Three legendary games that changed PC gaming forever

Mass Effect Legendary Edition, the exceptional 4K remaster of the original trilogy that we said 'looks great' and 'absolutely flies on modern hardware', is now 90% off | PC Gamer

George Lucas’ prequel trilogy gave Star Wars a great gift, and Disney Plus’ Obi-Wan Kenobi and Ahsoka squandered it. Bring back the Jedi’s sick flips.

Star Wars has abandoned the prequel trilogy’s greatest gift: sick flips

The writer of The New York Trilogy, Leviathan and 4 3 2 1 – known for his stylised postmodernist fiction – has died from complications of lung cancer

Paul Auster, American author of The New York Trilogy, dies aged 77 | Books | The Guardian

US author Paul Auster, best known for writing the New York trilogy of novellas, has died at the age of 77.

The Bookseller - News - Paul Auster, author of the New York trilogy, dies

The writer was best known for The New York Trilogy and gained cult status in the 1980s and 90s.

Paul Auster: Author of The New York Trilogy dies aged 77 - BBC News

Paul Auster&#x27;s work was celebrated around the world and translated into dozens of languages. He worked on a typewriter and once described writing as a &quot;kind of illness&quot;.

Paul Auster, author of New York Trilogy and 4 3 2 1, dies aged 77 | Ents & Arts News | Sky News

TROOLOGY

troology

Startups Of The Year 2023: Asia Winners

189 Stories To Learn About Reading Books

Let's learn about via these 36 free stories. They are ordered by most time reading created on HackerNoon. Visit the to find the most read stories about any technology. Analysis /Learn Repo 1. ChatGPT’s Facebook-Like Rise: Don’t Get Hooked on Big Tech’s Bait Again GPT-4, arguably the most powerful AI model ever, has just been released. Does it matter? Probably not as much as ChatGPT, probably not for the reasons you think 2. How Data Analysis Helps Unveil the Truth of Coronavirus These days we are all scared of the new airborne contagious coronavirus (2019-nCoV). Even if it is a tiny cough or low fever, it might underlie a lethargic symptom. However, what is the real truth? 3. 5 Prominent Big Data Analytics Tools to Learn in 2020 Data, data and data. This seems to be what our world is swimming and immersing in. Why? The answer is simple: simply everything we use, such as mobile phones, and with it, all that it has, such as the social media, churn out unimaginable amounts of data. 4. What Is “Suckers List” And Why Texas Drivers Sued Its Creator A joint investigation by The Markup and Consumer Reports was cited in a lawsuit over discriminatory car insurance rates 5. How BTC Futures CME Manipulate the price of Bitcoin On December 15, two years ago, the largest American company  CME Group launched  Bitcoin futures .  
CME is the only one who provides the opportunity to trade bitcoin futures in the United States and they are the first to enter the territory of derivatives instrument for the cryptocurrency market and now they have no competitors. CME  futures are one of the most manipulative instruments on the cryptocurrency market. And the opening and closing of their futures suggests when the bottom will be. On December 15, 2017 two contracts were opened: 3 months until March 29: opening price $20,631 - closing price $7,070; 6 months until June 29: opening price $20,631 - closing price $5,800. 6. Using Artificial Intelligence and Big Data To Deliver Your Pitch A good pitch tells the story of your idea. From its inception to its present form and everything in between. Utilising multimedia, graphs and visuals is a good way to keep your audience engaged and up to speed. Most fundamentally, using data is important for both your audience and your idea. 7. Should i Build An Analyzer for C# or .NET? edit: Unlike the conclusion of the post below, I based the analyzer on C# CaaS and not .NET IL. Its now available to use https://devsnicket.com/eunice/#csharp. The latter doesn't preserve the order of members of a class either in the dll or pdb. Although, after looking at some open source C# projects, this capability wasn't used by default. 8. The Top Crypto Loan Services With The Lowest Interest Rates Ever since its tumultuous stint in 2018, the crypto market has been on the rise. Experts believe that this is due to the emergence of new and formidable financial applications that promise to change the way we access financial services. At the heart of this turnaround is the budding crypto lending sector currently pushing new paradigms in the DeFI and CeFI terrains. 9. Bitcoin Mining Still Requires Too Much Energy Bitcoin is far less popular now than it was in its heyday, when the world’s leading cryptocurrency was regularly dominating international headlines and leading newcomers into the burgeoning crypto-marketplace. After it caught on like wildfire amongst non-traditional investors, the Bitcoin hype also generated a slew of headlines and controversies surrounding the immense amount of energy that’s needed to mine - or generate - new tokens. These days, that controversy has largely died down, but many people still wonder if Bitcoin mining still presents a serious new global warming problem. 10. The Liquidity Crisis - Part One: BTC Bitcoin supply is becoming scarce in the face of demand - Here's what you need to know. 11. Analyzing Blockchain Business Models In the Blockchain Model, we want to understand how the protocol connects the value model to everything else. The rules of the Blockchain make it compelling. 12. What is Data Analytics and How It Can  Be Used WHAT IS DATA ANALYTICS? 13. Predict Customer Churn With Machine Learning, Data Science and Survival Analysis Predicting customer churn is very important because businesses have limited resources and cannot afford to lose customers if they want to stay profitable. 14. Choosing a Computing Method: a Serverless SWOT Analysis If you ever find yourself deciding for or against serverless the following tries to make the decision easier for you. 15. How To Use the Exit Waterfall Tool in the Cap Table In this article, we will learn about the exit waterfall tool in the cap table. 16. How to Conduct a Content Audit for Your Nonprofit Website A website is like a garden which needs to be maintained, pruned and groomed. 17. The Housing Shortage is the Next Target for Blockchain Development A player from the blockchain and cryptocurrency space takes on real estate development and the housing shortage. Let's take a look. 18. The Hunger Games As A Roadmap For Revolution Remember the time when the dystopian triology genre was taking the world by storm? 19. Crypto Casinos vs. Blockchain Casinos: Fairspin Attempt to Challenge Online Casinos with TruePlay For a few years, the growth of online casinos has been surprising. According to some projected estimates, the online gambling market is projected to reach $102.97 billion by 2025. The figures that are managed by this industry have been amazing. 20. Using Monte Carlo to Explain Why You Don't Win Daily Fantasy Baseball Games Use Monte Carlo simulation to understand the risk in fantasy baseball.  Learn why optimizing a lineup is a tall order. 21. Can Bitcoin Go Even Higher? An overview of bitocin's bull cycle using on-chain analysis. Can bitcoin go higher? Read more to find out. 22. Can BuildBox Challenge The Might Of The Unity Game Engine? As someone who has used both Buildbox and Unity to make games, I read with extreme interest a post by Venturebeat about the announcement of AppOnboard's new CEO, Mike Seavers. This could be one of the biggest wins for the company ever since Jonathan Zweig and AppOnboard acquired Buildbox last year. Zweig himself is an experienced entrepreneur, having co-founded AdColony in 2008 (sold to Opera for $350m in 2014) and co-founded AppOnboard in 2016. AppOnboard has raised $35m from investors, according to Crunchbase, and the new appointment could lead to new funding rounds when the time is right. This is going to lead to a very interesting David vs Goliath battle between AppOnboard's Buildbox and Unity Technologies. 23. How to Be Systematic When Testing Marketing Hypotheses The main thing for a successful marketing strategy is to keep track of it systematically: test hypotheses, learn from best practices, and constantly improve. 24. How-to: Website Benchmarks Time to first byte, first contentful paint, DNS response time, round-trip time, and the list goes on and on. With all of these metrics, how are you supposed to know which are the most important ones that you should monitor? 25. Prioritising Security Vulnerabilities with CVSS 3.1 [An Overview] With thousands of security vulnerabilities reported each month in products ranging from hardware devices to firmware to popular software apps, how does one prioritise what needs the most attention? From a business and project management perspective, it makes sense to, first and foremost, allocate engineering and/or risk assessment resources to the most severe vulnerabilities that need immediate patching. 26. Competitive Analysis Crash Course No business booms in isolation, neither does any fail. Every founder aiming to share with the world the amazing products they’ve built and solutions they provide must vie for customers with a dozen of other businesses and startups providing the same or similar products. Competitive Analysis gives businesses the unique ability to understand the ins and outs of how their competition works– it provides them with opportunities and inefficiencies in the aspect of competition they can take advantage of. Staying on top of industry trends in order to make sure your products are leading the competition is important and with well-done competitive analyses, business owners and managers can more easily identify opportunities to innovate, market their companies, improve their products or services, and stand out, despite the competition. 27. Best Libraries That Will Assist You In EDA: 2021 Edition Exploratory Data Analysis (EDA) is an essential step in the data science project lifecycle. Here are the top 10 python tools for EDA. 28. [Analysis] News Distribution of Telegram's GRAM Token Sale on Liquid Cryptocurrency Exchange On June 11, Techcrunch.com and the Liquid cryptocurrency exchange website published news that the sale of the alleged Telegram tokens will start on July 10. The newsworthy event has caught buzz within the cryptocurrency community, and based on it we have decided to explore the channels and the news flash sequence, namely which sites are most effective and what tools can be used to make the newsworthy event even more noticeable. 29. Convert Formatted Text Into a Data Structure Using Parsing Parsing is a process of converting formatted text into a data structure. A data structure type can be any suitable representation of the information engraved in the source text. 30. Make Software Great Again: Can Open Source be Ethical and Fair? Is there a way to go beyond open source, and have ethical, fair software in a cloud-first world? This is what some people in the open source community think. 31. Allstate's Car Insurance Algorithm: How Insurance Algorithm Squeezes Big Spenders Seven years ago, Allstate Corporation told Maryland regulators it was time to update its auto insurance rates. The insurer said its new, sophisticated risk analysis showed it was charging nearly all of its 93,000 Maryland customers outdated premiums. Some of the old rates were off by miles. One 36-year-old man from Prince George’s County, Md., who Allstate said in public records should have been paying $3,750 every six months, was instead being charged twice that, more than $7,500. Other customers were paying hundreds or thousands of dollars less than they should have been, based on Allstate’s new calculation of the risk that they would file a claim. 32. How to Surf the Dark Web safely and Monitor Traffic? The Onion Routing (Tor) act as a global gateway by enabling anonymity communication on the internet through various hidden channels network. [33. Here's How I Used The Golden Ratio To Make Winning Investments During The COVID19 Pandemic](https://hackernoon.com/heres-how-i-used-the-golden-ratio-to-make-winning-investments-during-the-covid19-pandemic-3n3v3zj0) When my organization decided to shift to work-from-home culture soon after coronavirus pandemic struck earlier this year, I've asked to work from home. I was happy to save a lot of time commuting to the office —and score some extra hours in each day to revisit and amplify my visual and UX Design skills. 34. How To Predict Water Pumps Failure in Tanzania using CatBoost Library It is based on the competition data. An example of data analysis,  insights from the data. The CatBoost library for the baseline model. High-score results. 35. Explaining Laravel Package Ecosystem Packages play a big role in the Laravel scene. How big and which packages?
To find it out we took the data from the Packagist.org API and filtered out the
Laravel packages. 36. Allstate's Car Insurance Algorithm: How Insurance Algorithm Was Analyzed State regulators and consumer advocacy groups have scrutinized Allstate Corporation’s use of big data and personalized pricing in the way it calculates how much the company charges its private auto insurance customers. Thank you for checking out the 36 most read stories about Analysis on HackerNoon. to find the most read stories about any technology. Visit the /Learn Repo

