The model, developed by Wanting Zhao, De Xia, and Yundong Zhang, is technically ambitious. The supply side is represented as a Watts-Strogatz small-world network of 200 firms, each choosing among four strategies: producing NEVs or internal combustion engine vehicles (ICEVs), combined with digital or traditional marketing. The demand side is a Barabási-Albert scale-free network of 1,000 consumers, whose connectivity mirrors real online social platforms—a few highly connected hubs and many sparsely linked users. The two layers are coupled functionally rather than through fixed one-to-one links: consumer adoption aggregates into market penetration, which drives firm payoffs, while firms’ technology and marketing choices determine the advertising exposure each consumer receives. Both populations update strategies asynchronously using the Fermi rule from statistical physics, comparing payoffs with connected neighbors under bounded rationality and noise.

Consumer utility in the model is a rich composite. It includes intrinsic vehicle value minus price, a purchase subsidy, a charging anxiety penalty that fades as NEV penetration grows, and a perceived technological risk term capturing worries about battery degradation and range. On top of this sit two distinctly digital ingredients. Marketing exposure is scaled by a consumer’s node degree, reflecting how platform algorithms target socially active users, and its utility follows a quadratic specification: exposure helps at moderate levels but triggers an information overload penalty that grows with the square of intensity. Social influence is borrowed from the Ising model of spin interactions, with a local bandwagon effect from neighbors plus a weighted term giving extra influence to hub nodes, whose degree serves as a proxy for their dissemination power.

The benchmark simulations, run for 150 periods and averaged over 100 Monte Carlo replications, produced a striking result: the steady-state distribution of NEV adoption is sharply bimodal. Most replications converge either to a low-adoption regime dominated by ICEVs or to a near-complete adoption regime, with very few outcomes in between. The Monte Carlo mean of roughly 0.53 is therefore an average across two divergent worlds, not a typical endpoint. Firm behavior mirrors this polarization in technology choice, yet digital marketing adoption stays close to one in almost every run—firms abandon traditional channels regardless of which technology wins.

The dependence on initial conditions follows a clean S-shaped curve. When the initial NEV adoption share is low, nearly all simulations stay stuck in the low-adoption regime. In an intermediate range, outcomes scatter widely and the probability of near-complete diffusion climbs steeply. At an initial adoption rate of 0.5, all 100 replications reach at least 95 percent NEV adoption. The mechanism is cross-layer reinforcement: a larger early user base strengthens pro-NEV social norms, eases charging anxiety as penetration rises, expands NEV demand, raises the payoff of NEV strategies for firms, and ultimately shifts the marketing exposure landscape further in NEVs’ favor. Early differences are amplified into divergent destinies.

Perhaps the most counterintuitive finding concerns digital marketing itself. Because ICEV-oriented firms initially dominate the market, they also hold the larger share of digital marketing exposure. Increasing the coefficient that converts marketing exposure into consumer utility therefore amplifies the incumbent’s advantage and suppresses NEV diffusion. Conversely, strengthening the information overload penalty—which hits the more heavily exposed ICEV option harder, quadratically—actually helps NEVs by eroding that exposure advantage. In other words, more advertising is not inherently good for the challenger technology; it depends entirely on who currently owns the attention. The same logic applies to social influence: stronger local conformity and stronger hub-node influence both reinforce whatever the majority already prefers, making it harder for NEVs to expand from a small base.

Government policy, by contrast, transmits asymmetrically through the two layers. The researchers standardized three instruments—a consumer purchase subsidy, a firm-side technology transition subsidy, and a carbon price—by scaling each proportionally from zero to twice its benchmark. The purchase subsidy, which directly improves consumer utility and demand, produced by far the strongest response, lifting steady-state adoption from about 0.10 when removed toward near-complete diffusion at double intensity. The technology transition subsidy and carbon price, which act only through firm payoffs, moved long-run outcomes far less. Policy effectiveness, the authors conclude, depends not just on intensity but on the channel through which a policy enters the coupled system.

Physical constraints proved to be hard boundaries that money can only partly buy down. Rising charging anxiety and perceived technological risk both suppressed consumer adoption and, through demand feedback, firm transition. Joint analyses showed that purchase subsidies can compensate for a weak initial market base, shifting the adoption threshold downward, and can partially offset charging constraints—but at sufficiently severe charging anxiety, even strong subsidies fail to sustain high adoption. Meanwhile, digital marketing utility and hub influence were found to act jointly, each amplifying the other’s incumbent-favoring effect. The lesson is sobering for policymakers: financial incentives cannot substitute for the infrastructure and reliability that make the technology genuinely attractive.

Robustness tests added an important nuance. Varying the firm network’s rewiring probability, neighborhood size, random seeds, and population size produced only modest changes, but the consumer network’s structure mattered substantially. Replacing the scale-free consumer layer with a small-world or Erdős-Rényi topology changed the quantitative diffusion level—near-complete diffusion emerged under a WS-WS configuration—without overturning the directional effects of the behavioral and policy parameters. Network structure, in other words, acts as a boundary condition on how large the effects are, even as the mechanisms themselves remain stable. This cautions against forecasting adoption from a single assumed social topology.

