Papers

In randomized controlled trials, ordinal outcomes typically improve statistical efficiency over binary outcomes. The treatment effect on an ordinal outcome is usually described by the odds ratio from a proportional odds model, but this summary measure lacks transparency with respect to its emphasis on the components of the ordinal outcome when proportional odds is violated. We propose various summary measures for ordinal outcomes that are fully transparent in this regard, including 'weighted geometric mean' odds ratios and relative risks, and 'weighted mean' risk differences. We also develop and evaluate efficient model-assisted Bayesian estimators for these population level summary measures based on non-proportional odds models that facilitate covariate adjustment with marginalization via the Bayesian bootstrap. We propose a weighting scheme that engenders appealing invariance properties, including to whether the ordinal outcome is ordered from best to worst versus worst to best. Using computer simulation, we show that comparative testing based on the proposed population level summary measures performs well relative to the conventional proportional odds approach. We also report an analysis of the COVID-OUT trial, which exhibits evidence of non-proportional odds.

This report investigates the extension of the Bayesian Subspace System Identification method proposed in our previous work to the Multiple-Input Multiple-Output (MIMO) case. We derive new equivariant priors and posterior distributions specifically suited for the MIMO framework. Numerical results utilizing the DAISY dataset are reported to validate the approach.

In case of sparse graphs, relation between the real eigenvalues of the non-backtracking matrix and those of the non-backtracking transition probability matrix is considered with respect to vertex clustering. For this purpose, the random walk along the non-backtracking graph is considered, the vertices of which are the bioriented edges, and the adjacency relation depends on whether the random walk goes through the oriented edges with the rule of “not going back in the next step”. This is encoded in the non-backtracking matrix that is the adjacency matrix of the non-backtracking graph. The structural real eigenvalues of the transition probability matrix are related to the constant multiples of the non-backtracking one, the concordance of which indicates the existence of a sparse stochastic block model behind the graph. “Inflation–deflation” techniques are also developed for clustering the vertices of the original graph together with real world applications.

Discrete random probability measures are central to Bayesian inference, particularly as priors for mixture modeling and clustering. A broad and unifying class is that of proper species sampling processes (SSPs), encompassing many Bayesian nonparametric priors. We show that any proper SSP admits an exact two-stage finite-mixture representation built from a latent truncation index and a simple reweighting of the atoms. For each realized truncation index, the representation has finitely many atoms, and averaging over the induced law of that index recovers the original SSP setwise. This yields at least two consequences: (i) an exact two-stage finite construction for arbitrary SSPs, without user-chosen truncation levels; and (ii) posterior inference in SSP mixture models via standard finite-mixture machinery, leading to tractable MCMC algorithms without ad hoc truncations. We explore these consequences by deriving explicit total-variation bounds for the approximation error when the truncation level is fixed, and by studying practical performance in mixture modeling, with emphasis on Dirichlet and geometric SSPs.

Proximal causal inference (PCI) has emerged as a promising framework for identifying and estimating causal effects in the presence of unobserved confounders. While many traditional causal inference methods rely on the assumption of no unobserved confounding, this assumption is likely often violated. PCI addresses this challenge by relying on an alternative set of assumptions regarding the relationships between treatment, outcome, and auxiliary variables that serve as proxies for unmeasured confounders. We review existing identification results, discuss the assumptions necessary for valid causal effect estimation via PCI, and compare different PCI estimation methods. We offer practical guidance on operationalizing PCI, with a focus on selecting and evaluating proxy variables using domain knowledge, measurement error perspectives, and negative control analogies. Through conceptual examples, we demonstrate tensions in proxy selection and discuss the importance of clearly defining the unobserved confounding mechanism. By bridging formal results with applied considerations, this work aims to demystify PCI, encourage thoughtful use in practice, and identify open directions for methodological development and empirical research.

In many sequential decision-making problems, researchers observe actions but not the rewards that drive behavior, yet still wish to evaluate and compare counterfactual policies. Inverse reinforcement learning (IRL) and dynamic discrete choice (DDC) models address this setting by positing an optimality model that links latent rewards to observed actions. Existing flexible IRL methods allow rich reward representations but typically do not provide valid inference, whereas classical DDC methods support inference only under restrictive parametric structure. We develop a semiparametric framework for debiased inverse reinforcement learning in maximum-entropy IRL and Gumbel-shock DDC models. Our key identification result is that the log-behavior policy can be treated as a pseudo-reward: it point-identifies policy value differences and, under a normalization constraint, the reward itself. This reduces inference on reward-dependent estimands to inference on smooth functionals of the behavior policy and transition kernel. We establish pathwise differentiability, derive efficient influence functions, and construct automatic debiased machine-learning estimators that permit flexible nuisance estimation while attaining $\sqrt{n}$-consistency, asymptotic normality, and semiparametric efficiency. The result is a computationally tractable framework for valid uncertainty quantification in flexible IRL and DDC models.

