EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6476 ISSN 1307-5543 – ejpam.com Published by New York Business Global Detecting of Rumours in Social Media via Neutrosophic Hesitance Distance Measures M. Kaviyarasu1,∗, R. Venitha1, Mohammed Alqahtani2 1 Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Avadi, Chennai, Tamil Nadu 600062, India 2 Department of Basic Sciences, College of Science and Theoretical Studies, Saudi Electronic University, P.O. Box 93499, Riyadh 11673, Saudi Arabia Abstract. This article introduces a new approach for digital misinformation detection with the application of Neutrosophic Hesitance Distance Measures. Including hesitance values in neutro- sophic similarity and distance measurement helps us to deal with uncertainty and vagueness better. Complement, max, min, and max-min operations are explained along with definitions. An example provides a glimpse of how Neutrosophic Set Hesitance Distance Measure enhances decision-making for detecting misinformation on social media. The findings identify the benefits of merging hesi- tance into neutrosophic fuzzy systems to achieve more precise and reliable analysis. 2020 Mathematics Subject Classifications: 03E72, 03B52, 28E10 Key Words and Phrases: Neutrosophic fuzzy set, Neutrosophic hesitance distance measure, Score function, Decision-making problem 1. Introduction Smarandache [1] offered Neutrosophic logic for dealing with the uncertainties, inde- terminacies, and inconsistences of existing information systems proficiently in order to present a powerful mathematical framework. Instead of standard logic frameworks, this system clearly specifies the levels of truth, indeterminacy, and falsity, which makes it ex- clusively compatible for the examination of imprecise and incomplete data sets. Based on this originated perceptions, Single-Valued Neutrosophic Sets (SVNSs) were established by Wang et al. [2], to enhance the computational modeling efficiency under uncertainty, where each one of the apparatuses of truth, indeterminacy, and falsity might quantified through particular real numbers. Majumdar & Samanta [3] formulated numerous concepts and distance measures, to measure how near the neutrosophic entities are to each other. These gears are vital ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6476 Email addresses: drkaviyarasum@veltech.edu.in (M. Kaviyarasu), venitha2002@gmail.com (R. Venitha), m.alqahtani@seu.edu.sa (M. Alqahtani) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 2 of 26 in determining the extent to a particular data point which is constant with expected information. Distance measure specially structured for interval in neutrosophic sets was developed by Ye ([4]), which is useful in uncertainty profound decision-making situations. After that, the philosophy of Hesitant Neutrosophic Sets (HNSs) was proposed, paves path for multiple value attribute of truth, indeterminacy, and falsity to arrest the real-world hesitation and vagueness in judgments (Peng & Dai, [5]). In medical diagnosis (Ye et al., [6]), cybersecurity analysis (Broumi et al., 2023), and decision-making with multiple criteria (Pramanik et al., [7]), these models elevate the interpretation of uncertain or inconclusive outputs. In order to detecting misinformation we utilize Neutrosophic Hesitance Distance Mea- sures (NHDMs) to measure the distance between authenticated information and possibly falsified virtual content. By exceeding binary true/false labels, such measures can be able to precise and captures of the uncertainty involved in such content (Shu et al., [8]) More- over, census said that the major contribution of propagation of online lies are from bot accounts (Shao et al., [9]), the spread of disinformation where caused because of emo- tionally charged content and hyperpartisan media outlets (Potthast et al., [10]). These contemplations upgrades of uncertainty that can be managed by neutrosophic models uniquely (Abdel-Basset et al., [11]). For detecting misinformation offers a robust mecha- nism for dealing with the uncertainty embedded in digital information networks, we utilizes the collabration of neutrosophic logic and hesitancy-based distance functions. This article demonstrate a new framework under these concepts for the improvisation of detection and classification of false content on social media. In recent years various Distance Measures applied in neutrosophic sets [12–15]. 