Core Concepts
Given the full topology of a network, it is possible to efficiently test if the network is evolving according to a local rule or is far from doing so.
Abstract
The paper focuses on testing the spreading behavior of networks with arbitrary topologies, specifically the 1-BP (1-bootstrap percolation) rule. The key insights are:
For the case of testing a single time step (T=2), the authors provide both upper and lower bounds on the query complexity. They show that a simple algorithm with query complexity O(Δ/ε) is optimal up to Δ = O(√n), where Δ is the maximum degree and n is the number of nodes. For larger Δ, they present a more complex adaptive algorithm with query complexity ˜O(√n/ε).
For the case of testing multiple time steps (T>2), the authors provide two algorithms. The first has query complexity O(ΔT-1/εT), which is useful when T = O(log Δn). The second has query complexity ˜O(|E|/εT), which is non-trivial when T = ω(√(Δ/ε) log n) (or T = ω(√Δ/ε) if the graph excludes a fixed minor).
The authors also provide lower bounds for both one-sided and two-sided error testers in the T=2 case, which match the upper bounds up to certain regimes of Δ.
The algorithms are designed to be as non-adaptive and time-conforming as possible, as these properties are desirable in the context of testing the evolution of large networks.