Invariant Theory for Hypothesis Testing on Graphs
Priebe, Carey
Johns Hopkins University, Applied Mathematics and Statistics 3400 North Charles Street Baltimore, Maryland 21218-2682, USA E-mail: cep@jhu.edu
Rukhin, Andrey
Naval Surface Warfare Center, Sensor Fusion Department 18444 Frontage Road Dahlgren, Virginia 22448, USA E-mail: andrey.rukhin@navy.mil
1 Introduction
Following the setting outlined in Priebe et al. [2011] we aim to detect anomalies within attributed graphs. In particular, let $\mathcal{V} = {1, \dots, n}$ be the fixed set of vertices and $\phi: (\nu_2) \to {0, 1, \dots, K}$ be an edge-attribution function. The graph on $\mathcal{V}$ is defined to be $G = (\mathcal{V}, \mathcal{E}_{\phi})$ where
We say that the edge $(u, v)$ has attribute $c \in {1, \dots, K}$ if $\phi(u, v) = c$. One can view the categorical edge attributes as some mode of the communication event between actors $u$ and $v$ (e.g., a topic label derived from the content of the communication).
The specific anomaly we aim to detect is the “chatter” alternative – a small (unspecified) subset of vertices with altered communication behavior in an otherwise homogeneous setting. Our inference task is to determine whether or not a graph $(\mathcal{V}, \mathcal{E}_{\phi})$ includes a subset of vertices $\mathcal{M} = {v_1, v_2, \dots, v_m}$ whose edge-connectivity within the subset exhibits a different behaviour than that found among the remaining vertices in the graph.
To this end we consider the problem of detecting chatter anomalies in a graph using hypothesis testing on a fusion of attributed graph invariants. In particular, the focus of this paper is analyzing and comparing the inferential power of the linear attribute fusion of the attributed $q$-clique invariant
where the sum is over the collection of partitions $P((q_2), K)$ of $(q_2)$ into $K$ non-negative parts, $W = {w_i}_{i \in P((q_2), K)}$ are the fusion weights, and the summand $h(u_1, \dots, u_q; c_1, \dots, c_K)$ indicates the event that the vertices $u_1, \dots, u_q$ are elements of a $q$-clique with $c_r$ edges of color $r$. Specifically, we consider the cases $q=2$ which yields the size fusion $T_2^W$ and $q=3$ which yields the triangle fusion $T_3^W$.
Our random graph model is motivated by the time series model found in Lee and Priebe [forthcoming]: for each vertex $v \in \mathcal{V}$ we assign a latent variable $\mathbf{X}^v = (X_1^v, \dots, X_d^v)$ drawn independently of all other vertices from some $d$-dimensional distribution. The edge-attribution function will be a random variable where the probability of an edge $(u, v)$ having attribute $c$ is defined to be a some predetermined function of the inner product of the latent variables. We assume that the edge attributes, conditioned on the latent variables, are independent. In this paper, we will assume that $\mathbf{X}^v \sim \text{Dirichlet}(\lambda_0^v, \dots, \lambda_K^v)$ and