The practical implications are pointed. Early-stage efforts should concentrate on building the initial adopter base, because that base determines which basin of attraction the market falls into. Demand-side instruments deserve priority when the goal is market expansion, with firm-side support and carbon pricing playing complementary roles. Automakers should target NEV-oriented exposure effectiveness rather than raw marketing intensity, avoiding the overload regime, and should focus social-influence strategies on well-connected early adopters who can seed favorable local norms. Above all, the study reframes the electric vehicle transition as a problem of coupled networks and path dependence: the technology that wins is not necessarily the better one, but the one that starts with enough users, enough attention, and enough infrastructure to trigger the self-reinforcing loop before the incumbent’s advantage hardens into permanence.

Subject of Research: Evolutionary game modeling of new energy vehicle diffusion in a dual-layer firm-consumer heterogeneous network

Article Title: Digital marketing and social network effects in new energy vehicle diffusion: evolutionary dynamics in a dual-layer heterogeneous network

Article References: Zhao, W., Xia, D., & Zhang, Y. (2026). Digital marketing and social network effects in new energy vehicle diffusion: evolutionary dynamics in a dual-layer heterogeneous network. Frontiers in Physics, 14, Article 1972563. https://doi.org/10.3389/fphy.2026.1972563

Image Credits: AI Generated

DOI: 10.3389/fphy.2026.1972563

Keywords: new energy vehicles, digital marketing, social network effects, evolutionary game theory, dual-layer network, complex networks, government subsidies, carbon pricing, charging anxiety, information overload, technology diffusion, scale-free networks

To understand why this matters, it helps to trace the lineage of the underlying mathematics. Fuzzy sets, introduced by Lotfi Zadeh in 1965, replaced the rigid yes-or-no logic of classical set theory with degrees of membership, allowing an element to belong to a set partially. Hesitant fuzzy sets, proposed by Vicenç Torra in 2010, went a step further by permitting several possible membership values for a single element, capturing the situation where a panel of experts offers conflicting assessments rather than a single consensus figure. Dual hesitant fuzzy sets then added a second dimension: alongside multiple possible membership degrees, they allow multiple possible non-membership degrees, so an evaluator can simultaneously express hesitation about how well an option performs and hesitation about how badly it fails.

The new paper builds on two prominent extensions of that dual structure. Probabilistic dual hesitant fuzzy sets attach probability weights to each possible membership and non-membership value, acknowledging that some expert opinions are more likely or better supported than others. Interval-valued dual hesitant fuzzy sets, by contrast, replace single numbers with intervals, which is useful when an assessor can only bound a judgment rather than pin it down. Each extension captures something the other misses. Probabilities encode the relative credibility of individual assessments; intervals encode the inherent imprecision of the assessments themselves. Yan’s contribution is the probabilistic interval-valued dual hesitant fuzzy set, or PIVDHFS, which fuses both properties into a single representation. In a PIVDHFS, each element carries a set of interval-valued membership degrees and a set of interval-valued non-membership degrees, and each of those interval values is associated with a probability. The result is a data structure that can model disagreement among experts, imprecision within each expert’s judgment, and the unequal reliability of different judgments, all simultaneously.

Representing information this way is only useful if you can compute with it, and that requires a way to measure how different two PIVDHFSs are. Distance measures are the workhorses of fuzzy decision theory: they underpin ranking methods, clustering, pattern recognition, and medical diagnosis applications. The paper defines the operational laws of PIVDHFSs and then derives a family of distance formulas. The probabilistic interval-valued dual hesitant fuzzy weighted distance treats each attribute with a fixed importance weight. The ordered weighted distance instead assigns weights to positions in an ordered sequence, so that the largest deviations can be emphasized or downplayed depending on the decision-maker’s risk posture. The hybrid weighted distance combines both ideas, weighting attributes and positions at the same time, and the generalized hybrid weighted distance adds a tunable parameter that lets the analyst adjust how aggressively large differences are penalized. Together these measures give decision-makers a flexible toolkit for comparing alternatives described in this rich uncertain format.

A second, often overlooked problem in multi-attribute decision-making is determining how much each criterion should matter in the first place. When attribute weights are unknown, subjective assignment invites bias, so objective weighting methods infer importance directly from the data. The study enhances one such method, CRITIC, which stands for criteria importance through intercriteria correlation. The original CRITIC, introduced by Diakoulaki and colleagues in 1995, assigns higher weights to criteria that carry more information, measured by standard deviation, and that are less redundant with other criteria, measured by correlation. Yan’s version incorporates the Spearman correlation coefficient, a rank-based measure of association, within the PIVDHFS context. Rank correlation is more robust than the Pearson coefficient when the underlying data are ordinal or non-linear, which is precisely the character of hesitant fuzzy assessments. The enhanced CRITIC therefore extracts objective weights from the structure of the probabilistic interval-valued data itself, reducing the influence of arbitrary human judgment on the weighting stage.