The framework of geometric extremes is based on the convergence of scaled sample clouds onto a limit set, characterized by a gauge function, with the shape of the limit set determining extremal dependence structures. While it is known that a blunt limit set implies asymptotic independence, the absence of bluntness can be linked to both asymptotic dependence and independence. Focusing on the bivariate case, under a truncated gamma modeling assumption with bounded angular density, we show that a “pointy” limit set implies asymptotic dependence, thus offering practical geometric criteria for identifying extremal dependence classes. Suitable models for the gauge function offer the ability to capture asymptotically independent or dependent data structures, without requiring prior knowledge of the true extremal dependence structure. The geometric approach thus offers a simple alternative to various parametric copula models that have been developed for this purpose in recent years. We consider two types of additively mixed gauge functions that provide a smooth interpolation between asymptotic dependence and asymptotic independence. We derive their explicit forms, explore their properties, and establish connections to the developed geometric criteria. Through a simulation study, we evaluate the effectiveness of the geometric approach with additively mixed gauge functions, comparing its performance to existing methodologies that account for both asymptotic dependence and asymptotic independence. The methodology is computationally efficient and yields reliable performance across various extremal dependence scenarios.

We study the problem of estimating the score function using both implicit score matching and denoising score matching. Assuming that the data distribution exhibiting a low-dimensional structure, we prove that implicit score matching is able not only to adapt to the intrinsic dimension, but also to achieve the same rates of convergence as denoising score matching in terms of the sample size. Furthermore, we demonstrate that both methods allow us to estimate log-density Hessians without the curse of dimensionality by simple differentiation. This justifies convergence of ODE-based samplers for generative diffusion models. Our approach is based on Gagliardo-Nirenberg-type inequalities relating weighted $L^2$-norms of smooth functions and their derivatives.

In order to describe the extremal behaviour of some stochastic process $X$, approaches from univariate extreme value theory are typically generalized to the spatial domain. In particular, generalized peaks-over-threshold approaches allow for the consideration of single extreme events. These can be flexibly defined as exceedances of a risk functional $r$, such as a spatial average, applied to $X$. Inference for the resulting limit process, the so-called $r$-Pareto process, requires the evaluation of $r(X)$ and thus the knowledge of the whole process $X$. In many practical applications, however, observations of $X$ are only available at scattered sites. To overcome this issue, we propose a two-step MCMC-algorithm in a Bayesian framework. In a first step, we sample from $X$ conditionally on the observations in order to evaluate which observations lead to $r$-exceedances. In a second step, we use these exceedances to sample from the posterior distribution of the parameters of the limiting $r$-Pareto process. Alternating these steps results in a full Bayesian model for the extremes of $X$. We show that, under appropriate assumptions, the probability of classifying an observation as $r$-exceedance in the first step converges to the desired probability. Furthermore, given the first step, the distribution of the Markov chain constructed in the second step converges to the posterior distribution of interest. The procedure is compared to the Bayesian version of the standard procedure in a simulation study.

The integration of data from multiple sources is increasingly used to achieve larger sample sizes and enhance population diversity. Our previous work established that, under random sampling from the same underlying population, integrating large incomplete datasets with summary-level data produces unbiased parameter estimates. In this study, we develop a novel statistical framework that enables the integration of summary-level data with information from heterogeneous data sources by leveraging auxiliary information. The proposed approach estimates study-specific sampling weights using this auxiliary information and calibrates the estimating equations to obtain the full set of model parameters. We evaluate the performance of the proposed method through simulation studies under various sampling designs and illustrate its application by reanalyzing U.S. cancer registry data combined with summary-level odds ratio estimates for selected colorectal cancer (CRC) risk factors, while relaxing the random sampling assumption.

