The work, led by Haotian Zhang of China Mobile Group Henan and Beijing Normal University, together with Qi Jia of Yuntu Zhixing and colleagues at IEIT Systems, Southeast University, and Beijing Normal University, addresses three long-standing weaknesses in existing geometry problem solvers. According to the authors, current methods struggle to capture the structural information embedded in diagrams, to align textual symbols with the visual primitives they refer to, and to cope with the sparse distribution of geometric primitives in typical figures. Each of these gaps can cause a model to miss the very relationships a problem depends on.

The core idea of the new system is a graph construction method that converts every diagram into a heterogeneous graph. In such a graph, nodes represent different kinds of geometric entities, for example points, line segments, circles, and angles, while edges encode the diverse relations that connect them, such as incidence, parallelism, or containment. Because the node and edge types differ, the representation is called heterogeneous, and it preserves far more of the diagram’s logical structure than a flat grid of pixels. The model can then reason over explicit geometric relationships rather than having to infer them implicitly from raw visual features.

Building on this construction method, the team created the Geometric Relation Heterogeneous Graph dataset, abbreviated Geo-RHG. The dataset pairs geometric diagrams with their corresponding textual descriptions and, crucially, with heterogeneous graph representations of each diagram. This triple structure gives researchers a resource for training and evaluating models that must understand both what a diagram shows and what the accompanying words say. The authors state that the dataset and code will be released at the project’s GitHub repository upon publication, making the resource available to the wider research community.

A second pillar of the approach is a cross-modal pre-training strategy designed to bridge the gap between language and vision. Text in a geometry problem refers to visual objects: a phrase such as ‘AB is parallel to CD’ names specific segments in the figure. The pre-training procedure explicitly aligns textual symbols with their corresponding visual primitives, teaching the model which words point to which parts of the graph. This grounding step is intended to eliminate a common failure mode in which a model understands the sentence and sees the diagram but never connects the two.

The third component is a hierarchical fusion mechanism that injects structural cues into visual features during the modality fusion stage. Rather than simply concatenating text embeddings and image embeddings, the system lets the graph-derived structural information refine the visual representation before and during fusion. The authors report that this integration enhances the model’s capacity to reason over geometric content and supports more accurate problem solving in multimodal contexts, because the fused features carry both appearance and relational information.

Extensive experiments described in the paper show that the proposed approach achieves superior performance on standard geometry problem solving benchmarks, outperforming existing baselines in both accuracy and generalization. Generalization matters particularly in this field: a solver that merely memorizes the problem templates of a training set is of little use in classrooms or examination settings, where problems vary widely in wording and figure style. The combination of explicit graph structure and cross-modal alignment appears to give the model a more transferable understanding of geometric content.

The new study sits within a research lineage that stretches back decades. Early efforts at automated geometry theorem proving date to the 1960s, when researchers such as Gelernter and colleagues built the first geometry theorem machines. Modern data-driven approaches began with systems like GeoSolver, which combined text and diagram interpretation, and later work harvested axiomatic knowledge from textbooks. More recently, benchmarks such as GeoQA, UniGeo, and augmented geometric question answering datasets have fueled progress, while frameworks like Inter-GPS introduced formal languages and symbolic reasoning, and GeoDRL applied reinforcement learning to deductive reasoning. Explainable solvers and layout-aware neural models have pushed the field further, yet the authors argue that none of these fully resolves the structural modeling and alignment problems their graph-based approach targets.

Technically, the system draws on a broad set of tools from modern deep learning. Vision encoders of the kind introduced in residual networks and vision transformers provide the raw visual features, while object detection architectures in the Faster R-CNN tradition help localize geometric primitives. Language understanding builds on pretrained models in the BERT and RoBERTa family. Graph convolutional networks, which were developed for modeling relational data, supply the machinery for propagating information across the heterogeneous graph, and the transformer attention mechanism underlies the cross-modal fusion. The authors also cite visual question answering research, including graph-structured representations for VQA and conditioned graph structures, as conceptual ancestors of their approach, alongside classical sequence models such as long short-term memory networks that once dominated multimodal tasks.

