We study the Susceptible-Infectious-Susceptible model on arbitrary networks. The well-established pair approximation treats neighboring pairs of nodes exactly while making a mean-field approximation for the rest of the network. We improve the method by expanding the state space dynamically, giving nodes a memory of when they last became susceptible. The resulting approximation is simple to implement and appears to be highly accurate, both in locating the epidemic threshold and in computing the quasistationary fraction of infected individuals above the threshold, for both finite graphs and infinite random graphs.

Gurau (2020) proposed a generalization of the trace of the matrix resolvent to tensors of higher order, and recent work has explored analogs of the Wigner semicircle and Marchenko-Pastur distributions from random matrix theory as well as aspects of free probability theory from this perspective. In particular, when evaluated with appropriate large random tensors, the limiting expectations of the coefficients of a series expansion of Gurau's resolvent trace give the moment sequences of probability measures analogous to the above distributions. We construct, on the other hand, individual deterministic tensors such that the same coefficients evaluated on those tensors do not give the moment sequence of any probability measure. Thus, the "spectral density" associated to Gurau's resolvent trace, while in a sense defined on average for certain random tensor ensembles, is not defined pointwise (unless perhaps as a signed measure) for all individual tensors.

In 2013, Bollobas, Mitsche, and Pra & lstrok;at gave upper and lower bounds for the likely metric dimension of random Erdos-Renyi graphs G(n,p) for a large range of expected degrees. However, their results only apply when d=pn=omega(log(5 )n), leaving open sparser random graphs with d=O(log(5 )n) or d=o(log5n). Here we provide upper and lower bounds on the likely metric dimension of G(n,p) in a range of d starting just above the connectivity transition, i.e., where d=clogn for some constant c>1, up to d=O(log(5 )n). Our lower bound technique is based on an entropic argument which is weaker but more general than the use of Suen's inequality by Bollobas, Mitsche, and Pra & lstrok;at, whereas our upper bound is similar to theirs.

Artificial intelligence (AI) is increasingly being adopted in most industries, and for applications such as note taking and checking grammar, there is typically not a cause for concern. However, when constitutional rights are involved, as in the justice system, transparency is paramount. While AI can assist in areas such as risk assessment and forensic evidence generation, its "black box" nature raises significant questions about how decisions are made and whether they can be contested. This paper explores the implications of AI in the justice system, emphasizing the need for transparency in AI decision-making processes to uphold constitutional rights and ensure procedural fairness. The piece advocates for clear explanations of AI's data, logic, and limitations, and calls for periodic audits to address bias and maintain accountability in AI systems.

We study the Susceptible-Infectious-Susceptible (SIS) model on arbitrary networks. The well-established pair approximation treats neighboring pairs of nodes exactly while making a mean field approximation for the rest of the network. We improve the method by expanding the state space dynamically, giving nodes a memory of when they last became susceptible. The resulting approximation is simple to implement and appears to be highly accurate, both in locating the epidemic threshold and in computing the quasi-stationary fraction of infected individuals above the threshold, for both finite graphs and infinite random graphs.

We study the problem of detecting and recovering a planted spanning tree M_n^* hidden within a complete, randomly weighted graph G_n. Specifically, each edge e has a non-negative weight drawn independently from P_n if e ∈ M_n^* and from Q_n otherwise, where P_n ≡ P is fixed and Q_n scales with n such that its density at the origin satisfies lim_n→∞ n Q'_n(0)=1. We consider two representative cases: when M_n^* is either a uniform spanning tree or a uniform Hamiltonian path. We analyze the recovery performance of the minimum spanning tree (MST) algorithm and derive a fixed-point equation that characterizes the asymptotic fraction of edges in M_n^* successfully recovered by the MST as n →∞. Furthermore, we establish the asymptotic mean weight of the MST, extending Frieze's ζ(3) result to the planted model. Leveraging this result, we design an efficient test based on the MST weight and show that it can distinguish the planted model from the unplanted model with vanishing testing error as n →∞. Our analysis relies on an asymptotic characterization of the local structure of the planted model, employing the framework of local weak convergence.

