Graph Theory Intro Math Circle
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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). -
On the Archimedean Or Semiregular Polyhedra
ON THE ARCHIMEDEAN OR SEMIREGULAR POLYHEDRA Mark B. Villarino Depto. de Matem´atica, Universidad de Costa Rica, 2060 San Jos´e, Costa Rica May 11, 2005 Abstract We prove that there are thirteen Archimedean/semiregular polyhedra by using Euler’s polyhedral formula. Contents 1 Introduction 2 1.1 RegularPolyhedra .............................. 2 1.2 Archimedean/semiregular polyhedra . ..... 2 2 Proof techniques 3 2.1 Euclid’s proof for regular polyhedra . ..... 3 2.2 Euler’s polyhedral formula for regular polyhedra . ......... 4 2.3 ProofsofArchimedes’theorem. .. 4 3 Three lemmas 5 3.1 Lemma1.................................... 5 3.2 Lemma2.................................... 6 3.3 Lemma3.................................... 7 4 Topological Proof of Archimedes’ theorem 8 arXiv:math/0505488v1 [math.GT] 24 May 2005 4.1 Case1: fivefacesmeetatavertex: r=5. .. 8 4.1.1 At least one face is a triangle: p1 =3................ 8 4.1.2 All faces have at least four sides: p1 > 4 .............. 9 4.2 Case2: fourfacesmeetatavertex: r=4 . .. 10 4.2.1 At least one face is a triangle: p1 =3................ 10 4.2.2 All faces have at least four sides: p1 > 4 .............. 11 4.3 Case3: threefacesmeetatavertes: r=3 . ... 11 4.3.1 At least one face is a triangle: p1 =3................ 11 4.3.2 All faces have at least four sides and one exactly four sides: p1 =4 6 p2 6 p3. 12 4.3.3 All faces have at least five sides and one exactly five sides: p1 =5 6 p2 6 p3 13 1 5 Summary of our results 13 6 Final remarks 14 1 Introduction 1.1 Regular Polyhedra A polyhedron may be intuitively conceived as a “solid figure” bounded by plane faces and straight line edges so arranged that every edge joins exactly two (no more, no less) vertices and is a common side of two faces. -
Graph Theory
1 Graph Theory “Begin at the beginning,” the King said, gravely, “and go on till you come to the end; then stop.” — Lewis Carroll, Alice in Wonderland The Pregolya River passes through a city once known as K¨onigsberg. In the 1700s seven bridges were situated across this river in a manner similar to what you see in Figure 1.1. The city’s residents enjoyed strolling on these bridges, but, as hard as they tried, no residentof the city was ever able to walk a route that crossed each of these bridges exactly once. The Swiss mathematician Leonhard Euler learned of this frustrating phenomenon, and in 1736 he wrote an article [98] about it. His work on the “K¨onigsberg Bridge Problem” is considered by many to be the beginning of the field of graph theory. FIGURE 1.1. The bridges in K¨onigsberg. J.M. Harris et al., Combinatorics and Graph Theory , DOI: 10.1007/978-0-387-79711-3 1, °c Springer Science+Business Media, LLC 2008 2 1. Graph Theory At first, the usefulness of Euler’s ideas and of “graph theory” itself was found only in solving puzzles and in analyzing games and other recreations. In the mid 1800s, however, people began to realize that graphs could be used to model many things that were of interest in society. For instance, the “Four Color Map Conjec- ture,” introduced by DeMorgan in 1852, was a famous problem that was seem- ingly unrelated to graph theory. The conjecture stated that four is the maximum number of colors required to color any map where bordering regions are colored differently. -
Archimedean Solids
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2008 Archimedean Solids Anna Anderson University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Anderson, Anna, "Archimedean Solids" (2008). MAT Exam Expository Papers. 4. https://digitalcommons.unl.edu/mathmidexppap/4 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Archimedean Solids Anna Anderson In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor July 2008 2 Archimedean Solids A polygon is a simple, closed, planar figure with sides formed by joining line segments, where each line segment intersects exactly two others. If all of the sides have the same length and all of the angles are congruent, the polygon is called regular. The sum of the angles of a regular polygon with n sides, where n is 3 or more, is 180° x (n – 2) degrees. If a regular polygon were connected with other regular polygons in three dimensional space, a polyhedron could be created. In geometry, a polyhedron is a three- dimensional solid which consists of a collection of polygons joined at their edges. The word polyhedron is derived from the Greek word poly (many) and the Indo-European term hedron (seat). -
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 -
Graph Mining: Social Network Analysis and Information Diffusion
Graph Mining: Social network analysis and Information Diffusion Davide Mottin, Konstantina Lazaridou Hasso Plattner Institute Graph Mining course Winter Semester 2016 Lecture road Introduction to social networks Real word networks’ characteristics Information diffusion GRAPH MINING WS 2016 2 Paper presentations ▪ Link to form: https://goo.gl/ivRhKO ▪ Choose the date(s) you are available ▪ Choose three papers according to preference (there are three lists) ▪ Choose at least one paper for each course part ▪ Register to the mailing list: https://lists.hpi.uni-potsdam.de/listinfo/graphmining-ws1617 GRAPH MINING WS 2016 3 What is a social network? ▪Oxford dictionary A social network is a network of social interactions and personal relationships GRAPH MINING WS 2016 4 What is an online social network? ▪ Oxford dictionary An online social network (OSN) is a dedicated website or other application, which enables users to communicate with each other by posting information, comments, messages, images, etc. ▪ Computer science : A network that consists of a finite number of users (nodes) that interact with each other (edges) by sharing information (labels, signs, edge directions etc.) GRAPH MINING WS 2016 5 Definitions ▪Vertex/Node : a user of the social network (Twitter user, an author in a co-authorship network, an actor etc.) ▪Edge/Link/Tie : the connection between two vertices referring to their relationship or interaction (friend-of relationship, re-tweeting activity, etc.) Graph G=(V,E) symbols meaning V users E connections deg(u) node degree: -
1 Bipartite Matching and Vertex Covers
