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## KS2 Maths (Line Graphs)

Subject: Mathematics

Age range: 7-11

Resource type: Worksheet/Activity

Last updated

16 January 2019

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These topic-focused SATs questions at the end of a unit will help to test and extend students’ understanding as well as helping them to prepare for SATs next year. These questions have fully-worked solutions which can be displayed on a whiteboard making feedback with students more efficient. Click 👉 tes.com/…/KS2-Maths-Questions… for similar-style compilations on the other KS2 topics. <hr> This particular compilation is from the STATISTICS strand and contains questions on Line Graphs . <hr> I have designed this compilation to be printed as an A4 or A5 booklet which is in the style of the actual SATs papers and is convenient for use in class or as homework. It can even be given to individual students if a parent is asking for ‘some more work’!

KEY POINTS:

- I have provided full answers, with comments and working where helpful.
- I have maintained the style of the actual SATs questions so that students can become comfortable with the way that SATs questions are presented.
- Most of the questions are from actual SATs papers, but I have also added questions so that this resource matches the requirements the current curriculum better than the older resources that are still in common use (note that many of the older resources of this type contain questions on topics which are no longer examined).
- I have spent a lot of time arranging the questions so that there is a general increase in difficulty as students work through them, and so that they fit on the pages better – this means less wasted space and significant paper-saving when printing 😃 <hr> 👍If you like this resource, then please rate it and/or leave a comment💬. If the rate-resource button on this page doesn’t work, then go to your ratings page by clicking 👉 www.tes.com/…/rate-resources…

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## Line Graph Worksheets

Line graph worksheets have ample practice skills to analyze, interpret and compare the data from the graphs. Exercises to draw line graphs and double line graphs with a suitable scale; labeling the axes; giving a title for the graph and more are included in these printable worksheets for students of grade 2 through grade 6. Try some of these worksheets for free!

Interpreting Line Graph: Easy

Line graph worksheet pdfs have three interesting scenarios with graphs for 2nd grade and 3rd grade kids. Read the line graph and answer the word problems in each worksheet.

- Download the set

Interpreting Line Graph: Moderate

Line graphs on different themes are shown in these printable worksheets. Interpret the data from the line graph and answer the questions.

Interpreting Line Graph: Difficult

Line graphs in these 4th grade and 5th grade worksheets represent more than ten data. Read and interpret the graph carefully to answer the questions.

Drawing Line Graph: Easy

In these pdf worksheets, data for which the graph to be represented are given. Read the data, plot points and draw lines to complete the graph.

Drawing Line Graph: Moderate

The number usage (given data) gradually increases in this level. Plot points on the graph to represent the data and join them to make a line graph.

Drawing Line Graph: Difficult

Numerous data are used in these worksheets. Attentively read the data and represent it on the grid to draw the line graph.

Title, Scale, Labeling Axis, and Graphing

Draw a line graph for the represented data. Make an appropriate scale, label the x axis, y axis and write your own title for the graph.

Reading Double Line Graph

In these worksheets two sets of data are compared. Both the data are represented as a double line graph. Read them and answer the questions.

Drawing Double Line Graph

Two sets of data are given. 6th grade students need to analyze the data, make a suitable scale and draw double line graph. Label the axes and give a suitable title for the graph.

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» Tally Marks

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## Line Graphs Practice Questions

Click here for questions, click here for answers, gcse revision cards.

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## KS2 Maths Investigations Based On Real Life In Primary School

Sophie Bessemer

It’s been a long week and it is time to hand out your latest ‘exciting’ KS2 maths investigations, carefully crafted problem solving investigations focused specifically on the work you’ve been doing this week.

But then you hear the the immortal words from your Year 6: ” What Does This Have To Do With The Real World?”

Any good teacher knows, of course, exactly how relevant maths is in the real world and how, without maths, modern society as we know it would never have existed.

The problem is, not all 11 year olds know it too – and you’re going to have a hard time convincing some of them.

