An Optimal Path Cover Algorithm for Cographs R
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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). -
K-Path Centrality: a New Centrality Measure in Social Networks
Centrality Metrics in Social Network Analysis K-path: A New Centrality Metric Experiments Summary K-Path Centrality: A New Centrality Measure in Social Networks Adriana Iamnitchi University of South Florida joint work with Tharaka Alahakoon, Rahul Tripathi, Nicolas Kourtellis and Ramanuja Simha Adriana Iamnitchi K-Path Centrality: A New Centrality Measure in Social Networks 1 of 23 Centrality Metrics in Social Network Analysis Centrality Metrics Overview K-path: A New Centrality Metric Betweenness Centrality Experiments Applications Summary Computing Betweenness Centrality Centrality Metrics in Social Network Analysis Betweenness Centrality - how much a node controls the flow between any other two nodes Closeness Centrality - the extent a node is near all other nodes Degree Centrality - the number of ties to other nodes Eigenvector Centrality - the relative importance of a node Adriana Iamnitchi K-Path Centrality: A New Centrality Measure in Social Networks 2 of 23 Centrality Metrics in Social Network Analysis Centrality Metrics Overview K-path: A New Centrality Metric Betweenness Centrality Experiments Applications Summary Computing Betweenness Centrality Betweenness Centrality measures the extent to which a node lies on the shortest path between two other nodes betweennes CB (v) of a vertex v is the summation over all pairs of nodes of the fractional shortest paths going through v. Definition (Betweenness Centrality) For every vertex v 2 V of a weighted graph G(V ; E), the betweenness centrality CB (v) of v is defined by X X σst (v) CB -
The On-Line Shortest Path Problem Under Partial Monitoring
Journal of Machine Learning Research 8 (2007) 2369-2403 Submitted 4/07; Revised 8/07; Published 10/07 The On-Line Shortest Path Problem Under Partial Monitoring Andras´ Gyor¨ gy [email protected] Machine Learning Research Group Computer and Automation Research Institute Hungarian Academy of Sciences Kende u. 13-17, Budapest, Hungary, H-1111 Tamas´ Linder [email protected] Department of Mathematics and Statistics Queen’s University, Kingston, Ontario Canada K7L 3N6 Gabor´ Lugosi [email protected] ICREA and Department of Economics Universitat Pompeu Fabra Ramon Trias Fargas 25-27 08005 Barcelona, Spain Gyor¨ gy Ottucsak´ [email protected] Department of Computer Science and Information Theory Budapest University of Technology and Economics Magyar Tudosok´ Kor¨ utja´ 2. Budapest, Hungary, H-1117 Editor: Leslie Pack Kaelbling Abstract The on-line shortest path problem is considered under various models of partial monitoring. Given a weighted directed acyclic graph whose edge weights can change in an arbitrary (adversarial) way, a decision maker has to choose in each round of a game a path between two distinguished vertices such that the loss of the chosen path (defined as the sum of the weights of its composing edges) be as small as possible. In a setting generalizing the multi-armed bandit problem, after choosing a path, the decision maker learns only the weights of those edges that belong to the chosen path. For this problem, an algorithm is given whose average cumulative loss in n rounds exceeds that of the best path, matched off-line to the entire sequence of the edge weights, by a quantity that is proportional to 1=pn and depends only polynomially on the number of edges of the graph. -
An Efficient Reachability Indexing Scheme for Large Directed Graphs
1 Path-Tree: An Efficient Reachability Indexing Scheme for Large Directed Graphs RUOMING JIN, Kent State University NING RUAN, Kent State University YANG XIANG, The Ohio State University HAIXUN WANG, Microsoft Research Asia Reachability query is one of the fundamental queries in graph database. The main idea behind answering reachability queries is to assign vertices with certain labels such that the reachability between any two vertices can be determined by the labeling information. Though several approaches have been proposed for building these reachability labels, it remains open issues on how to handle increasingly large number of vertices in real world graphs, and how to find the best tradeoff among the labeling size, the query answering time, and the construction time. In this paper, we introduce a novel graph structure, referred to as path- tree, to help labeling very large graphs. The path-tree cover is a spanning subgraph of G in a tree shape. We show path-tree can be generalized to chain-tree which theoretically can has smaller labeling cost. On top of path-tree and chain-tree index, we also introduce a new compression scheme which groups vertices with similar labels together to further reduce the labeling size. In addition, we also propose an efficient incremental update algorithm for dynamic index maintenance. Finally, we demonstrate both analytically and empirically the effectiveness and efficiency of our new approaches. Categories and Subject Descriptors: H.2.8 [Database management]: Database Applications—graph index- ing and querying General Terms: Performance Additional Key Words and Phrases: Graph indexing, reachability queries, transitive closure, path-tree cover, maximal directed spanning tree ACM Reference Format: Jin, R., Ruan, N., Xiang, Y., and Wang, H. -
PDF Reference