1.1. Motivation In today’s world, the prompt diffusion of rumours threatens the lucidity of society, political stability, and public trust. Traditional methods for detecting disinformation rely mostly on probabilistic models, statistical analysis, or machine learning algorithms. Never- theless, these tend to flop when provoked with imprecise or deceptive content particularly when input data is half-done, contradictory, or deceitful. Social media has high potential to create an impact on political debate and also makes this problem more challenging, as web-based discussions frequently consist of uncertain, imprecise, or half-formed opin- ions. This type of human imprecision and partial belief are quiet hard to capture through conventional binary or fuzzy logical systems. To handle such complications more excellently, Neutrosophic Logic delivers a more subtle approach by recognising the truth, indeterminacy, and falsity as separate and con- comitant dimensions. Through this, Single-Valued Neutrosophic Sets (SVNSs) and Hesi- tant Neutrosophic Sets (HNSs) have developed into impact means of representing hesitant and multi-valued judgments pervasive features in relation on online. M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 3 of 26 1.2. Novelty This research offers a new method to detect false political rumour using Neutrosophic Hesitance Distance Measures (NHDMs). In contrast to traditional methods depending on static labels or confidence levels, the new model involves varying levels of hesitation in information’s truth, indeterminacy, and falsity aspects. The major contributions are: • Formulation and application of Neutrosophic Hesitance Distance Measures intended to determine similarity between possibly deceiving content and established fact checks with heightened sensitivity towards hesitation and doubt. • An adaptive detection model that can identify misinformation with degrees of certainty and ambiguity, instead of strict binary classification. • Validation of the method on actual social media datasets, illustrating its capability to outperform conventional detection methods in situations where uncertainty, hesi- tancy, and partial truth prevail. • This method fills the gap between uncertainty mathematical modeling and real-world misinformation detection, presenting a more interpretable and human-oriented solu- tion towards fighting fake news and deceptive information in online communities. 1.3. Structure and Contribution of this paper The paper begins with an Introduction that establishes the need for advanced dis- tance measures in uncertain environments and introduces neutrosophic hesitant fuzzy sets as a solution. In Preliminaries, foundational concepts such as fuzzy sets, neutrosophic sets, and hesitancy are discussed to provide theoretical background. The core of the pa- per lies in Section 3, where novel Neutrosophic Fuzzy Distance Measures are proposed and mathematically formulated. These measures are then applied in Section 4 through a structured Decision-Making Process, including numerical examples that demonstrate their practical relevance. Section 5 outlines the Advantages of the proposed distance measures, emphasizing their ability to handle uncertainty and hesitation more effectively than ex- isting approaches. The paper concludes with Section 6, summarizing key findings and suggesting future research directions for extending the proposed models to more complex decision-making systems. 2. Preliminaries Definition 1. [1] An NS D is defined as:D = {⟨d, νD(d), σD(d), µD(d)⟩ : d ∈ B}, νD(d) : B → [0, 1] denote the truth grade value, σD(d) : B → [0, 1] denote the indeterminacy grade value and µD(d) : B → [0, 1] denote the falsity grade value, where satisfies the condition 0 ≤ (νD(d)) + (σD(d)) + (µD(d)) ≤ 3. M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 4 of 26 Definition 2. [1] A NFS is characterised by a grade value νD(d) = TD(d), σD(d) = ID(d) and µD(d) = FD(d) The Hesitance degree of NFS is defined by hD(d) = {|(1−TD(d))|, |(1− ID(d))|, |(1− FD(d))|} written as D ={⟨x, νD(d), σD(d), µD(d)⟩ : x ∈ B}, D =TD(d), ID(d), FD(d) Definition 3. Let Dn, n = 1, 2, ..., N be N NFS defined on B and νD(d) = TD(d), σD(d) = ID(d), µD(d) = FD(d) their grade value. The Neutrosophic fuzzy (NF) cartesian product of Dn represented by ×n i=1Di = D1 ×D2 × · · · ×Dnis defined by the function ν×n i=1Di(d) =T×n i=1Di(d) = min{TD1(d), TD2(d), · · · , TDn(d)} σ×n i=1Di(d) =I×n i=1Di(d) = min{ID1(d), ID2(d), · · · , IDn(d)} µ×n i=1Di(d) =F×n i=1Di(d) = max{FD1(d), FD2(d), · · · , FDn(d)} Definition 4. [2] Let D1 and D2 be any two NFS on B then the intersection of two NFS is written as D1 ∩D2 =TD1(d) ∩ TD2(d) = min[TD1(d), TD2(d)] D1 ∩D2 =ID1(d) ∩ ID2(d) = min[ID1(d), ID2(d)] D1 ∩D2 =FD1(d) ∩ FD2(d) = max[FD1(d), FD2(d)] Definition 5. [2] Let D1 and D2 be any two NFS on B then the union of two NFS is written as D1 ∪D2 =TD1(d) ∪ TD2(d) = max[TD1(d), TD2(d)] D1 ∪D2 =ID1(d) ∪ ID2(d) = max[ID1(d), ID2(d)] D1 ∪D2 =FD1(d) ∪ FD2(d) = min[FD1(d), FD2(d)] Definition 6. [2] Let D be any two NFS on B then the complement of NFS D is written as Dc =TDc(d) = [1− TDc(d)] Dc =IDc(d) = [1− IDc(d)] Dc =FDc(d) = [1− FDc(d)] Theorem 1. For any these NFS D1, D2, D3, D4 on S. The standard intersection, union and complement satisfy the following i. D1 ∩D2 = D2 ∩D1 ii. D1 ∪D2 = D2 ∪D1 iii. D1 ∩ (D2 ∩D3) = (D1 ∩D2) ∩D3 