36 Stories To Learn About Analysis

Remember the time when the genre was taking the world by storm? dystopian triology The writers were churning out these books so fast that one can only wonder what were they even writing. But to be fair, the stories of all of them were pretty similar with different atmosphere. I mean, there is not much difference between Maze Runner triology and when you cut it to bare. Divergent Then there was series, and other triologies  that came and went. Few were amazing, like The Chaos Walking by Patrick Ness, but even the best of the best falls into the trap of the Matrix effect. The Selection Shatter Me The Matrix effect is when the first part of a triology is good, but the last two are mediocre. For right now, fantasy is leading (courtesy of ). Six of Crows However, among all the Dystopia, Hunger Games triology is most popular, and it should be because it is an excellent book. I would like to be bold here and say that the Hunger Games series is the reason for the popularity of the Dystopian Triologies. (When I say Dystopian Fiction, I mean Young Adult Fiction one, not 1984 type of books.) Hunger Games and Revolution The Hunger Games series is about many things, but the thing I want to discuss here is the revolution and the circle of revolution. - Pic When the first time  I read the Hunger Games, I read it as a book, like any other book. Later, like a few months later, I watched the movie, I find a scene in the book that I think of grave importance. After the game was over, the President Coin (self-made) asks the remaining Victors if they would like to begin the Hunger Games again. This time with the kids of the people of the Capitol, as the punishment. The same thing the Capitol was doing with the Districts. However, one could say that Civilians of the Capitol were not responsible for the yearly massacre. But they did not do anything to stop it, so in a way, they were behind the idea of it. When the time comes, District was angry because of the years and years of the oppression, and President Coin was right when she said that people need a way to release their anger. Pic Or else, what could be the alternative? The suffering and miseries of the District people would not let them live peacefully with the Capitol. Sooner or later a Civil War (if you call it a civil War) would begin, and the newly made country would be in flames. As much I do not want to be on anyone's side, but then I admit that people need to release their anger healthily. When people become slaves of technology and form a routine life, their wild side starts to cool down. But it never disappears. Humans are naturally wild animals, however years of modern civilisation may have trained as to be modest and discipline, but there will always remain a barbaric caveman inside of us. has shown that humans are still as violent as before. Studies We can see that in the Hunger Games. People were furious and sick of the ruler class. They wanted to be a ruler, to show that if they can be violent while slaving, they could do much more while ruling. When the Coin does the Voting for the 76th Hunger Games, the result was in support of Hunger Games. Well, we can assume it is all part of the plan of Katniss and Haymitch, but making plan was never the strong game of the Katniss Everdeen. So what her to go with the Hunger Games? May we will never know. We can only speculate her reasoning. The death of her father has impacted her a lot. She volunteered herself for Hunger Game to protect her sister, but she failed to protect her. She might be thinking herself as a failure, and she does not have anything left to loose. May be in that moment the hatred towards the everything was enough to make her such a wrong decision. But one thing was important here, there was supposed to be 76th Hunger Games. Victors said YES to it. The existence of idea is itself a strong suggestion that people wanted the revenge. It is like having a Second World War after the First World War, but, again, the second World War happened. It might be the most stupid and valiant affair of the human history. However, we cannot deny the existence of spiteful behaviour of people, more important, when they are hurting for a long time. As soon as the District came to the Powers, they wanted the revenge, the Hunger Games and massacre of the Capitol. Now rewind to 75 years back. What do we know about how the Hunger Games started?? All we know is the Capitol was oppressing the Districts. District revolted and a Civil War broke. It was the . First Rebellion There is too much history into it, but the basic is that Capitol won, with various means. However, District 13 somehow take the hold of Capitol's nuclear power and got his independence based on power. The other District also surrendered when District 13 left the fight. To teach the lesson to the 12 Districts, Capitol begins the Hunger Games. The idea was to show them that even the strongest of them is nothing in front of Capitol. After 75 years, the District catches fire and Revolution Begins. It all begins from the 74th Hunger Games when Katniss and Peeta try to swallow those berries and go directly against the order of the Capitol in front of the live audience. A light of hope, even the slight one burn brighter than the Dark. Katniss Everdeen and Peeta Mellarc gave them that hope, and they were not alone. A revolution is not an instantaneous event, it is a result of years of progress. It is itself a progress. The district started going against the Capitol one by one. In the Catching Fire, the series comes to the full glory and displays how symbols, tunes, and signals become the pathway to freedom. There is the story in English, . The Berlin forced France to teach the German instead of French in the school. The teacher in the story tells the student that this is the last lesson of the French class and they will not learn the French language anymore. The Last Lesson Until you have your language, you have your freedom. The Languages, words, and symbols united the people against the capitol. I think the biggest mistake of the Capitol was not controlling the language here. Though they try to fight off the tune, and the three-finger salute, but it does not work well when the masses started doing it. Coming back to the point, the first Hunger Games begins to teach the Districts a lesson for their revolution. But there is more to it. If you oppress the people, they will be angry. And you cannot keep a group of angry people together in one place peacefully. They will soon start fighting. The only way is to make them feel helpless and hopeless. Also, give them a distraction to vent out their anger. That was also the reason for the Hunger Games. The district would fight each other, be angry at each other, and then they will kill each other. While the Capitol would sponsor them and show them how tiny and unimportant they are. It will also keep them away from uniting. An excellent plan covering everything. So, what went wrong? The Civilians grew closer to the Victors, and the Capitol have to keep the Civilians happy. In the 74th Hunger Games, people see it live, that a pair from District 12 went against the Capitol and Capitol has to accept the display of live dissent. (Pressure of the Show Fans) It is how a revolution starts. The anger turns to hopelessness, and a little ray of Hope shows that there is a way to get out of it. People begin to gather. I am not declining the help of District 13 here. The district 13 came forward after looking at the Katniss's brave attempt of going against the Capitol. So, all in all, it was the hope of being successful that starts the revolution. As the saying is, the oppressor always digs his own grave. Now, from the hope, the early revolutionary begins to rise. Then the masses, and then even the people of Capitol helps the District. Later after the fight was won, and everything was as normal as possible, President Coin asks for voting for the 76th Hunger Games. At the moment she was thinking the same thing the Capitol thought at the time of first Hunger Games. A foolproof plan to keep everything in control. It is the best way to solve the civil war. A purposeful distraction that also gradually decrease the anger of Districts. This is the circle of revolution. If the Games begin, after the 75 year, the Capitol will revolt. The table will turn. We do not know what happen to the World. Do they have democracy, republic, or what?? But one thing is clear. If anyone has to look for a book that tells the story of a revolution in an amazing manner, Hunger Games is the series you should read. Hunger Games is a fucking case study is what it is.