Diagram parsing itself has advanced rapidly in recent years, with work on plane geometry diagram parsing and alignment-based relation learning for diagram parsing showing that the geometric content of figures can be extracted with increasing fidelity. The new system builds on this progress but goes a step further by making the relational structure the central object of representation rather than a byproduct of parsing. The authors suggest that this shift is what allows their model to handle the sparse, symbol-heavy nature of geometric figures, where a handful of lines and labels carry all the meaning and background pixels carry almost none.

The implications extend beyond geometry homework. Multimodal reasoning systems that must combine images and language appear in visual question answering, document understanding, scientific figure analysis, and educational technology. A method that explicitly structures visual content as a graph and aligns it with text could transfer to any domain where the image is not a photograph but a structured drawing: circuit diagrams, flowcharts, chemical structures, or architectural plans. The hierarchical fusion idea, in which structural cues refine visual features before fusion with language, may prove useful wherever relational structure is the key to understanding an image.

For the field of automatic problem solving, the study offers a concrete recipe: represent the diagram as a heterogeneous graph, pre-train the model to align symbols with primitives, and fuse modalities hierarchically with structure in mind. The strong benchmark results reported by the team suggest that explicit structure, long championed in symbolic AI, and large-scale neural learning are not rivals but complements. As the Geo-RHG dataset and code become publicly available, other researchers will be able to test that proposition, refine the graph construction methods, and push multimodal mathematical reasoning closer to the fluency of a well-trained human geometer.

The article, ‘A geometry problem solver based on heterogeneous graph,’ appears in Applied Intelligence, volume 56, article number 439, published on 22 September 2026. The work was received on 12 November 2025 and accepted on 7 September 2026, and its authors span industry and academia in China, reflecting the growing collaboration between technology companies and universities on foundational problems in artificial intelligence. With benchmarks falling and resources opening up, geometry problem solving is shaping up as one of the clearest windows into how machines can learn to see, read, and reason as one.

Subject of Research: A multimodal AI system that solves geometry problems by representing diagrams as heterogeneous graphs and aligning them with text.

Article Title: A geometry problem solver based on heterogeneous graph

Article References: Zhang, H., Jia, Q., Liu, L., Xu, C., Jin, L., Pan, Y., Chen, F., Wang, Y., & Liu, Y. (2026). A geometry problem solver based on heterogeneous graph. Applied Intelligence, 56(15), Article 439. https://doi.org/10.1007/s10489-026-07471-z

Image Credits: AI Generated

DOI: 10.1007/s10489-026-07471-z

Keywords: geometry problem solving, heterogeneous graph, multimodal learning, cross-modal pre-training, Geo-RHG dataset, graph neural network, visual reasoning, mathematical reasoning, AI benchmark, feature representation, automatic problem solving, Applied Intelligence

Social influence estimation is a cornerstone problem for platforms that rely on word-of-mouth dynamics. Identifying influential users underpins targeted advertising, viral marketing campaigns, and behavior prediction models. Prior work has long suggested that similarity between users is a significant driver of influence: people tend to trust, imitate, and adopt behaviors from those who resemble them. Traditional approaches measured that resemblance through structural metrics such as common neighbors, Jaccard coefficients, or path-based distances within the social graph. More recent methods turned to representation learning, embedding users into continuous vector spaces where proximity reflects similarity. Yet, as the authors point out, these techniques capture only limited facets of user characteristics, typically ignoring the rich geographic dimension that location-based social networks uniquely provide.

The study addresses this gap with two progressive contributions. The first, called NSH, short for Node Embedding and Spatial Homophily, is a hybrid, unsupervised method that linearly combines embedding-based structural similarity with spatial homophily, the well-documented tendency of geographically close users to be socially connected and behaviorally alike. NSH is deliberately simple and interpretable: it takes a user similarity score derived from network embeddings and blends it with a spatial homophily score computed from check-in co-location patterns. The result is a transparent similarity measure that serves as a strong baseline for analyzing the relationship between similarity and influence, and one that practitioners can compute without heavy machinery.