M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 5 of 26 iv. D1 ∪ (D2 ∪D3) = (D1 ∪D2) ∪D3 v. D1 ∩ (D2 ∪D3) = (D1 ∩D2) ∪ (D1 ∩D3) vi. D1 ∪ (D2 ∩D3) = (D1 ∪D2) ∩ (D1 ∪D3) vii. (D1 ∩D2) c = Dc 1 ∪Dc 2 viii. (D1 ∪D2) c = Dc 1 ∩Dc 2 Proof. i. D1 ∩D2 =TD1(d) ∩ TD2(d) = min[TD1(d), TD2(d)] =min[TD2(d), TD1(d)] = TD2(d) ∩ TD1(d) =D2 ∩D1 D1 ∩D2 =ID1(d) ∩ ID2(d) = min[ID1(d), ID2(d)] =min[ID2(d), ID1(d)] = ID2(d) ∩ ID1(d) =D2 ∩D1 D1 ∩D2 =FD1(d) ∩ FD2(d) = max[FD1(d), FD2(d)] =max[FD2(d), FD1(d)] = FD2(d) ∩ FD1(d) =D2 ∩D1 ii. D1 ∪D2 =TD1(d) ∪ TD2(d) = max[TD1(d), TD2(d)] =max[TD2(d), TD1(d)] = TD2(d) ∪ TD1(d) =D2 ∪D1 D1 ∪D2 =ID1(d) ∪ ID2(d) = max[ID1(d), ID2(d)] =max[ID2(d), ID1(d)] = ID2(d) ∪ ID1(d) =D2 ∪D1 D1 ∪D2 =FD1(d) ∪ FD2(d) = min[FD1(d), FD2(d)] =min[FD2(d), FD1(d)] = FD2(d) ∪ FD1(d) =D2 ∪D1 iii. D1 ∩ (D2 ∩D3) =TD1(d) ∩ TD2∩D3(d) =min[TD1(d),min[TD2(d), TD3(d)]] =min[TD1(d), TD2(d), TD3(d)] =TD1∩D2(d) ∩ TD3(d) M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 6 of 26 =(D1 ∩D2) ∩D3 D1 ∩ (D2 ∩D3) =ID1(d) ∩ ID2∩D3(d) =min[ID1(d),min[ID2(d), ID3(d)]] =min[ID1(d), ID2(d), ID3(d)] =ID1∩D2(d) ∩ ID3(d) =(D1 ∩D2) ∩D3 D1 ∩ (D2 ∩D3) =FD1(d) ∩ FD2∩D3(d) =max[FD1(d),max[FD2(d), FD3(d)]] =max[FD1(d), FD2(d), FD3(d)] =FD1∩D2(d) ∩ FD3(d) =(D1 ∩D2) ∩D3 iv. D1 ∪ (D2 ∪D3) =TD1(d) ∪ TD2∪D3(d) =max[TD1(d),max[TD2(d), TD3(d)]] =max[TD1(d), TD2(d), TD3(d)] =TD1∪D2(d) ∪ TD3(d) =(D1 ∪D2) ∪D3 D1 ∪ (D2 ∪D3) =ID1(d) ∪ ID2∪D3(d) =max[ID1(d),max[ID2(d), ID3(d)]] =max[ID1(d), ID2(d), ID3(d)] =ID1∪D2(d) ∪ ID3(d) =(D1 ∪D2) ∪D3 D1 ∪ (D2 ∪D3) =FD1(d) ∪ FD2∪D3(d) =min[FD1(d),min[FD2(d), FD3(d)]] =min[FD1(d), FD2(d), FD3(d)] =FD1∪D2(d) ∪ FD3(d) =(D1 ∪D2) ∪D3 v. To prove the distributive law of intersection over union, we take some cases. Case 1 TD1(d) ≥ TD2(d) ≥ TD3(d), ID1(d) ≥ ID2(d) ≥ ID3(d), FD1(d) ≥ FD2(d) ≥ FD3(d) D1 ∩ (D2 ∪D3) =TD1(d) ∩ TD2∪D3(d) =min[TD1(d),max[TD2(d), TD3(d)]] M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 7 of 26 =min[TD1(d), TD2(d)] =TD2(d) (1) (D1 ∩D2) ∪ (D1 ∩D3) =TD1∩D2(d) ∪ TD1∩D3(d) =max[min[TD1(d), TD2(d)],min[TD1(d), TD3(d)]] =max[TD2(d), TD3(d)] =TD2(d) (2) D1 ∩ (D2 ∪D3) =ID1(d) ∩ ID2∪D3(d) =min[ID1(d),max[ID2(d), ID3(d)]] =min[ID1(d), ID2(d)] =ID2(d) (3) (D1 ∩D2) ∪ (D1 ∩D3) =ID1∩D2(d) ∪ ID1∩D3(d) =max[min[ID1(d), ID2(d)],min[ID1(d), ID3(d)]] =max[ID2(d), ID3(d)] =ID2(d) (4) D1 ∩ (D2 ∪D3) =FD1(d) ∩ FD2∪D3(d) =max[FD1(d),min[FD2(d), FD3(d)]] =max[FD1(d), FD3(d)] =FD1(d) (5) (D1 ∩D2) ∪ (D1 ∩D3) =FD1∩D2(d) ∪ FD1∩D3(d) =min[max[FD1(d), FD2(d)],max[FD1(d), FD3(d)]] =min[FD1(d), FD1(d)] =FD1(d) (6) From (1),(2),(3),(4),(5) and (6) we have, D1 ∩ (D2 ∪D3) = (D1 ∩D2) ∪ (D1 ∩D3) case 2 TD1(d) ≤ TD2(d) ≤ TD3(d), ID1(d) ≤ ID2(d) ≤ ID3(d), FD1(d) ≤ FD2(d) ≤ FD3(d) D1 ∩ (D2 ∪D3) =TD1(d) ∩ TD2∪D3(d) =min[TD1(d),max[TD2(d), TD3(d)]] =min[TD1(d), TD3(d)] =TD1(d) (7) (D1 ∩D2) ∪ (D1 ∩D3) =TD1∩D2(d) ∪ TD1∩D3(d) =max[min[TD1(d), TD2(d)],min[TD1(d), TD3(d)]] =max[TD1(d), TD1(d)] M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 8 of 26 =TD1(d) (8) D1 ∩ (D2 ∪D3) =ID1(d) ∩ ID2∪D3(d) =min[ID1(d),max[ID2(d), ID3(d)]] =min[ID1(d), ID3(d)] =ID1(d) (9) (D1 ∩D2) ∪ (D1 ∩D3) =ID1∩D2(d) ∪ ID1∩D3(d) =max[min[ID1(d), ID2(d)],min[ID1(d), ID3(d)]] =max[ID1(d), ID1(d)] =ID1(d) (10) D1 ∩ (D2 ∪D3) =FD1(d) ∩ FD2∪D3(d) =max[FD1(d),min[FD2(d), FD3(d)]] =max[FD1(d), FD2(d)] =FD2(d) (11) (D1 ∩D2) ∪ (D1 ∩D3) =FD1∩D2(d) ∪ FD1∩D3(d) =min[max[FD1(d), FD2(d)],max[FD1(d), FD3(d)]] =min[FD2(d), FD2(d)] =FD2(d) (12) From (7),(8),(9),(10),(11) and (12) we have, D1 ∩ (D2 ∪D3) = (D1 ∩D2) ∪ (D1 ∩D3) vi. To prove the distributive law of intersection over union, we take some cases. Case 1 TD1(d) ≥ TD2(d) ≥ TD3(d), ID1(d) ≥ ID2(d) ≥ ID3(d), FD1(d) ≥ FD2(d) ≥ FD3(d) D1 ∪ (D2 ∩D3) =TD1(d) ∪ TD2∩D3(d) =max[TD1(d),min[TD2(d), TD3(d)]] =max[TD1(d), TD3(d)] =TD1(d) (13) (D1 ∪D2) ∩ (D1 ∪D3) =TD1∪D2(d) ∩ TD1∪D3(d) =min[max[TD1(d), TD2(d)],max[TD1(d), TD3(d)]] =min[TD1(d), TD1(d)] =TD1(d) (14) D1 ∪ (D2 ∩D3) =ID1(d) ∪ ID2∩D3(d) M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 9 of 26 =max[ID1(d),min[ID2(d), ID3(d)]] =max[ID1(d), ID3(d)] =ID1(d) (15) (D1 ∪D2) ∩ (D1 ∪D3) =ID1∪D2(d) ∩ ID1∪D3(d) =min[max[ID1(d), ID2(d)],max[ID1(d), ID3(d)]] =min[ID1(d), ID1(d)] =ID1(d) (16) D1 ∪ (D2 ∩D3) =FD1(d) ∪ FD2∩D3(d) =min[FD1(d),max[FD2(d), FD3(d)]] =min[FD1(d), FD2(d)] =FD2(d) (17) (D1 ∪D2) ∩ (D1 ∪D3) =FD1∪D2(d) ∩ FD1∪D3(d) =max[min[FD1(d), FD2(d)],min[FD1(d), FD3(d)]] =max[FD2(d), FD3(d)] =FD2(d) (18) From (13),(14),(15),(16),(17) and (18) we have D1 ∪ (D2 ∩D3) = (D1 ∪D2) ∩ (D1 ∪D3) Case 2 TD1(d) ≤ TD2(d) ≤ TD3(d), ID1(d) ≤ ID2(d) ≤ ID3(d), FD1(d) ≤ FD2(d) ≤ FD3(d) D1 ∪ (D2 ∩D3) =TD1(d) ∪ TD2∩D3(d) =max[TD1(d),min[TD2(d), TD3(d)]] =max[TD1(d), TD2(d)] =TD2(d) (19) (D1 ∪D2) ∩ (D1 ∪D3) =TD1∪D2(d) ∩ TD1∪D3(d) =min[max[TD1(d), TD2(d)],max[TD1(d), TD3(d)]] =min[TD2(d), TD3(d)] =TD2(d) (20) D1 ∪ (D2 ∩D3) =ID1(d) ∪ ID2∩D3(d) =max[ID1(d),min[ID2(d), ID3(d)]] =max[ID1(d), ID2(d)] =ID2(d) (21) (D1 ∪D2) ∩ (D1 ∪D3) =ID1∪D2(d) ∩ ID1∪D3(d) =min[max[ID1(d), ID2(d)],max[ID1(d), ID3(d)]] M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 10 of 26 =min[ID2(d), ID3(d)] =ID2(d) (22) D1 ∪ (D2 ∩D3) =FD1(d) ∪ FD2∩D3(d) =min[FD1(d),max[FD2(d), FD3(d)]] =min[FD1(d), FD3(d)] =FD1(d) (23) (D1 ∪D2) ∩ (D1 ∪D3) =FD1∪D2(d) ∩ FD1∪D3(d) =max[min[FD1(d), FD2(d)],min[FD1(d), FD3(d)]] =max[FD1(d), FD1(d)] =FD1(d) (24) From (19),(20),(21),(22),(23) and (24) we have, D1 ∪ (D2 ∩D3) =(D1 ∪D2) ∩ (D1 ∪D3) vii. (D1 ∩D2) c =T(D1∩D2)c(d) =[1−min(TD1(d), TD2(d))] =max[(1− TD1(d)), (1− TD2(d))] =max[TDc 1 (d), TDc 2 (d)] =max[TDc 2 (d), TDc 1 (d)] =Dc 2 ∪Dc 1 (D1 ∩D2) c =I(D1∩D2)c(d) =[1−min(ID1(d), ID2(d))] =max[(1− ID1(d)), (1− ID2(d))] =max[IDc 1 (d), IDc 2 (d)] =max[IDc 2 (d), IDc 1 (d)] =Dc 2 ∪Dc 1 (D1 ∩D2) c =F(D1∩D2)c(d) =[1−max(FD1(d), FD2(d))] =min[(1− FD1(d)), (1− FD2(d))] =min[FDc 1 (d), FDc 2 (d)] =min[FDc 2 (d), FDc 1 (d)] =Dc 2 ∪Dc 1 viii. (D1 ∪D2) c =T(D1∪D2)c(d) M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 11 of 26 =[1−max(TD1(d), TD2(d))] =min[(1− TD1(d)), (1− TD2(d))] =min[TDc 1 (d), TDc 2 (d)] =min[TDc 2 (d), TDc 1 (d)] =Dc 2 ∩Dc 1 (D1 ∪D2) c =I(D1∪D2)c(d) =[1−max(ID1(d), ID2(d))] =min[(1− ID1(d)), (1− ID2(d))] =min[IDc 1 (d), IDc 2 (d)] =min[IDc 2 (d), IDc 1 (d)] =Dc 2 ∩Dc 1 (D1 ∪D2) c =F(D1∪D2)c(d) =[1−min(FD1(d), FD2(d))] =max[(1− FD1(d)), (1− FD2(d))] =max[FDc 1 (d), FDc 2 (d)] =max[FDc 2 (d), FDc 1 (d)] =Dc 2 ∩Dc 1 Definition 7. Let D1 and D2 be two NFSs on B and then NF product of D1 and D2 is D1 ◦D2 defined by D1 ◦D2 = TD1(d) ◦ TD2(d) = [TD1(d).TD2(d)] D1 ◦D2 = ID1(d) ◦ ID2(d) = [ID1(d).ID2(d)] D1 ◦D2 = FD1(d) ◦ FD2(d) = [FD1(d).FD2(d)] 3. Neutrosophic Fuzzy Distance Measures In decision-making and fuzzy logic, the hesitance degree is a key concept signifying un- certainty or indecision accompanied with a certain selection or preference. In scenarios like preferences or similarities are not sternly defined but exist along a continuum then the hes- itance values become particularly relevant when we applied it in conjunction with Distance Measures (DMs). In decision-making methods involving multiple criteria, hesitance values helps justify the uncertainty in the importance or increment of each norm. In clustering algorithms, Integrating DMs with hesitance values are advantageous for controling data points that doesn’t belong completely to a single cluster but display a degree of member- ship through multiple clusters. Likewise, in string and pattern matching, hesitance values indicate uncertainty in toning, it is also useful in applications like record linkage and infor- mation retrieval, where limited matches or ambiguous relationships may occur. In order M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 12 of 26 to captures uncertainty within the specific problem domain accurately, it is mandatory to deliver a suitable metric when we incorporate Dms with hesitance values. This frequently involves leveraging fuzzy logic or related techniques to manage imprecision and uncertainty in data. In this study Neutrosophic Distance Measures(NDMs) creating a massive impact in decision-making, pattern recognition, medical diagnosis, human-computer interaction, and preference modelling. 3.1. Neutrosophic Hesitance Distance Measures Definition 8. The Hesitance distance measure of NFSs is a function dH : D(B)×D(B) → [0, 1] with the condition for the collection of NFSs D1, D2, D3 ∈ D(B) i. 0 ≤ dH(D1, D2) ≤ 1, dH(D1, D2) = 0 ⇐⇒ D1 = D2. ii. dH(D1, D2) = dH(D2, D1). iii. dH(D1, D3) ≤ dH(D1, D2) + dH(D2, D3). The Hesitance DMs is defined as: dH(D1, D2) = n∑ i=1 [ |TD1(di)− TD2(di)| +|hD1(di)− hD2(di)| ] 2n (25) dH(D1, D2) = n∑ i=1 [ |ID1(di)− ID2(di)| +|hD1(di)− hD2(di)| ] 2n (26) dH(D1, D2) = n∑ i=1 [ |FD1(di)− FD2(di)| +|hD1(di)− hD2(di)| ] 2n (27) Example 1. Let us take D1 = 0.5, 0.7, 0.9 Da + 0.2, 0.5, 0.2 Db + 0.6, 0.7, 0.4 Dc D2 = 0.7, 0.6, 0.5 Da + 0.8, 0.8, 0.9 Db + 0.4, 0.6, 0.3 Dc we know that, hD(d) = {|(1− TD(d))|, |(1− ID(d))|, |(1− FD(d))|} then for truth, dH(D1, D2) =  (|0.5− 0.7|+ |0.5− 0.3|+) (|0.2− 0.8|+ |0.8− 0.2|+) (|0.6− 0.4|+ |0.4− 0.6|)  2(3) M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 13 of 26 = [0.2 + 0.2 + 0.6 + 0.6 + 0.2 + 0.2] 6 =0.333 for indeterminacy, dH(D1, D2) =  (|0.7− 0.6|+ |0.3− 0.4|+) (|0.5− 0.8|+ |0.5− 0.2|+) (|0.7− 0.6|+ |0.3− 0.4|+)  2(3) = [0.1 + 0.1 + 0.3 + 0.3 + 0.1 + 0.1] 6 =0.166 for falsity, dH(D1, D2) =  (|0.9− 0.5|+ |0.1− 0.5|+) (|0.2− 0.9|+ |0.8− 0.1|+) (|0.4− 0.3|+ |0.6− 0.7|+)  2(3) = [0.4 + 0.4 + 0.7 + 0.7 + 0.1 + 0.1] 6 =0.400 Theorem 2. The function dH defined in (25),(26) and (27) is a distance function. Proof. i. The condition dH(D1, D2) ≤ 0 is obviously true, Then dH(D1, D2) = n∑ i=1 [ |TD1(di)− TD2(di)|+ |hD1(di)− hD2(di)| ] 2n = n∑ i=1 [1 + 1] 2n =1 dH(D1, D2) = n∑ i=1 [ |ID1(di)− ID2(di)|+ |hD1(di)− hD2(di)| ] 2n = n∑ i=1 [1 + 1] 2n M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 14 of 26 =1 dH(D1, D2) = n∑ i=1 [ |FD1(di)− FD2(di)|+ |hD1(di)− hD2(di)| ] 2n = n∑ i=1 [1 + 1] 2n =1 Therefore 0 ≤ dH(D1, D2) ≤ 1. ii. dH(D1, D2) = n∑ i=1 [ |TD1(di)− TD2(di)|+ |hD1(di)− hD2(di)| ] 2n = n∑ i=1 [ |TD2(di)− TD1(di)|+ |hD2(di)− hD1(di)| ] 2n =dH(D2, D1) dH(D1, D2) = n∑ i=1 [ |ID1(di)− ID2(di)|+ |hD1(di)− hD2(di)| ] 2n = n∑ i=1 [ |ID2(di)− ID1(di)|+ |hD2(di)− hD1(di)| ] 2n =dH(D2, D1) dH(D1, D2) = n∑ i=1 [ |FD1(di)− FD2(di)|+ |hD1(di)− hD2(di)| ] 2n = n∑ i=1 [ |FD2(di)− FD1(di)|+ |hD2(di)− hD1(di)| ] 2n =dH(D2, D1) hence it is true. iii. dH(D1, D3) = n∑ i=1 [ |TD1(di)− TD3(di)|+ |hD1(di)− hD3(di)| ] 2n M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 15 of 26 = n∑ i=1 [ |TD1(di)− TD2(di) + TD2(di)− TD3(di)| +|hD1(di)− hD2(di) + hD2(di)− hD3(di)| ] 2n ≤ n∑ i=1 [ |TD1(di)− TD2(di)|+ |TD2(di)− TD3(di)| +|hD1(di)− hD2(di)|+ |hD2(di)− hD3(di)| ] 2n = n∑ i=1 [ |TD1(di)− TD3(di)|+ |hD1(di)− hD3(di)| ] 2n + n∑ i=1 [ |TD3(di)− TD2(di)|+ |hD3(di)− hD2(di)| ] 2n ≤dH(D1, D2) + dH(D2, D3)dH(D1, D3) = n∑ i=1 [ |ID1(di)− ID3(di)|+ |hD1(di)− hD3(di)| ] 2n = n∑ i=1 [ |ID1(di)− ID2(di) + ID2(di)− ID3(di)| +|hD1(di)− hD2(di) + hD2(di)− hD3(di)| ] 2n ≤ n∑ i=1 [ |ID1(di)− ID2(di)|+ |ID2(di)− ID3(di)| +|hD1(di)− hD2(di)|+ |hD2(di)− hD3(di)| ] 2n = n∑ i=1 [ |ID1(di)− ID3(di)|+ |hD1(di)− hD3(di)| ] 2n + n∑ i=1 [ |ID3(di)− ID2(di)|+ |hD3(di)− hD2(di)| ] 2n ≤dH(D1, D2) + dH(D2, D3) dH(D1, D3) = n∑ i=1 [ |FD1(di)− FD3(di)|+ |hD1(di)− hD3(di)| ] 2n = n∑ i=1 [ |FD1(di)− FD2(di) + FD2(di)− FD3(di)| +|hD1(di)− hD2(di) + hD2(di)− hD3(di)| ] 2n ≤ n∑ i=1 [ |FD1(di)− FD2(di)|+ |FD2(di)− FD3(di)| +|hD1(di)− hD2(di)|+ |hD2(di)− hD3(di)| ] 2n M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 16 of 26 = n∑ i=1 [ |FD1(di)− FD3(di)|+ |hD1(di)− hD3(di)| ] 2n + n∑ i=1 [ |FD3(di)− FD2(di)|+ |hD3(di)− hD2(di)| ] 2n ≤dH(D1, D2) + dH(D2, D3) Definition 9. Let dH be the Hesitance measure of NFSs. Then, the complement Hesitance distance measure dH((D1) c, (D2) c) of two NFSs is defined as dH((D1) c, (D2) c) = n∑ i=1 [ |(1− TD1(di))− (1− TD2(di))| +|(1− hD1(di))− (1− hD2(di))| ] 2n dH((D1) c, (D2) c) = n∑ i=1 [ |(1− ID1(di))− (1− ID2(di))| +|(1− hD1(di))− (1− hD2(di))| ] 2n dH((D1) c, (D2) c) = n∑ i=1 [ |(1− FD1(di))− (1− FD2(di))| +|(1− hD1(di))− (1− hD2(di))| ] 2n Definition 10. Let dH be the Hesitance measure of NFSs. Then, the max Hesitance distance measure of two NFSs D1 and D2 is denoted by dH(D1∪D2, D2) or dH(D1, D2∪D2) and satisfies the function dH(D1 ∪D2, D2) = n∑ i=1 [ |(TD1(di) ∨ TD2(di))− TD2(di)| +|(hD1(di) ∨ hD2(ri))− hD2(di)| ] 2n dH(D1 ∪D2, D2) = n∑ i=1 [ |(ID1(di) ∨ ID2(di))− ID2(di)| +|(hD1(di) ∨ hD2(ri))− hD2(di)| ] 2n dH(D1 ∪D2, D2) = n∑ i=1 [ |(FD1(di) ∨ FD2(di))− FD2(di)| +|(hD1(di) ∨ hD2(ri))− hD2(di)| ] 2n and dH(D1, D2 ∪D2) = n∑ i=1 [ |TD1(di)− (TD2(di) ∨ TD2(di))| +|hD1(di)− (hD2(di) ∨ hD2(di))| ] 2n M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 17 of 26 dH(D1, D2 ∪D2) = n∑ i=1 [ |ID1(di)− (ID2(di) ∨ ID2(di))| +|hD1(di)− (hD2(di) ∨ hD2(di))| ] 2n dH(D1, D2 ∪D2) = n∑ i=1 [ |FD1(di)− (FD2(di) ∨ FD2(di))| +|hD1(di)− (hD2(di) ∨ hD2(di))| ] 2n Definition 11. Let dH be the Hesitance measure of NFSs. Then, the min Hesitance distance measure of two NFSs D1 and D2 is denoted by dH(D1∩D2, D2) or dH(D1, D2∩D2) and satisfies the function dH(D1 ∩D2, D2) = n∑ i=1 [ |(TD1(di) ∧ TD2(di))− TD2(di)| +|(hD1(di) ∧ hD2(di))− hD2(di)| ] 2n dH(D1 ∩D2, D2) = n∑ i=1 [ |(ID1(di) ∧ ID2(di))− ID2(di)| +|(hD1(di) ∧ hD2(di))− hD2(di)| ] 2n dH(D1 ∩D2, D2) = n∑ i=1 [ |(FD1(di) ∧ FD2(di))− FD2(di)| +|(hD1(di) ∧ hD2(di))− hD2(di)| ] 2n and dH(D1, D2 ∩D2) = n∑ i=1 [ |TD1(di)− (TD2(di) ∧ TD2(di))| +|hD1(di)− (hD2(di) ∧ hD2(di))| ] 2n dH(D1, D2 ∩D2) = n∑ i=1 [ |ID1(di)− (ID2(di) ∧ ID2(di))| +|hD1(di)− (hD2(di) ∧ hD2(di))| ] 2n dH(D1, D2 ∩D2) = n∑ i=1 [ |FD1(di)− (FD2(di) ∧ FD2(di))| +|hD1(di)− (hD2(di) ∧ hD2(di))| ] 2n Definition 12. Let dH be the Hesitance measure of NFSs. Then, the max-min Hesitance distance measure of two NFSs D1 and D2 is denoted by dH(D1∪D2, D1∩D2) or dH(D1∩ D2, D1 ∪D2) and satisfies the function dH(D1 ∪D2, D1 ∩D2) = n∑ i=1 [ |(TD1(di) ∨ TD2(di), TD1(di) ∧ TD2(di))| +|((hD1(di) ∨ hD2(di), hD1(di) ∧ hD2(di))| ] 2n dH(D1 ∪D2, D1 ∩D2) = n∑ i=1 [ |(ID1(di) ∨ ID2(di), ID1(di) ∧ ID2(di))| +|((hD1(di) ∨ hD2(di), hD1(di) ∧ hD2(di))| ] 2n M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 18 of 26 dH(D1 ∪D2, D1 ∩D2) = n∑ i=1 [ |(FD1(di) ∨ FD2(di), FD1(di) ∧ FD2(di))| +|((hD1(di) ∨ hD2(di), hD1(di) ∧ hD2(di))| ] 2n and dH(D1 ∩D2, D1 ∪D2) = n∑ i=1 [ |(TD1(di) ∧ TD2(di), TD1(di) ∨ TD2(di))| +|((hD1(di) ∧ hD2(di), hD1(di) ∨ hD2(di))| ] 2n dH(D1 ∩D2, D1 ∪D2) = n∑ i=1 [ |(ID1(di) ∧ ID2(di), ID1(di) ∨ ID2(di))| +|((hD1(di) ∧ hD2(di), hD1(di) ∨ hD2(di))| ] 2n dH(D1 ∩D2, D1 ∪D2) = n∑ i=1 [ |(FD1(di) ∧ FD2(di), FD1(di) ∨ FD2(di))| +|((hD1(di) ∧ hD2(di), hD1(di) ∨ hD2(di))| ] 2n 4. Decision-Making Process for Neutrosophic Distance Measure with Hesitance Values The rapid exponential growth