Centrality in Networks: Finding the Most Important Nodes
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Directed Graph Algorithms
Directed Graph Algorithms CSE 373 Data Structures Readings • Reading Chapter 13 › Sections 13.3 and 13.4 Digraphs 2 Topological Sort 142 322 143 321 Problem: Find an order in which all these courses can 326 be taken. 370 341 Example: 142 Æ 143 Æ 378 Æ 370 Æ 321 Æ 341 Æ 322 Æ 326 Æ 421 Æ 401 378 421 In order to take a course, you must 401 take all of its prerequisites first Digraphs 3 Topological Sort Given a digraph G = (V, E), find a linear ordering of its vertices such that: for any edge (v, w) in E, v precedes w in the ordering B C A F D E Digraphs 4 Topo sort – valid solution B C Any linear ordering in which A all the arrows go to the right F is a valid solution D E A B F C D E Note that F can go anywhere in this list because it is not connected. Also the solution is not unique. Digraphs 5 Topo sort – invalid solution B C A Any linear ordering in which an arrow goes to the left F is not a valid solution D E A B F C E D NO! Digraphs 6 Paths and Cycles • Given a digraph G = (V,E), a path is a sequence of vertices v1,v2, …,vk such that: ›(vi,vi+1) in E for 1 < i < k › path length = number of edges in the path › path cost = sum of costs of each edge • A path is a cycle if : › k > 1; v1 = vk •G is acyclic if it has no cycles. -
Approximating Network Centrality Measures Using Node Embedding and Machine Learning
Approximating Network Centrality Measures Using Node Embedding and Machine Learning Matheus R. F. Mendon¸ca,Andr´eM. S. Barreto, and Artur Ziviani ∗† Abstract Extracting information from real-world large networks is a key challenge nowadays. For instance, computing a node centrality may become unfeasible depending on the intended centrality due to its computational cost. One solution is to develop fast methods capable of approximating network centralities. Here, we propose an approach for efficiently approximating node centralities for large networks using Neural Networks and Graph Embedding techniques. Our proposed model, entitled Network Centrality Approximation using Graph Embedding (NCA-GE), uses the adjacency matrix of a graph and a set of features for each node (here, we use only the degree) as input and computes the approximate desired centrality rank for every node. NCA-GE has a time complexity of O(jEj), E being the set of edges of a graph, making it suitable for large networks. NCA-GE also trains pretty fast, requiring only a set of a thousand small synthetic scale-free graphs (ranging from 100 to 1000 nodes each), and it works well for different node centralities, network sizes, and topologies. Finally, we compare our approach to the state-of-the-art method that approximates centrality ranks using the degree and eigenvector centralities as input, where we show that the NCA-GE outperforms the former in a variety of scenarios. 1 Introduction Networks are present in several real-world applications spread among different disciplines, such as biology, mathematics, sociology, and computer science, just to name a few. Therefore, network analysis is a crucial tool for extracting relevant information. -
Graph Varieties Axiomatized by Semimedial, Medial, and Some Other Groupoid Identities
Discussiones Mathematicae General Algebra and Applications 40 (2020) 143–157 doi:10.7151/dmgaa.1344 GRAPH VARIETIES AXIOMATIZED BY SEMIMEDIAL, MEDIAL, AND SOME OTHER GROUPOID IDENTITIES Erkko Lehtonen Technische Universit¨at Dresden Institut f¨ur Algebra 01062 Dresden, Germany e-mail: [email protected] and Chaowat Manyuen Department of Mathematics, Faculty of Science Khon Kaen University Khon Kaen 40002, Thailand e-mail: [email protected] Abstract Directed graphs without multiple edges can be represented as algebras of type (2, 0), so-called graph algebras. A graph is said to satisfy an identity if the corresponding graph algebra does, and the set of all graphs satisfying a set of identities is called a graph variety. We describe the graph varieties axiomatized by certain groupoid identities (medial, semimedial, autodis- tributive, commutative, idempotent, unipotent, zeropotent, alternative). Keywords: graph algebra, groupoid, identities, semimediality, mediality. 2010 Mathematics Subject Classification: 05C25, 03C05. 1. Introduction Graph algebras were introduced by Shallon [10] in 1979 with the purpose of providing examples of nonfinitely based finite algebras. Let us briefly recall this concept. Given a directed graph G = (V, E) without multiple edges, the graph algebra associated with G is the algebra A(G) = (V ∪ {∞}, ◦, ∞) of type (2, 0), 144 E. Lehtonen and C. Manyuen where ∞ is an element not belonging to V and the binary operation ◦ is defined by the rule u, if (u, v) ∈ E, u ◦ v := (∞, otherwise, for all u, v ∈ V ∪ {∞}. We will denote the product u ◦ v simply by juxtaposition uv. Using this representation, we may view any algebraic property of a graph algebra as a property of the graph with which it is associated. -
Networkx: Network Analysis with Python