The second and central innovation is GeoSocial2Vec, an end-to-end unified embedding model designed to learn a single representation for each user by intrinsically merging social ties and spatial co-visitation patterns. Rather than computing two separate similarity scores and averaging them, GeoSocial2Vec constructs a heterogeneous graph in which users, social links, and locations coexist as nodes of different types. An adaptive random walk mechanism then traverses this combined graph, generating sequences of nodes that alternate between social hops and spatial transitions. These walks are fed into a skip-gram style embedding procedure, so that the resulting vectors encode both who a user knows and where a user goes within one coherent geometric space. The adaptive component weights different transition types dynamically, allowing the model to balance social and spatial evidence according to the structure of the data rather than a fixed heuristic.

This design reflects a broader shift in representation learning toward heterogeneous graph embeddings, where multiple node and edge types are embedded jointly instead of being projected into separate spaces. In the context of location-based social networks, the payoff is substantial. Two users who never interact directly but repeatedly visit the same neighborhoods at similar times will land close together in the GeoSocial2Vec embedding, capturing a latent affinity that pure network metrics would miss. Conversely, two friends who live in different cities and never share locations will still be pulled together by their social tie, something a purely geographic model would overlook. The embedding, in effect, learns the full socio-spatial signature of each user.

To test whether these richer similarity measures actually track influence, the researchers evaluated the relationship between user similarity and social influence through Pearson correlation and regression analysis on geographically filtered segments of three real-world location-based social networks: Gowalla, Brightkite, and Foursquare. These datasets, drawn from check-in services that were popular in the early 2010s, remain standard benchmarks in this field because they pair explicit friendship graphs with millions of timestamped location records. Filtering the data geographically ensured that the spatial component of the analysis was meaningful, focusing on regions dense enough for co-visitation patterns to be statistically robust.

The results were striking. Both proposed methods, in which spatial homophily is integrated with embedding-based similarity, produced a significantly stronger correlation with observed influence patterns than classical approaches that rely on structure alone. The GeoSocial2Vec framework achieved the highest accuracy across the benchmarks, demonstrating that intrinsically coupled socio-spatial learning outperforms post-hoc fusion of separate signals. In other words, it matters not just that social and geographic information are both considered, but that they are woven together during representation learning itself, allowing the model to capture interactions between the two dimensions that a linear combination cannot express.

At the same time, the authors are careful not to declare NSH obsolete. While GeoSocial2Vec offers the best predictive performance, the NSH framework retains clear advantages in interpretability and computational simplicity. Because NSH is a linear fusion of two comprehensible scores, an analyst can inspect exactly how much of a predicted influence relationship stems from network proximity versus geographic overlap. This transparency matters in business applications where stakeholders must justify targeting decisions, and in settings where training a deep heterogeneous embedding model is impractical due to limited compute or data volume. The two frameworks thus occupy complementary niches: GeoSocial2Vec as the accuracy-oriented state of the art, and NSH as a practical, explainable tool for everyday analysis.

The implications extend well beyond academic benchmarking. For advertisers on check-in-driven platforms, influence estimation that accounts for where people go, not just who they know, could sharpen the selection of seed users in viral campaigns, ensuring that promotional content spreads through communities that are both socially and geographically primed to receive it. For behavior prediction, the fused embeddings offer a way to anticipate adoption of products, venues, or even public health behaviors by modeling the twin channels of social contagion and spatial co-presence. Urban planners and mobility researchers may also find the technique useful, since the same embeddings encode movement regularities that correlate with social structure.

The study also contributes to a long-running scientific conversation about homophily, the principle that similar individuals connect and influence one another. Spatial homophily has been documented repeatedly in location-based social networks, where the probability of friendship decays with physical distance and shared places act as meeting grounds for ties. By demonstrating that this geographic dimension measurably strengthens the link between similarity and influence, the research quantifies how much predictive power has been left on the table by purely structural methods. It suggests that future influence models, and by extension the recommendation and marketing systems built on them, should treat location not as metadata but as a first-class signal in the learning process.