The Hunger Games As A Roadmap For Revolution

Understanding the Generalization Performance of GNNs: Topology Awareness and Future Directions

Authors:
(1) Junwei Su, Department of Computer Science, the University of Hong Kong and jwsu@cs.hku.hk;
(2) Chuan Wu, Department of Computer Science, the University of Hong Kong and cwu@cs.hku.hk. Table of Links Abstract and 1 Introduction 2 Related Work 3 Framework 4 Main Results 5 A Case Study on Shortest-Path Distance 6 Conclusion and Discussion, and References 7 Proof of Theorem 1 8 Proof of Theorem 2 9 Procedure for Solving Eq. (6) 10 Additional Experiments Details and Results 11 Other Potential Applications 4 Main Results In this section, we present the main results derived from our proposed framework for studying the generalization and fairness of GNNs with respect to different structural groups in a graph. We start by demonstrating the relationship between the generalization performance of the model and the structural distance between the subgroups and the training set (Theorem 1). We then show that such a structural relation can induce disparities in the generalization performance across different subgroups (Theorem 2). Finally, we connect these theoretical results to the recent deep learning spline line theory [2] to provide additional insights. Detailed proof of the results presented in this paper can be found in the supplementary material. 4.1 Generalization Performance and Structural Distance Theorem 1 states that the generalization risk of a classifier with respect to a structural subgroup depends on the empirical training loss of the classifier, the structural distance between the subgroup and the training set, and the distortion (topology awareness) of GNNs. Such a relation provides a new perspective on understanding the link between topology awareness and generalization performance in GNNs. Prior research in this area has mainly focused on the connection between GNNs and the Weisfeiler-Lehman algorithm, emphasizing the ability of GNNs to differentiate various graph structures. However, Theorem 1 highlights a direct link between topology awareness and generalization performance, and a potential issue of fairness in GNNs, as it shows that the model’s performance on different structural subgroups is influenced by their distance to the training set. Next, we extend the result of Theorem 1 to analyze the relation between the generalization of different structural subgroups and their distance to the training set. 4.2 Unfair Generalization Performance on Structural Subgroups Theorem 2 states that when comparing two subgroups, if the difference in their structural distance to the training set is larger than the threshold (δα), then GNNs exhibit unfair generalization performance with respect to these groups. Specifically, the theorem states that GNNs generalize better on subgroups that are structurally closer to the training set, which leads to accuracy disparities among different subgroups. Additionally, the theorem notes that the topology awareness of GNNs (measured by the distortion rate α) plays a key role in determining the strength of this accuracy disparity. This has important implications for the use of GNNs in critical applications, as it suggests that increased topology awareness in GNNs can lead to improved generalization performance but also a greater potential for unfairness in terms of accuracy disparities among different structural subgroups. As such, it is important to carefully consider the trade-offs between topology awareness and fairness when designing new GNNs and applying them to real-world problems. 4.3 Connection with Deep Learning Spline Theory Our theoretical findings reveal that when considering the performance of the model from the viewpoint of the embedding space, the training set creates a “valley” effect on generalization performance, with the training vertex at the bottom of the valley. Representations that are closer to the embedding of the training set perform better, while those farther away perform worse. This pattern has an interesting connection with deep learning spline theory [2]. This theory suggests that in DNNs, the learning process for classification divides the input/latent space into small circles (referred to as power diagrams), where representations within each circle are grouped, and representations closer to the centre of the circle are predicted with more confidence. Our findings suggest that in GNNs, the mapping of test data to these power diagrams is related to the structural similarity between test and training data in the graph. This connection provides further validation for our derived results. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Junwei Su, Department of Computer Science, the University of Hong Kong and jwsu@cs.hku.hk; (2) Chuan Wu, Department of Computer Science, the University of Hong Kong and cwu@cs.hku.hk. Authors: Authors: (1) Junwei Su, Department of Computer Science, the University of Hong Kong and jwsu@cs.hku.hk; (2) Chuan Wu, Department of Computer Science, the University of Hong Kong and cwu@cs.hku.hk. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Related Work 2 Related Work 3 Framework 3 Framework 4 Main Results 4 Main Results 5 A Case Study on Shortest-Path Distance 5 A Case Study on Shortest-Path Distance 6 Conclusion and Discussion, and References 6 Conclusion and Discussion, and References 7 Proof of Theorem 1 7 Proof of Theorem 1 8 Proof of Theorem 2 8 Proof of Theorem 2 9 Procedure for Solving Eq. (6) 9 Procedure for Solving Eq. (6) 10 Additional Experiments Details and Results 10 Additional Experiments Details and Results 11 Other Potential Applications 11 Other Potential Applications 4 Main Results In this section, we present the main results derived from our proposed framework for studying the generalization and fairness of GNNs with respect to different structural groups in a graph. We start by demonstrating the relationship between the generalization performance of the model and the structural distance between the subgroups and the training set (Theorem 1). We then show that such a structural relation can induce disparities in the generalization performance across different subgroups (Theorem 2). Finally, we connect these theoretical results to the recent deep learning spline line theory [2] to provide additional insights. Detailed proof of the results presented in this paper can be found in the supplementary material. 4.1 Generalization Performance and Structural Distance Theorem 1 states that the generalization risk of a classifier with respect to a structural subgroup depends on the empirical training loss of the classifier, the structural distance between the subgroup and the training set, and the distortion (topology awareness) of GNNs. Such a relation provides a new perspective on understanding the link between topology awareness and generalization performance in GNNs. Prior research in this area has mainly focused on the connection between GNNs and the Weisfeiler-Lehman algorithm, emphasizing the ability of GNNs to differentiate various graph structures. However, Theorem 1 highlights a direct link between topology awareness and generalization performance, and a potential issue of fairness in GNNs, as it shows that the model’s performance on different structural subgroups is influenced by their distance to the training set. Next, we extend the result of Theorem 1 to analyze the relation between the generalization of different structural subgroups and their distance to the training set. 4.2 Unfair Generalization Performance on Structural Subgroups Theorem 2 states that when comparing two subgroups, if the difference in their structural distance to the training set is larger than the threshold (δα), then GNNs exhibit unfair generalization performance with respect to these groups. Specifically, the theorem states that GNNs generalize better on subgroups that are structurally closer to the training set, which leads to accuracy disparities among different subgroups. Additionally, the theorem notes that the topology awareness of GNNs (measured by the distortion rate α) plays a key role in determining the strength of this accuracy disparity. This has important implications for the use of GNNs in critical applications, as it suggests that increased topology awareness in GNNs can lead to improved generalization performance but also a greater potential for unfairness in terms of accuracy disparities among different structural subgroups. As such, it is important to carefully consider the trade-offs between topology awareness and fairness when designing new GNNs and applying them to real-world problems. 4.3 Connection with Deep Learning Spline Theory Our theoretical findings reveal that when considering the performance of the model from the viewpoint of the embedding space, the training set creates a “valley” effect on generalization performance, with the training vertex at the bottom of the valley. Representations that are closer to the embedding of the training set perform better, while those farther away perform worse. This pattern has an interesting connection with deep learning spline theory [2]. This theory suggests that in DNNs, the learning process for classification divides the input/latent space into small circles (referred to as power diagrams), where representations within each circle are grouped, and representations closer to the centre of the circle are predicted with more confidence. Our findings suggest that in GNNs, the mapping of test data to these power diagrams is related to the structural similarity between test and training data in the graph. This connection provides further validation for our derived results. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

