Topologic

topologic

Topological sorting refers to the linear ordering of vertices in a graph such that for every directed edge , of vertex to , always comes before . ab a b a, b Above is an image showing a directed acyclic graph (edge 0 and edge 1 are directed at edge 2, edge 2 is directed at edge 3 and edge 3 is directed at edge 4). from: https://generalducky.github.io/DAG/ : For the image above, there are two possible topological sort orders for this graph which would be: 1, 0, 2, 3, 4 or 0, 1, 2, 3, 4 . Example of topological sorting would be: task scheduling (what task should proceed before the order), pre-requisite problems (what course or problems should be solved in order to solve an higher problem) — (adapted from ) Typical applications geekforgeeks Our approach to solving this problem would involve using the . let’s get down to business already 😊 depth-first search First, we would create a module, import from rust’s standard library into our module, and define a ( ) struct name it , which defines what our graph would like: Graph hashMap Directed Acyclic Graph DAG DAG Graph { std::collections::HashMap; { graph: <HashMap< , < >>,
  }
} mod use //import hashmap from std lib pub struct DAG // define the DAG struct Option u8 Vec u8 Next, we will define a method on the struct call it that will create a graph from the argument passed into it. 1. The method accepts an argument which is a vector of tuples containing values. 2. Creates a hashMap ( ) that would contain the edges and vectors in our graph. 3. Loops through the argument and populates the with the information on the graph 4. Move the into the struct, wrapping it in theOption type as defined on the struct with it’s key as . 5. and then calls the method. DAG new new graph_info u8 adjacency_map graph_info adjacency_map adjacency_map Graph graph get_topological_order Graph { std::collections::HashMap; {
    graph: <HashMap< , < >>,
  } DAG { (graph_info: <( , )>) -> < > { adjacency_map: HashMap< , < >> = HashMap::new(); value graph_info { source_vertex = & adjacency_list.entry(value. ).or_insert( []);
                source_vertex.push(value. );
            } the_graph = DAG {
                graph: (adjacency_map),
            }; the_graph.get_topological_order();
        }
} mod use pub struct DAG Option u8 Vec u8 impl pub fn new Vec u8 u8 Vec u8 let mut u8 Vec u8 for in let mut 0 vec! 1 let Some return The method basically stores the order of the graph in a stack ( 1. Create a stack (in this case, we would use a vector but only use it as a stack) data-structure called , a vector of values. 2. Loop through the keys to get the order using the method. 3. Once the order is obtained, we would reverse the stack. get_topological_order LIFO i.e last in, first out). stack u8 graph get_order std::collections::HashMap; {
    graph: <HashMap< , < >>,
  } DAG { (graph_info: <( , )>) -> < > {...} (& ) -> < > { source_nodes = .graph.as_ref().unwrap().keys(); stack: < > = []; node source_nodes { .get_order(node, & stack);
            }
            stack.reverse(); ( , stack); stack;
        }
} use pub struct DAG Option u8 Vec u8 impl pub fn new Vec u8 u8 Vec u8 pub fn get_topological_order self Vec u8 let self let mut Vec u8 vec! for in self mut println! "THE STACK!! {:?}" return The method accepts the Using the node (current key) passed to it, the method gets all the values of such key, which are edges that this edge points to and does the same of all of such edges points to. Adding them to the stack. if such node does not point anywhere and has not been visited, i.e. it is not on the stack., it is pushed into the stack. get_order node (current key, borrowed u8 value), stack(mutable vector of u8 values) . get_order Graph { std::collections::HashMap; {
    graph: <HashMap< , < >>,
  } DAG { (graph_info: <( , )>) -> < > {...} (& )-> < > {
            ...
            stack.reverse(); ( , stack); stack;
        } (& , node: & , stack: & < >) { receiving_nodes = .graph.as_ref().unwrap().get(node); receiving_nodes != { value receiving_nodes.unwrap() { .get_order(value, stack);
                    }
                } !stack.contains(node) {
                    stack.push(*node);
                }
        }
} mod use pub struct DAG Option u8 Vec u8 impl pub fn new Vec u8 u8 Vec u8 pub fn get_topological_order self Vec u8 println! "THE STACK!! {:?}" return pub fn get_order self u8 mut Vec u8 let self if None for in self if The result of our function should be one of the two vectors here: [1, 0, 2, 3, 4] or [0, 1, 2, 3, 4] . If we passed the argument as: vec![(0, 2), (1, 2), (2, 3), (3, 4)] () { Graphs; Graphs::DAG::new( [( , ), ( , ), ( , ), ( , )]);
} Graphs { std::collections::HashMap; {
        graph: <HashMap< , < >>>,
    } DAG { (graph_info: <( , )>) -> < > { adjacency_list: HashMap< , < >> = HashMap::new(); graph = graph_info.get( ..); value graph.unwrap() { source_vertex = & adjacency_list.entry(value. ).or_insert( []);
                source_vertex.push(value. );
            } the_graph = DAG {
                graph: (adjacency_list),
            }; the_graph.get_topological_order();
        } (& ) -> < > { source_nodes = .graph.as_ref().unwrap().keys(); stack: < > = []; node source_nodes { .get_order(node, & stack); }
            stack.reverse(); ( , stack); stack;
        } (& , node: & , stack: & < >) { receiving_nodes = .graph.as_ref().unwrap().get(node); receiving_nodes != { value receiving_nodes.unwrap() { .get_order(value, stack);
                }
            } !stack.contains(node) {
                stack.push(*node);
            } }
    }
} pub fn execute use // Graphs::DAG::new(vec![(4, 3), (1, 2), (4, 1), (3, 1)]); // Graphs::DAG::new(vec![(1, 2), (1, 4), (2, 3), (2, 4), (4, 3), (4, 5)]); vec! 0 2 1 2 2 3 3 4 mod use pub struct DAG Option u8 Vec u8 impl pub fn new Vec u8 u8 Vec u8 // DirectedGraph { graph: None } let mut u8 Vec u8 let 0 for in let mut 0 vec! 1 let Some return pub fn get_topological_order self Vec u8 let self let mut Vec u8 vec! // let mut visited: HashMap<u8, bool> = HashMap::new(); for in // if visited.get(node) == None { self mut // } println! "THE STACK!! {:?}" return pub fn get_order self u8 mut Vec u8 let self // if visited.get(node) == None { if None for in self if // } Going a step forward to test our output: () { Graphs; result = Graphs::DAG::new( [( , ), ( , ), ( , ), ( , )]); ( , result);
} Graphs { std::collections::HashMap; {
        graph: <HashMap< , < >>>,
    } DAG { (graph_info: <( , )>) -> < > { adjacency_list: HashMap< , < >> = HashMap::new(); graph = graph_info.get( ..); value graph.unwrap() { source_vertex = & adjacency_list.entry(value. ).or_insert( []);
                source_vertex.push(value. );
            } the_graph = DAG {
                graph: (adjacency_list),
            }; the_graph.get_topological_order();
        } (& ) -> < > { source_nodes = .graph.as_ref().unwrap().keys(); stack: < > = []; node source_nodes { .get_order(node, & stack); }
            stack.reverse(); ( , stack); stack;
        } (& , node: & , stack: & < >) { receiving_nodes = .graph.as_ref().unwrap().get(node); receiving_nodes != { value receiving_nodes.unwrap() { .get_order(value, stack);
                }
            } !stack.contains(node) {
                stack.push(*node);
            } }
    }
} test { super::Graphs; () { result = Graphs::DAG::new( [( , ), ( , ), ( , ), ( , )]); (result.pop(), ( )); (result.pop(), ( )); (result.pop(), ( ));
    }
} pub fn execute use // Graphs::DAG::new(vec![(4, 3), (1, 2), (4, 1), (3, 1)]); // Graphs::DAG::new(vec![(1, 2), (1, 4), (2, 3), (2, 4), (4, 3), (4, 5)]); let vec! 0 2 1 2 2 3 3 4 println! "THE RESULT >>>  {:?}" mod use pub struct DAG Option u8 Vec u8 impl pub fn new Vec u8 u8 Vec u8 // DirectedGraph { graph: None } let mut u8 Vec u8 let 0 for in let mut 0 vec! 1 let Some return pub fn get_topological_order self Vec u8 let self let mut Vec u8 vec! // let mut visited: HashMap<u8, bool> = HashMap::new(); for in // if visited.get(node) == None { self mut // } println! "THE STACK!! {:?}" return pub fn get_order self u8 mut Vec u8 let self // if visited.get(node) == None { if None for in self if // } #[cfg(test)] mod use #[test] fn test_topological_order let mut vec! 0 2 1 2 2 3 3 4 assert_eq! Some 4 assert_eq! Some 3 assert_eq! Some 2 find me on twitter [ @tolumide ] and github: [ tolumide-ng ]