With distances and weights in hand, the framework turns to ranking. TOPSIS, the technique for order of preference by similarity to ideal solution, is one of the most widely used methods in operations research. Its logic is intuitive: the best alternative is the one that is simultaneously closest to a hypothetical positive ideal solution and farthest from a negative ideal one. The paper extends TOPSIS to operate directly on PIVDHFSs, using the newly developed distance measures to compute each alternative’s separation from the ideal and anti-ideal points. The extended PIVDHF-CRITIC-TOPSIS pipeline thus proceeds in three stages: encode expert evaluations as PIVDHFSs, derive objective attribute weights with the Spearman-enhanced CRITIC method, and rank alternatives by their relative closeness computed through the new distance family.

The framework was put to the test on a problem with genuine industrial stakes: digital supplier selection for a new energy vehicle manufacturer in China. Digital suppliers provide the software platforms, data services, and connected-technology capabilities that modern electric vehicle production increasingly depends on, and choosing among them involves criteria that are hard to quantify precisely, such as technological maturity, data security posture, service responsiveness, and long-term partnership potential. Evaluations from multiple experts naturally conflict, and confidence in each assessment varies. The PIVDHFS format captured this layered uncertainty, and the extended framework produced a complete ranking of candidate suppliers. The author validated the result in two ways. A sensitivity analysis varied key parameters, including the generalized distance parameter and the derived attribute weights, to check whether the ranking was an artifact of particular settings. A comparative study then applied alternative decision methods to the same data to confirm that the proposed approach yields consistent and scientifically rational outcomes.

What makes this work notable in the broader landscape of decision science is its position at the confluence of several research threads that had previously advanced separately. Probabilistic dual hesitant fuzzy sets have been applied to risk evaluation, sustainable supplier selection, and rainfall analysis, while interval-valued dual hesitant fuzzy sets have supported emergency response evaluation and aggregation-based decision models. Distance measures for dual hesitant structures have found uses ranging from image segmentation to medical diagnosis. By unifying the probabilistic and interval-valued extensions and wiring them into an objective weighting scheme and a classical ranking method, the new framework offers a template that other researchers can adapt. The same architecture could, in principle, support green supplier selection, renewable energy assessment, medical diagnosis, or any domain where expert judgment is hesitant, imprecise, and unevenly reliable.

The practical implications extend to the fast-growing field of digital supply chain management. Recent literature on supplier selection in digital manufacturing has emphasized resilience, cyber risk, and data-driven simulation, and machine learning approaches have been proposed for modeling supplier performance under disruption. Yet the front end of those pipelines, the structured elicitation of expert judgment about candidates, still relies heavily on decision frameworks that assume crisp numbers or single-layer fuzziness. A representation that preserves probabilities, intervals, and dual hesitation through every stage of weighting and ranking could feed cleaner, more honest inputs into downstream analytics. For industries such as electric vehicles, where supply chains span continents and technology cycles are measured in months, the difference between a robust ranking and a fragile one can translate into millions in procurement outcomes.

There are, of course, limits worth noting. The framework’s sophistication comes at the cost of computational and elicitation overhead: asking experts to supply interval-valued memberships, non-memberships, and probabilities for every attribute of every alternative is demanding, and the quality of the output depends on the quality of those inputs. The study itself reports validation through a single real-world case, supplemented by sensitivity and comparative analyses, which is a solid but not exhaustive evidence base. Still, the paper represents a careful piece of mathematical engineering, defining operational laws, deriving a graded family of distance measures, enhancing an objective weighting method with a rank correlation coefficient, and demonstrating the whole pipeline on a decision that manufacturers actually face. As uncertain, expert-mediated judgments continue to pervade high-stakes industrial and environmental choices, tools that quantify hesitation rather than papering over it are likely to become standard equipment in the decision scientist’s toolkit.

Subject of Research: Multi-attribute decision-making using probabilistic interval-valued dual hesitant fuzzy sets with distance measures and an extended TOPSIS framework

Article Title: Multi-attribute decision-making based on probabilistic interval-valued dual hesitant fuzzy sets and extended TOPSIS

Article References: Yan, L. (2026). Multi-attribute decision-making based on probabilistic interval-valued dual hesitant fuzzy sets and extended TOPSIS. International Journal of Machine Learning and Cybernetics, 17(10), Article 486. https://doi.org/10.1007/s13042-026-03294-z

Image Credits: AI Generated

DOI: 10.1007/s13042-026-03294-z

Keywords: fuzzy sets, multi-attribute decision-making, TOPSIS, CRITIC method, distance measures, dual hesitant fuzzy sets, supplier selection, uncertainty modeling, operations research, new energy vehicles, Spearman correlation, decision science