Toward a Confidence Set for

In order to build a confidence set for the entire graph structure, we can use the same simulation method on other graphs. For example, we could consider a graph that we call “graph 25” or 𝒢25, since it is the HIV graph with 25 randomly-chosen edges removed. A

picture is available in Figure6.3.

As discussed in Section 4.2.3, there are a number of different statistics that one might consider. Here, we consider two-sided tests for 𝑊𝒢25, 𝑊𝒢HIV, and 𝑊𝒢25 −𝑊𝒢HIV, i.e. the edges within statistic computed with respect to 𝒢₂₅, the edges within statistic computed with respect to the HIV graph, and the difference of these statistics. The results are shown in Figure 6.4.

There are a number of things to observe. First, using the statistics 𝑊𝒢25 and𝑊𝒢HIV lead to confidence sets for 𝛽 of approximately [1.148,10] and [1.27347,10] respectively. On the other hand, the difference 𝑊𝒢25 −𝑊𝒢HIV leads to the confidence interval [0,10]. This is unsurprising, since the 𝒢₂₅ and 𝒢_HIV are not too different.

We can repeat this procedure on a different graph,𝒢₁₀₀, which has 100 edges removed from the HIV graph uniformly at random. An image may be found in Figure6.5, and note that

𝒢100 is a subgraph of 𝒢25. The results of the analysis are shown in Figure6.6.

The major difference for 𝒢₁₀₀ is that no value of 𝛽 is included in the confidence set for the difference statistic 𝑊𝒢100 −𝑊𝒢HIV. This is likely due to there being many edges in 𝒢HIV between infected vertices that do not exist when simulating the distribution using 𝒢100. In

Figure 6.3The graph𝒢25 used in computing the joint confidence set of (𝒢25, 𝛽). This graph was formed from the HIV graph by removing 25 edges uniformly at random. As in Figure2.1, black vertices are infected, white vertices are uninfected, and gray vertices are censored.

0.0 0.2 0.4 0.0 2.5 5.0 7.5 10.0 β p−v alue W Difference G HIV

Figure 6.4The 𝑝-values as a function of 𝛽 for the HIV graph using the two-statistic test with different test statistics. The statistics are𝑊𝒢25,𝑊𝒢HIV, and𝑊𝒢25−𝑊𝒢HIV, and the𝑝-values for

these are given by the black line, the light gray line, and the dark graph line respectively.

short, for the values of 𝛽 from [0,10], we have found a graph that is not in the confidence set of graphs given by the observed infection.

Figure 6.5The graph𝒢100used in computing the joint confidence set of (𝒢100, 𝛽). This graph was formed from the HIV graph by removing 100 edges uniformly at random. As in Figure2.1, black vertices are infected, white vertices are uninfected, and gray vertices are censored.

0.0 0.2 0.4 0.0 2.5 5.0 7.5 10.0 β p−v alue W Difference G HIV

Figure 6.6The 𝑝-values as a function of 𝛽 for the HIV graph using the two-statistic test with different test statistics. The statistics are𝑊𝒢100,𝑊𝒢HIV, and 𝑊𝒢100−𝑊𝒢HIV, and the 𝑝-values

for these are given by the black line, the light gray line, and the dark graph line respectively.

set for this particular infection. This provides a very general method of graph testing at the expense of requiring many simulations.

Part III

Permutation

7.

Theory

In this chapter, we cover the primary contribution of this dissertation: tests based on permutation. Note that we restrict our inquiry to the case where the edges are unweighted, or equivalently𝑤(𝑢, 𝑣) = 1.

7.1

Permutation-Invariant Statistics

A natural statistic to consider for the purpose of graph testing is the likelihood ratio. For the stochastic spreading model, the likelihood ratio is often difficult to compute and depends on𝛽 in a nontrivial manner, making the theoretical derivations somewhat challenging. Our main focus will be on a class of statistics that are invariant under a group of permutations, which allow us to perform permutation testing based on symmetries in the graph sets. We first introduce some terminology regarding permutations and group actions, and then introduce a class of invariant statistics that will be central to our analysis.

7.1.1 Permutations and Group Actions

Recall that agraph automorphism 𝒢= (𝒱,ℰ) is an element𝜑of the permutation group 𝑆𝑛

such that (𝑢, 𝑣)∈ ℰ if and only if (𝜑(𝑢), 𝜑(𝑣))∈ ℰ. For simple hypotheses, we denote the automorphism groups of 𝒢₀ and 𝒢₁ by Π₀ = Aut(𝒢₀) and Π₁= Aut(𝒢₁), respectively. We also need to define theaction of a permutation on vertices, graphs, and infections. The action of a permutation𝜋 on a vertex𝑢 is simply the image𝜋(𝑢). This is easily extended to tuples and subsets of vertices by applying𝜋 to the underlying vertices. A specific example is the action on edges of the graph:

𝜋ℰ ={(𝜋(𝑢), 𝜋(𝑣)) : (𝑢, 𝑣)∈ ℰ}. The action of 𝜋 on a graph 𝒢= (𝒱,ℰ) is then defined to be

𝜋𝒢:= (𝜋𝒱, 𝜋ℰ) = (𝒱, 𝜋ℰ).

Another natural extension is to define the action of a set of permutations on a set of graphs: ΠG={𝜋𝒢 :𝜋∈Π and𝒢 ∈G}.

If G𝑖 = 𝑆𝑛{𝒢𝑖}, we say that hypothesis 𝑖 corresponds to a hypothesis of a particular

graphtopology, since all node labelings are included in the set. We also define the action ΠΘ_𝑖 = ΠG_𝑖×𝐴𝑖. Finally, we define the action of a permutation 𝜋 on an infection𝐽:

𝜋𝐽 :=(︁𝐽_𝜋−1₍₁₎, . . . , 𝐽_𝜋−1_(𝑛)

)︁

.

In other words, the infection status of the image vertex 𝜋(𝑢) is the infection status of 𝑢 under𝐽.

7.1.2 Invariant Statistics

The theory presented in our paper applies to the following class of statistics:

Definition 7.1.1. SupposeΠ is a subgroup of 𝑆𝑛. A statistic 𝑆 is Π-invariant if𝑆(𝐽) =𝑆(𝜋𝐽) for any𝐽 ∈_I𝑘,𝑐 and 𝜋∈Π.

In our permutation test, we will compute the edges-within statistic with respect to the graph

𝒢₁ appearing in the alternative hypothesis in the case of a simple test, so we reject 𝐻0 when

𝑊1(𝐽) :=𝑊𝒢1(𝐽) exceeds a certain threshold. We derive the invariance of the statistic 𝑊 under the permutation group Π = Aut(𝒢) in Chapter 8.