Topological Sorting of a Directed Acyclic Graph in Rust Using DFS

How K-SIF and SIF Revolutionize Anomaly Detection in Complex Datasets

Authors:
(1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France;
(2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France;
(3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Table of Links Abstract and 1. Introduction 2. Background & Preliminaries 2.1. Functional Isolation Forest 2.2. The Signature Method 3. Signature Isolation Forest Method 4. Numerical Experiments 4.1. Parameters Sensitivity Analysis 4.2. Advantages of (K-)SIF over FIF 4.3. Real-data Anomaly Detection Benchmark 5. Discussion & Conclusion, Impact Statements, and References Appendix A. Additional Information About the Signature B. K-SIF and SIF Algorithms C. Additional Numerical Experiments 2.2. The Signature Method The signature of a path is a sequence of iterated integrals that captures important information about the path’s geometric and topological features (Lyons et al., 2007; Fermanian, 2021). Furthermore, the signature of X is defined as the infinite collection of coordinate signature One may refer to Lee and Oberhauser (2023) for an account of the untruncated kernel signature. See also Section A in the Appendix for further details about the signature and its properties. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France; (2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France; (3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Authors: Authors: (1) Guillaume Staerman, INRIA, CEA, Univ. Paris-Saclay, France; (2) Marta Campi, CERIAH, Institut de l’Audition, Institut Pasteur, France; (3) Gareth W. Peters, Department of Statistics & Applied Probability, University of California Santa Barbara, USA. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2. Background & Preliminaries 2. Background & Preliminaries 2.1. Functional Isolation Forest 2.1. Functional Isolation Forest 2.2. The Signature Method 2.2. The Signature Method 3. Signature Isolation Forest Method 3. Signature Isolation Forest Method 4. Numerical Experiments 4. Numerical Experiments 4.1. Parameters Sensitivity Analysis 4.1. Parameters Sensitivity Analysis 4.2. Advantages of (K-)SIF over FIF 4.2. Advantages of (K-)SIF over FIF 4.3. Real-data Anomaly Detection Benchmark 4.3. Real-data Anomaly Detection Benchmark 5. Discussion & Conclusion, Impact Statements, and References 5. Discussion & Conclusion, Impact Statements, and References Appendix A. Additional Information About the Signature A. Additional Information About the Signature B. K-SIF and SIF Algorithms B. K-SIF and SIF Algorithms C. Additional Numerical Experiments C. Additional Numerical Experiments 2.2. The Signature Method The signature of a path is a sequence of iterated integrals that captures important information about the path’s geometric and topological features (Lyons et al., 2007; Fermanian, 2021). Furthermore, the signature of X is defined as the infinite collection of coordinate signature Furthermore, the signature of X is defined as the infinite collection of coordinate signature X One may refer to Lee and Oberhauser (2023) for an account of the untruncated kernel signature. See also Section A in the Appendix for further details about the signature and its properties. 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