Community structure in social and collaborative networks often emerges from a complex interplay between structural mechanisms, such as degree heterogeneity and leader-driven attraction, and homophily on node attributes. Existing community detection methods typically focus on these dimensions in isolation, limiting their ability to recover interpretable communities in presence of such mechanisms. In this paper, we propose AttDeCoDe, an attribute-driven extension of a density-based community detection framework, developed to analyse networks where node characteristics play a central role in group formation. Instead of defining density purely from network topology, AttDeCoDe estimates node-wise density in the attribute space, allowing communities to form around attribute-based community representatives while preserving structural connectivity constraints. This approach naturally captures homophily-driven aggregation while remaining sensitive to leader influence. We evaluate the proposed method through a simulation study based on a novel generative model that extends the degree-corrected stochastic block model by incorporating attribute-driven leader attraction, reflecting key features of collaborative research networks. We perform an empirical application to research collaboration data from the Horizon programmes, where organisations are characterised by project-level thematic descriptors. Both results show that AttDeCoDe offers a flexible and interpretable framework for community detection in attributed networks achieving competitive performance relative to topology-based and attribute-assisted benchmarks.

Spatial graphs are particular graphs for which the nodes are localized in space (e.g., public transport network, molecules, branching biological structures). In this work, we consider the problem of spatial graph reduction, that aims to find a smaller spatial graph (i.e., with less nodes) with the same overall structure as the initial one. In this context, performing the graph reduction while preserving the main topological features of the initial graph is particularly relevant, due to the additional spatial information. Thus, we propose a topological spatial graph coarsening approach based on a new framework that finds a trade-off between the graph reduction and the preservation of the topological characteristics. The coarsening is realized by collapsing short edges. In order to capture the topological information required to calibrate the reduction level, we adapt the construction of classical topological descriptors made for point clouds (the so-called persistent diagrams) to spatial graphs. This construction relies on the introduction of a new filtration called triangle-aware graph filtration. Our coarsening approach is parameter-free and we prove that it is equivariant under rotations, translations and scaling of the initial spatial graph. We evaluate the performances of our method on synthetic and real spatial graphs, and show that it significantly reduces the graph sizes while preserving the relevant topological information.

Subgroup analysis evaluates treatment effects across multiple sub-populations. When subgroups are defined by latent memberships inferred from imperfect measurements, the analysis typically involves two inter-connected models, a latent class model and a subgroup outcome model. The classical one-stage framework, which models the joint distribution of the two models, may be infeasible with observational data containing many confounders. The two-stage framework, which first estimates the latent class model and then performs subgroup analysis using estimated latent memberships, can accommodate potential confounders but may suffer from bias issues due to misclassification of latent subgroup memberships. This paper focuses on latent subgroups inferred from binary item responses and addresses when and how a valid two-stage latent subgroup analysis can be made with observational data. We investigate the maximum misclassification rate that a valid two-stage framework can tolerate. Introducing a spectral method perspective, we propose a two-stage approach to achieve the desired misclassification rate with the blessing of many item responses. Our method accommodates high-dimensional confounders, is computationally efficient and robust to noninformative items. In observational studies, our methods lead to consistent estimation and valid inference on latent subgroup effects. We demonstrate its merit through simulation studies and an application to educational assessment data.

This article studies the robust version of persistent homology based on trimming methodology to capture the geometric feature through support of the data in presence of outliers. Precisely speaking, the proposed methodology works when the outliers lie outside the main data cloud as well as inside the data cloud. In the course of theoretical study, it is established that the Bottleneck distance between the proposed robust version of persistent homology and its population analogue can be made arbitrary small with a certain rate for a sufficiently large sample size. The practicability of the methodology is shown for various simulated data and bench mark real data associated with cellular biology.

Electoral boundaries in Malaysia are not publicly available in machine-readable form. This prevents rigorous analysis of geography-centric issues such as malapportionment and gerrymandering, and constrains spatial perspectives on electoral outcomes. We present the second component of the Malaysian Election Corpus (MECo), an open-access collection of digital electoral boundaries covering all 19 approved delimitation exercises in Malaysia's history, from the first set of Malayan boundaries in 1954 until the 2019 Sabah delimitation. We also auto-generate election-time maps for all federal and state elections up to 2025, and include equal-area and electorate-weighted cartograms to support deeper geospatial analysis. This is the first complete, publicly-available, and machine-readable record of Malaysia's electoral boundaries, and fills a critical gap in the country's electoral data infrastructure.

