IntroductionSpam webpage detection has become an important research problem due to the rapid growth of manipulative hyperlink structures in complex web networks. Traditional centrality measures often fail to identify spam webpages effectively because they focus on isolated structural characteristics and overlook higher-order neighborhood interactions and spam propagation behavior. To address these limitations, this study proposes a Hybrid Influence Gravity Model (HIGM) for detecting influential spam webpages in directed weighted web graphs.MethodsThe proposed HIGM integrates weighted degree, improved weighted k-shell decomposition, Laplacian centrality, neighborhood structural influence, average shortest-path variation, and clustering-based reduction into a unified gravity-based model. The framework incorporates hyperlink direction, weighted interactions, and multi-hop influence to capture structural relationships and spam propagation behavior. Its performance is evaluated using Naive Bayes, Support Vector Machine (SVM), AdaBoost Classification, and AdaBoost Regression on directed web graphs and the real-world WEBSPAM-UK2006 and WEBSPAM-UK2007 datasets. Additionally, an ablation study is conducted to examine the contributions of individual HIGM components, and sensitivity analysis is performed to investigate the effects of HIGM parameter variations. Machine-learning validation is also conducted using HIGM alone, PageRank alone, and the combined HIGM + PageRank model.ResultsExperimental evaluations demonstrate the effectiveness of HIGM in distinguishing spam webpages from trustworthy webpages compared with traditional centrality measures, including degree, closeness, betweenness, eigenvector, Laplacian, Degree K-shell Eigenvector (DKE), and DKE-based Gravity Model (DKEGM) centralities. The detection performance is assessed using accuracy, precision, recall, and F1-score across the evaluated datasets and machine-learning models. The ablation study evaluates the contributions of individual components, while the sensitivity analysis examines the influence of parameter variations on the model's performance. The comparative validation further evaluates the individual and combined contributions of HIGM and PageRank to spam webpage detection.DiscussionThe findings highlight the importance of integrating local and global structural information, hyperlink direction, weighted interactions, and gravity-based influence for identifying influential spam webpages in complex web networks. The ablation study and sensitivity analysis provide additional insight into the contributions of the HIGM components and the effects of parameter selection. Furthermore, comparing HIGM alone, PageRank alone, and their combination helps assess the effectiveness of the proposed model and the complementary role of PageRank. Overall, HIGM provides a unified framework for spam webpage detection in directed weighted web graphs.
In a fuzzy graph (FG), A membership value ranging from 0 to 1 is allocated to every vertex and edge, indicating the extent to which they participate in the graph. This idea is elaborated upon in an intuitionistic fuzzy graph (IFG), offering a more sophisticated method of addressing uncertainty through providing vertices and edges membership, non-membership, and hesitation values. Hesitancy fuzzy graphs (HFGs), which show the degrees of truthfulness, falsehood, and abstention, extend the idea of the IFG. The Zagreb indices (ZIs) are topological descriptors in graph theory (GT) defining the structural aspects of a graph utilizing vertex degrees. This study enhances the notion of fuzzy ZIs in FGs to hesitancy fuzzy Zagreb indices (HFZIs) in the context of HFGs. This study demonstrates the hesitancy fuzzy Zagreb matrix (HFZM) and the hesitancy fuzzy Zagreb energy (HFZE) of HFGs with the aid of illustrated examples. The lower and upper limits of the HFZM and HFZE have been established. To demonstrate their relevance, these HFGs were used to solve a decision-making (DM) problem, specifically identifying the ideal location for launching a Zagreb energy (ZE) based firm. Along with that, the ZE of the spherical fuzzy Zagreb matrix (SFZM) and the HFZM are compared.