Unveiling Path Signatures: A Key to Geometric Insights in Data

Onthulling van padhandtekeninge: 'n Sleutel tot meetkundige insigte in data

Descubriendo las firmas de trayectoria: una clave para obtener información geométrica en los datos

Revelación de sinaturas de camiños: unha clave para coñecer os datos xeométricos

Mengungkap Tanda Jalur: Kunci Wawasan Geometris dalam Data

パスシグネチャの解明: データの幾何学的洞察への鍵

Жол қолтаңбаларын ашу: деректердегі геометриялық түсініктердің кілті

경로 서명 공개: 데이터에서 기하학적 통찰력의 열쇠

Famoahana sonia lalana: lakile ho an'ny fahitana geometrika amin'ny angona

د لارې لاسلیکونه افشا کول: په ډیټا کې د جیومیټریک بصیرت لپاره کلیدي

Kuzarura Nzira Masiginicha: Kiyi yeGeometric Insights muData

Zbulimi i nënshkrimeve të rrugës: Një çelës për njohuritë gjeometrike në të dhëna

راستے کے دستخطوں کی نقاب کشائی: ڈیٹا میں جیومیٹرک بصیرت کی کلید

What is the Signature Isolation Forest? 

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 

Authors:
(1) Revathy B S, Raman Research Institute, Bengaluru, 560080, Karnataka, India and Corresponding author;
(2) Victor Mukherjee, Department of Physical Sciences, Indian Institute of Science Education and Research Berhampur, Berhampur, 760010, Odisha, India;
(3) Uma Divakaran, Department of Physics, Indian Institute of Technology Palakkad, Palakkad, 678623, Kerala, India. Table of Links Abstract and 1 Introduction 2 Free fermionic model 3 Many body quantum Otto cycle 4 Universal scalings in work output 5 Transverse Ising model as working medium 6 Conclusion and References 6 Conclusion We construct a many body quantum Otto cycle with a WM that undergoes a quantum phase transition. The non-unitary strokes of the cycle are powered by finite temperature baths, while the unitary strokes involve driving the WM close to the critical point. This driving leads to nonadiabatic excitations which can be quantified using relative excess energy that follows universal scalings with the rate of driving as well as the temperature of the cold bath. The excess energy can be linked to the output work of the engine which thus manifests the universal scalings shown by the excess energy. Notably, we show that higher values of the cold bath temperature TC allows one to operate the engine close to the adiabatic limit for lower values of τ2 ≈ τmin, which further follows universal scaling relations. This raises interesting questions regarding the importance of control methods such as shortcuts to adiabaticity [41], or bath engineering [23], for finite temperature quantum heat engines. Furthermore, our results for one-dimensional transverse Ising model WM suggest the existence of an optimal value of the cold bath temperature TC > 0, for operating the QHE with high work output at high power. These counterintuitive results stem from the dominance of thermal fluctuations over quantum fluctuations in finite-temperature quantum critical heat engines, for higher bath temperatures. Acknowledgements. R.B.S. and U.D. acknowledge the use of HPC facility Chandra at IIT Palakkad. U.D. acknowledges support from SERB (SPG/2022/000708). V.M. acknowledges support from SERB through MATRICS (Project No. MTR/2021/000055) and a Seed Grant from IISER Berhampur. Data Availability Statement. Any data that support the findings of this study are included within the article. References [1] Jochen Gemmer, Mathias Michel, and G¨unter Mahler. Quantum thermodynamics: Emergence of thermodynamic behavior within composite quantum systems, volume 784. Springer, 2009. [2] Robert Alicki and Ronnie Kosloff. Introduction to Quantum Thermodynamics: History and Prospects. Springer International Publishing, 2018. [3] Sai Vinjanampathy and Janet Anders. Quantum thermodynamics. Contemporary Physics, 57:545, 2016. [4] Johannes Roßnagel, Samuel T. Dawkins, Karl N. Tolazzi, Obinna Abah, Eric Lutz, Ferdinand Schmidt-Kaler, and Kilian Singer. A single-atom heat engine. Science, 352(6283):325–329, 2016. [5] D. von Lindenfels, O. Grab, C. T. Schmiegelow, V. Kaushal, J. Schulz, Mark T. Mitchison, John Goold, F. Schmidt-Kaler, and U. G. Poschinger. Spin heat engine coupled to a harmonic-oscillator flywheel. Phys. Rev. Lett., 123:080602, Aug 2019. [6] John P. S. Peterson, Tiago B. Batalh˜ao, Marcela Herrera, Alexandre M. Souza, Roberto S. Sarthour, Ivan S. Oliveira, and Roberto M. Serra. Experimental characterization of a spin quantum heat engine. Phys. Rev. Lett., 123:240601, Dec 2019. [7] James Klatzow, Jonas N. Becker, Patrick M. Ledingham, Christian Weinzetl, Krzysztof T. Kaczmarek, Dylan J. Saunders, Joshua Nunn, Ian A. Walmsley, Raam Uzdin, and Eilon Poem. Experimental demonstration of quantum effects in the operation of microscopic heat engines. Phys. Rev. Lett., 122:110601, Mar 2019. [8] Obinna Abah and Eric Lutz. Optimal performance of a quantum otto refrigerator. Europhysics Letters, 113(6):60002, apr 2016. [9] Andreas Hartmann, Victor Mukherjee, Glen Bigan Mbeng, Wolfgang Niedenzu, and Wolfgang Lechner. Multi-spin counterdiabatic driving in many-body quantum otto refrigerators. Quantum, 4:377, 2020. [10] Thao P. Le, Jesper