Annotated Hypergraphs: Models and Applications
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From Static to Temporal Network Theory: Applications to Functional Brain Connectivity
METHODS From static to temporal network theory: Applications to functional brain connectivity William Hedley Thompson1, Per Brantefors1, and Peter Fransson1 1Department of Clinical Neuroscience, Karolinska Institutet, Stockholm, Sweden Keywords: Resting-state, Temporal network theory, Temporal networks, Functional connectome, Dynamic functional connectivity Downloaded from http://direct.mit.edu/netn/article-pdf/1/2/69/1091947/netn_a_00011.pdf by guest on 28 September 2021 ABSTRACT an open access journal Network neuroscience has become an established paradigm to tackle questions related to the functional and structural connectome of the brain. Recently, interest has been growing in examining the temporal dynamics of the brain’s network activity. Although different approaches to capturing fluctuations in brain connectivity have been proposed, there have been few attempts to quantify these fluctuations using temporal network theory. This theory is an extension of network theory that has been successfully applied to the modeling of dynamic processes in economics, social sciences, and engineering article but it has not been adopted to a great extent within network neuroscience. The objective of this article is twofold: (i) to present a detailed description of the central tenets of temporal network theory and describe its measures, and; (ii) to apply these measures to a resting-state fMRI dataset to illustrate their utility. Furthermore, we discuss the interpretation of temporal network theory in the context of the dynamic functional brain connectome. All the temporal network measures and plotting functions described in this article are freely available as the Python Citation: Thompson, W. H., Brantefors, package Teneto. P., & Fransson, P. (2017). From static to temporal network theory: Applications to functional brain connectivity. -
Optimal Subgraph Structures in Scale-Free Configuration
Submitted to the Annals of Applied Probability OPTIMAL SUBGRAPH STRUCTURES IN SCALE-FREE CONFIGURATION MODELS , By Remco van der Hofstad∗ §, Johan S. H. van , , Leeuwaarden∗ ¶ and Clara Stegehuis∗ k Eindhoven University of Technology§, Tilburg University¶ and Twente Universityk Subgraphs reveal information about the geometry and function- alities of complex networks. For scale-free networks with unbounded degree fluctuations, we obtain the asymptotics of the number of times a small connected graph occurs as a subgraph or as an induced sub- graph. We obtain these results by analyzing the configuration model with degree exponent τ (2, 3) and introducing a novel class of op- ∈ timization problems. For any given subgraph, the unique optimizer describes the degrees of the vertices that together span the subgraph. We find that subgraphs typically occur between vertices with specific degree ranges. In this way, we can count and characterize all sub- graphs. We refrain from double counting in the case of multi-edges, essentially counting the subgraphs in the erased configuration model. 1. Introduction Scale-free networks often have degree distributions that follow power laws with exponent τ (2, 3) [1, 11, 22, 34]. Many net- ∈ works have been reported to satisfy these conditions, including metabolic networks, the internet and social networks. Scale-free networks come with the presence of hubs, i.e., vertices of extremely high degrees. Another property of real-world scale-free networks is that the clustering coefficient (the probability that two uniformly chosen neighbors of a vertex are neighbors themselves) decreases with the vertex degree [4, 10, 24, 32, 34], again following a power law. -
Detecting Statistically Significant Communities
1 Detecting Statistically Significant Communities Zengyou He, Hao Liang, Zheng Chen, Can Zhao, Yan Liu Abstract—Community detection is a key data analysis problem across different fields. During the past decades, numerous algorithms have been proposed to address this issue. However, most work on community detection does not address the issue of statistical significance. Although some research efforts have been made towards mining statistically significant communities, deriving an analytical solution of p-value for one community under the configuration model is still a challenging mission that remains unsolved. The configuration model is a widely used random graph model in community detection, in which the degree of each node is preserved in the generated random networks. To partially fulfill this void, we present a tight upper bound on the p-value of a single community under the configuration model, which can be used for quantifying the statistical significance of each community analytically. Meanwhile, we present a local search method to detect statistically significant communities in an iterative manner. Experimental results demonstrate that our method is comparable with the competing methods on detecting statistically significant communities. Index Terms—Community Detection, Random Graphs, Configuration Model, Statistical Significance. F 1 INTRODUCTION ETWORKS are widely used for modeling the structure function that is able to always achieve the best performance N of complex systems in many fields, such as biology, in all possible scenarios. engineering, and social science. Within the networks, some However, most of these objective functions (metrics) do vertices with similar properties will form communities that not address the issue of statistical significance of commu- have more internal connections and less external links. -
