \(i\)’s radius-\(r\) ego-network is the vector \(\mathbf Z_{E_i^{(r)}}\). A convenient exposure mapping can retain that local vector exactly, or map it to a more compact description. For instance, it might record \(i\)’s own treatment and the number of treated units in \(N_i^{(r)}\).

An ego-network exposure condition is a specified value of such a mapping. It describes a feature of the assignment around \(i\), not a new treatment assigned to \(i\). In particular, two different full network assignments can satisfy the same ego-network exposure condition.

Worked Example: Finding a Radius-One Exposure

Consider a hypothetical five-unit path, \(1-2-3-4-5\), and focus on unit 3. Under radius one, its neighbors are the units one edge away. Suppose the assignment, listed in unit order, is \(\mathbf z=(1,0,1,0,1)\).

The distances from unit 3 to units 2 and 4 are both one; the distances to units 1 and 5 are both two. Thus \(N_3^{(1)}=\{2,4\}\) and \(E_3^{(1)}=\{2,3,4\}\). The local assignment is \(z_3=1\) for unit 3 and \((z_2,z_4)=(0,0)\) for its neighbors. An exact radius-one exposure can therefore be written as \(g_3^{(1)}(\mathbf z)=(1;(0,0))\).

A coarser exposure could retain unit 3’s treatment and count its treated radius-one neighbors: \(G_3^{(1)}(\mathbf z)=(z_3,\sum_{j\in N_3^{(1)}}z_j)=(1,0+0)=(1,0)\). Units 1 and 5 are outside this radius, so their treatment values do not enter either mapping.

For example, the different assignment \(\mathbf z'=(0,0,1,0,0)\) has the same local exposure for unit 3. If radius one is sufficient for its outcome, then assignments \(\mathbf z\) and \(\mathbf z'\) imply the same potential outcome for unit 3. The example identifies the condition and its implication; it does not establish that radius one is substantively adequate in a real network.

Exposure Mappings and the Locality Assumption

Write \(Y_i(\mathbf z)\) for unit \(i\)’s potential outcome under the entire assignment \(\mathbf z\). This notation allows interference: in general, \(Y_i(\mathbf z)\) can vary when another unit’s treatment changes. A radius defines which entries of the assignment are considered local, but it does not by itself prove that the other entries are irrelevant.

Definition: A radius-\(r\) exposure mapping for unit \(i\) is a function \(g_i^{(r)}\) that assigns an exposure label to each full treatment assignment using treatments within \(E_i^{(r)}\). The radius-\(r\) exposure-sufficiency assumption is that there exists a function \(Y_i^{(r)}\) such that, for every assignment \(\mathbf z\), \(Y_i(\mathbf z)=Y_i^{(r)}(g_i^{(r)}(\mathbf z))\). Thus, any two assignments with the same mapped exposure give unit \(i\) the same potential outcome.

The assumption has two parts worth keeping separate. First, assignments outside the chosen ego-network cannot affect the outcome once local assignment is fixed. Second, the mapping itself must retain enough information about the local assignment. If the mapping records the exact vector on \(E_i^{(r)}\), the second requirement imposes no additional compression within that ego-network. If the mapping records only a count, it also assumes that local patterns with the same count have equivalent effects for this outcome.

An exact local mapping and a coarser one answer different modeling needs. An exact mapping can distinguish which neighbors are treated, but may produce many exposure conditions. A count or treated proportion reduces the number of conditions, but merges assignments. The coarser mapping is appropriate only when the distinction between merged patterns is irrelevant to the potential outcome under the assumed model.

Worked Example: Radius Two Adds Indirectly Connected Units

Return to the path \(1-2-3-4-5\), now focusing on unit 3 at radius two. Suppose \(\mathbf z=(1,0,1,1,0)\). Units 1 and 5 are each two edges from unit 3, so \(N_3^{(2)}=\{1,2,4,5\}\), while \(E_3^{(2)}\) contains all five units.

The exact local assignment, ordered as \((z_3;z_1,z_2,z_4,z_5)\), is \((1;1,0,1,0)\). There are two treated radius-two neighbors, so a count-based mapping that includes own treatment is \((z_3,\sum_{j\in N_3^{(2)}}z_j)=(1,1+0+1+0)=(1,2)\). This assignment has a different radius-one count exposure, \((1,1)\), because unit 4 is the only treated radius-one neighbor.

Suppose, as a hypothetical model, unit 3’s outcome is known to depend on assignments only within two edges. Then changing treatment at a unit beyond distance two would leave its potential outcome unchanged. In this five-unit path there is no such unit; in a larger network, the same radius-two assumption would exclude effects from more distant units. The assumption does not follow merely from counting two steps in the graph.

What a Coarser Exposure Assumes

A radius and an exposure summary are separate choices. The radius determines which units are eligible to contribute to exposure; the summary determines how their treatment values are represented. For example, a radius-one mapping might use the exact vector of neighbor treatments, the number treated, the proportion treated, or an indicator for whether at least one neighbor is treated. Each retains different information.

The consequence of summarizing can be stated precisely. Suppose \(g_i^{(r)}\) is the exact local assignment and \(G_i^{(r)}\) is a coarser mapping computed from it. For \(G_i^{(r)}\) to suffice, there must be a function \(m_i\) such that \(Y_i^{(r)}(g)=m_i(G_i^{(r)}(g))\) for every local assignment \(g\). In other words, \(Y_i^{(r)}\) must be constant across all local assignments that the coarser mapping groups together.

