Community structure in networks naturally arises in various applications. But while the topic has received significant attention for static networks, the literature on community structure in temporally evolving networks is more scarce. In particular, there are currently no statistical methods available to test for the presence of community structure in a sequence of networks evolving over time. In this work, we propose a simple yet powerful test using e-values, an alternative to p-values that is more flexible in certain ways. Specifically, an e-value framework retains valid testing properties even after combining dependent information, a relevant feature in the context of testing temporal networks. We apply the proposed test to synthetic and real-world networks, demonstrating various features inherited from the e-value formulation and exposing some of the inherent difficulties of testing on temporal networks.

This paper overviews an overlapping generations financial cash flow valuation model that evaluates the financial sustainability of English NHS Trusts. It quantifies the financial sustainability constraints related to societal demographic shifts that affect their ability to maintain service delivery quality. The financial model computes a new long-term financial sustainability performance metric based on the notion of the Social Return on Investment (“SROI”). The measure evaluates the financial sustainability of two English acute care hospital foundation Trusts. Significant generational imbalances are identified for both sample NHS Trusts, both within different cohorts of existing generations, and between existing and future generational cohorts. Suggestions are provided for implementing these ideas and expanding actuaries’ expertise and skill sets.

The tail behavior of aggregates of heavy-tailed random vectors is known to be determined by the so-called principle of ‘one large jump’, be it for finite sums, random sums, or Lévy processes. We establish that, in fact, a more general principle is at play. Assuming that the random vectors are multivariate regularly varying on various subcones of the positive orthant $[0,\infty)^d$, first we show that their aggregates are also multivariate regularly varying on these subcones. This allows us to approximate certain tail probabilities rendered asymptotically negligible under classical regular variation. Second, we discover that depending on the structure of a particular tail event, the tail behavior of the aggregates may be characterized by more than a single large jump. Finally, we illustrate a similar phenomenon for regularly varying multivariate Lévy processes, establishing as well a relationship between regular variation of a multivariate Lévy process and multivariate regular variation of its Lévy measure on different subcones. The applicability of these results in financial and insurance risk management is discussed.

The Bayesian approach offers a systematic framework for updating finite element (FE) models and quantifying the remaining uncertainty given measured data. However, an inappropriate formulation of the probabilistic model can compromise accuracy. This paper presents an improved hierarchical Bayesian method for FE model updating by formulating the likelihood function in a fully probabilistic manner and incorporating time-varying stiffness parameters. A key methodological novelty lies in latent variable treatment of unmeasured mode shapes within the Bayesian hierarchy, yielding the joint inference of structural parameters and modal quantities in a fully generative manner without explicit eigenvalue decomposition. Furthermore, the geometric nature of mode shapes is rigorously respected by constraining them to the unit hypersphere using a Bingham distribution. A Metropolis-within-Gibbs sampling algorithm is developed to approximate the posterior distribution, with QR and Cholesky decompositions ensuring computational efficiency and accuracy. Three case studies, including synthetic, lab, and field test data, validate the effectiveness of the proposed approach. The updated model can be used as a reference model for structural damage detection and condition assessment in structural health monitoring.

Health insurers systematically underinvest in prevention. Programme costs are immediate but claims benefits accrue over years, and actuaries have lacked a formal mechanism to translate behavioural intervention evidence into pricing-ready claims adjustments. This paper introduces the Behavioural Adjustment Factor (BAF), a multiplicative actuarial framework that quantifies the claims impact of behavioural interventions by decomposing reach, efficacy, clinical translation, and durability into a single pricing-ready construct. To the best of the author’s knowledge, the BAF is the first actuarial framework to decompose behavioural intervention impact into condition-specific claims projections suitable for pricing and reserving. Drawing on randomised controlled trial evidence, the framework distinguishes interventions that generate reliable claims savings from those that do not. Programme architecture is shown to matter more than incentive magnitude, and the distinction between disease management and general lifestyle programmes emerges as the principal axis along which actuarial expectations should diverge. A worked hypertension example illustrates how the four BAF components combine to produce a defensible claims-adjustment range, and a sensitivity analysis highlights the dominant role of effect persistence. The framework provides confidence intervals, Monte Carlo integration for Solvency II capital modelling, a milestone-based pilot funding structure, and a clear pathway from international evidence to UK-calibrated practice.

We study Stackelberg equilibria (Bowley optima) in a monopolistic centralized sequential-move insurance market, with a profit-maximizing insurer who sets premia using a distortion premium principle, and a single policyholder who seeks to minimize a distortion risk measure. We show that equilibria are characterized as follows: In equilibrium, the optimal indemnity function exhibits a layer-type structure, providing full insurance over any loss layer on which the policyholder is more pessimistic than the insurer’s pricing functional about tail losses; and no insurance coverage over loss layers on which the policyholder is less pessimistic than the insurer’s pricing functional about tail losses. In equilibrium, the optimal pricing distortion function is determined by the policyholder’s degree of risk aversion, whereby prices never exceed the policyholder’s marginal willingness to insure tail losses. Moreover, we show that both the insurance coverage and the insurer’s expected profit increase with the policyholder’s degree of risk aversion. Additionally, and echoing recent work in the literature, we show that equilibrium contracts are Pareto efficient, but they do not induce a welfare gain to the policyholder. Conversely, any Pareto-optimal contract that leaves no welfare gain to the policyholder can be obtained as an equilibrium contract. Finally, we consider a few examples of interest that recover some existing results in the literature as special cases of our analysis.