Levinsen, Kavan Modi, Meera M. Parish, and Felix A. Pollock. Spinchain model of a many-body quantum battery. Phys. Rev. A, 97:022106, Feb 2018. [11] Francesco Campaioli, Felix A. Pollock, and Sai Vinjanampathy. Quantum batteries - review chapter. arXiv: Quantum Physics, 2018. [12] Victor Montenegro, Marco G. Genoni, Abolfazl Bayat, and Matteo G. A. Paris. Quantum metrology with boundary time crystals. Communications Physics, 6(1):304, Oct 2023. [13] Karl Joulain, Jeremie Drevillon, Younes Ezzahri, and Jose Ordonez-Miranda. Quantum thermal transistor. Phys. Rev. Lett., 116:200601, May 2016. [14] Nikhil Gupt, Srijan Bhattacharyya, Bikash Das, Subhadeep Datta, Victor Mukherjee, and Arnab Ghosh. Floquet quantum thermal transistor. Phys. Rev. E, 106:024110, Aug 2022. [15] H. T. Quan, Yu-xi Liu, C. P. Sun, and Franco Nori. Quantum thermodynamic cycles and quantum heat engines. Phys. Rev. E, 76:031105, Sep 2007. [16] Ronnie Kosloff and Yair Rezek. The quantum harmonic otto cycle. Entropy, 19(4), 2017. [17] M. Campisi and R. Fazio. The power of a critical heat engine. Nat. Commun., 7:11895, 2016. [18] Thomas Fogarty and Thomas Busch. A many-body heat engine at criticality. Quantum Science and Technology, 6(1):015003, nov 2020. [19] Giulia Piccitto, Michele Campisi, and Davide Rossini. The ising critical quantum otto engine. New Journal of Physics, 2022. [20] Yu-Han Ma, Shan-He Su, and Chang-Pu Sun. Quantum thermodynamic cycle with quantum phase transition. Phys. Rev. E, 96:022143, Aug 2017. [21] Watanabe G. Yu YC Guan XW Chen, YY. and del Campo. An interaction-driven many particles quantum heat engine and its universal behavior. npj Quantum Inf, 5, 2019. [22] Revathy B.S, Victor Mukherjee, Uma Divakaran, and Adolfo del Campo. Universal finite-time thermodynamics of many-body quantum machines from kibble-zurek scaling. Phys. Rev. Res., 2:043247, 2020. [23] Revathy B.S, Victor Mukherjee, and Uma Divakaran. Bath engineering enhanced quantum critical engines. Entropy, 24(10), 2022. [24] Revathy B.S, Harsh Sharma, and Uma Divakaran. Improving performance of quantum heat engines by free evolution. arXiv preprint arXiv:2302.07003, 2023. [25] Jacek Dziarmaga. Dynamics of a quantum phase transition: Exact solution of the quantum ising model. Phys. Rev. Lett., 95:245701, Dec 2005. [26] Victor Mukherjee, Uma Divakaran, Amit Dutta, and Diptiman Sen. Quenching dynamics of a quantum xy spin- 1 2 chain in a transverse field. Phys. Rev. B, 76:174303, Nov 2007. [27] K. Sengupta, Diptiman Sen, and Shreyoshi Mondal. Exact results for quench dynamics and defect production in a two-dimensional model. Phys. Rev. Lett., 100:077204, Feb 2008. [28] Maximilian Keck, Simone Montangero, Giuseppe E Santoro, Rosario Fazio, and Davide Rossini. Dissipation in adiabatic quantum computers: lessons from an exactly solvable model. New Journal of Physics, 19(11):113029, nov 2017. [29] Souvik Bandyopadhyay, Sudarshana Laha, Utso Bhattacharya, and Amit Dutta. Exploring the possibilities of dynamical quantum phase transitions in the presence of a markovian bath. Scientific Reports, 8(1):11921, 2018. [30] Bogdan Damski and Wojciech H. Zurek. Adiabatic-impulse approximation for avoided level crossings: From phase-transition dynamics to landau-zener evolutions and back again. Phys. Rev. A, 73:063405, Jun 2006. [31] Anatoli Polkovnikov, Krishnendu Sengupta, Alessandro Silva, and Mukund Vengalattore. Colloquium: Nonequilibrium dynamics of closed interacting quantum systems. Rev. Mod. Phys., 83:863–883, Aug 2011. [32] Adolfo Del Campo and Wojciech H Zurek. Universality of phase transition dynamics: Topological defects from symmetry breaking. International Journal of Modern Physics A, 29(08):1430018, 2014. [33] Amit Dutta, Gabriel Aeppli, Bikas K Chakrabarti, Uma Divakaran, Thomas F Rosenbaum, and Diptiman Sen. Quantum phase transitions in transverse field spin models: from statistical physics to quantum information. Cambridge University Press, 2015. [34] A. Polkovnikov and V. Gritsev. Breakdown of the adiabatic limit in low-dimensional gapless systems. Nature Physics, 4:477, June 2008. [35] C. De Grandi, V. Gritsev, and A. Polkovnikov. Quench dynamics near a quantum critical point: Application to the sine-gordon model. Phys. Rev. B, 81:224301, Jun 2010. [36] C. De Grandi, V. Gritsev, and A. Polkovnikov. Quench dynamics near a quantum critical point. Phys. Rev. B, 81:012303, Jan 2010. [37] Shusa Deng, Gerardo Ortiz, and Lorenza Viola. Dynamical critical scaling and effective thermalization in quantum quenches: Role of the initial state. Phys. Rev. B, 83:094304, 2011. [38] Pierre Pfeuty. The one-dimensional ising model with a transverse field. Annals of Physics, 57(1):79 – 90, 1970. [39] J. E. Bunder and Ross H. McKenzie. Effect of disorder on quantum phase transitions in anisotropic xy spin chains in a transverse field. Phys. Rev. B, 60:344–358, Jul 1999. [40] Elliott Lieb, Theodore Schultz, and Daniel Mattis. Two soluble models of an antiferromagnetic chain. Annals of Physics, 16(3):407 – 466, 1961. [41] Andreas Hartmann, Victor Mukherjee, Wolfgang Niedenzu, and Wolfgang Lechner. Many-body quantum heat engines with shortcuts to adiabaticity. Phys. Rev. Res., 2:023145, May 2020. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Revathy B S, Raman Research Institute, Bengaluru, 560080, Karnataka, India and Corresponding author; (2) Victor Mukherjee, Department of Physical Sciences, Indian Institute of Science Education and Research Berhampur, Berhampur, 760010, Odisha, India; (3) Uma Divakaran, Department of Physics, Indian Institute of Technology Palakkad, Palakkad, 678623, Kerala, India. Authors: Authors: (1) Revathy B S, Raman Research Institute, Bengaluru, 560080, Karnataka, India and Corresponding author; (2) Victor Mukherjee, Department of Physical Sciences, Indian Institute of Science Education and Research Berhampur, Berhampur, 760010, Odisha, India; (3) Uma Divakaran, Department of Physics, Indian Institute of Technology Palakkad, Palakkad, 678623, Kerala, India. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Free fermionic model 2 Free fermionic model 3 Many body quantum Otto cycle 3 Many body quantum Otto cycle 4 Universal scalings in work output 4 Universal scalings in work output 5 Transverse Ising model as working medium 5 Transverse Ising model as working medium 6 Conclusion and References 6 Conclusion and References 6 Conclusion We construct a many body quantum Otto cycle with a WM that undergoes a quantum phase transition. The non-unitary strokes of the cycle are powered by finite temperature baths, while the unitary strokes involve driving the WM close to the critical point. This driving leads to nonadiabatic excitations which can be quantified using relative excess energy that follows universal scalings with the rate of driving as well as the temperature of the cold bath. The excess energy can be linked to the output work of the engine which thus manifests the universal scalings shown by the excess energy. Notably, we show that higher values of the cold bath temperature TC allows one to operate the engine close to the adiabatic limit for lower values of τ2 ≈ τmin, which further follows universal scaling relations. This raises interesting questions regarding the importance of control methods such as shortcuts to adiabaticity [41], or bath engineering [23], for finite temperature quantum heat engines. Furthermore, our results for one-dimensional transverse Ising model WM suggest the existence of an optimal value of the cold bath temperature TC > 0, for operating the QHE with high work output at high power. These counterintuitive results stem from the dominance of thermal fluctuations over quantum fluctuations in finite-temperature quantum critical heat engines, for higher bath temperatures. Acknowledgements . R.B.S. and U.D. acknowledge the use of HPC facility Chandra at IIT Palakkad. U.D. acknowledges support from SERB (SPG/2022/000708). V.M. acknowledges support from SERB through MATRICS (Project No. MTR/2021/000055) and a Seed Grant from IISER Berhampur. Acknowledgements Data Availability Statement. Any data that support the findings of this study are included within the article. Data Availability Statement. References [1] Jochen Gemmer, Mathias Michel, and G¨unter Mahler. Quantum thermodynamics: Emergence of thermodynamic behavior within composite quantum systems, volume 784. Springer, 2009. [2] Robert Alicki and Ronnie Kosloff. Introduction to Quantum Thermodynamics: History and Prospects. Springer International Publishing, 2018. [3] Sai Vinjanampathy and Janet Anders. Quantum thermodynamics. Contemporary Physics, 57:545, 2016. [4] Johannes Roßnagel, Samuel T. Dawkins, Karl N. Tolazzi, Obinna Abah, Eric Lutz, Ferdinand Schmidt-Kaler, and Kilian Singer. A single-atom heat engine. Science, 352(6283):325–329, 2016. [5] D. von Lindenfels, O. Grab, C. T. Schmiegelow, V. Kaushal, J. Schulz, Mark T. Mitchison, John Goold, F. Schmidt-Kaler, and U. G. Poschinger. Spin heat engine coupled to a harmonic-oscillator flywheel. Phys. Rev. Lett., 123:080602, Aug 2019. [6] John P. S. Peterson, Tiago B. Batalh˜ao, Marcela Herrera, Alexandre M. Souza, Roberto S. Sarthour, Ivan S. Oliveira, and Roberto M. Serra. Experimental characterization of a spin quantum heat engine. Phys. Rev. Lett., 123:240601, Dec 2019. [7] James Klatzow, Jonas N. Becker, Patrick M. Ledingham, Christian Weinzetl, Krzysztof T. Kaczmarek, Dylan J. Saunders, Joshua Nunn, Ian A. Walmsley, Raam Uzdin, and Eilon Poem. Experimental demonstration of quantum effects in the operation of microscopic heat engines. Phys. Rev. Lett., 122:110601, Mar 2019. [8] Obinna Abah and Eric Lutz. Optimal performance of a quantum otto refrigerator. Europhysics Letters, 113(6):60002, apr 2016. [9] Andreas Hartmann, Victor Mukherjee, Glen Bigan Mbeng, Wolfgang Niedenzu, and Wolfgang Lechner. Multi-spin counterdiabatic driving in many-body quantum otto refrigerators. Quantum, 4:377, 2020. [10] Thao P. Le, Jesper Levinsen, Kavan Modi, Meera M. Parish, and Felix A. Pollock. Spinchain model of a many-body quantum battery. Phys. Rev. A, 97:022106, Feb 2018. [11] Francesco Campaioli, Felix