Are the Different Layers of a Social Network Conveying the Same
Manivannan et al. REGULAR ARTICLE Are the different layers of a social network conveying the same information? Ajaykumar Manivannan1, W. Quin Yow1, Roland Bouffanais1y and Alain Barrat2,3*y *Correspondence: [email protected] Abstract 1Singapore University of Technology and Design, 8 Comprehensive and quantitative investigations of social theories and phenomena Somapah Road, 487372, Singapore increasingly benefit from the vast breadth of data describing human social 2 Aix Marseille Univ, Universit´ede relations, which is now available within the realm of computational social science. Toulon, CNRS, CPT, Marseille, France Such data are, however, typically proxies for one of the many interaction layers 3Data Science Laboratory, composing social networks, which can be defined in many ways and are typically Institute for Scientific Interchange composed of communication of various types (e.g., phone calls, face-to-face (ISI) Foundation, Torino, Italy Full list of author information is communication, etc.). As a result, many studies focus on one single layer, available at the end of the article corresponding to the data at hand. Several studies have, however, shown that y Equal contributor these layers are not interchangeable, despite the presence of a certain level of correlations between them. Here, we investigate whether different layers of interactions among individuals lead to similar conclusions with respect to the presence of homophily patterns in a population|homophily represents one of the widest studied phenomenon in social networks. To this aim, we consider a dataset describing interactions and links of various nature in a population of Asian students with diverse nationalities, first language and gender. -
Robot Vision: Projective Geometry
Robot Vision: Projective Geometry Ass.Prof. Friedrich Fraundorfer SS 2018 1 Learning goals . Understand homogeneous coordinates . Understand points, line, plane parameters and interpret them geometrically . Understand point, line, plane interactions geometrically . Analytical calculations with lines, points and planes . Understand the difference between Euclidean and projective space . Understand the properties of parallel lines and planes in projective space . Understand the concept of the line and plane at infinity 2 Outline . 1D projective geometry . 2D projective geometry ▫ Homogeneous coordinates ▫ Points, Lines ▫ Duality . 3D projective geometry ▫ Points, Lines, Planes ▫ Duality ▫ Plane at infinity 3 Literature . Multiple View Geometry in Computer Vision. Richard Hartley and Andrew Zisserman. Cambridge University Press, March 2004. Mundy, J.L. and Zisserman, A., Geometric Invariance in Computer Vision, Appendix: Projective Geometry for Machine Vision, MIT Press, Cambridge, MA, 1992 . Available online: www.cs.cmu.edu/~ph/869/papers/zisser-mundy.pdf 4 Motivation – Image formation [Source: Charles Gunn] 5 Motivation – Parallel lines [Source: Flickr] 6 Motivation – Epipolar constraint X world point epipolar plane x x’ x‘TEx=0 C T C’ R 7 Euclidean geometry vs. projective geometry Definitions: . Geometry is the teaching of points, lines, planes and their relationships and properties (angles) . Geometries are defined based on invariances (what is changing if you transform a configuration of points, lines etc.) . Geometric transformations -
COMBINATORICS, Volume
http://dx.doi.org/10.1090/pspum/019 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume XIX COMBINATORICS AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island 1971 Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society Held at the University of California Los Angeles, California March 21-22, 1968 Prepared by the American Mathematical Society under National Science Foundation Grant GP-8436 Edited by Theodore S. Motzkin AMS 1970 Subject Classifications Primary 05Axx, 05Bxx, 05Cxx, 10-XX, 15-XX, 50-XX Secondary 04A20, 05A05, 05A17, 05A20, 05B05, 05B15, 05B20, 05B25, 05B30, 05C15, 05C99, 06A05, 10A45, 10C05, 14-XX, 20Bxx, 20Fxx, 50A20, 55C05, 55J05, 94A20 International Standard Book Number 0-8218-1419-2 Library of Congress Catalog Number 74-153879 Copyright © 1971 by the American Mathematical Society Printed in the United States of America All rights reserved except those granted to the United States Government May not be produced in any form without permission of the publishers Leo Moser (1921-1970) was active and productive in various aspects of combin• atorics and of its applications to number theory. He was in close contact with those with whom he had common interests: we will remember his sparkling wit, the universality of his anecdotes, and his stimulating presence. This volume, much of whose content he had enjoyed and appreciated, and which contains the re• construction of a contribution by him, is dedicated to his memory. CONTENTS Preface vii Modular Forms on Noncongruence Subgroups BY A. O. L. ATKIN AND H. P. F. SWINNERTON-DYER 1 Selfconjugate Tetrahedra with Respect to the Hermitian Variety xl+xl + *l + ;cg = 0 in PG(3, 22) and a Representation of PG(3, 3) BY R. -
A Novel Neuronal Network Approach to Express Network Motifs
