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ABSTRACT A random algebraic graph is defined by a group with a uniform distribution over it and a connection with expectation satisfying . The random graph with vertex set is formed as follows. First, independent variables are sampled uniformly from . Then, vertices are connected with probability . This model captures random geometric graphs over the sphere, torus, and hypercube; certain instances of the stochastic block model; and random subgraphs of Cayley graphs. The main question of interest to the current paper is: when is a random algebraic graph statistically and/or computationally distinguishable from ? Our results fall into two categories. (1) Geometric. We focus on the case and use Fourier‐analytic tools. We match and extend the following results from the prior literature: For hard threshold connections, we match for , and for ‐Lipschitz connections we extend the results of when to the non‐monotone setting. (2) Algebraic. We provide evidence for an exponential statistical‐computational gap. Consider any finite group and let be a set of elements formed by including each set of the form independently with probability Let be the distribution of random graphs formed by taking a uniformly random induced subgraph of size of the Cayley graph . Then, and are statistically indistinguishable with high probability over if and only if . However, low‐degree polynomial tests fail to distinguish and with high probability over when 

{"Abstract":["For a set M of m elements, we define a decreasing chain of classes of normalized monotone-increasing valuation functions from 2^M to ℝ_{≥ 0}, parameterized by an integer q ∈ [2,m]. For a given q, we refer to the class as q-partitioning. A valuation function is subadditive if and only if it is 2-partitioning, and fractionally subadditive if and only if it is m-partitioning. Thus, our chain establishes an interpolation between subadditive and fractionally subadditive valuations. We show that this interpolation is smooth (q-partitioning valuations are "nearly" (q-1)-partitioning in a precise sense, Theorem 6), interpretable (the definition arises by analyzing the core of a cost-sharing game, à la the Bondareva-Shapley Theorem for fractionally subadditive valuations, Section 3.1), and non-trivial (the class of q-partitioning valuations is distinct for all q, Proposition 3).\r\nFor domains where provable separations exist between subadditive and fractionally subadditive, we interpolate the stronger guarantees achievable for fractionally subadditive valuations to all q ∈ {2,…, m}. Two highlights are the following:\r\n1) An Ω ((log log q)/(log log m))-competitive posted price mechanism for q-partitioning valuations. Note that this matches asymptotically the state-of-the-art for both subadditive (q = 2) [Paul Dütting et al., 2020], and fractionally subadditive (q = m) [Feldman et al., 2015]. \r\n2) Two upper-tail concentration inequalities on 1-Lipschitz, q-partitioning valuations over independent items. One extends the state-of-the-art for q = m to q < m, the other improves the state-of-the-art for q = 2 for q > 2. Our concentration inequalities imply several corollaries that interpolate between subadditive and fractionally subadditive, for example: 𝔼[v(S)] ≤ (1 + 1/log q)Median[v(S)] + O(log q). To prove this, we develop a new isoperimetric inequality using Talagrand’s method of control by q points, which may be of independent interest.\r\nWe also discuss other probabilistic inequalities and game-theoretic applications of q-partitioning valuations, and connections to subadditive MPH-k valuations [Tomer Ezra et al., 2019]."]}