in social media has not only changed the nature of in- formation sharing but has also speeded up the dissemination of misinformation, especially in the arena of political debate., Twitter, Facebook and Instagram are some of the plat- forms that feature huge amounts of content produced by humans as well as bots. Though these networks offer real-time information and citizens’ participation, they are also fertile ground for disinformation operations, particularly on occasions that require sensitivity like elections, referendums, or government policy launches. Conventional machine learning approaches, though great, tend to be weak at dealing with indeterminacy, vagueness, and linguistic hesitations—typical traits in political texts where messages are usually speculative or intentionally misleading. To solve these prob- lems, the current research proposes the application of Neutrosophic Hesitance Distance Measures (NHDM), which involve truth-membership (T), indeterminacy-membership (I), falsity-membership (F), and hesitance (H) for every piece of content. These values permit greater modeling of tweets in which truth values are unknown or partially known. The case study applies a simulated environment where the set of tweets 4.1. Algorithm Step 1: Consider a set of i alternatives, denoted as (αi, βi, γi) = {(α1, β1, γ1), (α2, β2, γ2), · · · , (αi, βi, γi)}, and a set of j attributes, rep- resented as Xj = (D1, D2, · · · , Xj). Additionally, let Z = {Z1, Z2, · · · , Zn} be a group of n experts, where each Zn corresponds to an expert participating in the decision-making process. Step 2: The attribute value assigned to an alternative (αi, βi, γi) for a specific criterionXj is expressed as Tij = xTij(Tj); Iij = xIij(Ij)andFij = xFij(Fj). This value represents M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 19 of 26 the assessment given by expert Zm regarding alternative (αi, βi, γi) in relation to criterion Xj . Consequently, the structured table derived from these evaluated values serves as a fundamental representation of the decision-making data. Step 3: Structure the table derived from these evaluated values serves as a fundamental representation of the decision-making data. Zi αi, βi, γi T1, I1, F1 T2, I2, F2 · · · Tj , Ij , Fj Z1 α1, β1, γ1 (xT1T1(T1), (xT1T2(T2), · · · (xT1Tj (Tj), xI1I1(I1), xI1I2(I2), xI1Ij (Ij), xF1F1(F1)) xF1F2(F2)) xF1Fj (Fj)) α2, β2, γ2 (xT2T1(T1), (xT2T2(T2), · · · (xT2Tj (Tj), xI2I1(I1), xI2I2(I2), xI2Ij (Ij), xF2F1(F1)) xF2F2(F2)) xF2Fj (Fj)) ... ... ... ... αi, βi, γi (xTiT1(T1), (xTiT2(T2), · · · (xTiTj (Tj), xIiI1(I1), xIiI2(I2), xIiIj (Ij), xFiF1(F1)) xFiF2(F2)) xFiFj (Fj)) Z2 α1, β1, γ1 (xT1T1(T1), (xT1T2(T2), · · · (xT1Tj (Tj), xI1I1(I1), xI1I2(I2), xI1Ij (Ij), xF1F1(F1)) xF1F2(F2)) xF1Fj (Fj)) α2, β2, γ2 (xT2T1(T1), (xT2T2(T2), · · · (xT2Tj (Tj), xI2I1(I1), xI2I2(I2), xI2Ij (Ij), xF2F1(F1)) xF2F2(F2)) xF2Fj (Fj)) ... ... ... ... αi, βi, γi (xTiT1(T1), (xTiT2(T2), · · · (xTiTj (Tj), xIiI1(I1), xIiI2(I2), xIiIj (Ij), xFiF1(F1)) xFiF2(F2)) xFiFj (Fj)) ... ... ... ... Zi α1, β1, γ1 (xT1T1(T1), (xT1T2(T2), · · · (xT1Tj (Tj), xI1I1(I1), xI1I2(I2), xI1Ij (Ij), xF1F1(F1)) xF1F2(F2)) xF1Fj (Fj)) α2, β2, γ2 (xT2T1(T1), (xT2T2(T2), · · · (xT2Tj (Tj), xI2I1(I1), xI2I2(I2), xI2Ij (Ij), xF2F1(F1)) xF2F2(F2)) xF2Fj (Fj)) ... ... ... ... αi, βi, γi (xTiT1(T1), (xTiT2(T2), · · · (xTiTj (Tj), xIiI1(I1), xIiI2(I2), xIiIj (Ij), xFiF1(F1)) xFiF2(F2)) xFiFj (Fj)) Table 1: Collection of attributes M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 20 of 26 Step 4: Construct distance matrix using the definition of Hesitance distance measure, dH(D1, D2) = ∑ i = 1n [ |TD1(Di)− TD2(Di)| +|hD1(Di)− hD2(Di)| ] 2n dH(D1, D2) = ∑ i = 1n [ |ID1(Di)− ID2(Di)| +|hD1(Di)− hD2(Di)| ] 2n dH(D1, D2) = ∑ i = 1n [ |FD1(Di)− FD2(Di)| +|hD1(Di)− hD2(Di)| ] 2n The distance matrix is demonstrted as [DM ]3×3 = (d(T (Z1) 1 , T (Z2) 1 ), d(I (Z1) 1 , I (Z2) 1 ), d(F (Z1) 1 , F (Z2) 1 )) (d(T (Z1) 1 , T (Z3) 1 ), d(I (Z1) 1 , I (Z3) 1 ), d(F (Z1) 1 , F (Z3) 1 )) (d(T (Z1) 2 , T (Z2) 2 ), d(I (Z1) 2 , I (Z2) 2 ), d(F (Z1) 2 , F (Z2) 2 )) (d(T (Z1) 2 , T (Z3) 2 ), d(I (Z1) 2 , I (Z3) 2 ), d(F (Z1) 2 , F (Z3) 2 )) (d(T (Z1) 3 , T (Z2) 3 ), d(I (Z1) 3 , I (Z2) 3 ), d(F (Z1) 3 , F (Z2) 3 )) (d(T (Z1) 3 , T (Z3) 3 ), d(I (Z1) 3 , I (Z3) 3 ), d(F (Z1) 3 , F (Z3) 3 )) (d(T (Z2) 1 , T (Z3) 1 ), d(I (Z2) 1 , I (Z3) 1 ), d(F (Z2) 1 , F (Z3) 1 )) (d(T (Z2) 2 , T (Z3) 2 ), d(I (Z2) 2 , I (Z3) 2 ), d(F (Z2) 2 , F (Z3) 2 )) (d(T (Z2) 3 , T (Z3) 3 ), d(I (Z2) 3 , I (Z3) 3 ), d(F (Z2) 3 , F (Z3) 3 ))  Step 5: Find score function S(Di) of alternative Di where i = 1, 2, · · · , n is defined as S(Di) = n∑ i=1 [ d(T (Zj) i , T (Zk) i ) n , d(I (Zj) i , I (Zk) i ) n , d(F (Zj) i , F (Zk) i ) n ] 4.2. Numerical Illustrations Let D = {D1 = Twitter,D2 = Facebook,D3 = Instagram} be a collection of three social medias for political debate. Let Y = {Y1, Y2, Y3} be a set of three