Group Testing With Correlation Under Edge-Faulty Graphs

In applications of group testing in networks, e.g. identifying individuals who are infected by a disease spread over a network, exploiting correlation among network nodes provides fundamental opportunities in reducing the number of tests needed. We model and analyze group testing on n correlated nodes whose interactions are specified by a graph G. We model correlation through an edge-faulty random graph formed from G in which each edge is dropped with probability <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1-r$ </tex-math></inline-formula>, and in the newly formed graph, all nodes in the same component have the same state. We consider three classes of graphs: cycles and trees, d-regular graphs and stochastic block models or SBM, and obtain lower and upper bounds on the number of tests needed to identify the defective nodes. Roughly speaking, we use correlation among the states of the nodes to transform the problem into that of a smaller graph with independent node states. This enhancement is quantified through the ratio of the diminished node count to the overall count of nodes, n; thus, a lower ratio signifies superior performance. The lower bounds are derived by illustrating a strong dependence of the number of tests needed on the expected number of components. In this regard, we establish a new approximation for the distribution of component sizes in “d-regular trees” which may be of independent interest and leads to a lower bound on the expected number of components in d-regular graphs. The upper bounds are found by forming dense subgraphs in which nodes are more likely to be in the same state. When G is a cycle or tree, we show an improvement by a factor of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\log (1/r)$ </tex-math></inline-formula>. For grid, a graph with almost <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2n$ </tex-math></inline-formula> edges, the improvement is by a factor of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(1-r) \log (1/r)$ </tex-math></inline-formula>, indicating drastic improvement compared to trees. When G has a larger number of edges, as in SBM, the improvement can scale in n.