Epidemic spreading is well understood when a disease propagates around a contact graph. In a stochastic susceptible–infected–susceptible setting, spectral conditions characterize whether the disease vanishes. However, modelling human interactions using a graph is a simplification which only considers pairwise relationships. This does not fully represent the more realistic case where people meet in groups. Hyperedges can be used to record higher order interactions, yielding more faithful and flexible models and allowing for the rate of infection of a node to depend on group size and also to vary as a nonlinear function of the number of infectious neighbours. We discuss different types of contagion models in this hypergraph setting and derive spectral conditions that characterize whether the disease vanishes. We study both the exact individual-level stochastic model and a deterministic mean field ODE approximation. Numerical simulations are provided to illustrate the analysis. We also interpret our results and show how the hypergraph model allows us to distinguish between contributions to infectiousness that (i) are inherent in the nature of the pathogen and (ii) arise from behavioural choices (such as social distancing, increased hygiene and use of masks). This raises the possibility of more accurately quantifying the effect of interventions that are designed to contain the spread of a virus.
Let M be a compact, unit volume, Riemannian manifold with boundary. We study the homology of a random Čech-complex generated by a homogeneous Poisson process in M. Our main results are two asymptotic threshold formulas, an upper threshold above which the Čech complex recovers the kth homology of M with high probability, and a lower threshold below which it almost certainly does not. These thresholds share the same leading term. This extends work of Bobrowski-Weinberger and Bobrowski-Oliveira who establish similar formulas when M has no boundary. The cases with and without boundary differ: the corresponding common leading terms for the upper and lower thresholds differ being log(n) when M is closed and (2−2∕𝑑) log(n) when M has boundary; here n is the expected number of sample points. Our analysis identifies a special type of homological cycle occurring close to the boundary.
Anchor-based techniques reduce the computational complexity of spectral clustering algorithms. Although empirical tests have shown promising results, there is currently a lack of theoretical support for the anchoring approach. We define a specific anchor-based algorithm and show that it is amenable to rigorous analysis, as well as being effective in practice. We establish the theoretical consistency of the method in an asymptotic setting where data is sampled from an underlying continuous probability distribution. In particular, we provide sharp asymptotic conditions for the number of nearest neighbors in the algorithm, which ensure that the anchor-based method can recover with high probability disjoint clusters that are mutually separated by a positive distance. We illustrate the performance of the algorithm on synthetic data and explain how the theoretical convergence analysis can be used to inform the practical choice of parameter scalings. We also test the accuracy and efficiency of the algorithm on two large scale real data sets. We find that the algorithm offers clear advantages over standard spectral clustering. We also find that it is competitive with the state-of-the-art LSC method of Chen and Cai (Twenty-Fifth AAAI Conference on Artificial Intelligence, 2011), while having the added benefit of a consistency guarantee.
Anchor-based techniques reduce the computational complexity of spectral clustering algorithms. Although empirical tests have shown promising results, there is currently a lack of theoretical support for the anchoring approach. We define a specific anchor-based algorithm and show that it is amenable to rigorous analysis, as well as being effective in practice. We establish the theoretical consistency of the method in an asymptotic setting where data is sampled from an underlying continuous probability distribution. In particular, we provide sharp asymptotic conditions for the algorithm parameters which ensure that the anchor-based method can recover with high probability disjoint clusters that are mutually separated by a positive distance. We illustrate the performance of the algorithm on synthetic data and explain how the theoretical convergence analysis can be used to inform the practical choice of parameter scalings. We also test the accuracy and efficiency of the algorithm on two large scale real data sets. We find that the algorithm offers clear advantages over standard spectral clustering. We also find that it is competitive with the state-of-the-art LSC method of Chen and Cai (Twenty-Fifth AAAI Conference on Artificial Intelligence, 2011), while having the added benefit of a consistency guarantee.
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