misleading facts, where Y1 = New policy announced by minister X is fake Y2 = Government to reduce fuel prices next month Y3 = Rumors suggest parliament dissolution next week M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 21 of 26 Zi Ti, Ii, Fi Y1 Y2 Y3 T1, I1, F1 (0.3, 0.4, 0.5) (0.3, 0.1, 0.2) (0.1, 0.3, 0.5) Zi T2, I2, F2 (0.6, 0.7, 0.8) (0.8, 0.9, 0.7) (0.6, 0.5, 0.7) T3, I3, F3 (0.8, 1.0, 0.9) (0.9, 0.2, 0.6) (0.8, 0.3, 0.9) T1, I1, F1 (0.2, 0.4, 0.6) (0.4, 0.5, 0.1) (0.2, 0.4, 0.6) Z2 T2, I2, F2 (0.7, 0.2, 0.8) (0.9, 1.0, 0.1) (0.8, 0.7, 0.6) T3, I3, F3 (0.8, 0.3, 0.5) (0.5, 0.9, 0.7) (0.3, 0.6, 0.9) T1, I1, F1 (0.4, 0.5, 0.3) (0.2, 0.4, 0.1) (0.5, 0.4, 0.3) Z3 T2, I2, F2 (0.7, 0.3, 0.6) (0.5, 0.3, 0.7) (0.7, 0.5, 0.4) T3, I3, F3 (0.3, 0.6, 0.9) (0.9, 0.5, 0.6) (0.8, 0.1, 0.2) Table 2: Evaluated values of alternatives d(T (Zi) 1 , T (Z2) 1 ) =  (|0.3− 0.2|+ |0.7− 0.8|)+ (|0.3− 0.4|+ |0.7− 0.6|)+ (|0.1− 0.2|+ |0.9− 0.8|)  3(2) = 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 6 = 0.100 d(I (Z1) 1 , I (Z2) 1 ) =  (|0.4− 0.4|+ |0.6− 0.6|)+ (|0.1− 0.5|+ |0.9− 0.5|)+ (|0.3− 0.4|+ |0.7− 0.6|)  3(2) = 0.4 + 0.4 + 0.1 + 0.1 6 = 0.166 d(F (Z1) 1 , F (Z2) 1 ) =  (|0.5− 0.6|+ |0.5− 0.4|)+ (|0.2− 0.1|+ |0.8− 0.9|)+ (|0.5− 0.6|+ |0.5− 0.4|)  3(2) = 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 6 = 0.100 d(T (Z1) 1 , T (Z3) 1 ) =  (|0.3− 0.4|+ |0.7− 0.6|)+ (|0.3− 0.2|+ |0.7− 0.8|)+ (|0.1− 0.5|+ |0.9− 0.5|)  3(2) = 0.1 + 0.1 + 0.1 + 0.1 + 0.4 + 0.4 6 = 0.200 d(I (Z1) 1 , I (Z3) 1 ) =  (|0.4− 0.5|+ |0.6− 0.5|)+ (|0.1− 0.4|+ |0.9− 0.6|)+ (|0.3− 0.4|+ |0.7− 0.6|)  3(2) M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 22 of 26 = 0.1 + 0.1 + 0.3 + 0.3 + 0.1 + 0.1 6 = 0.166 d(F (Z1) 1 , F (Z3) 1 ) =  (|0.5− 0.3|+ |0.5− 0.7|)+ (|0.2− 0.1|+ |0.8− 0.9|)+ (|0.5− 0.3|+ |0.5− 0.7|)  3(2) = 0.2 + 0.2 + 0.1 + 0.1 + 0.2 + 0.2 6 = 0.166 d(T (Z2) 1 , T (Z3) 1 ) =  (|0.2− 0.4|+ |0.8− 0.6|)+ (|0.4− 0.2|+ |0.6− 0.8|)+ (|0.2− 0.5|+ |0.8− 0.5|)  3(2) = 0.2 + 0.2 + 0.2 + 0.2 + 0.3 + 0.3 6 = 0.150 d(I (Z2) 1 , I (Z3) 1 ) =  (|0.4− 0.5|+ |0.6− 0.5|)+ (|0.5− 0.4|+ |0.5− 0.6|)+ (|0.4− 0.4|+ |0.6− 0.6|)  3(2) = 0.1 + 0.1 + 0.1 + 0.1 6 = 0.066 d(F (Z2) 1 , F (Z3) 1 ) =  (|0.6− 0.3|+ |0.4− 0.7|)+ (|0.1− 0.1|+ |0.9− 0.9|)+ (|0.6− 0.3|+ |0.4− 0.7|)  3(2) = 0.3 + 0.3 + 0.3 + 0.3 6 = 0.200 Similarly, d(T (Z1) 2 , T (Z2) 2 ) = 0.133 d(T (Z2) 2 , T (Z3) 2 ) = 0.166 d(T (Z1) 3 , T (Z3) 3 ) = 0.166 d(I (Z1) 2 , I (Z2) 2 ) = 0.266 d(I (Z2) 2 , I (Z3) 2 ) = 0.333 d(I (Z1) 3 , I (Z3) 3 ) = 0.300 d(F (Z1) 2 , F (Z2) 2 ) = 0.233 d(F (Z2) 2 , F (Z3) 2 ) = 0.333 d(F (Z1) 3 , F (Z3) 3 ) = 0.233 d(T (Z1) 2 , T (Z3) 2 ) = 0.166 d(T (Z1) 3 , T (Z2) 3 ) = 0.383 d(T (Z2) 3 , T (Z3) 3 ) = 0.466 d(I (Z1) 2 , I (Z3) 2 ) = 0.333 d(I (Z1) 3 , I (Z2) 3 ) = 0.566 d(I (Z2) 3 , I (Z3) 3 ) = 0.400 d(F (Z1) 2 , F (Z3) 2 ) = 0.166 d(F (Z1) 3 , F (Z2) 3 ) = 0.166 d(F (Z2) 3 , F (Z3) 3 ) = 0.400 M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 23 of 26 [DM ]3×3 =  (d(T (Z1) 1 , T (Z2) 1 ), d(I (Z1) 1 , I (Z2) 1 ), d(F (Z1) 1 , F (Z2) 1 )) (d(T (Z1) 1 , T (Z3) 1 ), d(I (Z1) 1 , I (Z3) 1 ), d(F (Z1) 1 , F (Z3) 1 )) (d(T (Z1) 2 , T (Z2) 2 ), d(I (Z1) 2 , I (Z2) 2 ), d(F (Z1) 2 , F (Z2) 2 )) (d(T (Z1) 2 , T (Z3) 2 ), d(I (Z1) 2 , I (Z3) 2 ), d(F (Z1) 2 , F (Z3) 2 )) (d(T (Z1) 3 , T (Z2) 3 ), d(I (Z1) 3 , I (Z2) 3 ), d(F (Z1) 3 , F (Z2) 3 )) (d(T (Z1) 3 , T (Z3) 3 ), d(I (Z1) 3 , I (Z3) 3 ), d(F (Z1) 3 , F (Z3) 3 )) (d(T (Z2) 1 , T (Z3) 1 ), d(I (Z2) 1 , I (Z3) 1 ), d(F (Z2) 1 , F (Z3) 1 )) (d(T (Z2) 2 , T (Z3) 2 ), d(I (Z2) 2 , I (Z3) 2 ), d(F (Z2) 2 , F (Z3) 2 )) (d(T (Z2) 3 , T (Z3) 3 ), d(I (Z2) 3 , I (Z3) 3 ), d(F (Z2) 3 , F (Z3) 3 ))  = (0.100, 0.166, 0.100) (0.200, 0.166, 0.166) (0.150, 0.066, 0.200) (0.133, 0.266, 0.233) (0.166, 0.333, 0.166) (0.166, 0.333, 0.333) (0.383, 0.566, 0.166) (0.166, 0.300, 0.233) (0.466, 0.400, 0.400)  S(Di) = 3∑ i=1 [ d(T (Zj) i , T (Zk) i ) 3 , d(I (Zj) i , I (Zk) i ) 3 , d(F (Zj) i , F (Zk) i ) 3 ] S(D1) = [ 0.100 + 0.200 + 0.150 3 , 0.166 + 0.166 + 0.066 3 , 0.100 + 0.166 + 0.200 3 ] S(D1) =(0.150, 0.132, 0.155) S(D2) = [ 0.133 + 0.166 + 0.166 3 , 0.266 + 0.333 + 0.333 3 , 0.233 + 0.166 + 0.333 3 ] S(D2) =(0.155, 0.310, 0.244) S(D3) = [ 0.383 + 0.166 + 0.466 3 , 0.566 + 0.300 + 0.400 3 , 0.166 + 0.233 + 0.400 3 ] S(D3) =(0.338, 0.422, 0.266) D1 ≤ D2 ≤ D3 since, D3 has the maximum value its detects that the most misleading rumours are spread- ing through instagram. 