On the Topology Awareness and Generalization Performance of Graph Neural Networks: Main Results

Authors:
(1) Junwei Su, Department of Computer Science, the University of Hong Kong and jwsu@cs.hku.hk;
(2) Chuan Wu, Department of Computer Science, the University of Hong Kong and cwu@cs.hku.hk. Table of Links Abstract and 1 Introduction 2 Related Work 3 Framework 4 Main Results 5 A Case Study on Shortest-Path Distance 6 Conclusion and Discussion, and References 7 Proof of Theorem 1 8 Proof of Theorem 2 9 Procedure for Solving Eq. (6) 10 Additional Experiments Details and Results 11 Other Potential Applications 3 Framework In this section, we introduce our proposed framework, which aims to investigate the intricate relationship between topology awareness and generalization performance in GNNs. The main idea of the framework is to use metric distortion to characterize the topology awareness of GNNs. With this characterization, we are able to show the relation between topology awareness of GNNs and their generalization performance. 3.1 GNN Preliminaries Let G = (V, E, X) be an undirected graph where V = {1, 2, ..., n} is the set of n nodes and E ⊆ V × V is the set of edges. Let X = {x1, x2, ..., xn} denote the set of node attribute vectors corresponding to all the nodes in V. More specifically, xv ∈ X is an m-dimensional vector which captures the attribute values of node v ∈ V. In addition, each vertex v ∈ V is associated with a node label yv ∈ {1, ..., c}, where c is the total number of classes. For our analysis, we focus on the transductive node classification setting [47], where the graph G are observed prior to learning and labels are only available for a small subset of nodes V0 ⊂ V. Without loss of generality, we treat G as fixed throughout our analysis and the randomness comes from the labels y. Given a small set of the labelled nodes, V0 ⊂ V, the task of node-level semi-supervised learning is to learn a classifier f, such that yˆv = f(v), from a hypothesis space F and perform it on the remaining unlabeled nodes. Let L(., .) 7→ R+ be a bounded continuous loss function. We denote by RL(f, V) the generalization risk of the classifier f with respect to L and V and it is defined as follows, In our case, the hypothesis space is captured by a GNN and an additional read-out function. GNNs are commonly used as an embedding function and adopt a neighbourhood aggregation scheme, where the representation of a node is learnt iteratively by aggregating representations of its neighbours. After k iterations of aggregation, a node’s representation captures the information within its k-hop neighbourhood in the graph. Formally, the k-th layer of a GNN can be written as follows, It should be noted that the distance function in Def. 2 is not symmetrical, and Eq. 4 captures the maximal structural difference of vertices in V1 to V2. To provide further insight into the concept of a structural subgroup, let’s consider a scenario where V0 ⊂ V represents the training set and ds(.) represents the shortest-path distance. In this context, we can partition the remaining vertices V\V0 into different structural subgroups based on their shortest-path distance to the nearest vertex in the training set. Specifically, we can define Vk = {v ∈ V\V0|Ds(v, V0) = k}. However, it is important to emphasize that our analysis does not assume a specific partition scheme, and the framework remains adaptable to various scenarios. 3.3 Topology Awareness To characterize and formalize the topology awareness of GNNs, we introduce the concept of distortion, which measures the degree to which the graph structure is preserved in the embedding space. Mathematically, the distortion of a function between two general metric spaces is defined as, The definition above captures the maximum expansion and minimum contraction between the two metric spaces. In the extreme case when α = 1, the mapping γ is the so-called isometric mapping that perfectly preserves the distance between any two points in the space. In other words, for any two points x and y in the domain of the function, their distance d(x, y) in the original metric space is equal to the distance d ′ (γ(x), γ(y)) in the target metric space (up to a constant scaling factor). If the distortion rate of the mapping is close to 1, then this mapping is said to have a low distortion rate. In the context of GNNs, evaluating distortion involves a straightforward comparison of embeddings in the embedding domain with the structural measures on the graph, as demonstrated in the case study. If the distortion rate between the graph domain and the embedding domain is close to 1, the resulting embedding would largely preserve the corresponding structural information from the graph space, thereby indicating high topology awareness. Consequently, distortion can be employed to measure the topology awareness of GNNs concerning the corresponding structural properties (e.g., shortest-path distance) measured in the graph domain. Consequently, distortion can be employed to measure the topology awareness of GNNs concerning the corresponding structural properties (e.g., shortest-path distance) measured in the graph domain. This concept of distortion is agnostic to specific structural configurations, enabling the study of diverse graph structures using appropriate distance functions. Moreover, its estimation can be readily achieved by comparing the embedding with the original structural measure, as illustrated later in the case study This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Junwei Su, Department of Computer Science, the University of Hong Kong and jwsu@cs.hku.hk; (2) Chuan Wu, Department of Computer Science, the University of Hong Kong and cwu@cs.hku.hk. Authors: Authors: (1) Junwei Su, Department of Computer Science, the University of Hong Kong and jwsu@cs.hku.hk; (2) Chuan Wu, Department of Computer Science, the University of Hong Kong and cwu@cs.hku.hk. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Related Work 2 Related Work 3 Framework 3 Framework 4 Main Results 4 Main Results 5 A Case Study on Shortest-Path Distance 5 A Case Study on Shortest-Path Distance 6 Conclusion and Discussion, and References 6 Conclusion and Discussion, and References 7 Proof of Theorem 1 7 Proof of Theorem 1 8 Proof of Theorem 2 8 Proof of Theorem 2 9 Procedure for Solving Eq. (6) 9 Procedure for Solving Eq. (6) 10 Additional Experiments Details and Results 10 Additional Experiments Details and Results 11 Other Potential Applications 11 Other Potential Applications 3 Framework In this section, we introduce our proposed framework, which aims to investigate the intricate relationship between topology awareness and generalization performance in GNNs. The main idea of the framework is to use metric distortion to characterize the topology awareness of GNNs. With this characterization, we are able to show the relation between topology awareness of GNNs and their generalization performance. 3.1 GNN Preliminaries Let G = (V, E, X) be an undirected graph where V = {1, 2, ..., n} is the set of n nodes and E ⊆ V × V is the set of edges. Let X = {x1, x2, ..., xn} denote the set of node attribute vectors corresponding to all the nodes in V. More specifically, xv ∈ X is an m-dimensional vector which captures the attribute values of node v ∈ V. In addition, each vertex v ∈ V is associated with a node label yv ∈ {1, ..., c}, where c is the total number of classes. For our analysis, we focus on the transductive node classification setting [47], where the graph G are observed prior to learning and labels are only available for a small subset of nodes V0 ⊂ V. Without loss of generality, we treat G as fixed throughout our analysis and the randomness comes from the labels y. Given a small set of the labelled nodes, V0 ⊂ V, the task of node-level semi-supervised learning is to learn a classifier f, such that yˆv = f(v), from a hypothesis space F and perform it on the remaining unlabeled nodes. Let L(., .) 7→ R+ be a bounded continuous loss function. We denote by RL(f, V) the generalization risk of the classifier f with respect to L and V and it is defined as follows, In our case, the hypothesis space is captured by a GNN and an additional read-out function. GNNs are commonly used as an embedding function and adopt a neighbourhood aggregation scheme, where the representation of a node is learnt iteratively by aggregating representations of its neighbours. After k iterations of aggregation, a node’s representation captures the information within its k-hop neighbourhood in the graph. Formally, the k-th layer of a GNN can be written as follows, It should be noted that the distance function in Def. 2 is not symmetrical, and Eq. 4 captures the maximal structural difference of vertices in V1 to V2. To provide further insight into the concept of a structural subgroup, let’s consider a scenario where V0 ⊂ V represents the training set and ds(.) represents the shortest-path distance. In this context, we can partition the remaining vertices V\V0 into different structural subgroups based on their shortest-path distance to the nearest vertex in the training set. Specifically, we can define Vk = {v ∈ V\V0|Ds(v, V0) = k}. However, it is important to emphasize that our analysis does not assume a specific partition scheme, and the framework remains adaptable to various scenarios. 3.3 Topology Awareness To characterize and formalize the topology awareness of GNNs, we introduce the concept of distortion, which measures the degree to which the graph structure is preserved in the embedding space. Mathematically, the distortion of a function between two general metric spaces is defined as, The definition above captures the maximum expansion and minimum contraction between the two metric spaces. In the extreme case when α = 1, the mapping γ is the so-called isometric mapping that perfectly preserves the distance between any two points in the space. In other words, for any two points x and y in the domain of the function, their distance d(x, y) in the original metric space is equal to the distance d ′ (γ(x), γ(y)) in the target metric space (up to a constant scaling factor). If the distortion rate of the mapping is close to 1, then this mapping is said to have a low distortion rate. In the context of GNNs, evaluating distortion involves a straightforward comparison of embeddings in the embedding domain with the structural measures on the graph, as demonstrated in the case study. If the distortion rate between the graph domain and the embedding domain is close to 1, the resulting embedding would largely preserve the corresponding structural information from the graph space, thereby indicating high topology awareness. Consequently, distortion can be employed to measure the topology awareness of GNNs concerning the corresponding structural properties (e.g., shortest-path distance) measured in the graph domain. Consequently, distortion can be employed to measure the topology awareness of GNNs concerning the corresponding structural properties (e.g., shortest-path distance) measured in the graph domain. This concept of distortion is agnostic to specific structural configurations, enabling the study of diverse graph structures using appropriate distance functions. Moreover, its estimation can be readily achieved by comparing the embedding with the original structural measure, as illustrated later in the case study This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Framework for Analyzing Topology Awareness and Generalization in Graph Neural Networks