In defence of 11 year olds, the curriculum – maths in particular – can sometimes feel all too distant from what’s ‘real’.

So the question then becomes, how do we show young learners how Maths intersects and dominates our day to day life?

How do we give our KS2 pupils maths investigations that inspire them, change their perceptions and help them to move beyond a fixed mindset to see maths problem solving as entirely relevant to what may come next in life?

If you’re just interested in maths investigations for Year 5 and Year 6 we’ve created jump to the end of the blog where they’re all listed by term.

## KS2 Maths Investigations: Problem solving in context

Benefits of maths investigations at ks2, 5 top tips for creating your own ks2 maths investigations, year 5 and year 6 maths investigations.

We believe that one of the answers is putting your maths problem solving activities into a context that your pupils can relate to.

We call this Topical Maths, and we’ve used this idea as the source for several of our most popular Year 5 and Year 6 maths problem solving resources, all offering the kind of KS2 maths investigations we know your pupils will love!

KS2 Topical Maths Problems

25 real world maths investigations to practise reasoning and problem solving based on primary school calendar events

There are lots of benefits of course, but the most important as far as we’re concerned are these:

- Pupils are required to talk and reason about their maths
- The maths problem solving investigations cement higher order reasoning skills and problem solving
- Starting early with Year 5 maths investigations you can support familiarity with the sorts of questions that come up in Year 6 SATs.

To encourage you to give these KS2 maths investigations a go, we’ll first look at the benefits and principles of introducing them for your reasoning and problem solving at Year 5 and Year 6, we’ll then give you some ideas for how you can create these problem solving activities for the rest of KS2 yourself.

We guarantee you’ll see your pupils’ reasoning and problems solving skills improve!

## 1. KS2 Maths Investigations Involve Pupils Talking and Reasoning

Getting pupils to verbalise their numerical reasoning has a knock-on effect on pupils’ overall reasoning skills, which is why the core element of our KS2 maths intervention is mathematical reasoning; asking pupils to explain not just what they’re doing, but why they’re doing it.

As a teacher of a large class, it can be difficult to provide the teacher time necessary for each pupil to verbalise to you their reasoning.

The inherently collaborative nature of year 5 and year 6 maths investigations gives pupils the opportunities to to reason out loud and work on their maths problem solving skills.

## 2. KS2 Maths Investigations Cement Higher Order Reasoning Skills

Our experience teaching thousands of primary school pupils maths every week has shown us that at KS2, even by Year 5 or Year 6, pupils often have good procedural understanding, but struggle with higher order problem solving questions.

The problem solving element to these topical maths investigations naturally improves reasoning skills in Year 5 and Year 6 pupils, as they are more likely on reaching an answer to have to think about not just how but why that answer is correct.

By setting topical maths investigations at KS2 as group-work or a whole class activity, you can ensure that all pupils get to experience this deep level of reasoning.

Many of our topical maths investigations are open ended, but if you’re teaching a fully mixed ability class, we’ve also created some low threshold high ceiling open ended maths investigations specifically for mixed ability classes.

## 3. KS2 Maths Investigations Give Early Exposure To SATs Style, Reasoning Questions

Most, if not all, schools will provide their pupils with exposure to reasoning via SATs-style questions, but this often comes hand in hand with exams and assessment.

Yet, it is equally important to get pupils reasoning and problem solving in a low stakes classroom setting or as a group.

Creating a learning environment where the types of problem solving questions found in SATs just become part of your lessons will help pupils feel comfortable with exam terminology, and ensures they are more at ease with being asked the same kind of question (say, multiplying and dividing fractions) in lots of different ways.

These maths problem solving investigations and downloadable resources enable you to include these type of SATs style questions in a way which is fun and confidence boosting.

## More problem solving and reasoning articles

- Ultimate Guide to Maths Problem Solving Techniques
- Maths Investigations: How To Develop Mathematical Reasoning
- 35 Year 6 Maths Reasoning Questions .