A. Pollock, and Sai Vinjanampathy. Quantum batteries - review chapter. arXiv: Quantum Physics, 2018. [12] Victor Montenegro, Marco G. Genoni, Abolfazl Bayat, and Matteo G. A. Paris. Quantum metrology with boundary time crystals. Communications Physics, 6(1):304, Oct 2023. [13] Karl Joulain, Jeremie Drevillon, Younes Ezzahri, and Jose Ordonez-Miranda. Quantum thermal transistor. Phys. Rev. Lett., 116:200601, May 2016. [14] Nikhil Gupt, Srijan Bhattacharyya, Bikash Das, Subhadeep Datta, Victor Mukherjee, and Arnab Ghosh. Floquet quantum thermal transistor. Phys. Rev. E, 106:024110, Aug 2022. [15] H. T. Quan, Yu-xi Liu, C. P. Sun, and Franco Nori. Quantum thermodynamic cycles and quantum heat engines. Phys. Rev. E, 76:031105, Sep 2007. [16] Ronnie Kosloff and Yair Rezek. The quantum harmonic otto cycle. Entropy, 19(4), 2017. [17] M. Campisi and R. Fazio. The power of a critical heat engine. Nat. Commun., 7:11895, 2016. [18] Thomas Fogarty and Thomas Busch. A many-body heat engine at criticality. Quantum Science and Technology, 6(1):015003, nov 2020. [19] Giulia Piccitto, Michele Campisi, and Davide Rossini. The ising critical quantum otto engine. New Journal of Physics, 2022. [20] Yu-Han Ma, Shan-He Su, and Chang-Pu Sun. Quantum thermodynamic cycle with quantum phase transition. Phys. Rev. E, 96:022143, Aug 2017. [21] Watanabe G. Yu YC Guan XW Chen, YY. and del Campo. An interaction-driven many particles quantum heat engine and its universal behavior. npj Quantum Inf, 5, 2019. [22] Revathy B.S, Victor Mukherjee, Uma Divakaran, and Adolfo del Campo. Universal finite-time thermodynamics of many-body quantum machines from kibble-zurek scaling. Phys. Rev. Res., 2:043247, 2020. [23] Revathy B.S, Victor Mukherjee, and Uma Divakaran. Bath engineering enhanced quantum critical engines. Entropy, 24(10), 2022. [24] Revathy B.S, Harsh Sharma, and Uma Divakaran. Improving performance of quantum heat engines by free evolution. arXiv preprint arXiv:2302.07003, 2023. [25] Jacek Dziarmaga. Dynamics of a quantum phase transition: Exact solution of the quantum ising model. Phys. Rev. Lett., 95:245701, Dec 2005. [26] Victor Mukherjee, Uma Divakaran, Amit Dutta, and Diptiman Sen. Quenching dynamics of a quantum xy spin- 1 2 chain in a transverse field. Phys. Rev. B, 76:174303, Nov 2007. [27] K. Sengupta, Diptiman Sen, and Shreyoshi Mondal. Exact results for quench dynamics and defect production in a two-dimensional model. Phys. Rev. Lett., 100:077204, Feb 2008. [28] Maximilian Keck, Simone Montangero, Giuseppe E Santoro, Rosario Fazio, and Davide Rossini. Dissipation in adiabatic quantum computers: lessons from an exactly solvable model. New Journal of Physics, 19(11):113029, nov 2017. [29] Souvik Bandyopadhyay, Sudarshana Laha, Utso Bhattacharya, and Amit Dutta. Exploring the possibilities of dynamical quantum phase transitions in the presence of a markovian bath. Scientific Reports, 8(1):11921, 2018. [30] Bogdan Damski and Wojciech H. Zurek. Adiabatic-impulse approximation for avoided level crossings: From phase-transition dynamics to landau-zener evolutions and back again. Phys. Rev. A, 73:063405, Jun 2006. [31] Anatoli Polkovnikov, Krishnendu Sengupta, Alessandro Silva, and Mukund Vengalattore. Colloquium: Nonequilibrium dynamics of closed interacting quantum systems. Rev. Mod. Phys., 83:863–883, Aug 2011. [32] Adolfo Del Campo and Wojciech H Zurek. Universality of phase transition dynamics: Topological defects from symmetry breaking. International Journal of Modern Physics A, 29(08):1430018, 2014. [33] Amit Dutta, Gabriel Aeppli, Bikas K Chakrabarti, Uma Divakaran, Thomas F Rosenbaum, and Diptiman Sen. Quantum phase transitions in transverse field spin models: from statistical physics to quantum information. Cambridge University Press, 2015. [34] A. Polkovnikov and V. Gritsev. Breakdown of the adiabatic limit in low-dimensional gapless systems. Nature Physics, 4:477, June 2008. [35] C. De Grandi, V. Gritsev, and A. Polkovnikov. Quench dynamics near a quantum critical point: Application to the sine-gordon model. Phys. Rev. B, 81:224301, Jun 2010. [36] C. De Grandi, V. Gritsev, and A. Polkovnikov. Quench dynamics near a quantum critical point. Phys. Rev. B, 81:012303, Jan 2010. [37] Shusa Deng, Gerardo Ortiz, and Lorenza Viola. Dynamical critical scaling and effective thermalization in quantum quenches: Role of the initial state. Phys. Rev. B, 83:094304, 2011. [38] Pierre Pfeuty. The one-dimensional ising model with a transverse field. Annals of Physics, 57(1):79 – 90, 1970. [39] J. E. Bunder and Ross H. McKenzie. Effect of disorder on quantum phase transitions in anisotropic xy spin chains in a transverse field. Phys. Rev. B, 60:344–358, Jul 1999. [40] Elliott Lieb, Theodore Schultz, and Daniel Mattis. Two soluble models of an antiferromagnetic chain. Annals of Physics, 16(3):407 – 466, 1961. [41] Andreas Hartmann, Victor Mukherjee, Wolfgang Niedenzu, and Wolfgang Lechner. Many-body quantum heat engines with shortcuts to adiabaticity. Phys. Rev. Res., 2:023145, May 2020. 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