Reverse mortgage markets remain small internationally, with bequest motives frequently cited as a key reason. We develop a new two-generation lifecycle simulation model to study the role of reverse mortgages in intergenerational financial planning, particularly as a tool for families to bring forward bequests. The model incorporates parental altruism by assuming parents derive utility not only from their own consumption and housing but also from their child’s current and future financial well-being. This extends traditional bequest models, which typically consider only the wealth transferred at death. The model accounts for house price risk, interest rate risk, investment risk, wage growth, health shocks, long-term care costs, private pensions, and means-tested public pensions. Using this framework, calibrated to Australian economic and policy settings, we compare the welfare gains from bequests and early bequests (inter vivos gifts) for homeowning parents and adult children seeking to purchase their first home. The results suggest that families across a range of wealth levels can experience substantial welfare gains when the parent uses a reverse mortgage both for retirement income and to gift the adult child a first home deposit. Early bequests funded through reverse mortgages increase overall family welfare compared to preserving home equity for a bequest or using a reverse mortgage solely for the parent’s consumption, particularly for middle-wealth households. A policy experiment shows that gifting limits reduce welfare gains for some families, but have a small overall impact. These findings suggest that policies encouraging informed use of reverse mortgages could improve intergenerational financial security.

We present a detailed study of the evolution of the number of connected components in sub-critical multiplicative random graph processes. We consider a model where edges appear independently after an exponential time at a rate equal to the product of the sizes of the vertices. We provide an explicit expression for the fluid limit of the number of connected components normalized by its initial value, when the time is smaller than the inverse of the sum of the square of the initial vertex sizes. We also identify the diffusion limit of the rescaled fluctuations around the fluid limit. This is applied to several examples. In the particular setting of the Erdős–Rényi graph process, we give the fluid limit of the normalized number of connected components, and the diffusion limit of the scaled fluctuations in the sub-critical regime, where the mean degree is between zero and one.

We study branching Brownian motion with absorption, in which particles undergo Brownian motions and are killed upon hitting the absorption barrier. We prove that the empirical distribution function of the maximum of this process converges almost surely to a randomly shifted Gumbel distribution.

This article studies latent space models for social network data in which actors are embedded on a hypersphere and link probabilities depend on angular similarity. In contrast to Euclidean embeddings, the spherical formulation provides a compact parameter space, stabilizes the linear predictor through bounded inner products, and offers a natural representation of directional and cyclic structure. For inference, we combine maximum likelihood estimation, used to obtain initial values for latent positions and model parameters, with geometry-aware Bayesian methods based on Metropolis–Hastings and Hamiltonian Monte Carlo algorithms, including a geodesic Hamiltonian scheme for manifold-constrained parameters. We conduct a systematic empirical comparison between Euclidean and spherical latent space models on a benchmark social network dataset, evaluating model fit, predictive performance, and interpretability. The results show that spherical representations provide competitive performance while offering a more constrained and geometrically interpretable structure. Overall, the paper clarifies the role of latent space geometry in network modeling and highlights the importance of geometry-aware inference in statistical analysis of relational data.

In England, Lyme disease (LD) surveillance is based on laboratory-confirmed cases. However, clinical and epidemiological characteristics of these cases are limited; therefore, we conducted an enhanced surveillance study of acute LD in England. Enhanced data were collected through an online questionnaire sent to all laboratory-confirmed acute cases of LD resident in England, with specimen dates between 1 April 2023 and 31 March 2024. The analysis included data from 511/1086 cases. Respondents were representative of age, sex, and region of national LD trends. A total of 57.3% of the respondents reported that they did not realize that they had been bitten by a tick. Among those who remembered a tick bite, 66.1% reported bites close to their home and only 10.6% happened abroad, and 28.5% reported that the tick bite could have occurred in a garden. Most respondents were residents of areas of higher socioeconomic status. Erythema migrans was noted by 70.5% of the respondents. It is important to raise awareness among both patients and clinicians that ticks can be found in urban and suburban parks and gardens, as well as ensuring that wildlife initiatives to increase biodiversity include information on tick prevention and tick checks.

In this paper, we analyze the distribution of the total overlapping time spent with other customers in the $\mathrm{M}_\lambda/\mathrm{M}_\mu/1$ queue with First-Come First-Served service discipline. We show that the Laplace–Stieltjes transform of the overlapping time reduces to an incomplete gamma function representation. We also calculate the transform of the joint distribution of the overlapping time and the number of overlaps. In addition, we prove a heavy-traffic limit for the total overlapping time with scaling $1/(1-\rho)^2$ to a $\operatorname{Weibull}(1/2,1/\mu)$ random variable. Finally, for the $\mathrm{M}_{\lambda}/\mathrm{G}/1$ case, we derive the first two moments of the overlapping time.