You don’t need to create your own problem solving investigations – the links at the end of this article should provide you with everything you need. However if you do want to have a go these were our principles and, judging by the number of visits. to the date related articles and downloads of these resources we get every year, they’re still very popular.

## 1. Date-based themes for maths problem solving activities

Nothing solidifies maths in the real world quite like the real world. Nearly, if not everyday of the year holds some significance to someone.

February? Pancake day, Valentines Day, and Fairtrade Fortnight.

March? Red Nose Day, World Book Day, and Holi.

Why not spice your lesson up and throw in some Pancake Day Maths for ratios, or Bonfire Night Maths for measurements. Capitalise on special celebrations throughout the world to excite and enthuse young learners.

For example these Christmas activities always prove popular with KS1 and KS2 or at a different time of year you could try these summer holiday maths investigations or any of these maths activities .

## 2. Trends and pop-culture KS2 maths investigations

Peers and pop culture hold huge sway over most pupils, and the reason for this is that as growing persons we want to fit in and find friends.

Nothing achieves this more effectively than mutual interest. As a teacher, utilise it – whether this is measuring the speed of explosions in the latest Transformer film, or totalling the high notes in Disney’s Moana – you’ll have pupils hooked in no time.

For the exceptionally savvy teacher, you might want to capitalise on the latest fads and trends within your school. How about measuring amounts via the infamous bottle trick, or examining angles through the lens of the dab?

## 3. Simple stuff engages pupils with maths problem solving and reasoning

Sometimes when teachers link maths back to real world issues, politics, and the universe at large, it can still feel a little dissonant for the younger pupils.

Don’t be afraid to stick with the simple stuff and the smaller aspects of the world.

Everybody needs to know how much change they’ll have left over after a bus ride home, everyone wants to know exactly how many chocolate bars they can gorge themselves on with two pounds, and everyone wants to know how many times they can go on the log-flume with five tickets.

Keeping it simple can be one of the most effective ways to engage pupils by showing them the mathematics they will employ in every-day life.

## 4. Cool factor for primary teachers – even in maths

Generally – note this is a generalisation – as a secondary teacher, one can spend eternity being uncool. Luckily primary school teachers get an easier rap, and KS2 pupils are willing to be ‘wowed’. All students can be ‘wowed’ under the right circumstances, but with younger, more malleable minds it can be somewhat easier.

Astronauts, magicians, superheroes, cute animals, cartoons, all carry the power of enthusiasm. They can be your secret weapon for making percentages fun – you’re not halving a number, you’re a magician halving a 167cm person in a special box, etc.

## 5. Make your maths investigations REALLY relevant

Relevance can be highly underrated when it comes to linking seemingly abstract topics to the real world.

One trick is to instead of distributing your problem solving activity sheets with strangers’ names and unrelatable allegories in the questions, why not make those names and allegories about your class.

Instead of a stranger gathering four apples and eating three, make your pupil.

Instead of apples, why not their favourite snack?

Letting students pick names in questions gives them greater agency in their learning and can be highly engaging.

Better still, putting their names/hobbies/likes/dislikes will not only show them that you know them, and that you care, but it will establish clear links between the work they are doing and the world proper.

Here’s our complete list of topical maths investigations for year 5 and year 6.

## Autumn Term maths investigations year 6 and year 5

- Autumn maths activities
- Halloween maths activities
- Bonfire Night maths activities
- Christmas maths activities

## Spring Term maths investigations year 6 and year 5

- Heart Month Months activities
- Shrove Tuesday Maths activities
- Pancake Day Maths activities
- World Book Day Maths activities
- International Women’s Day Maths activities
- British Science Week Maths activities
- Holi Maths activities
- Easter/Lent Maths activities

## Summer term maths investigations year 6 and year 5

- Share-a-Story Month activities
- FA Cup Maths activities
- Walk to School Week activities
- Ramadan Maths activities
- Child Safety Week activities

And if that’s not enough we’ve even got maths activities for Year 5 and Year 6 for events you’re likely to celebrate in primary school but don’t come round every year…

- Red Nose Day Maths activities
- World Cup Maths activities
- Election Maths
- Jubilee Maths activities

We update these blog posts every year so keep an eye on your calendar, and let us know how you get on @thirdspacetweet.