Authors:
(1) Junwei Su, Department of Computer Science, the University of Hong Kong and jwsu@cs.hku.hk;
(2) Chuan Wu, Department of Computer Science, the University of Hong Kong and cwu@cs.hku.hk. Table of Links Abstract and 1 Introduction 2 Related Work 3 Framework 4 Main Results 5 A Case Study on Shortest-Path Distance 6 Conclusion and Discussion, and References 7 Proof of Theorem 1 8 Proof of Theorem 2 9 Procedure for Solving Eq. (6) 10 Additional Experiments Details and Results 11 Other Potential Applications 2 Related Work 2.1 Topology Awareness of GNNs Modern GNN architectures take inspiration from the Weisfeiler-Lehman isomorphism test [12,19], which employs the graph structure to propagate information. Consequently, a significant portion of existing literature on GNNs concentrates on understanding their ability to differentiate various graph structures, known as the expressive power of GNNs [28–30, 44, 48, 50]. Inspired by the expressiveness studies, it is commonly believed that increasing topology awareness is universally beneficial and many studies focus on enabling GNNs to preserve more structural properties in the learned representation [29, 33, 48]. Nevertheless, the precise ramifications of heightened topology awareness on GNNs’ generalization performance remain shrouded in ambiguity. The intricate variations within structures across different domains and tasks further compound this complexity, underscoring the necessity for a unified framework—akin to the one proposed in this paper—to shed light on these intricacies. 2.2 Generalization of GNNs Relatively few studies focus on understanding the generalization performance of GNNs in node-level tasks. [3, 9, 18, 23, 25, 34, 39, 41, 45] extend the conventional statistical learning or information-theoretic framework to GNNs, providing different generalization bounds based on concepts such as Vapnik–Chervonenkis dimension, mutual information, and algorithm stability. In light of this, despite empirical indications highlighting the critical role of GNNs’ topology awareness in generalization performance [20,29,48], there remains limited understanding of the relationship between topology awareness of GNNs and their generalization. Our proposed framework addresses this gap by utilizing the distortion concept in the approximate metric embedding to connect the topology awareness of GNNs and their generalization performance. 2.3 Active Learning in Graph Active Learning (AL) seeks to enhance model performance within a limited labeling budget by identifying the most representative samples for annotation [32]. The AL methodology generally involves two phases: 1) selecting a small, preliminary subset of samples for labeling; 2) utilizing this labeled subset to iteratively discover the most informative sample for future labeling. The initial selection of an effective subset represents the cold-start problem in AL [11]. While AL has proven effective in settings with independent and identically distributed (i.i.d.) data, such as images and text, applying it to Graph Neural Networks (GNNs) poses unique challenges. These challenges stem from the necessity to account for node interdependencies and the computational complexity of iterative sample selection and retraining, especially when considering multi-hop neighbors. Recent AL methodologies for GNNs, such as AGE [5], FeatProp [42], and GraphPart [26], aim to overcome these obstacles by leveraging the graph structure and initial node features to boost efficiency and efficacy, although their success may significantly hinge on the quality of the initial features. In this work, we focus on the cold-start problem of graph active learning and introduce a novel approach (based on our analysis) that relies exclusively on graph structure, thus eliminating the reliance on feature quality at the onset of learning. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Junwei Su, Department of Computer Science, the University of Hong Kong and jwsu@cs.hku.hk; (2) Chuan Wu, Department of Computer Science, the University of Hong Kong and cwu@cs.hku.hk. Authors: Authors: (1) Junwei Su, Department of Computer Science, the University of Hong Kong and jwsu@cs.hku.hk; (2) Chuan Wu, Department of Computer Science, the University of Hong Kong and cwu@cs.hku.hk. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Related Work 2 Related Work 3 Framework 3 Framework 4 Main Results 4 Main Results 5 A Case Study on Shortest-Path Distance 5 A Case Study on Shortest-Path Distance 6 Conclusion and Discussion, and References 6 Conclusion and Discussion, and References 7 Proof of Theorem 1 7 Proof of Theorem 1 8 Proof of Theorem 2 8 Proof of Theorem 2 9 Procedure for Solving Eq. (6) 9 Procedure for Solving Eq. (6) 10 Additional Experiments Details and Results 10 Additional Experiments Details and Results 11 Other Potential Applications 11 Other Potential Applications 2 Related Work 2.1 Topology Awareness of GNNs Modern GNN architectures take inspiration from the Weisfeiler-Lehman isomorphism test [12,19], which employs the graph structure to propagate information. Consequently, a significant portion of existing literature on GNNs concentrates on understanding their ability to differentiate various graph structures, known as the expressive power of GNNs [28–30, 44, 48, 50]. Inspired by the expressiveness studies, it is commonly believed that increasing topology awareness is universally beneficial and many studies focus on enabling GNNs to preserve more structural properties in the learned representation [29, 33, 48]. Nevertheless, the precise ramifications of heightened topology awareness on GNNs’ generalization performance remain shrouded in ambiguity. The intricate variations within structures across different domains and tasks further compound this complexity, underscoring the necessity for a unified framework—akin to the one proposed in this paper—to shed light on these intricacies. 2.2 Generalization of GNNs Relatively few studies focus on understanding the generalization performance of GNNs in node-level tasks. [3, 9, 18, 23, 25, 34, 39, 41, 45] extend the conventional statistical learning or information-theoretic framework to GNNs, providing different generalization bounds based on concepts such as Vapnik–Chervonenkis dimension, mutual information, and algorithm stability. In light of this, despite empirical indications highlighting the critical role of GNNs’ topology awareness in generalization performance [20,29,48], there remains limited understanding of the relationship between topology awareness of GNNs and their generalization. Our proposed framework addresses this gap by utilizing the distortion concept in the approximate metric embedding to connect the topology awareness of GNNs and their generalization performance. 2.3 Active Learning in Graph Active Learning (AL) seeks to enhance model performance within a limited labeling budget by identifying the most representative samples for annotation [32]. The AL methodology generally involves two phases: 1) selecting a small, preliminary subset of samples for labeling; 2) utilizing this labeled subset to iteratively discover the most informative sample for future labeling. The initial selection of an effective subset represents the cold-start problem in AL [11]. While AL has proven effective in settings with independent and identically distributed (i.i.d.) data, such as images and text, applying it to Graph Neural Networks (GNNs) poses unique challenges. These challenges stem from the necessity to account for node interdependencies and the computational complexity of iterative sample selection and retraining, especially when considering multi-hop neighbors. Recent AL methodologies for GNNs, such as AGE [5], FeatProp [42], and GraphPart [26], aim to overcome these obstacles by leveraging the graph structure and initial node features to boost efficiency and efficacy, although their success may significantly hinge on the quality of the initial features. In this work, we focus on the cold-start problem of graph active learning and introduce a novel approach (based on our analysis) that relies exclusively on graph structure, thus eliminating the reliance on feature quality at the onset of learning. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Exploring Topology Awareness, Generalization, and Active Learning in Graph Neural Networks