Technically, the work sits at the intersection of network embedding, heterogeneous graph learning, and random-walk-based representation methods. The adaptive random walk at the heart of GeoSocial2Vec echoes the logic of metapath-guided walks in heterogeneous graphs, but with a twist: transition probabilities are tuned to the data, letting the model decide how often to follow social edges versus spatial co-visit links. This adaptivity is what allows a single embedding space to serve both dimensions faithfully, and it is likely the key reason the framework outperformed the linear NSH fusion. The evaluation methodology, combining Pearson correlation with regression analysis across three independent platforms, adds confidence that the findings are not an artifact of one dataset’s quirks.

Limitations and open questions remain, as they do in any study of this kind. The benchmark datasets are aging, and modern platforms may exhibit different spatial dynamics, particularly as privacy changes and check-in behavior declines in some services. Influence itself is a latent quantity, inferred from observed diffusion patterns rather than measured directly, so all correlation results inherit the assumptions of the influence estimation pipeline. The authors also note that the published version is an early-access, peer-reviewed accepted manuscript carrying a permanent DOI, subject to final editorial edits. Still, the core message is robust: when it comes to understanding who moves whom in location-based social networks, knowing both the social graph and the map beats knowing either alone.

As location-aware platforms continue to blend commerce, mobility, and social connection, methods like GeoSocial2Vec point toward a future in which influence is predicted not from a single flattened network, but from the rich, coupled reality of how people relate and where they roam. The research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors, and the authors declare no competing interests.

Subject of Research: Estimating social influence in location-based social networks using embedding-based similarity and spatial homophily

Article Title: Investigating social influence using embedding-based similarity and spatial homophily on the location based social networks

Article References: Akhavan-Hejazi, Z. S., Ghobaei-Arani, M., & Sajjadi, Z. S. (2026). Investigating social influence using embedding-based similarity and spatial homophily on the location based social networks. Journal of Big Data. https://doi.org/10.1186/s40537-026-01546-x

Image Credits: AI Generated

DOI: 10.1186/s40537-026-01546-x

Keywords: social influence, location-based social networks, node embeddings, spatial homophily, GeoSocial2Vec, heterogeneous graph, random walk, Gowalla, Brightkite, Foursquare, representation learning, targeted advertising

The study, led by Dezhi Zhi and colleagues at Northeast Forestry University in Harbin, China, tackles a problem that has grown acute as spatial transcriptomics has swept through plant science. Spatial transcriptomics allows researchers to measure gene expression across intact tissue sections, preserving the physical layout of cells within a leaf, root, or stem. That spatial context is exactly what plant biologists need to understand how vascular stem cells, epidermal cells, and meristematic pools coordinate their behavior. But the raw data alone does not reveal which of the thousands of possible ligand–receptor combinations are actually at work in a given tissue.

Existing tools for inferring cell–cell communication were largely built for animal single-cell data, where physical proximity is a reasonable proxy for signaling potential. In plants, that assumption breaks down. Cell walls, plasmodesmata, and the intricate local architecture of tissues mean that spatial adjacency is a weak and sometimes misleading indicator of effective communication. On top of this, plant ligand–receptor resources are complicated by massively expanded gene families, mappings derived from sequence homology rather than direct experiment, and highly uneven levels of experimental validation across candidate pairs.

PlantCCC approaches the problem as a graph-learning challenge. The framework takes a plant ligand–receptor database as its candidate search space and then builds a directed heterogeneous graph that connects cells, genes, and candidate communication edges. Rather than treating every neighboring cell pair as equally likely to exchange signals, PlantCCC applies expression-gated spatial weighting, a mechanism that scales the influence of spatial proximity according to whether the relevant genes are actually expressed. This allows the model to distinguish between cells that merely coexist in a tissue region and cells whose molecular profiles suggest an active signaling relationship.