Embedding graphs in a geographical or latent space, i.e. inferring locations for vertices in Euclidean space or on a smooth manifold or submanifold, is a common task in network analysis, statistical inference, and graph visualization. We consider the classic model of random geometric graphs where n points are scattered uniformly in a square of area n, and two points have an edge between them if and only if their Euclidean distance is less than r. The reconstruction problem then consists of inferring the vertex positions, up to the symmetries of the square, given only the adjacency matrix of the resulting graph. We give an algorithm that, if r=nα for any 0<α<1/2, with high probability reconstructs the vertex positions with a maximum error of O(nβ) where β=1/2−(4/3)α, until α≥3/8 where β=0 and the error becomes O(logn). This improves over earlier results, which were unable to reconstruct with error less than r. Our method estimates Euclidean distances using a hybrid of graph distances and short-range estimates based on the number of common neighbors. We extend our results to the surface of the sphere in R3 and to hypercubes in any constant fixed dimension.

Grigoriev (2001) and Laurent (2003) independently showed that the sum-of-squares hierarchy of semidefinite programs does not exactly represent the hypercube $\{\pm 1\}^n$ until degree at least $n$ of the hierarchy. Laurent also observed that the pseudomoment matrices her proof constructs appear to have surprisingly simple and recursively structured spectra as $n$ increases. While several new proofs of the Grigoriev-Laurent lower bound have since appeared, Laurent's observations have remained unproved. We give yet another, representation-theoretic proof of the lower bound, which also yields exact formulae for the eigenvalues of the Grigoriev-Laurent pseudomoments. Using these, we prove and elaborate on Laurent's observations. Our arguments have two features that may be of independent interest. First, we show that the Grigoriev-Laurent pseudomoments are a special case of a Gram matrix construction of pseudomoments proposed by Bandeira and Kunisky (2020). Second, we find a new realization of the irreducible representations of the symmetric group corresponding to Young diagrams with two rows, as spaces of multivariate polynomials that are multiharmonic with respect to an equilateral simplex.

We consider the question of whether thermodynamic macrostates are objective consequences of dynamics, or subjective reflections of our ignorance of a physical system. We argue that they are both; more specifically, that the set of macrostates forms the unique maximal partition of phase space which 1) is consistent with our observations (a subjective fact about our ability to observe the system) and 2) obeys a Markov process (an objective fact about the system's dynamics). We review the ideas of computational mechanics, an information-theoretic method for finding optimal causal models of stochastic processes, and argue that macrostates coincide with the ``causal states'' of computational mechanics. Defining a set of macrostates thus consists of an inductive process where we start with a given set of observables, and then refine our partition of phase space until we reach a set of states which predict their own future, i.e. which are Markovian. Macrostates arrived at in this way are provably optimal statistical predictors of the future values of our observables.

Many problems in high-dimensional statistics appear to have a statistical-computational gap: a range of values of the signal-to-noise ratio where inference is information-theoretically possible, but (conjecturally) computationally intractable. A canonical such problem is Tensor PCA, where we observe a tensor Y consisting of a rank-one signal plus Gaussian noise. Multiple lines of work suggest that Tensor PCA becomes computationally hard at a critical value of the signal's magnitude. In particular, below this transition, no low-degree polynomial algorithm can detect the signal with high probability; conversely, various spectral algorithms are known to succeed above this transition. We unify and extend this work by considering tensor networks, orthogonally invariant polynomials where multiple copies of Y are "contracted" to produce scalars, vectors, matrices, or other tensors. We define a new set of objects, tensor cumulants, which provide an explicit, near-orthogonal basis for invariant polynomials of a given degree. This basis lets us unify and strengthen previous results on low-degree hardness, giving a combinatorial explanation of the hardness transition and of a continuum of subexponential-time algorithms that work below it, and proving tight lower bounds against low-degree polynomials for recovering rather than just detecting the signal. It also lets us analyze a new problem of distinguishing between different tensor ensembles, such as Wigner and Wishart tensors, establishing a sharp computational threshold and giving evidence of a new statistical-computational gap in the Central Limit Theorem for random tensors. Finally, we believe these cumulants are valuable mathematical objects in their own right: they generalize the free cumulants of free probability theory from matrices to tensors, and share many of their properties, including additivity under additive free convolution.