4.3. Comparison Analysis Historically, models based on probabilistic logic, binary classification, or machine learn- ing techniques that assume deterministic and well-structured input data have been used to detect digital disinformation. These methods frequently fail to address the underly- ing contradiction, ambiguity, and vagueness seen in actual social media posts. Current methods typically oversimplify reluctant and doubtful viewpoints, which results in low efficacy when categorizing information that is speculative or partially true. The suggested work, on the other hand, presents a unique framework based on NHDMs, which can be used to describe untruth, truth, indeterminacy, and especially the hesitancy that comes M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 24 of 26 with human interpretation. Through the use of Hesitant Neutrosophic Sets, the suggested approach successfully reflects the inherent hesitancy and doubt present in user generated internet content. NHDM provide a more nuanced representation of information by allow- ing multiple values for each component of truth, indeterminacy, and falsehood in contrast to previous models that reduce complicated uncertainty to single-valued metrics. A well- defined set of operations and axioms for hesitance distance measurements, such as union, intersection, complement, max-min, and score functions, enhances the mathematical rigor of the suggested method and allows for reliable comparison and clustering of ambiguous data. Additionally, the suggested decision-making framework uses multi-criteria decision-making to adaptively rank alternative pieces of information based on estimated hesitance distances, whereas existing systems frequently employ rigid decision thresholds or predefined labels. This method works especially well in situations involving misinformation where the truth is ambiguous or changing. The model’s better interpretability and practicality are demon- strated by applying it to a dataset of political misinformation from social media sites like Facebook, Instagram, and Twitter. All things considered, our study closes a large gap be- tween theoretical uncertainty modeling and practical disinformation detection, providing a more flexible, interpretable, and human-aligned approach than traditional techniques. Criteria DT LSTM Fuzzy Logic Bayesian NHDM (Proposed) Uncertainty Handling ✗ ✓ ✓ ✓ ✓✓✓ Indeterminacy Handling ✗ ✗ ✗ ✗ ✓✓✓ Hesitancy Representation ✗ ✗ ✗ ✗ ✓✓✓ Interpretability ✓✓✓ ✗ ✓✓ ✓ ✓✓✓ Data Requirements ✓✓✓ ✗✗ ✓✓ ✓ ✓✓ Performance on Noisy Data ✗ ✓ ✓ ✓ ✓✓✓ Table 3. Comparison of Rumor Detection Models 5. Advantages of Neutrosophic Hesitance Distance Measures Hesitance distance measures in neutrosophic fuzzy systems afford a potential tool for controlling uncertainty in decision-making. These measures improving decision analysis by integrating hesitancy, truth, falsity, and indeterminacy, paves way for a lot of inclusive and realistic representation of uncertainty. By obviously enumerating the level of hesitation in expert evaluations, they empower decision-makers to capture the degree of uncertainty as- sociated with different alternatives, leading to more informed and confident choices. One of the core advantages of hesitance distance measures in neutrosophic fuzzy systems is their tractability in modelling uncertainty. Decision-makers can allocate different levels of hesitancy to various criteria, effectively reflecting the dynamic nature of real-world deci- sion problems. This malleability makes them particularly useful in situations where likings are ambiguous, inconsistent, or influenced by multiple factors. Moreover, integrating hes- itance values into distance measures agrees for a more coarse assessment of alternatives, M. Kaviyarasu, R. Venitha, M. Alqahtani / Eur. J. Pure Appl. Math, 18 (3) (2025), 6476 25 of 26 warranting that uncertainty is not over streamlined but rather treated with the required depth. This capability is necessary in applications where multiple contradictory attitudes exist, such as medical diagnosis, risk assessment, and intelligent decision support systems. Experts can achieve greater accuracy, transparency, and adaptability in their evaluations, by leveraging hesitance-based distance measures in neutrosophic fuzzy decision-making organizations, leading to better strategic outcomes in highly uncertain environments. 6. Conclusion The proposed framework demonstrates that incorporating hesitance values into neutro- sophic distance measures significantly enhances the capacity to detect digital misinforma- tion. By formally defining operations such as complement, max, min, and max-min within this extended neutrosophic environment, the method offers a more nuanced approach to handling uncertainty and vagueness. The example illustrates the practical potential of this technique in social media contexts, showing that decision-making processes become more accurate and robust. Overall, this study highlights the value of neutrosophic hesitance distance measures as an effective tool for improving the reliability of misinformation detec- tion systems. Future directions of proposed work is Beyond misinformation detection, the proposed method can be explored in domains such as medical diagnosis, risk assessment, and sentiment analysis, where uncertainty and ambiguity are prominent. 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