We develop and analyse a multidomain Chebyshev integral-tau method for occupation-time and cumulative-Parisian options, priced through a two-dimensional degenerate-parabolic equation in log-price and accumulated occupation time in which the monitoring level switches the occupation clock on and off. Treating the monitoring level and the strike as element interfaces places every spatial singularity on an element face, so each open element carries analytic data. The diffusion uses an integration-preconditioned, primitive-matrix tau discretisation with pentadiagonal blocks closed by finitely many boundary and transmission tau rows, and the resulting bordered-banded systems are solved in linear time per step by a Sherman–Morrison–Woodbury correction. The occupation coupling is integrated exactly through a Strang splitting in which the clock transport reduces to an index shift, while the diffusion is advanced by a backward-Euler-damped, L-stable TR-BDF2 scheme that is second order yet free of the oscillations to which Crank–Nicolson is prone on non-smooth data. We prove geometric coefficient decay on aligned elements, an exact integration-preconditioning identity, finite-rank tau closure, dissipativity of the coupled operator (hence a contraction semigroup), the A-acceptability and L-stability of TR-BDF2, and—conditionally on a left half-plane, diagonalisable discrete generator—unconditional power boundedness of the discrete propagator. We also establish spectral and fully discrete second-order convergence and a linear-complexity solver whose well-posedness follows from a direct independence argument for the interface functionals. Numerical experiments support these predictions: with identical market data and polynomial degrees, alignment turns algebraic spatial convergence into geometric and reduces the European-call error by a factor of 1.4 × 107 at degree N = 48; TR-BDF2 is second order where Crank–Nicolson is only first order on the kinked payoff and yields a Gamma about two orders of magnitude more accurate at coarse steps; the bordered-banded solve is up to 92 times faster than dense factorization at s = 2050 degrees of freedom; and the occupation-time price agrees with finite-difference and Monte-Carlo references to within Monte-Carlo error. The discrete propagator eigenvalues lie in the unit disk uniformly in N in the tests reported here; this certifies power boundedness of GN, not by itself the left half-plane location of σ(AN).MSC 202065M70; 65N35; 65M12; 91G20; 91G60.JELC63; G13.
IntroductionHuman papillomavirus (HPV) is the most common sexually transmitted infection and the cause of virtually all cervical cancer, yet elimination is complicated by two features of real vaccination programmes: available vaccines are imperfect, leaving vaccinated individuals partially susceptible, and naturally acquired immunity wanes. We ask whether reducing the basic reproduction number below unity is sufficient for elimination when both mechanisms are present.MethodsWe formulate and analyse a six-compartment ( SEIRVC ) deterministic model combining an imperfect vaccine and waning immunity with explicit progression to an HPV-related cancer class. We establish positivity, boundedness, and an invariant region; derive R0 by the next-generation matrix method; characterize the equilibria and their stability using Routh–Hurwitz criteria, Lyapunov functions, and the geometric approach of Li and Muldowney; and analyse the bifurcation structure, supported by sensitivity indices and simulation.ResultsThe reproduction number decomposes additively into a susceptible and a vaccine-breakthrough route, and the disease-free equilibrium is locally asymptotically stable precisely when R0 < 1, and globally so under an explicit condition. Reducing the endemic equilibria to the roots of a quadratic shows that a leaky vaccine together with waning immunity produces a backward bifurcation, so that R0 < 1 is necessary but not sufficient for elimination and control must clear a turning point R0c<1. Above threshold the unique endemic equilibrium can lose stability through a Hopf bifurcation, giving sustained oscillations whose amplitude is magnified in the cancer class, which varies by a factor of order 102 within a cycle under constant parameters. Both phenomena require both mechanisms and vanish when the vaccine becomes perfectly protective or immunity permanent, the limit in which global stability of the endemic equilibrium is proved; two-parameter continuation shows the two regimes to be disjoint alternatives.DiscussionSensitivity analysis gives sgnΥβ = sgn(θ−λ H) so scaling up an insufficiently protective vaccine raises R0, while the waning rate has zero elasticity yet determines both non-classical regimes. Threshold quantities alone are therefore an unreliable guide to HPV control, and surveillance must track prevalence over time.