Authors:
(1) Junwei Su, Department of Computer Science, the University of Hong Kong and jwsu@cs.hku.hk;
(2) Chuan Wu, Department of Computer Science, the University of Hong Kong and cwu@cs.hku.hk. Table of Links Abstract and 1 Introduction 2 Related Work 3 Framework 4 Main Results 5 A Case Study on Shortest-Path Distance 6 Conclusion and Discussion, and References 7 Proof of Theorem 1 8 Proof of Theorem 2 9 Procedure for Solving Eq. (6) 10 Additional Experiments Details and Results 11 Other Potential Applications Abstract Many computer vision and machine learning problems are modelled as learning tasks on graphs, where graph neural networks (GNNs) have emerged as a dominant tool for learning representations of graphstructured data. A key feature of GNNs is their use of graph structures as input, enabling them to exploit the graphs’ inherent topological properties—known as the topology awareness of GNNs. Despite the empirical successes of GNNs, the influence of topology awareness on generalization performance remains unexplored, particularly for node-level tasks that diverge from the assumption of data being independent and identically distributed (I.I.D.). The precise definition and characterization of the topology awareness of GNNs, especially concerning different topological features, are still unclear. This paper introduces a comprehensive framework to characterize the topology awareness of GNNs across any topological feature. Using this framework, we investigate the effects of topology awareness on GNN generalization performance. Contrary to the prevailing belief that enhancing the topology awareness of GNNs is always advantageous, our analysis reveals a critical insight: improving the topology awareness of GNNs may inadvertently lead to unfair generalization across structural groups, which might not be desired in some scenarios. Additionally, we conduct a case study using the intrinsic graph metric, the shortest-path distance, on various benchmark datasets. The empirical results of this case study confirm our theoretical insights. Moreover, we demonstrate the practical applicability of our framework by using it to tackle the cold start problem in graph active learning. 1 Introduction Many problems in computer vision and machine learning are modeled as learning tasks on graphs. For example, in semantic segmentation, graphs model the relationships between different image regions, enhancing accuracy and context-aware segmentation. Graph neural networks (GNNs) have emerged as a dominant class of machine learning models specifically designed for learning representations of graph-structured data. They have demonstrated considerable success in addressing a wide range of graph-related problems in various domains such as chemistry [10], biology [37], social networking [6, 22], scene graph generation [46, 51] and visual relationship detection [24,43,49]. A defining characteristic of GNNs is their use of a spatial approach through message passing on the graph structure for feature aggregation. This enables GNNs to preserve structural information or dependencies (referred to as topology awareness) from the underlying graph structure, allowing them to be highly effective in tasks such as node classification. Fig. 1 illustrates the overall learning process of GNNs. Despite their practicality and potential, there remains a lack of theoretical understanding about GNNs, particularly in the semi-supervised node classification setting where the dependencies among the data differ significantly from other machine learning models [25]. In this setting, the goal is to leverage relations, as captured by the graph structure, among the data and a small set of labelled nodes to predict labels for the remaining nodes. Most of the existing theoretical studies of GNNs have focused on the connection between the message-passing mechanism of GNNs and the Weisfeiler-Lehman isomorphism test [19], aiming to understand the ability of GNNs to differentiate different graph structures in the learned representations, known as the expressive power of GNNs. Inspired by the expressiveness studies, it is commonly believed that increasing topology awareness is universally beneficial and many studies focus on enabling GNNs to preserve more structural properties in the learned representation [29, 33, 48]. However, as GNNs become more reliant on and sensitive (aware) of graph structure as input, they may exhibit different generalization performances towards certain structural subgroups (distinct data subsets grouped by structural similarity to the training set) within the data. The quantification of GNN generalization across distinct structural subgroups is termed structural subgroup generalization [25]. Such considerations are vital in GNN application and development. For instance, within protein-protein interaction networks, these structural subgroups could represent different molecular complexes, influencing the accuracy of interaction predictions. Similarly, understanding how the topology awareness of GNNs influences generalization is essential when devising sampling strategies for training. The extent to which the generalization performance of GNNs is influenced by specific structural features of graph data is critical in deciding the composition of training datasets. Despite its importance, an understanding of the relationship between the topology awareness of GNNs and its structural subgroup generalization is still lacking. Furthermore, characterizing the topology awareness of GNNs poses a challenge, especially considering that different domains and tasks may prioritize distinct structural aspects. Therefore, a versatile framework is needed to assess the topology awareness of GNNs in relation to various structures. To address this gap, in this paper, we propose a novel framework based on approximate metric embedding to study the relationship between structural subgroup generalization and topology awareness of GNNs in the context of semisupervised node classification. The proposed framework allows for the investigation of the structural subgroup generalization of GNNs with respect to different structural subgroups. More concretely, the main contributions of this work are summarized as follows. 1. We propose a novel, structure-agnostic framework using approximate metric embedding to examine the interplay between GNNs’ structural subgroup generalization and topology awareness. This framework is versatile, accommodating various structural measures like shortest-path distance, and requires only the corresponding structural measure. Its simplicity in estimating key factors makes it applicable and generalizable to a wide range of scenarios. 2. Through formal analysis within our framework, we establish a clear link between GNN topology awareness and their generalization performance (Theorem 1). We also demonstrate that while enhanced topology awareness boosts GNN expressiveness, it can result in uneven generalization performance, favouring subgroups more structurally similar to the training set (Theorem 2). Such structural property can be harmful (causing unfairness issues) or useful (informing design decisions) depending on the scenario. This challenges the prevailing belief that increased topology awareness universally benefits GNNs [29, 33, 48], emphasizing the importance of considering the relationship between topology awareness and generalization performance. 3. We validate our framework through a case study on shortest-path distance, highlighting its practicality and relevance. The results corroborate our theoretical findings, showing that GNNs with heightened awareness of shortestpath distances excel in classifying vertex groups closer to the training set. Moreover, we demonstrate how our findings can be applied to mitigate the cold start problem in graph active learning [11,15], highlighting the practical implications of our framework and results. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Junwei Su, Department of Computer Science, the University of Hong Kong and jwsu@cs.hku.hk; (2) Chuan Wu, Department of Computer Science, the University of Hong Kong and cwu@cs.hku.hk. Authors: Authors: (1) Junwei Su, Department of Computer Science, the University of Hong Kong and jwsu@cs.hku.hk; (2) Chuan Wu, Department of Computer Science, the University of Hong Kong and cwu@cs.hku.hk. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Related Work 2 Related Work 3 Framework 3 Framework 4 Main Results 4 Main Results 5 A Case Study on Shortest-Path Distance 5 A Case Study on Shortest-Path Distance 6 Conclusion and Discussion, and References 6 Conclusion and Discussion, and References 7 Proof of Theorem 1 7 Proof of Theorem 1 8 Proof of Theorem 2 8 Proof of Theorem 2 9 Procedure for Solving Eq. (6) 9 Procedure for Solving Eq. (6) 10 Additional Experiments Details and Results 10 Additional Experiments Details and Results 11 Other Potential Applications 11 Other Potential Applications Abstract Many computer vision and machine learning problems are modelled as learning tasks on graphs, where graph neural networks (GNNs) have emerged as a dominant tool for learning representations of graphstructured data. A key feature of GNNs is their use of graph structures as input, enabling them to exploit the graphs’ inherent topological properties—known as the topology awareness of GNNs. Despite the empirical successes of GNNs, the influence of topology awareness on generalization performance remains unexplored, particularly for node-level tasks that diverge from the assumption of data being independent and identically distributed (I.I.D.). The precise definition and characterization of the topology awareness of GNNs, especially concerning different topological features, are still unclear. This paper introduces a comprehensive framework to characterize the topology awareness of GNNs across any topological feature. Using this framework, we investigate the effects of topology awareness on GNN generalization performance. Contrary to the prevailing belief that enhancing the topology awareness of GNNs is always advantageous, our analysis reveals a critical insight: improving the topology awareness of GNNs may inadvertently lead to unfair generalization across structural groups, which might not be desired in some scenarios. Additionally, we conduct a case study using the intrinsic graph metric, the shortest-path distance, on various benchmark datasets. The empirical results of this case study confirm our theoretical insights. Moreover, we demonstrate the practical applicability of our framework by using it to tackle the cold start problem in graph active learning. 1 Introduction Many problems in computer vision and machine learning are modeled as learning tasks on graphs. For example, in semantic segmentation, graphs model the relationships between different image regions, enhancing accuracy and context-aware segmentation. Graph neural networks (GNNs) have emerged as a dominant class of machine learning models specifically designed for learning representations of graph-structured data. They have demonstrated considerable success in addressing a wide range of graph-related problems in various domains such as chemistry [10], biology [37], social networking [6, 22], scene graph generation [46, 51] and visual relationship detection [24,43,49]. A defining characteristic of GNNs is their use of a spatial approach through message passing on the graph structure for feature aggregation. This enables GNNs to preserve structural information or dependencies (referred to as topology awareness) from the underlying graph structure, allowing them to be highly effective in tasks such as node classification. Fig. 1 illustrates the overall learning process of GNNs. Despite their practicality and potential, there remains a lack of theoretical understanding about GNNs, particularly in the semi-supervised node classification setting where the dependencies among the data differ significantly from other machine learning models [25]. In this setting, the goal is to leverage relations, as captured by the graph structure, among the data and a small set of labelled nodes to predict labels for the remaining nodes. Most of the existing theoretical studies of GNNs have focused on the connection between the message-passing mechanism of GNNs and the Weisfeiler-Lehman isomorphism test [19], aiming to understand the ability of GNNs to differentiate different graph structures in the learned representations, known as the expressive power of GNNs. Inspired by the expressiveness studies, it is commonly believed that increasing topology awareness is universally beneficial and many studies focus on enabling GNNs to preserve more structural properties in the learned representation [29, 33, 48]. However, as GNNs become more reliant on and sensitive (aware) of graph structure as input, they may exhibit different generalization performances towards certain structural subgroups (distinct data subsets grouped by structural similarity to the training set) within the data. The quantification of GNN generalization across distinct structural subgroups is termed structural subgroup generalization [25]. Such considerations are vital in GNN application and development. For instance, within protein-protein interaction networks, these structural subgroups could represent different molecular complexes, influencing the accuracy of interaction predictions. Similarly, understanding how the topology awareness of GNNs influences generalization is essential when devising sampling strategies for training. The extent to which the generalization performance of GNNs is influenced by specific structural features of graph data is critical in deciding the composition of training datasets. Despite its importance, an understanding of the relationship between the topology awareness of GNNs and its structural subgroup generalization is still lacking. Furthermore, characterizing the topology awareness of GNNs poses a challenge, especially considering that different domains and tasks may prioritize distinct structural aspects. Therefore, a versatile framework is needed to assess the topology awareness of GNNs in relation to various structures. To address this gap, in this paper, we propose a novel framework based on approximate metric embedding to study the relationship between structural subgroup generalization and topology awareness of GNNs in the context of semisupervised node classification. The proposed framework allows for the investigation of the structural subgroup generalization of GNNs with respect to different structural subgroups. More concretely, the main contributions of this work are summarized as follows. 1. We propose a novel, structure-agnostic framework using approximate metric embedding to examine the interplay between GNNs’ structural subgroup generalization and topology awareness. This framework is versatile, accommodating various structural measures like shortest-path distance, and requires only the corresponding structural measure. Its simplicity in estimating key factors makes it applicable and generalizable to a wide range of scenarios. 2. Through formal analysis within our framework, we establish a clear link between GNN topology awareness and their generalization performance (Theorem 1). We also demonstrate that while enhanced topology awareness boosts GNN expressiveness, it can result in uneven generalization performance, favouring subgroups more structurally similar to the training set (Theorem 2). Such structural property can be harmful (causing unfairness issues) or useful (informing design decisions) depending on the scenario. This challenges the prevailing belief that increased topology awareness universally benefits GNNs [29, 33, 48], emphasizing the importance of considering the relationship between topology awareness and generalization performance. 3. We validate our framework through a case study on shortest-path distance, highlighting its practicality and relevance. The results corroborate our theoretical findings, showing that GNNs with heightened awareness of shortestpath distances excel in classifying vertex groups closer to the training set. Moreover, we demonstrate how our findings can be applied to mitigate the cold start problem in graph active learning [11,15], highlighting the practical implications of our framework and results. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Understanding Topology Awareness in Graph Neural Networks