ORF 523 Lecture 6 Princeton University Instructor: A.A. Ahmadi Scribe: G. Hall Any typos should be emailed to a a [email protected]. In this lecture, we will cover an application of LP strong duality to combinatorial optimiza- tion: • Bipartite matching • Vertex covers • K¨onig'stheorem • Totally unimodular matrices and integral polytopes. 1 Bipartite matching and vertex covers Recall that a bipartite graph G = (V; E) is a graph whose vertices can be divided into two disjoint sets such that every edge connects one node in one set to a node in the other. Definition 1 (Matching, vertex cover). A matching is a disjoint subset of edges, i.e., a subset of edges that do not share a common vertex. A vertex cover is a subset of the nodes that together touch all the edges. (a) An example of a bipartite (b) An example of a matching (c) An example of a vertex graph (dotted lines) cover (grey nodes) Figure 1: Bipartite graphs, matchings, and vertex covers 1 Lemma 1. The cardinality of any matching is less than or equal to the cardinality of any vertex cover. This is easy to see: consider any matching. Any vertex cover must have nodes that at least touch the edges in the matching. Moreover, a single node can at most cover one edge in the matching because the edges are disjoint. As it will become clear shortly, this property can also be seen as an immediate consequence of weak duality in linear programming. Theorem 1 (K¨onig). If G is bipartite, the cardinality of the maximum matching is equal to the cardinality of the minimum vertex cover. -
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 Enduring Error
Version June 5, 2008 Branko Grünbaum: An enduring error 1. Introduction. Mathematical truths are immutable, but mathematicians do make errors, especially when carrying out non-trivial enumerations. Some of the errors are "innocent" –– plain mis- takes that get corrected as soon as an independent enumeration is carried out. For example, Daublebsky [14] in 1895 found that there are precisely 228 types of configurations (123), that is, collections of 12 lines and 12 points, each incident with three of the others. In fact, as found by Gropp [19] in 1990, the correct number is 229. Another example is provided by the enumeration of the uniform tilings of the 3-dimensional space by Andreini [1] in 1905; he claimed that there are precisely 25 types. However, as shown [20] in 1994, the correct num- ber is 28. Andreini listed some tilings that should not have been included, and missed several others –– but again, these are simple errors easily corrected. Much more insidious are errors that arise by replacing enumeration of one kind of ob- ject by enumeration of some other objects –– only to disregard the logical and mathematical distinctions between the two enumerations. It is surprising how errors of this type escape detection for a long time, even though there is frequent mention of the results. One example is provided by the enumeration of 4-dimensional simple polytopes with 8 facets, by Brückner [7] in 1909. He replaces this enumeration by that of 3-dimensional "diagrams" that he inter- preted as Schlegel diagrams of convex 4-polytopes, and claimed that the enumeration of these objects is equivalent to that of the polytopes. -
A Multigraph Approach to Social Network Analysis
1 Introduction Network data involving relational structure representing interactions between actors are commonly represented by graphs where the actors are referred to as vertices or nodes, and the relations are referred to as edges or ties connecting pairs of actors. Research on social networks is a well established branch of study and many issues concerning social network analysis can be found in Wasserman and Faust (1994), Carrington et al. (2005), Butts (2008), Frank (2009), Kolaczyk (2009), Scott and Carrington (2011), Snijders (2011), and Robins (2013). A common approach to social network analysis is to only consider binary relations, i.e. edges between pairs of vertices are either present or not. These simple graphs only consider one type of relation and exclude the possibility for self relations where a vertex is both the sender and receiver of an edge (also called edge loops or just shortly loops). In contrast, a complex graph is defined according to Wasserman and Faust (1994): If a graph contains loops and/or any pairs of nodes is adjacent via more than one line the graph is complex. [p. 146] In practice, simple graphs can be derived from complex graphs by collapsing the multiple edges into single ones and removing the loops. However, this approach discards information inherent in the original network. In order to use all available network information, we must allow for multiple relations and the possibility for loops. This leads us to the study of multigraphs which has not been treated as extensively as simple graphs in the literature. As an example, consider a network with vertices representing different branches of an organ- isation. -
5 Graph Theory
last edited March 21, 2016 5 Graph Theory Graph theory – the mathematical study of how collections of points can be con- nected – is used today to study problems in economics, physics, chemistry, soci- ology, linguistics, epidemiology, communication, and countless other fields. As complex networks play fundamental roles in financial markets, national security, the spread of disease, and other national and global issues, there is tremendous work being done in this very beautiful and evolving subject. The first several sections covered number systems, sets, cardinality, rational and irrational numbers, prime and composites, and several other topics. We also started to learn about the role that definitions play in mathematics, and we have begun to see how mathematicians prove statements called theorems – we’ve even proven some ourselves. At this point we turn our attention to a beautiful topic in modern mathematics called graph theory. Although this area was first introduced in the 18th century, it did not mature into a field of its own until the last fifty or sixty years. Over that time, it has blossomed into one of the most exciting, and practical, areas of mathematical research. Many readers will first associate the word ‘graph’ with the graph of a func- tion, such as that drawn in Figure 4. Although the word graph is commonly Figure 4: The graph of a function y = f(x). used in mathematics in this sense, it is also has a second, unrelated, meaning. Before providing a rigorous definition that we will use later, we begin with a very rough description and some examples.