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Since 2013 these personalised one to 1 lessons have helped over 150,000 primary and secondary students become more confident, able mathematicians.

Learn how the programmes are aligned to maths mastery teaching or request a personalised quote for your school to speak to us about your school’s needs and how we can help.

## Related articles

Maths Problem Solving: Engaging Your Students And Strengthening Their Mathematical Skills

Free Year 7 Maths Test With Answers And Mark Scheme: Mixed Topic Questions

What Is A Number Square? Explained For Primary School Teachers, Parents & Pupils

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## FREE Guide to Maths Mastery

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Ideal for running staff meetings on mastery or sense checking your own approach to mastery.

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## I See Problem-Solving - UKS2

The ebook i see problem-solving - uks2 helps all children to learn how to solve multi-step maths questions. questions are broken down step-by-step and represented visually to build understanding, whilst extension tasks and reasoning prompts allow children to explore ideas at greater depth. the ultimate resource for teaching problem-solving in y5/6.

I See Problem-Solving - UKS2 Sample provides 5 free sample tasks. For each task there is a main question , a support prompt and explain and extend features to deepen the challenge. The worked examples model the solutions to the main tasks step-by-step - download below for free. The 58 tasks span all areas of the UKS2 maths curriculum. The purchase price is £30 (£25 + £5 VAT) for the PDF digital download.

I See Problem-Solving - UKS2: Worked Examples (PowerPoint)

I See Problem-Solving - UKS2: Worked Examples (PDF)

In these blogs I explain how the task prompts provide extra scaffold to give children additional help , and how extra layers of challenge can be added. Each task starts with a main question, like the examples below:

Children can use the support prompt if they need some extra help to answer the main question. The support prompt may give a suggested starting point, represent the question visually or address a possible misconception. It helps children to understand and access the task. Here are the support prompts for the above examples:

The explain features then provide an opportunity for children to reason based on the initial task. Prompts may ask children to explain a mistake, spot a pattern or make a link with a similar question. Children have to explain their opinion, deepening their understanding of the main question.

The extend feature gives a more challenging, related question for the children to solve. The mathematical structure of these tasks are similar to the initial prompt, usually with one added layer of complexity. Extend tasks may require children to make a generalisation, think laterally or find all possible answers to a question.

The 58 tasks of I See Problem-Solving - UKS2 cover all areas of the UKS2 curriculum. It corresponds to US grade 4&5 and Australian school years 5&6. The sale price is £30. When ordered, the PDF file is emailed direct to your inbox for your use.

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## Line Graph Problem Solving (Year 6)

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Children can consolidate their understanding of line graphs and apply it in a problem-solving context with this worksheet. They will interpret the data showing time and distance on the graph and use the information to answer the questions that follow.

Answers are included.

An equivalent foundation level worksheet is also available.

- Key Stage: Key Stage 2
- Subject: Maths
- Topic: Using Data
- Topic Group: Statistics
- Year(s): Year 6
- Media Type: PDF
- Resource Type: Worksheet
- Last Updated: 24/10/2023
- Resource Code: M2WAT395
- Curriculum Point(s): Interpret and construct pie charts and line graphs and use these to solve problems.