NetworkX Reference Release 1.0 Aric Hagberg, Dan Schult, Pieter Swart January 08, 2010 CONTENTS 1 Introduction 1 1.1 Who uses NetworkX?..........................................1 1.2 The Python programming language...................................1 1.3 Free software...............................................1 1.4 Goals...................................................1 1.5 History..................................................2 2 Overview 3 2.1 NetworkX Basics.............................................3 2.2 Nodes and Edges.............................................4 3 Graph types 9 3.1 Which graph class should I use?.....................................9 3.2 Basic graph types.............................................9 4 Operators 129 4.1 Graph Manipulation........................................... 129 4.2 Node Relabeling............................................. 134 4.3 Freezing................................................. 135 5 Algorithms 137 5.1 Boundary................................................. 137 5.2 Centrality................................................. 138 5.3 Clique.................................................. 142 5.4 Clustering................................................ 145 5.5 Cores................................................... 147 5.6 Matching................................................. 148 5.7 Isomorphism............................................... 148 5.8 PageRank................................................. 161 5.9 HITS.................................................. -
Single-Source Bottleneck Path Algorithm Faster Than Sorting For
Single-Source Bottleneck Path Algorithm Faster than Sorting for Sparse Graphs 1 1 1 1 Ran Duan ∗ , Kaifeng Lyu † , Hongxun Wu ‡ , and Yuanhang Xie § 1Institute for Interdisciplinary Information Sciences, Tsinghua University Abstract In a directed graph G = (V, E) with a capacity on every edge, a bottleneck path (or widest path) between two vertices is a path maximizing the minimum capacity of edges in the path. For the single-source all-destination version of this problem in directed graphs, the previous best algorithm runs in O(m+n log n) (m = E and n = V ) time, by Dijkstra search with Fibonacci heap [Fredman and Tarjan 1987]. We| improve| this| time| bound to O(m√log n), thus it is the first algorithm which breaks the time bound of classic Fibonacci heap when m = o(n√log n). It is a Las-Vegas randomized approach. By contrast, the s-t bottleneck path has an algorithm with k running time O(mβ(m,n)) [Chechik et al. 2016], where β(m,n) = min k 1 : log( ) n m . { ≥ ≤ n } 1 Introduction The bottleneck path problem is a graph optimization problem finding a path between two ver- tices with the maximum flow, in which the flow of a path is defined as the minimum capacity of edges on that path. The bottleneck problem can be seen as a mathematical formulation of many network routing problems, e.g. finding the route with the maximum transmission speed between two nodes in a network, and it has many other applications such as digital compositing [FGA98]. It is also the important building block of other algorithms, such as the improved Ford-Fulkerson algorithm [EK72, FF56], and k-splittable flow algorithm [BKS02]. -
Dynamic Programming
Dynamic Programming Comp 122, Fall 2004 Review: the previous lecture • Principles of dynamic programming: optimization problems, optimal substructure property, overlapping subproblems, trade space for time, implementation via bottom-up/memoization. • Example of Match Game. • Example of Fibinacci: from Divide and Conquer (exponential time complexity) to Dynamic Programming (O(n) time and space) , then to more efficient dynamic programming (constant space, O(logn) time) • Example of LCS in O(m n) time and space. Today we will show how to reduce the space complexity of LCS to O(n) and in future lectures we will show how to reduce the time complexity of LCS. Comp 122, Spring 2004 Longest Common Subsequence • Problem: Given 2 sequences, X = x1,...,xm and Y = y1,...,yn, find a common subsequence whose length is maximum. X: springtime Y: printing LCS(X,Y): printi Comp 122, Spring 2004 0 if empty or empty, c[, ] c[ prefix, prefix ]1 if end() end( ), max(c[ prefix, ],c[, prefix ]) if end() end( ). p r i n t i n g •Keep track of c[,] in a table of nm 0 0 0 0 0 0 0 0 0 entries: s 0 0 0 0 0 0 0 0 0 •top/down: increasing row order p 0 1 1 1 1 1 1 1 1 r 0 1 2 2 •within each row left-to-right: increasing column order i n g t i m Comp 122,e Spring 2004 0 if empty or empty, c[, ] c[ prefix, prefix ]1 if end() end( ), max(c[ prefix, ],c[, prefix ]) if end() end( ). -
Longest Path in a Directed Acyclic Graph (DAG)
Longest path in a directed acyclic graph (DAG) Mumit Khan CSE 221 April 10, 2011 The longest path problem is the problem of finding a simple path of maximal length in a graph; in other words, among all possible simple paths in the graph, the problem is to find the longest one. For an unweighted graph, it suffices to find the longest path in terms of the number of edges; for a weighted graph, one must use the edge weights instead. To explain the process of developing this algorithm, we’ll first develop an algorithm for computing the single-source longest path for an unweighted directed acyclic graph (DAG), and then generalize that to compute the longest path in a DAG, both unweighted or weighted. We’ve already seen how to compute the single-source shortest path in a graph, cylic or acyclic — we used BFS to compute the single-source shortest paths for an unweighted graph, and used Dijkstra (non-negative edge weights only) or Bellman-Ford (negative edge weights allowed) for a weighted graph without negative cycles. If we needed the shortest path between all pairs, we could always run the single-source shortest path algorithm using each vertex as the source; but of course there is Floyd-Warshall ready to do just that. The use of dynamic programming or a greedy algorithm is made possible by the fact the shortest path problem has optimal substructure — the subpaths in the shortest path between two vertices must themselves be the shortest ones! When it comes to finding the longest path however, we find that the problem does not have optimal substructure, and we lose the ability to use dynamic programming (and of course greedy strategy as well!). -