NetworkX: Network Analysis with Python Salvatore Scellato Full tutorial presented at the XXX SunBelt Conference “NetworkX introduction: Hacking social networks using the Python programming language” by Aric Hagberg & Drew Conway Outline 1. Introduction to NetworkX 2. Getting started with Python and NetworkX 3. Basic network analysis 4. Writing your own code 5. You are ready for your project! 1. Introduction to NetworkX. Introduction to NetworkX - network analysis Vast amounts of network data are being generated and collected • Sociology: web pages, mobile phones, social networks • Technology: Internet routers, vehicular flows, power grids How can we analyze this networks? Introduction to NetworkX - Python awesomeness Introduction to NetworkX “Python package for the creation, manipulation and study of the structure, dynamics and functions of complex networks.” • Data structures for representing many types of networks, or graphs • Nodes can be any (hashable) Python object, edges can contain arbitrary data • Flexibility ideal for representing networks found in many different fields • Easy to install on multiple platforms • Online up-to-date documentation • First public release in April 2005 Introduction to NetworkX - design requirements • Tool to study the structure and dynamics of social, biological, and infrastructure networks • Ease-of-use and rapid development in a collaborative, multidisciplinary environment • Easy to learn, easy to teach • Open-source tool base that can easily grow in a multidisciplinary environment with non-expert users -
1 Hamiltonian Path
6.S078 Fine-Grained Algorithms and Complexity MIT Lecture 17: Algorithms for Finding Long Paths (Part 1) November 2, 2020 In this lecture and the next, we will introduce a number of algorithmic techniques used in exponential-time and FPT algorithms, through the lens of one parametric problem: Definition 0.1 (k-Path) Given a directed graph G = (V; E) and parameter k, is there a simple path1 in G of length ≥ k? Already for this simple-to-state problem, there are quite a few radically different approaches to solving it faster; we will show you some of them. We’ll see algorithms for the case of k = n (Hamiltonian Path) and then we’ll turn to “parameterizing” these algorithms so they work for all k. A number of papers in bioinformatics have used quick algorithms for k-Path and related problems to analyze various networks that arise in biology (some references are [SIKS05, ADH+08, YLRS+09]). In the following, we always denote the number of vertices jV j in our given graph G = (V; E) by n, and the number of edges jEj by m. We often associate the set of vertices V with the set [n] := f1; : : : ; ng. 1 Hamiltonian Path Before discussing k-Path, it will be useful to first discuss algorithms for the famous NP-complete Hamiltonian path problem, which is the special case where k = n. Essentially all algorithms we discuss here can be adapted to obtain algorithms for k-Path! The naive algorithm for Hamiltonian Path takes time about n! = 2Θ(n log n) to try all possible permutations of the nodes (which can also be adapted to get an O?(k!)-time algorithm for k-Path, as we’ll see). -
Emerging Topics in Internet Technology: a Complex Networks Approach
1 Emerging Topics in Internet Technology: A Complex Networks Approach Bisma S. Khan 1, Muaz A. Niazi *,2 COMSATS Institute of Information Technology, Islamabad, Pakistan Abstract —Communication networks, in general, and internet technology, in particular, is a fast-evolving area of research. While it is important to keep track of emerging trends in this domain, it is such a fast-growing area that it can be very difficult to keep track of literature. The problem is compounded by the fast-growing number of citation databases. While other databases are gradually indexing a large set of reliable content, currently the Web of Science represents one of the most highly valued databases. Research indexed in this database is known to highlight key advancements in any domain. In this paper, we present a Complex Network-based analytical approach to analyze recent data from the Web of Science in communication networks. Taking bibliographic records from the recent period of 2014 to 2017 , we model and analyze complex scientometric networks. Using bibliometric coupling applied over complex citation data we present answers to co-citation patterns of documents, co-occurrence patterns of terms, as well as the most influential articles, among others, We also present key pivot points and intellectual turning points. Complex network analysis of the data demonstrates a considerably high level of interest in two key clusters labeled descriptively as “social networks” and “computer networks”. In addition, key themes in highly cited literature were clearly identified as “communication networks,” “social networks,” and “complex networks”. Index Terms —CiteSpace, Communication Networks, Network Science, Complex Networks, Visualization, Data science. -
Network Centrality) (100 Points)