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## Learning to solve graph metric dimension problem based on graph contrastive learning

- Published: 15 November 2023
- Volume 53 , pages 30300–30318, ( 2023 )

## Cite this article

- Jian Wu ORCID: orcid.org/0000-0002-1706-2967 1 , 2 ,
- Li Wang 1 ,
- Weihua Yang 3 ,
- Haixia Zhao 4 ,
- Rui Wang 2 ,
- Jianji Cao 2 &
- Fuhong Wei 5

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Deep learning has been widely used to solve graph and combinatorial optimization problems. However, proper model deployment is critical for training a model and solving all problems. Existing frameworks mainly use reinforcement learning to learn to solve combinatorial optimization problems, in which a partial solution of the problem is regarded as an environmental state and each vertex of the corresponding graph is regarded as an action. As a result, using the sample data in model training effectively is challenging for different graphs. This study proposes a sampling-based, data-driven and distributed independent graph learning framework, based on decoupling graph structure learning and problem solving processes. To some extent, it facilitates industrial applications. Specifically, the framework consists of two independent parts: extracting graph structure and learning to solve the problem. Under this framework, the graph contrastive learning(GCL) is used to finish the graph structure learning process. Then by means of state-value aggregation on all of nodes in graphs, a global reinforcement learning method is established to learn to solve the graph problem, associated with repair policies to get improvement of performance. Experiments on synthetic graph datasets show that the graph contrastive learning is beneficial or has some advantages for training stability and improving the accuracy of solving the graph problem, and that the repair policies are stable for solution search. However, it also demonstrates that the graph neural network is not necessarily needed in the process of learning to solve the graph problem. Moreover, learning to solve MDP still has some challenges, such as decreasing learning performance with increasing edge existence probability of graphs, and it is unknown what kind of reward function is appropriate for solving MDP.

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Agasucci V, Grani G, Lamorgese L (2023) Solving the train dispatching problem via deep reinforcement learning. J Rail Trans Plan Manag 26:100394

Google Scholar

Apicella A, Isgrò F, Pollastro A et al (2023) Adaptive filters in Graph Convolutional Neural Networks. Patt Recognit 144:109867

Article Google Scholar

Chen T, Chen X, Chen W et al (2022) Learning to Optimize: A Primer and A Benchmark. J Mach Learn Res 23:1–59

MathSciNet Google Scholar

Dai H, Khalil E B, Zhang Y et al. (2017) Learning combinatorial optimization algorithms over graphs. Adv Neural Inf Process Syst

Eroh L, Kang CX, Yi E (2020) The connected metric dimension at a vertex of a graph. Theor Comput Sci 806:53–69

Article MathSciNet Google Scholar

Uno Fang, Li Jianxin Lu, Xuequan, et al (2023) Robust image clustering via context-aware contrastive graph learning. Patt Recognit 138:109340

Geneson J (2020) Metric dimension and pattern avoidance in graphs. Discret Appl Math 284:1–7

Hagberg A A, National L A, Alamos L et al. (2008) Exploring Network Structure, Dynamics, and Function using NetworkX. In Proceedings of the 7th python in science conference (SciPy2008). Gel Varoquaux, Travis Vaught, and Jarrod Millman (Eds), Pasadena, CA USA, pp 11–15

Hassani K, Khasahmadi AH (2020) Contrastive multi-view representation learning on graphs. In: Proceedings of the international conference on machine learning. PMLR, 4116–4126

Kallestad J, Hasibi R, Hemmati A et al (2023) A general deep reinforcement learning hyperheuristic framework for solving combinatorial optimization problems. Eur J Operat Res 309(1):446-468

Latifpour MH, Mills MS, Miri MA (2022) Combinatorial optimization with photonics-inspired clock models. Commun Phys 5:104

Kun Lei, Peng Guo, Yi Wang et al (2022) Solve routing problems with a residual edge-graph attention neural network. Neurocomputing 508:79–98

Liang H, Du X, Zhu B et al (2023) Graph contrastive learning with implicit augmentations. Neural Netw 163:156–164

Li Q, Chen W, Fang Z et al (2023) A multi-view contrastive learning for heterogeneous network embedding. Sci Rep 13:6732

Li S, Han L, Wang Y et al (2023) GCL: Contrastive learning instead of graph convolution for node classification. Neurocomputing 551:126491

Li W, Guo C, Liu Y et al (2023) Rumor source localization in social networks based on infection potential energy. Inf Sci 634:172–188