Sampling based on score diffusions has led to striking empirical results, and has attracted considerable attention from various research communities. It depends on availability of (approximate) Stein score functions for various levels of additive noise. We show how in some generality, the availability of scores allows the general problem to be “reduced” to sampling from an adaptively constructed sequence of $K$ strongly log-concave (SLC) sub-problems. The reduction is simple, constructive and algorithm-independent, so that any SLC sampler can be used as a subroutine. Various bounds on score-based sampling complexity follow directly: for instance, high-accuracy SLC samplers yield $\tilde{\mathcal{O}}(K \sqrt{d} \operatorname{polylog}(1/\varepsilon))$ guarantees for accuracy $\varepsilon$ in dimension $d$, where randomized midpoint SLC schemes yield $\tilde{\mathcal{O}}(K d^{1/3} \operatorname{poly}(1/\varepsilon))$ guarantees. When the original distribution itself is SLC, we prove that $K \leq 1 + \log_2(\kappa)$, thereby obtaining the first efficient procedure with logarithmic dependence on condition number $\kappa$; for general distributions, the quantity $K$ depends on the geometry of score Hessian across the trajectory. Our analysis is direct and simple, involving techniques and insights complementary to those in standard analyses of discretized diffusions.

Machine learning systems appear stochastic but are deterministically random, as seeded pseudorandom number generators produce identical realisations across repeated executions. Standard evaluation practice typically treats runs across alternatives as independent and does not exploit shared sources of randomness. This paper analyses the statistical structure of comparative evaluation under shared random seeds. Under this design, competing systems are evaluated using identical seeds, inducing matched stochastic realisations and yielding strict variance reduction whenever outcomes are positively correlated at the seed level. We demonstrate these effects using an extended learning-based multi-agent economic simulator, where paired evaluation exposes systematic differences in aggregate and distributional outcomes that remain statistically inconclusive under independent evaluation at fixed budgets.

Although conformal prediction provides robust marginal coverage guarantees, achieving reliable conditional coverage for specific inputs remains challenging. While exact distribution-free conditional coverage is impossible with finite samples, recent work has focused on improving the conditional coverage of standard conformal procedures. Distinct from approaches that target relaxed notions of conditional coverage, we directly target the mean squared error of conditional coverage by refining the quantile regression components that underpin many conformal methods. Leveraging a Taylor expansion, we derive a sharp surrogate objective for quantile regression: a density-weighted pinball loss, where the weights are given by the conditional density of the nonconformity score evaluated at the true quantile. We propose a three-headed quantile network that estimates these weights via finite differences using auxiliary quantile levels at $1-\alpha \pm \delta$, subsequently fine-tuning the central quantile by optimizing the weighted loss. We provide a theoretical analysis with exact non-asymptotic guarantees characterizing the resulting excess risk. Extensive experiments on diverse high-dimensional real-world datasets demonstrate remarkable improvements in conditional coverage performance.

We introduce a new goodness-of-fit test for count data on $\mathbb{N}$ for the Zeta distribution with unknown parameter. The test is built on a Stein-type characterization that uses, as Stein operator, the infinitesimal generator of a birth-death process whose stationary distribution is Zeta. The resulting $L^2$-type statistic is shown to be omnibus consistent, and we establish the limit null behavior as well as the validity of the associated parametric bootstrap procedure. In a Monte Carlo simulation study, we compare the proposed test with the only existing Zeta-specific procedure of Meintanis (2009), as well as with more general competitors based on empirical distribution functions, kernel Stein discrepancies and other Stein-type characterizations.

In this paper, we construct a class of stochastic interpolation neural network operators (SINNOs) with random coefficients activated by sigmoidal functions. We establish their boundedness, interpolation accuracy, and approximation capabilities in the mean square sense, in probability, as well as path-wise within the space of second-order stochastic (random) processes \( L^2(\Omega, \mathcal{F},\mathbb{P}) \). Additionally, we provide quantitative error estimates using the modulus of continuity of the processes. These results highlight the effectiveness of SINNOs for approximating stochastic processes with potential applications in COVID-19 case prediction.