In the name of God. Sharif University of Technology Analysis of Biological Networks CE 558 Spring 2020 Dr. H.R. Rabiee Homework 3 (Network Centrality) (100 points) 1. Compute centrality of the nodes with the most and least centrality for the following network according to the following measures: • Degree • Eccentricity • Closeness • Shortest Path Betweenness 1 2. For a complete (m,n)-bipartite graph compute Katz centrality measure with α = 2mn for each node and determine which nodes have the most centrality. ( (m, n)-bipartite graph consists of two independent partitions with m and n nodes each) 3. (a) For a cycle graph prove that we don't have a central node. (The centrality of all nodes are the same). Prove it for following centrality measures. • Degree • Eccentricity • Closeness • Shortest Path Betweenness • Katz • PageRank (b) Prove that in a graph that has a full-cycle automorphism, there is no central measure. Prove it for an arbitrary centrality measure. (Note that an appropriate centrality measure only depends on the structure of the graph and not the node labels) (c) for n > 2, find a graph with n nodes that has an automorphism but the centrality of the nodes are not all equal for some measure. 1 4. Prove that for any d-regular graph, PageRank centrality measure approaches to nm CKatz as the number of steps approaches to infinity, where n is the number of nodes, m is the number of steps and CKatz is 1 the Katz centrality measure with α = d . 5. (Fast Algorithm to Calculate Shortest-Path-Betweenness Centrality Measure) Consider an arbitrary undirected graph G = (V; E). -
Measuring Homophily
Measuring Homophily Matteo Cristani, Diana Fogoroasi, and Claudio Tomazzoli University of Verona fmatteo.cristani, diana.fogoroasi.studenti, [email protected] Abstract. Social Network Analysis is employed widely as a means to compute the probability that a given message flows through a social net- work. This approach is mainly grounded upon the correct usage of three basic graph- theoretic measures: degree centrality, closeness centrality and betweeness centrality. We show that, in general, those indices are not adapt to foresee the flow of a given message, that depends upon indices based on the sharing of interests and the trust about depth in knowledge of a topic. We provide new definitions for measures that over- come the drawbacks of general indices discussed above, using Semantic Social Network Analysis, and show experimental results that show that with these measures we have a different understanding of a social network compared to standard measures. 1 Introduction Social Networks are considered, on the current panorama of web applications, as the principal virtual space for online communication. Therefore, it is of strong relevance for practical applications to understand how strong a member of the network is with respect to the others. Traditionally, sociological investigations have dealt with problems of defining properties of the users that can value their relevance (sometimes their impor- tance, that can be considered different, the first denoting the ability to emerge, and the second the relevance perceived by the others). Scholars have developed several measures and studied how to compute them in different types of graphs, used as models for social networks. This field of research has been named Social Network Analysis. -
Modeling Customer Preferences Using Multidimensional Network Analysis in Engineering Design
Modeling customer preferences using multidimensional network analysis in engineering design Mingxian Wang1, Wei Chen1, Yun Huang2, Noshir S. Contractor2 and Yan Fu3 1 Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA 2 Science of Networks in Communities, Northwestern University, Evanston, IL 60208, USA 3 Global Data Insight and Analytics, Ford Motor Company, Dearborn, MI 48121, USA Abstract Motivated by overcoming the existing utility-based choice modeling approaches, we present a novel conceptual framework of multidimensional network analysis (MNA) for modeling customer preferences in supporting design decisions. In the proposed multidimensional customer–product network (MCPN), customer–product interactions are viewed as a socio-technical system where separate entities of `customers' and `products' are simultaneously modeled as two layers of a network, and multiple types of relations, such as consideration and purchase, product associations, and customer social interactions, are considered. We first introduce a unidimensional network where aggregated customer preferences and product similarities are analyzed to inform designers about the implied product competitions and market segments. We then extend the network to a multidimensional structure where customer social interactions are introduced for evaluating social influence on heterogeneous product preferences. Beyond the traditional descriptive analysis used in network analysis, we employ the exponential random graph model (ERGM) as a unified statistical -
Models for Networks with Consumable Resources: Applications to Smart Cities Hayato Montezuma Ushijima-Mwesigwa Clemson University, [email protected]