Quantum Critical Engine at Finite Temperatures: Conclusion and References

Efficient Detection of Defects in Magnetic Labyrinthine Patterns: Conclusion and References

Authors:
(1) Vinicius Yu Okubo, B.S. in electrical engineering from the University of São Paulo in 2022 and currently, he is pursuing his M.S. in electrical engineering at the University of São Paulo;
(2) Kotaro Shimizu, B.S. degree in Physics from Waseda University, Japan, in 2019 and M.S. degree in Physics from the University of Tokyo, Japan, 2021 and He has been pursuing his Ph.D. in Physics as a JSPS research fellowship for young scientists in the University of Tokyo since 2021;
(3) B.S. Shivaram, received his B.S. degree in Physics, Chemistry and Mathematics from Bangalore University, India, in 1977 and the M.S. degree in Physics from the Indian Institute of Technology, Madras, India, in 1979 and his Ph.D. in experimental condensed matter physics from Northwestern University, Evanston, Illinois in 1984;
(4) Hae Yong Kim, He received the B.S. and M.S. degrees (with distinctions) in computer science and the Ph.D. degree in electrical engineering from the Universidade de São Paulo (USP), Brazil, in 1988, 1992 and 1997, respectively. Table of Links Abstract and I Introduction II. Related Works III Methodology IV Experiments and Results V Conclusion and References IV. EXPERIMENTS AND RESULTS A. PERFORMANCE EVALUATION Evaluating TM-CNN poses a certain challenge because there are no benchmarks or published results for direct comparison. So, we defined that junction and terminal detections reviewed by human observers as the gold standard and compared the algorithms against this standard. For testing purposes, we annotated 5 independent images from varying experiments, using the same procedure as the training images (subsection III-E). False detections were excluded since they do not contribute to the detection evaluation. In total, the test set contained 3,551 junctions and 3,653 terminals. B. RESULTS FROM A PHYSICS PERSPECTIVE In Bi:YIG films, it is known that a perfect stripe pattern is the most energetically favorable and stable configuration of magnetic moments [4]. In this structure, the magnetic moments exhibit stripes aligning straight in one direction throughout the entire structure, resulting in large spatial coherence. Meanwhile, the experimentally obtained structures often manifest labyrinthine patterns, as illustrated by dark and bright stripes in Fig. 1. In these labyrinthine patterns, stripes are aligned within small domains but propagate in different directions between different domains, and thus the spatial coherence of stripes is smaller than the perfect stripe. Notably, junctions and terminals are defects emerging in the formation of these labyrinthine patterns, intricately linked to the growth of spatial coherence in the system. Introducing such defects into the perfect stripe structure leads to the bending and branching of the initially aligned stripes, eventually reducing spatial coherence accompanied by the transformation to labyrinthine patterns. Indeed, a plethora of defects are observed in the labyrinthine patterns shown in Fig. 2. Conversely, the elimination of these defects plays a crucial role in the transition from a quenched state to an annealed state, where the system increases the spatial coherence. Each of these defects cannot be eliminated alone by continuous deformation of the stripes due to their topological properties. They can be removed through the pair annihilation. For instance, in Fig. 2, terminals are branched from junctions and they can be removed pairwisely by reducing the branching distance. These defects, known as topological defects, are widely recognized for playing a crucial role in the formation of periodic structures. By detecting the positions of defects, the evolution of the labyrinthine patterns would be systematically quantified. Specifically in the context of magnetic labyrinthine patterns, a much smaller dataset was used to detect these defects manually [4]. Recently, persistent homology has been used to extract the topological features of the labyrinthine patterns in a systematic way [38]. However, it is still difficult to detect the positions of defects and classify them into junctions and terminals. In this article, we achieved the systematic detection. To quantify the evolution of the labyrinthine patterns, we applied the TM-CNN technique to the 444 images and investigated how the number of defects and their locations changes step by step. Fig. 6 illustrates the step dependence of the number of junctions and terminals during the demagnetization process. Results from 6 different experimental runs were averaged to estimate the number of defects. In the initial stages of the demagnetization process, there are approximately 800 defects. The number of defects sharply decreases around step 5 and remains nearly unchanged from step 10 onward. Consequently, the transition to the quenched state is inferred to start before step 5 and be completed by step 10. The reduction in the number of defects during the demagnetization process is compatible with the naive expectation based on the physical argument of a decrease in the number of defects associated with the increased spatial coherence of the stripes. Furthermore, the number of junctions and terminals are close, which is consistent with the topological argument that junctions and terminals are paired. Further quantification of the evolution of the labyrinthine patterns based on the spatial distribution of the defects (and not only on their number) is discussed in another article [6]. This paper is available on arxiv under CC BY 4.0 DEED license. Authors: (1) Vinicius Yu Okubo, B.S. in electrical engineering from the University of São Paulo in 2022 and currently, he is pursuing his M.S. in electrical engineering at the University of São Paulo; (2) Kotaro Shimizu, B.S. degree in Physics from Waseda University, Japan, in 2019 and M.S. degree in Physics from the University of Tokyo, Japan, 2021 and He has been pursuing his Ph.D. in Physics as a JSPS research fellowship for young scientists in the University of Tokyo since 2021; (3) B.S. Shivaram, received his B.S. degree in Physics, Chemistry and Mathematics from Bangalore University, India, in 1977 and the M.S. degree in Physics from the Indian Institute of