Depth-First Search, Topological Sort
And, for the hous is crinkled to and fro, And hath so queinte weyes for to go— For hit is shapen as the mase is wroght— Therto have I a remedie in my thoght, That, by a clewe of twyne, as he hath goon, The same wey he may returne anoon, Folwing alwey the threed, as he hath come. — Geoffrey Chaucer, The Legend of Good Women (c. 1385) Una lámpara ilustraba el andén, pero las caras de los niños quedaban en la zona de la sombra. Uno me interrogó: ¿Usted va a casa del doctor Stephen Albert?. Sin aguardar contestación, otro dijo: La casa queda lejos de aquí, pero usted no se perderá si toma ese camino a la izquierda y en cada encrucijada del camino dobla a la izquierda. — Jorge Luis Borges, “El jardín de senderos que se bifurcan” (1941) “Com’è bello il mondo e come sono brutti i labirinti!” dissi sollevato. “Come sarebbe bello il mondo se ci fosse una regola per girare nei labirinti,” rispose il mio maestro. — Umberto Eco, Il nome della rosa (1980) Chapter 6 Depth-First Search Status: Beta. Could add more, but don’t. In the previous chapter, we considered a generic algorithm—whatever-first search— for traversing arbitrary graphs, both undirected and directed. In this chapter, we focus on a particular instantiation of this algorithm called depth-first search, and primarily on the behavior of this algorithm in directed graphs. Rather than using an explicit stack, depth-first search is normally implemented recursively as follows: DFS(v): if v is unmarked mark v for each edge v w DFS(w) We can make this algorithm slightly faster (in practice) by checking whether a node is marked before we recursively explore it. -
Generalized Pseudoforest Deletion: Algorithms and Uniform Kernel⋆
Generalized Pseudoforest Deletion: Algorithms and Uniform Kernel? Geevarghese Philip1, Ashutosh Rai2, Saket Saurabh2;3?? 1 Chennai Mathematical Institute, Chennai, India. [email protected] 2 The Institute of Mathematical Sciences, HBNI, Chennai, India. fashutosh|[email protected] 3 University of Bergen, Bergen, Norway. Abstract. Feedback Vertex Set (FVS) is one of the most well stud- ied problems in the realm of parameterized complexity. In this problem we are given a graph G and a positive integer k and the objective is to test whether there exists S ⊆ V (G) of size at most k such that G − S is a forest. Thus, FVS is about deleting as few vertices as possible to get a forest. The main goal of this paper is to study the following interesting problem: How can we generalize the family of forests such that the nice structural properties of forests and the interesting algorithmic proper- ties of FVS can be extended to problems on this class? Towards this we define a graph class, Fl, that contains all graphs where each connected component can transformed into forest by deleting at most l edges. The class F1 is known as pseudoforest in the literature and we call Fl as l-pseudoforest. We study the problem of deleting k vertices to get into Fl, l-pseudoforest Deletion, in the realm of parameterized complex- ity. We show that l-pseudoforest Deletion admits an algorithm with k O(1) 2 running time cl n and admits a kernel of size f(l)k . Thus, for every fixed l we have a kernel of size O(k2). -
Measures of Centrality
Measures of centrality Measures of centrality Background Centrality Complex Networks measures Degree centrality CSYS/MATH 303, Spring, 2011 Closeness centrality Betweenness Eigenvalue centrality Hubs and Authorities Prof. Peter Dodds References Department of Mathematics & Statistics Center for Complex Systems Vermont Advanced Computing Center University of Vermont Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. 1 of 28 Measures of Outline centrality Background Centrality measures Degree centrality Closeness centrality Background Betweenness Eigenvalue centrality Hubs and Authorities References Centrality measures Degree centrality Closeness centrality Betweenness Eigenvalue centrality Hubs and Authorities References 2 of 28 Measures of How big is my node? centrality Background Centrality measures I Basic question: how ‘important’ are specific nodes Degree centrality Closeness centrality and edges in a network? Betweenness Eigenvalue centrality I An important node or edge might: Hubs and Authorities 1. handle a relatively large amount of the network’s References traffic (e.g., cars, information); 2. bridge two or more distinct groups (e.g., liason, interpreter); 3. be a source of important ideas, knowledge, or judgments (e.g., supreme court decisions, an employee who ‘knows where everything is’). I So how do we quantify such a slippery concept as importance? I We generate ad hoc, reasonable measures, and examine their utility... 3 of 28 Measures of Centrality centrality Background Centrality measures Degree centrality I One possible reflection of importance is centrality. Closeness centrality Betweenness I Presumption is that nodes or edges that are (in some Eigenvalue centrality Hubs and Authorities sense) in the middle of a network are important for References the network’s function.