Clemson University TigerPrints All Dissertations Dissertations 12-2018 Models for Networks with Consumable Resources: Applications to Smart Cities Hayato Montezuma Ushijima-Mwesigwa Clemson University, [email protected] Follow this and additional works at: https://tigerprints.clemson.edu/all_dissertations Recommended Citation Ushijima-Mwesigwa, Hayato Montezuma, "Models for Networks with Consumable Resources: Applications to Smart Cities" (2018). All Dissertations. 2284. https://tigerprints.clemson.edu/all_dissertations/2284 This Dissertation is brought to you for free and open access by the Dissertations at TigerPrints. It has been accepted for inclusion in All Dissertations by an authorized administrator of TigerPrints. For more information, please contact [email protected]. Models for Networks with Consumable Resources: Applications to Smart Cities A Dissertation Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Computer Science by Hayato Ushijima-Mwesigwa December 2018 Accepted by: Dr. Ilya Safro, Committee Chair Dr. Mashrur Chowdhury Dr. Brian Dean Dr. Feng Luo Abstract In this dissertation, we introduce different models for understanding and controlling the spreading dynamics of a network with a consumable resource. In particular, we consider a spreading process where a resource necessary for transit is partially consumed along the way while being refilled at special nodes on the network. Examples include fuel consumption of vehicles together with refueling stations, information loss during dissemination with error correcting nodes, consumption of ammunition of military troops while moving, and migration of wild animals in a network with a limited number of water-holes. We undertake this study from two different perspectives. First, we consider a network science perspective where we are interested in identifying the influential nodes, and estimating a nodes’ relative spreading influence in the network. -
RESEARCH PAPER . Text Information Aggregation with Centrality Attention
SCIENCE CHINA Information Sciences . RESEARCH PAPER . Text Information Aggregation with Centrality Attention Jingjing Gong, Hang Yan, Yining Zheng, Qipeng Guo, Xipeng Qiu* & Xuanjing Huang School of Computer Science, Fudan University, Shanghai 200433, China Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China fjjgong, hyan11, ynzheng15, qpguo16, xpqiu, [email protected] Abstract A lot of natural language processing problems need to encode the text sequence as a fix-length vector, which usually involves aggregation process of combining the representations of all the words, such as pooling or self-attention. However, these widely used aggregation approaches did not take higher-order relationship among the words into consideration. Hence we propose a new way of obtaining aggregation weights, called eigen-centrality self-attention. More specifically, we build a fully-connected graph for all the words in a sentence, then compute the eigen-centrality as the attention score of each word. The explicit modeling of relationships as a graph is able to capture some higher-order dependency among words, which helps us achieve better results in 5 text classification tasks and one SNLI task than baseline models such as pooling, self-attention and dynamic routing. Besides, in order to compute the dominant eigenvector of the graph, we adopt power method algorithm to get the eigen-centrality measure. Moreover, we also derive an iterative approach to get the gradient for the power method process to reduce both memory consumption and computation requirement. Keywords Information Aggregation, Eigen Centrality, Text Classification, NLP, Deep Learning Citation Jingjing Gong, Hang Yan, Yining Zheng, Qipeng Guo, Xipeng Qiu, Xuanjing Huang. -
Received Citations As a Main SEO Factor of Google Scholar Results Ranking
RECEIVED CITATIONS AS A MAIN SEO FACTOR OF GOOGLE SCHOLAR RESULTS RANKING Las citas recibidas como principal factor de posicionamiento SEO en la ordenación de resultados de Google Scholar Cristòfol Rovira, Frederic Guerrero-Solé and Lluís Codina Nota: Este artículo se puede leer en español en: http://www.elprofesionaldelainformacion.com/contenidos/2018/may/09_esp.pdf Cristòfol Rovira, associate professor at Pompeu Fabra University (UPF), teaches in the Depart- ments of Journalism and Advertising. He is director of the master’s degree in Digital Documenta- tion (UPF) and the master’s degree in Search Engines (UPF). He has a degree in Educational Scien- ces, as well as in Library and Information Science. He is an engineer in Computer Science and has a master’s degree in Free Software. He is conducting research in web positioning (SEO), usability, search engine marketing and conceptual maps with eyetracking techniques. https://orcid.org/0000-0002-6463-3216 [email protected] Frederic Guerrero-Solé has a bachelor’s in Physics from the University of Barcelona (UB) and a PhD in Public Communication obtained at Universitat Pompeu Fabra (UPF). He has been teaching at the Faculty of Communication at the UPF since 2008, where he is a lecturer in Sociology of Communi- cation. He is a member of the research group Audiovisual Communication Research Unit (Unica). https://orcid.org/0000-0001-8145-8707 [email protected] Lluís Codina is an associate professor in the Department of Communication at the School of Com- munication, Universitat Pompeu Fabra (UPF), Barcelona, Spain, where he has taught information science courses in the areas of Journalism and Media Studies for more than 25 years.