Ma F, Liu Z-M, Yang L et al (2021) Source localization in large-scale asynchronous sensor networks. Digit Signal Process 109:102920

Mazyavkina N, Sviridov S, Ivanov S, Burnaev E (2021) Reinforcement learning for combinatorial optimization: A survey. Comput Operat Res 134:105400

Danas MM (2023) The difference between several metric dimension graph invariants. Discret Appl Math 332:1–6

Mohseni N, McMahon PL, Byrnes T (2022) Ising machines as hardware solvers of combinatorial optimization problems. Nat Rev Phys 4:363–379

Nie KR, Xu KX (2023) Mixed metric dimension of some graphs. Appl Math Comput 442:127737

Padhye V, Lakshmanan K (2023) A deep actor critic reinforcement learning framework for learning to rank. Neurocomputing 547:126314

Pinto PC, Thiran P, Vetterli M (2012) Locating the Source of Diffusion in Large-Scale Networks. Phys Rev Lett 109(6):068702

Qin W, Zhuang Z, Huang Z, Huang H (2021) A novel reinforcement learning-based hyper-heuristic for heterogeneous vehicle routing problem. Comput Ind Eng 156:107252

Ribeiro LFR, Saverese PHP, Figueiredo DR (2017) struc2vec: Learning Node Representations from Structural Identity. In: Proceedings of the 23rd ACM SIGKDD International conference on knowledge discovery and data mining 385–394

Mashkaria S, Ódor G, Thiran P (2022) On the robustness of the metric dimension of grid graphs to adding a single edge. Discret Appl Math 316:1–27

Schulman J, Filip W, Dhariwal P et al. (2017) Proximal policy optimization algorithm. Mach Learn

Shen Y, Sun Y, Li X et al (2023) Adaptive solution prediction for combinatorial optimization. Eur J Operat Res 309(3):1392–1408

Staudt CL, Sazonovs A, Meyerhenke H (2015) NetworKit: A Tool Suite for Large-scale Complex Network Analysis. Netw Sci 2015

Tran VP, Garratt MA, Kasmarik K et al (2022) Multi-gas source localization and mapping by flocking robots. Inf Fusion 91:665–680

Wang H, Fu T, Du Y et al (2023) Scientific discovery in the age of artificial intelligence. Nature 620:47–60

Wang Q, Lai KH, Tang CL (2023) Solving combinatorial optimization problems over graphs with BERT-Based Deep Reinforcement Learning. Inf Sci 619:930–946

Wang Q, Tang C (2021) Deep reinforcement learning for transportation network combinatorial optimization: A survey. Knowl-Based Syst 233:107526

Wang Z, Sun C, Rui X et al (2021) Localization of multiple diffusion sources based on overlapping community detection. Knowl-Based Syst 226:106613

Wu L, Lin H, Gao Z et al (2023) Self-supervised Learning on Graphs: Contrastive, Generative, or Predictive. IEEE Trans Knowl Data Eng 35(1):857–876

Wu J, Zhao HX, Yang WH (2020) Computing Partition Metric Dimension of Graphs Based on Genetic Algorithm. Acta Math Appl Sin 43(6):1013–1028

Wu J, Wang L, Yang W (2022) Learning to compute the metric dimension of graphs. Appl Math Comput 432:127350

Wu ZH, Pan SR, Chen FW et al (2021) A Comprehensive Survey on Graph Neural Networks. IEEE Trans Neural Netw Learn Syst 32(1):4–24

Yan D, Weng J, Huang S et al (2022) Deep reinforcement learning with credit assignment for combinatorial ptimization. Patt Recognit 124:108466

You Y, Chen T, Sui Y et al. (2020) Graph Contrastive Learning with Augmentations. NeurIPS

Zhang Z, Sun S, Ma G et al (2023) Line graph contrastive learning for link prediction. Patt Recognit 140:109537

Zhao J, Cheong KH (2023) Early identification of diffusion source in complex networks with evidence theory. Inf Sci 642:119061