As wireless communication applications evolve from traditional multipath environments to high-mobility scenarios like unmanned aerial vehicles, multiplexing techniques have advanced accordingly. Traditional single-carrier frequency-domain equalization (SC-FDE) and orthogonal frequency-division multiplexing (OFDM) have given way to emerging orthogonal time-frequency space (OTFS) and affine frequency-division multiplexing (AFDM). These approaches exploit specific channel structures to diagonalize or sparsify the effective channel, thereby enabling low-complexity detection. However, their reliance on these structures significantly limits their robustness in dynamic, real-world environments. To address these challenges, this paper studies a random multiplexing technique that is decoupled from the physical channels, enabling its application to arbitrary norm-bounded and spectrally convergent channel matrices. Random multiplexing achieves statistical fading-channel ergodicity for transmitted signals by constructing an equivalent input-isotropic channel matrix in the random transform domain. It guarantees the asymptotic replica MAP bit-error rate (BER) optimality of AMP-type detectors for linear systems with arbitrary norm-bounded, spectrally convergent channel matrices and signaling configurations, under the unique fixed point assumption. A low-complexity cross-domain memory AMP (CD-MAMP) detector is considered, leveraging the sparsity of the time-domain channel and the randomness of the equivalent channel. Optimal power allocations are derived to minimize the replica MAP BER and maximize the replica constrained capacity of random multiplexing systems. The optimal coding principle and replica constrained-capacity optimality of CD-MAMP detector are investigated for random multiplexing systems. Additionally, the versatility of random multiplexing in diverse wireless applications is explored.

The reconstruction of time-dependent Robin coefficients is a challenging inverse heat transfer problem due to its inherent ill-posedness. This paper introduces a hierarchical Bayesian approach integrated with a persistent homology (PH) prior for robust coefficient estimation. By quantifying the birth and death of topological features, the PH-based prior provides a global structural constraint that transcends local derivative based penalties. Numerical experiments show that this topological perspective allows for the preservation of complex temporal profiles without the typical staircase distortions of total variation (TV) priors or the excessive blurring of Gaussian models. A key feature of our framework is the hierarchical implementation, which yields an automated, data-driven selection of hyperparameters. The results demonstrate that while PH-based inference yields competitive accuracy compared to TV regularization, it offers superior performance in preserving the multiscale characteristics of the Robin coefficient, providing a robust alternative for convective heat transfer diagnostics

This paper establishes the Local Asymptotic Normality (LAN) property for the mixed fractional Brownian motion under high-frequency observations with Hurst index $H \in (0, 3/4)$. The simultaneous estimation of the volatility and the Hurst index encounters a degeneracy problem in the Fisher information matrix.

To what extent are users surveilled on the web, by what technologies, and by whom? We answer these questions by combining passively observed, anonymized browsing data of a large, representative sample of Americans with domain-level data on tracking from Blacklight. We find that nearly all users ($ > 99\%$) encounter at least one ad tracker or third-party cookie over the observation window. More invasive techniques like session recording, keylogging, and canvas fingerprinting are less widespread, but over half of the users visited a site employing at least one of these within the first 48 hours of the start of tracking. Linking trackers to their parent organizations reveals that a single organization, usually Google, can track over $50\%$ of web activity of more than half the users. Demographic differences in exposure are modest and often attenuate when we account for browsing volume. However, disparities by age and race remain, suggesting that what users browse, not just how much, shapes their surveillance risk.

This paper introduces a novel quasi-likelihood extension of the generalised Kendall \(\tau_{a}\) estimator, together with an extension of the Kemeny metric and its associated covariance and correlation forms. The central contribution is to show that the U-statistic structure of the proposed coefficient \(\tau_{\kappa}\) naturally induces a quasi-maximum likelihood estimation (QMLE) framework, yielding consistent Wald and likelihood ratio test statistics. The development builds on the uncentred correlation inner-product (Hilbert space) formulation of Emond and Mason (2002) and resolves the associated sub-Gaussian likelihood optimisation problem under the \(\ell_{2}\)-norm via an Edgeworth expansion of higher-order moments. The Kemeny covariance coefficient \(\tau_{\kappa}\) is derived within a novel likelihood framework for pairwise comparison-continuous random variables, enabling direct inference on population-level correlation between ranked or weakly ordered datasets. Unlike existing approaches that focus on marginal or pairwise summaries, the proposed framework supports sample-observed weak orderings and accommodates ties without information loss. Drawing parallels with Thurstone's Case V latent ordering model, we derive a quasi-likelihood-based tie model with analytic standard errors, generalising classical U-statistics. The framework applies to general continuous and discrete random variables and establishes formal equivalence to Bradley-Terry and Thurstone models, yielding a uniquely identified linear representation with both analytic and likelihood-based estimators.