The architecture combines several techniques from modern deep learning. Residual spatial expression enhancement sharpens the gene expression profiles of individual cells using information from their spatial neighborhoods. Spatially aware multi-head graph attention lets the model weigh different neighbors differently when aggregating information, while self-supervised contrastive learning helps the framework learn robust representations without requiring labeled training examples of true interactions. Together, these components allow PlantCCC to score candidate ligand–receptor edges in a way that integrates expression, spatial adjacency, and tissue context simultaneously.

To test whether the framework could separate genuine signaling from mere coincidence, the researchers constructed a semi-synthetic benchmark using an Arabidopsis leaf single-cell Stereo-seq dataset as a realistic spatial background. Into this background they injected known interaction components, creating TRUE pairs that contained a genuine communication signal, alongside CONFOUNDER pairs that showed tissue co-localization alone. PlantCCC successfully distinguished the two categories, demonstrating that it can detect an injected interaction component rather than simply rewarding pairs of cells that happen to occupy the same neighborhood. The framework also remained comparatively robust under dropout perturbation, a common artifact in single-cell and spatial data in which genes are spuriously recorded as unexpressed.

The researchers then applied PlantCCC to real biological questions, beginning with poplar stem datasets. Because a curated ligand–receptor resource for poplar was not directly available, the team derived a candidate set for Populus through homology mapping from Arabidopsis entries. Despite this added layer of uncertainty, PlantCCC prioritized candidate ligand–receptor axes that were consistent with the known architecture of the poplar stem, including the organization of meristematic cell pools within the secondary vascular tissue, and with prior experimental evidence for the corresponding signaling modules.

As an independent validation, the team turned to a publicly available 10x Genomics Visium HD dataset of Arabidopsis thaliana, using Arabidopsis PlantPhoneDB entries as the candidate search space. PlantPhoneDB is a manually curated pan-plant database of ligand–receptor pairs, and grounding the analysis in its experimentally supported entries gave the results a firmer biological footing. Once again, the top-ranked candidate communication axes aligned with tissue architecture, spatial expression patterns, and established knowledge of plant signaling pathways, suggesting that the framework’s rankings reflect genuine biology rather than computational artifacts.

The significance of this work extends beyond a single algorithm. Plant development depends on countless short-range peptide signals and receptor kinases: the CLAVATA pathway limits stem cell proliferation in shoot meristems, the PXY–CLE41 pair controls the rate and orientation of vascular cell division, FERONIA-mediated signaling maintains cell-wall integrity during salt stress, and peptide hormones such as phytosulfokine regulate cell expansion. Tools that can reliably prioritize which of these candidate axes are active in a specific tissue, at a specific developmental stage, could accelerate the discovery of new regulatory mechanisms in wood formation, defense responses, and organ development.

PlantCCC is also designed with interpretability and reproducibility in mind. The study analyzed four publicly available spatial transcriptomic datasets, and all implementation code and analysis scripts are openly available in a GitHub repository, covering everything from data preprocessing and homology mapping to model training, inference, and visualization. For a field where computational results can be difficult to reproduce, this openness lowers the barrier for other laboratories to apply the framework to their own crops, forest trees, or model species. As spatial transcriptomics continues to drive a new era in plant research, frameworks like PlantCCC offer a way to move from maps of where genes are expressed to mechanistic hypotheses about how plant cells actually coordinate their lives.

Subject of Research: A computational framework for inferring ligand–receptor cell–cell communication from plant spatial transcriptomics data.

Article Title: PlantCCC prioritizes context-specific candidate ligand–receptor communication patterns in plant spatial transcriptomics

Article References: Zhi, D., Wang, L., Guan, X., Chen, W., & Chen, K. (2026). PlantCCC prioritizes context-specific candidate ligand–receptor communication patterns in plant spatial transcriptomics. Plant Molecular Biology, 116(5), Article 93. https://doi.org/10.1007/s11103-026-01758-y

Image Credits: AI Generated

DOI: 10.1007/s11103-026-01758-y

Keywords: spatial transcriptomics, plant cell–cell communication, ligand–receptor pairs, graph attention network, graph contrastive learning, PlantPhoneDB, Arabidopsis thaliana, poplar stem, expression-gated spatial weighting, heterogeneous graph, PlantCCC, plant signaling