Zhu T, Shi X, Xu X, Cao J (2023) An accelerated end-to-end method for solving routing problems. Neural Netw 164:535–545

Zhu Y, Xu Y, Yu F et al. (2020) Deep Graph Contrastive Representation Learning. ICML

Zhu Y, Xu Y, Yu F, et al. (2021) Graph Contrastive Learning with Adaptive Augmentation. WWW ’21: Proceedings of the Web Conference 2021,2069–2080

hang Y, Bai R, Qu R, et al (2022) A deep reinforcement learning based hyper-heuristic for combinatorial optimisation with uncertainties. Eur J Operat Res 300(2):418–427

Zhu Y, Xu Y, Liu Q, et al. (2021) An Empirical Study of Graph Contrastive Learning. NeurlPS

Download references

## Acknowledgements

This work is supported by the Regional Innovation and Development Joint Fund of NSFC (No.U22A20167) and National key research and development program of China (No.2021YFB3300503); National Natural Science Foundation of China (Nos. 12102236); Natural Science Foundation of Shanxi Province (No.20210302124258; No.20210302123097; No.202103021224287); Philosophy and Social Science Planning Project of Shanxi Province(No. 2022YJ075). We thank the editors and reviewers for their valuable suggestions in revising this paper.

## Author information

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School of Computer Science and Technology (School of Big Data), Taiyuan University of Technology, No.209 University Street, Jinzhong, Shanxi Province, 030600, China

Jian Wu & Li Wang

School of Applied Mathematics, Shanxi University of Finance and Economics, No.140 Wucheng Street, Taiyuan, 030006, Shanxi Province, China

Jian Wu, Rui Wang & Jianji Cao

School of Mathematics, Taiyuan University of Technology, No.209 University Street, Jinzhong, Jingzhong, 030600, China

Weihua Yang

School of Statistic, Shanxi University of Finance and Economics, No.140 Wucheng Street, Taiyuan, Shanxi Province, 030006, China

Haixia Zhao

School of Science, Inner Mongolia University of Science and Technology, No.7 Aldean Street, Baotou, Inner Mongolia Province, 014010, China

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## Contributions

Jian Wu developed methodology, code, and algorithms to solve the problem and prepare the original draft. Li Wang and Weihua Yang offered essential advice for completing this work, revising the draft, and adjusting the critical idea. Rui Wang and Jianji Cao generated original data to perform experiments and run the baselines. Fuhong Wei reviewed this work and developed the methods. Haixia Zhao developed codes in the 10-fold cross-validation experiment.

## Corresponding authors

Correspondence to Li Wang or Weihua Yang .

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## Appendix A: model setup

1.1 a.1 gcn-nf model.

The GNN-NF model consist of one graph convolutional network(GCN) layer and a projection layer. Moreover, the projection layer is a two-layer perceptron. Specifically, the input, hidden and output dimensions of the GCN model are 128,64,64 respectively. The input, hidden and output dimensions of the projection layer are 64,64,1 respectively. The ReLU activation function is used in the model. In the output layer of actor network, the Sigmoid() activation function is adopted. And in the output layer of critic network, no activation function is used.

## 1.2 A.2 MLP-NF model

The MLP-NF model is a two-layer perceptron, of which the input, hidden and output dimensions are 128,64,1 respectively. The ReLU activation function is used in the model. In the output layer of actor network, the Sigmoid() activation function is adopted. On the contrary, no activation function is used in the critic network’s output layer.

## 1.3 A.3 GCL model

In the graph contrastive learning, one GCL model is constructed. And the model are composed by a two-layer GIN graph neural network and a two-layer perceptron. In GIN module, the input, hidden and output dimensions are 64,32,32 respectively. In perceptron module, the input, hidden and output dimensions are 64,64,64 respectively. The ReLU activation function is used in the model. The representations from this model vill be sent to the graph classifier.