G_i^{(r)}(g)=G_i^{(r)}(g') \quad\Longrightarrow\quad Y_i^{(r)}(g)=Y_i^{(r)}(g')

This implication is the substantive restriction behind a count-based or category-based exposure condition. A count mapping treats two assignments with the same number of treated neighbors as equivalent even if different neighbors are treated. That may be plausible when neighbors are interchangeable for the outcome; it may fail when network position, relationship type, or the identity of a particular neighbor matters.

Worked Example: When a Treated-Neighbor Count Is Sufficient

Suppose a hypothetical unit \(i\) has three radius-one neighbors, \(a\), \(b\), and \(c\). Consider two assignments with \(z_i=0\): in the first, \((z_a,z_b,z_c)=(1,1,0)\); in the second, \((z_a,z_b,z_c)=(1,0,1)\). Both have two treated neighbors, but they are different exact local assignments.

For illustration, assume this unit’s local potential outcome follows the specified model \(Y_i^{(1)}=5+2z_i+1.5\sum_{j\in N_i^{(1)}}z_j\), measured in hypothetical score points. Under the first assignment, the calculation is \(5+2(0)+1.5(1+1+0)=5+3=8\) points. Under the second, it is \(5+2(0)+1.5(1+0+1)=5+3=8\) points.

Under this model, the count mapping is sufficient: both local assignments map to \((0,2)\) and give the same potential outcome, 8 points. The equality follows from the model’s use of the neighbor-treatment sum rather than neighbor identities. If the model instead gave neighbor \(a\) a different effect from neighbor \(c\), equal counts would not necessarily imply equal outcomes, and the count mapping would be too coarse.

A Short Justification of the Exposure Implication

The key implication of exposure sufficiency follows directly from its definition. It is useful because it makes clear what can and cannot be concluded when two full assignments share a local exposure.

Match the exposure.
Take two full assignments \(\mathbf z\) and \(\mathbf z'\) for which \(g_i^{(r)}(\mathbf z)=g_i^{(r)}(\mathbf z')=e\).

Apply exposure sufficiency.
By the assumed representation, \(Y_i(\mathbf z)=Y_i^{(r)}(e)\) and \(Y_i(\mathbf z')=Y_i^{(r)}(e)\).

Compare outcomes.
Both potential outcomes equal the same quantity, so \(Y_i(\mathbf z)=Y_i(\mathbf z')\). This conclusion is conditional on the mapping and locality assumptions; it is not guaranteed by graph distance alone.

The same reasoning applies to a coarser mapping only if the potential outcome is constant over every group of local assignments that mapping combines. Without that condition, equal coarse exposure labels do not justify equal potential outcomes.

Choosing and Interpreting the Radius

Selecting a radius requires subject-matter reasoning about how treatments could affect outcomes through the network. A short radius may omit relevant indirect pathways; a larger radius includes more units and can create a more complex exposure description. The radius should be defined using the network and time window relevant to the causal question. A connection that appears only after treatment, for example, should not automatically be treated as part of a fixed pre-treatment ego-network.

The radius also does not dictate that every unit within it has the same role. An exposure mapping may preserve unit-specific positions or edge types, if those features are part of the specified network and needed to describe treatment exposure. Conversely, a count deliberately discards those distinctions. The mapping should be described clearly enough that one can determine which full assignments receive the same exposure label.

These definitions are useful because causal effects under interference must refer to well-defined exposure conditions. “Two treated neighbors within one step,” for example, is more precise than “high peer treatment” only if the radius, neighbor set, and treatment summary are specified. Even then, an exposure label defines a condition; it does not alone establish that the condition is observed often enough for estimation or that its causal effect is identified.

Common Mistakes

Assuming that a radius proves locality. A distance cutoff defines a neighborhood; it does not demonstrate that more distant treatments have no effect. Treating the radius as sufficient requires an explicit substantive assumption.
Equating a count with the full local assignment. Two assignments can have the same number of treated neighbors but different treated neighbors. A count mapping treats them as equivalent only when that loss of identity is acceptable for the outcome model.
Calling a local exposure the full assignment. An ego-network exposure describes a selected local feature of \(\mathbf z\). It generally does not identify which treatments were assigned outside the radius.
Changing the radius without changing the definition. A unit two edges away is outside a radius-one neighbor set but inside a radius-two neighbor set. The exposure condition must state which radius it uses.

Key takeaway: An interference radius selects the units whose treatments enter a local exposure mapping. The mapping may preserve the ego-network treatment pattern or summarize it, but potential outcomes depend only on that mapped exposure only under an explicit exposure-sufficiency assumption.

Check Your Understanding

Answer each question by identifying both the definition being used and the assumption, if any, needed for the conclusion.

In a path network, how does a unit’s radius-two neighbor set differ from its radius-one neighbor set?
Can two different full treatment assignments give the same exact radius-one exposure for a unit? Explain which assignment entries may differ.
What additional assumption is required to use the number of treated neighbors instead of their exact treatment vector?
If two assignments have the same mapped exposure, what does exposure sufficiency imply about the unit’s potential outcomes under those assignments?
Why does choosing a radius not by itself show that treatments beyond it have no effect?