Comprensión de la conciencia topológica en redes neuronales gráficas: impactos en la generalización y la estructura

Konprann konsyantizasyon Topoloji nan rezo neral graf: enpak sou jeneralizasyon ak estriktirèl 

グラフニューラルネットワークにおけるトポロジー認識の理解：一般化と構造への影響

График мэдрэлийн сүлжээн дэх топологийн мэдлэгийг ойлгох нь: ерөнхий ойлголт ба бүтцийн байдалд үзүүлэх нөлөө 

ग्राफ न्यूरल नेटवर्कमा टोपोलोजी जागरूकता बुझ्दै: सामान्यीकरण र संरचनात्मकमा प्रभावहरू 

Go kwešiša Temogo ya Topology ka go Dinetweke tša Neural tša Kerafo: Ditlamorago go Kakaretšo le Sebopego 

په ګراف عصبي شبکو کې د توپولوژي پوهاوی پوهیدل: په عمومي کولو او جوړښت باندې اغیزې 

Kunzwisisa Kuzivikanwa kweTopology muGraph Neural Networks: Mhedzisiro paGeneralization uye Chimiro. 

การทำความเข้าใจการรับรู้เกี่ยวกับโทโพโลยีในเครือข่ายประสาทเทียมแบบกราฟ: ผลกระทบต่อการสรุปทั่วไปและโครงสร้าง

Pag-unawa sa Topology Awareness sa Graph Neural Networks: Mga Epekto sa Generalization at Structural 

گراف نیورل نیٹ ورکس میں ٹوپولوجی بیداری کو سمجھنا: عمومی اور ساختی اثرات پر 

Grafik neyron tarmoqlarida topologiyadan xabardorlikni tushunish: umumlashtirish va strukturaga ta'siri 

If you were to ask anyone whether they preferred the original Ace Attorney trilogy or the Apollo Justice trilogy, there’s a very good chance they would say the former. The first 3 games were so amazing that they were extremely hard to follow up. However, the Apollo Justice games (which consist of Ace Attorney 4,5, and 6) are still very good games, and I would encourage anyone who hasn’t played them yet to stop reading this article immediately and play them this instant. They’re filled to the brim with interesting cases, villains, and brand-new protagonists such as Athena Cykes and, of course, Apollo Justice himself. So today, I’m doing the near-impossible and choosing the top 5 Apollo Justice cases. Warning: Spoilers Ahead. The 5 Best Cases From the Apollo Justice Trilogy 5. Turnabout for Tomorrow 5-5 4. Turnabout Revolution 6-5 3. Turnabout Trump 4-1 2. Turnabout Succession 4-4 1. Turnabout Storyteller 6-4 5. Turnabout for Tomorrow 5-5 Throughout the fifth game, Dual Destinies, we know that something traumatic happened to newcomer Athena Cykes. She was keeping a big secret, and in Turnabout for Tomorrow, everything gets revealed. Cykes’ past, prosecutor Simon Blackquill’s connection to all of it, and the mysterious murderer of 2 people all come to light. Turnabout for Tomorrow is the final case of Dual Destinies, and it serves as a fitting end that ties up all the loose ends in a neat bow. The case also has an incredibly shocking twist that I did not see coming. 4. Turnabout Revolution 6-5 Speaking of wrapping things up and shocking twists, the last case in Spirit of Justice brings that in spades. The final case in every Ace Attorney game always goes hard, but probably none so than Turnabout Revolution. It’s long with twist after twist that you don’t see coming, and just when you think you have it figured out, they throw another twist at you. But apart from the craziness of the case, another reason why Turnabout Revolution is great is because it provides closure for Apollo Justice. The once very nervous rookie defense attorney is finally allowed to spread his wings and become a new and improved man. 3. Turnabout Trump 4-1 Shocking twists are usually reserved for the end of the game, but in Apollo Justice, they come out swinging with them. Apollo and his mentor, Kristoph Gavin, have to defend this strange, disheveled man who is accused of murder. Little does Apollo know that their client is the former Ace Attorney himself, Phoenix Wright. And if that wasn’t crazy enough, it turns out that the real murderer was helping us all along. That’s right, in the end, our friend and mentor Gavin is revealed to be the mastermind behind the murder. Apollo Justice has one of the strongest starts in the history of the series, and that’s why it’s one of the best cases in the Apollo Justice trilogy. 2. Turnabout Succession 4-4 When we meet Phoenix Wright at the beginning of the game, we know that he’s not the same man that we last saw in Trials and Tribulations. He lost his reputation, and most importantly, he lost his job as a defense attorney; he was disbarred. But we didn’t really know what happened until Turnabout Succession when everything was revealed. But not only do we see what happened, we experience it ourselves. We play as Phoenix Wright during his last trial when he’s framed for forging evidence, we find out who was behind it all, and we see Apollo Justice rise to the occasion in the present day. And events that we had no idea were important finally make their connections known. For example, the events and characters in cases 4-1 and 4-2 are revealed to actually be a huge part of the story, and it retroactively makes those cases that much better. 1. Turnabout Storyteller 6-4 I love the final cases in Ace Attorney games because it is the culmination of the overall story that they plant from the very beginning. However, I love Turnabout Storyteller for the exact opposite reason. This case has Athena Cykes defending a young man with the help of prosecutor Simon Blackquill. Case 6-4 has nothing to do with the overarching story, none of these characters will probably appear again in future installments, and in the grand scheme of things that were currently going on in the Ace Attorney world, it didn’t seem that big or urgent. But that’s what’s so great about it. The judicial system didn’t ride on the outcome of the case, nor did an entire kingdom’s future hang in the balance. It’s a simple story about young up-and-comer Athena trying to clear somebody of murder, without the help of Phoenix Wright or Apollo Justice. It’s just her, Widget, and Blackquill. And it turned out to be amazing. It’s relatively obvious who the killer is from the get-go, but I didn’t care. The story and the characters were intriguing, and best of all, the chemistry between Athena and Blackquill was amazing. Athena, the ambitious and eager student vs Blackquill as the reluctant but helpful mentor. This case is so good that it’s by far my favorite in the Apollo Justice trilogy, and I think this single case proves that Athena Cykes is an interesting character enough to warrant her being the star of her own game. Read More: The New Tomb Raider Games in Order Persona 3 Remake Officially Announced: Everything We Know So Far The Kingsman Movies in Order Also published here Feature image source

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