In addition, in order to carry out the GCL learning, the graph level graph augmentations are adopted. Specifically, the data augmentations can be written as [ 47 ]:

\(aug1 \!=\! A.RandomChoice([A.RWSampling(num\_seeds\) \(=\!1000, walk\_length\!=\!10), A.NodeDropping(pn=\) \(0.1), A.FeatureMasking(pf\!=\!0.1), A.EdgeRemoving\) \((pe=0.1),],1)\) ;

\(aug2 \!=\! A.RandomChoice([A.RWSampling(num\_seeds\) \(=1000, walk\_length\!=\!10), A.NodeDropping(pn=\) \(0.1), A.FeatureMasking(pf\!=\!0.1), A.EdgeRemoving\) \((pe=0.1)],1)\) ,

where the PyGCL library is used to finish the graph augmentations.

## 1.4 A.4 GCN-F model

This model, named GCN sampler, is used to learn to solve MDP (metric dimension problem of graphs) on a set of graphs. The input of the GCN sampler is the node representations generated by GCL model. This sampler’s architecture is the same as GCN-NF model.

## 1.5 A.5 MLP-F model

This model, named MLP sampler, is used to learn to solve MDP (metric dimension problem of graphs) on a set of graphs. The input of the MLP sampler is the node representations generated by GCL model. This sampler’s architecture is the same as MLP-NF model.

## Appendix B: training and evaluation setup

The environment for running is: Intel(R) Core(TM) i7-8750H CPU @ 2.20GHz RAM 16 GB. The PyGCL library, Gurobi, Networkx, networkit and Pytorch software are used.

## 1.1 B.1 GCL training and evaluation

In the GCL training, Adam optimization algorithm is used, where the learning rate is 0.01, and the DualBranchContrast contrastive model with InfoNCE loss function is adopted [ 47 ]. In order to evaluate the quality of GCL learning, the node representations are sent to a graph classifier, where a graph pre-task of graph classification is given. In this paper, the performance of GCL learning is evaluated on test data by SVMEvaluator as a classifier in PyGCL library [ 47 ].

## 1.2 B.2 LS training and evaluation

In LS training process, the sampler is trained to learn to solve MDP on a set of graphs, named train data, and then it is evaluated on test data. Note that, we use PPO reinforcement learning, an unsupervised learning method, to train the sampler. Furthermore, we train the sampler in a manner of mini-batch with batch size 32.

For training of GCN-NF and MLP-NF samplers, the feature for each of the nodes in a graph is \({\textbf {1}}_{1\times d}\) vector with all of the entries are 1. On contrary, for training of GCN-F and MLP-F samplers, the node features in a graph are generated by the well-trained GCL model, named GNN encoder with its frozen parameters.

The Adam optimization algorithm is used to train the samplers, where the learning rate for actor and critic network is 0.01. The number of line search for samplers is 1. The parameter \(\lambda \) for computing the advantage function is selected in \(\{0.001,0.1,0.2,0.3,0.5\}\) . The parameter for clip function is selected in \(\{0.1,0.2\}\) . We run the training process 1000 episodes on each train data, with 50 as the early stop threshold value. In the reward function, parameters \(\alpha _1=2\) , \(\alpha _2=2\) , \(\beta =3\) . The discount factor for calculating the expected reward is 0.98.

In the process of evaluation, we use the sampler to solve MDP on all of the train data and test data. Then we report the performance of sampler on train data and test data respectively. Where the relative ratio on each of the graphs is computed. Then the average relative ratio is calculated on all of the datasets. Particularly, the number of iteration for add-repair policy and reduce-repair policy is 10.

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## About this article

Wu, J., Wang, L., Yang, W. et al. Learning to solve graph metric dimension problem based on graph contrastive learning. Appl Intell 53 , 30300–30318 (2023). https://doi.org/10.1007/s10489-023-05130-1

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Published : 15 November 2023

Issue Date : December 2023

DOI : https://doi.org/10.1007/s10489-023-05130-1

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