Technology, Madras, India, in 1979 and his Ph.D. in experimental condensed matter physics from Northwestern University, Evanston, Illinois in 1984; (4) Hae Yong Kim, He received the B.S. and M.S. degrees (with distinctions) in computer science and the Ph.D. degree in electrical engineering from the Universidade de São Paulo (USP), Brazil, in 1988, 1992 and 1997, respectively. Authors: Authors: (1) Vinicius Yu Okubo, B.S. in electrical engineering from the University of São Paulo in 2022 and currently, he is pursuing his M.S. in electrical engineering at the University of São Paulo; (2) Kotaro Shimizu, B.S. degree in Physics from Waseda University, Japan, in 2019 and M.S. degree in Physics from the University of Tokyo, Japan, 2021 and He has been pursuing his Ph.D. in Physics as a JSPS research fellowship for young scientists in the University of Tokyo since 2021; (3) B.S. Shivaram, received his B.S. degree in Physics, Chemistry and Mathematics from Bangalore University, India, in 1977 and the M.S. degree in Physics from the Indian Institute of Technology, Madras, India, in 1979 and his Ph.D. in experimental condensed matter physics from Northwestern University, Evanston, Illinois in 1984; (4) Hae Yong Kim, He received the B.S. and M.S. degrees (with distinctions) in computer science and the Ph.D. degree in electrical engineering from the Universidade de São Paulo (USP), Brazil, in 1988, 1992 and 1997, respectively. Table of Links Abstract and I Introduction Abstract and I Introduction II. Related Works II. Related Works III Methodology III Methodology IV Experiments and Results IV Experiments and Results V Conclusion and References V Conclusion and References IV. EXPERIMENTS AND RESULTS A. PERFORMANCE EVALUATION A. PERFORMANCE EVALUATION Evaluating TM-CNN poses a certain challenge because there are no benchmarks or published results for direct comparison. So, we defined that junction and terminal detections reviewed by human observers as the gold standard and compared the algorithms against this standard. For testing purposes, we annotated 5 independent images from varying experiments, using the same procedure as the training images (subsection III-E). False detections were excluded since they do not contribute to the detection evaluation. In total, the test set contained 3,551 junctions and 3,653 terminals. B. RESULTS FROM A PHYSICS PERSPECTIVE B. RESULTS FROM A PHYSICS PERSPECTIVE In Bi:YIG films, it is known that a perfect stripe pattern is the most energetically favorable and stable configuration of magnetic moments [4]. In this structure, the magnetic moments exhibit stripes aligning straight in one direction throughout the entire structure, resulting in large spatial coherence. Meanwhile, the experimentally obtained structures often manifest labyrinthine patterns, as illustrated by dark and bright stripes in Fig. 1. In these labyrinthine patterns, stripes are aligned within small domains but propagate in different directions between different domains, and thus the spatial coherence of stripes is smaller than the perfect stripe. Notably, junctions and terminals are defects emerging in the formation of these labyrinthine patterns, intricately linked to the growth of spatial coherence in the system. Introducing such defects into the perfect stripe structure leads to the bending and branching of the initially aligned stripes, eventually reducing spatial coherence accompanied by the transformation to labyrinthine patterns. Indeed, a plethora of defects are observed in the labyrinthine patterns shown in Fig. 2. Conversely, the elimination of these defects plays a crucial role in the transition from a quenched state to an annealed state, where the system increases the spatial coherence. Each of these defects cannot be eliminated alone by continuous deformation of the stripes due to their topological properties. They can be removed through the pair annihilation. For instance, in Fig. 2, terminals are branched from junctions and they can be removed pairwisely by reducing the branching distance. These defects, known as topological defects, are widely recognized for playing a crucial role in the formation of periodic structures. By detecting the positions of defects, the evolution of the labyrinthine patterns would be systematically quantified. Specifically in the context of magnetic labyrinthine patterns, a much smaller dataset was used to detect these defects manually [4]. Recently, persistent homology has been used to extract the topological features of the labyrinthine patterns in a systematic way [38]. However, it is still difficult to detect the positions of defects and classify them into junctions and terminals. In this article, we achieved the systematic detection. To quantify the evolution of the labyrinthine patterns, we applied the TM-CNN technique to the 444 images and investigated how the number of defects and their locations changes step by step. Fig. 6 illustrates the step dependence of the number of junctions and terminals during the demagnetization process. Results from 6 different experimental runs were averaged to estimate the number of defects. In the initial stages of the demagnetization process, there are approximately 800 defects. The number of defects sharply decreases around step 5 and remains nearly unchanged from step 10 onward. Consequently, the transition to the quenched state is inferred to start before step 5 and be completed by step 10. The reduction in the number of defects during the demagnetization process is compatible with the naive expectation based on the physical argument of a decrease in the number of defects associated with the increased spatial coherence of the stripes. Furthermore, the number of junctions and terminals are close, which is consistent with the topological argument that junctions and terminals are paired. Further quantification of the evolution of the labyrinthine patterns based on the spatial distribution of the defects (and not only on their number) is discussed in another article [6]. 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

Efficient Detection of Defects in Magnetic Labyrinthine Patterns: Experiments and Results

Where does In-context Translation Happen in Large Language Models: Conclusion

10 More Patterns to Solve 1000 More LeetCode Problems - Part II

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