We present a physics-inspired method for inferring dynamic rankings in directed temporal networks - networks in which each directed and timestamped edge reflects the outcome and timing of a pairwise interaction. The inferred ranking of each node is real-valued and varies in time as each new edge, encoding an outcome like a win or loss, raises or lowers the node's estimated strength or prestige, as is often observed in real scenarios including sequences of games, tournaments, or interactions in animal hierarchies. Our method works by solving a linear system of equations and requires only one parameter to be tuned. As a result, the corresponding algorithm is scalable and efficient. We test our method by evaluating its ability to predict interactions (edges' existence) and their outcomes (edges' directions) in a variety of applications, including both synthetic and real data. Our analysis shows that in many cases our method's performance is better than existing methods for predicting dynamic rankings and interaction outcomes.

The m × n king graph consists of all locations on an m × n chessboard, where edges are legal moves of a chess king. represents a square on a chessboard and each edge is a legal move. Let P_m × n(z) denote its domination polynomial, i.e., ∑_S ⊆ V z^|S| where the sum is over all dominating sets S. We prove that P_m × n(-1) = (-1)^⌈ m/2⌉⌈ n/2⌉. In particular, the number of dominating sets of even size and the number of odd size differs by ± 1. sets is always odd. This property does not hold for king graphs on a cylinder or a torus, or for the grid graph. But it holds for d-dimensional kings, where P_n_1× n_2×⋯× n_d(-1) = (-1)^⌈ n_1/2⌉⌈ n_2/2⌉⋯⌈ n_d/2⌉.

Studies of dynamics on temporal networks often represent the network as a series of "snapshots," static networks active for short durations of time. We argue that successive snapshots can be aggregated if doing so has little effect on the overlying dynamics. We propose a method to compress network chronologies by progressively combining pairs of snapshots whose matrix commutators have the smallest dynamical effect. We apply this method to epidemic modeling on real contact tracing data and find that it allows for significant compression while remaining faithful to the epidemic dynamics.

Many studies of pretrial rearrest, including validations of risk assessment instruments, lump multiple types and severities of crimes together. Using a dataset of over 15,000 felony defendants who were released pretrial over a four-year period in New Mexico, we measure rearrest rates for specific types and severity of crime, and compare these with the risk scores provided by the widely-used Public Safety Assessment (PSA) developed by Arnold Ventures. Our data classifies both the original charge and new charges, if any, by severity (1st through 4th degree felony, misdemeanor, and petty misdemeanor) and by type (violent, drug, property, public order, and DWI).We find that the rates of rearrest for serious crimes during pretrial release are lower than overall rearrest rates suggest. Across all PSA score categories, about 1/3 of rearrests are for misdemeanors or petty misdemeanors. About 2/3 of rearrests are for felonies, most of which are fourth degree. Rearrest for 1st or 2nd degree felonies is very rare—less than 0.1% and 1% respectively—even among defendants whose initial charge is severe. We argue that policymakers who decide how to translate risk scores into recommended conditions of release, and judges who consider PSA scores or other risk assessments as factors in release decisions, should be provided with this richer picture of pretrial crime. We also urge that future validation studies of the PSA be carried out at this level of specificity rather than simply reporting rates of NCA (New Criminal Activity) and NVCA (New Violent Criminal Activity) in each score category.

We describe and analyze a simple protocol for $n$ parties that implements a randomness beacon: a sequence of high entropy values, continuously emitted at regular intervals, with sub-linear communication per value. The algorithm can tolerate a $(1-\epsilon)/2$ fraction of the $n$ players to be controlled by an adaptive adversary that may deviate arbitrarily from the protocol. The randomness mechanism relies on verifiable random functions (VRF), modeled as random functions, and effectively stretches an initial $\lambda$ -bit seed to an arbitrarily long public sequence so that (i) with overwhelming probability in k-the security parameter-each beacon value has high min-entropy conditioned on the full history of the algorithm, and (ii) the total work and communication required per value is $O(k)$ cryptographic operations. The protocol can be directly applied to provide a qualitative improvement in the security of several proof-of-stake blockchain algorithms, rendering them safe from “grinding” attacks.