This article investigates the least squares estimators (LSE) for the unknown parameters in stochastic differential equations (SDEs) that are affected by Lévy noise, particularly when the sample paths are sparse. Specifically, given $n$ sparsely observed curves related to this model, we derive the least squares estimators for the unknown parameters: the drift coefficient, the diffusion coefficient, and the jump-diffusion coefficient. We also establish the asymptotic rate of convergence for the proposed LSE estimators. Additionally, in the supplementary materials, the proposed methodology is applied to a benchmark dataset of functional data/curves, and a small simulation study is conducted to illustrate the findings.

Non-parametric correlation coefficients have been widely used for analysing arbitrary random variables upon common populations, when requiring an explicit error distribution to be known is an unacceptable assumption. We examine an \(\ell_{2}\) representation of a correlation coefficient (Emond and Mason, 2002) from the perspective of a statistical estimator upon random variables, and verify a number of interesting and highly desirable mathematical properties, mathematically similar to the Whitney embedding of a Hilbert space into the \(\ell_{2}\)-norm space. In particular, we show here that, in comparison to the traditional Spearman (1904) \(\rho\), the proposed Kemeny \(\rho_{\kappa}\) correlation coefficient satisfies Gauss-Markov conditions in the presence or absence of ties, thereby allowing both discrete and continuous marginal random variables. We also prove under standard regularity conditions a number of desirable scenarios, including the construction of a null hypothesis distribution which is Student-t distributed, parallel to standard practice with Pearson's r, but without requiring either continuous random variables nor particular Gaussian errors. Simulations in particular focus upon highly kurtotic data, with highly nominal empirical coverage consistent with theoretical expectation.

Consider the task of generating samples from a tilted distribution of a random vector whose underlying distribution is unknown, but samples from it are available. This finds applications in fields such as finance and climate science, and in rare event simulation. In this article, we discuss the asymptotic efficiency of a self-normalized importance sampler of the tilted distribution. We provide a sharp characterization of its accuracy, given the number of samples and the degree of tilt. Our findings reveal a surprising dichotomy: while the number of samples needed to accurately tilt a bounded random vector increases polynomially in the tilt amount, it increases at a super polynomial rate for unbounded distributions.

Generative artificial intelligence (GenAI) is shifting from conversational assistants toward agentic systems – autonomous decision-making systems that sense, decide, and act within operational workflows. This shift creates an autonomy paradox: as GenAI systems are granted greater operational autonomy, they should, by design, embody more formal structure, more explicit constraints, and stronger tail-risk discipline. We argue that stochastic generative models can be fragile in operational domains unless paired with mechanisms that provide verifiable feasibility, robustness to distribution shift, and stress testing under high-consequence scenarios. To address this challenge, we develop a conceptual framework for assured autonomy grounded in operations research (OR), built on two complementary approaches. First, flow-based generative models frame generation as deterministic transport characterized by an ordinary differential equation, enabling auditability, constraint-aware generation, and connections to optimal transport, robust optimization, and sequential decision control. Second, operational safety is formulated through an adversarial robustness lens: decision rules are evaluated against worst-case perturbations within uncertainty or ambiguity sets, making unmodeled risks part of the design. This framework clarifies how increasing autonomy shifts OR's role from solver to guardrail to system architect, with responsibility for control logic, incentive protocols, monitoring regimes, and safety boundaries. These elements define a research agenda for assured autonomy in safety-critical, reliability-sensitive operational domains.

Influence Maximization (IM) seeks to identify a small set of seed nodes in a social network to maximize expected information spread under a diffusion model. While community-based approaches improve scalability by exploiting modular structure, they typically assume independence between communities, overlooking inter-community influence$\unicode{x2014}$a limitation that reduces effectiveness in real-world networks. We introduce Community-IM++, a scalable framework that explicitly models cross-community diffusion through a principled heuristic based on community-based diffusion degree (CDD) and a progressive budgeting strategy. The algorithm partitions the network, computes CDD to prioritize bridging nodes, and allocates seeds adaptively across communities using lazy evaluation to minimize redundant computations. Experiments on large real-world social networks under different edge weight models show that Community-IM++ achieves near-greedy influence spread at up to 100 times lower runtime, while outperforming Community-IM and degree heuristics across budgets and structural conditions. These results demonstrate the practicality of Community-IM++ for large-scale applications such as viral marketing, misinformation control, and public health campaigns, where efficiency and cross-community reach are critical.