Introduction to NeuraBASE: A Novel Neuronal Network Approach to Express Network Motifs Robert Hercus Choong-Ming Chin Kim-Fong Ho Introduction Since the discovery by Alon et al. [1-3], network motifs are now at the forefront of unravelling complex networks, ranging from biological issues to managing the traffic on the Internet. Network motifs are used, in general, as a technique to detect recurring patterns featured in networks. This is based on the assumption that information flows in distinct networks and, hence, common causal associations between nodal networks can be surmised statistically. In essence, the motif discovery algorithm [1-4] begins with a list of variable-information of a particular network and, their connections to other network variables. An analysis is subsequently made on the most prevalent causal relationships in the network based on the frequency of occurrences. These are then compared with a randomised network with the same number of variables and directed edges. Finally, the algorithm lists out statistically significant patterns of associations that occur more frequently than they would at random. In the paper by Alon et al. [2], it was reported that most of the networks analysed demonstrate identical motifs, even though they belong to disparate network families. For example, the said literature discovered that gene regulation in genetics, neuronal connectivity network and electronic circuit systems share two identical network motifs, which involved less than four variables. This observation implies that some similarities exist in these three network architectures. Besides the MFinder tool [1], researchers have also developed other network motif discovery tools. However, MAVisto [6] was found to be as computationally costly with long run-time periods as the MFinder tool uses the same network search for the same subgraph sizes. -
Temporal Network Metrics and Their Application to Real World Networks
Temporal network metrics and their application to real world networks John Kit Tang Robinson College University of Cambridge 2011 This dissertation is submitted for the degree of Doctor of Philosophy Declaration This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except where specifically indicated in the text. This dissertation does not exceed the regulation length of 60 000 words, including tables and footnotes. Summary The analysis of real social, biological and technological networks has attracted a lot of attention as technological advances have given us a wealth of empirical data. Classic studies looked at analysing static or aggregated networks, i.e., networks that do not change over time or built as the results of aggregation of information over a certain period of time. Given the soaring collections of measurements related to very large, real network traces, researchers are quickly starting to realise that connections are inherently varying over time and exhibit more dimensionality than static analysis can capture. This motivates the work in this dissertation: new tools for temporal complex network analysis are required when analysing real networks that inherently change over time. Firstly, we introduce the temporal graph model and formalise the notion of shortest temporal paths, used extensively in graph theory, and show that as static graphs ignore the time order of contacts, the available links are overestimated and the true shortest paths are underestimated. In addition, contrary to intuition, we find that slowly evolving graphs can be efficient for information dissemination due to small-world behaviour in temporal graphs. -
Processes on Complex Networks. Percolation
Chapter 5 Processes on complex networks. Percolation 77 Up till now we discussed the structure of the complex networks. The actual reason to study this structure is to understand how this structure influences the behavior of random processes on networks. I will talk about two such processes. The first one is the percolation process. The second one is the spread of epidemics. There are a lot of open problems in this area, the main of which can be innocently formulated as: How the network topology influences the dynamics of random processes on this network. We are still quite far from a definite answer to this question. 5.1 Percolation 5.1.1 Introduction to percolation Percolation is one of the simplest processes that exhibit the critical phenomena or phase transition. This means that there is a parameter in the system, whose small change yields a large change in the system behavior. To define the percolation process, consider a graph, that has a large connected component. In the classical settings, percolation was actually studied on infinite graphs, whose vertices constitute the set Zd, and edges connect each vertex with nearest neighbors, but we consider general random graphs. We have parameter ϕ, which is the probability that any edge present in the underlying graph is open or closed (an event with probability 1 − ϕ) independently of the other edges. Actually, if we talk about edges being open or closed, this means that we discuss bond percolation. It is also possible to talk about the vertices being open or closed, and this is called site percolation. -
Performance Evaluation of Genetic Algorithm and Simulated Annealing in Solving Kirkman Schoolgirl Problem
FUOYE Journal of Engineering and Technology (FUOYEJET), Vol. 5, Issue 2, September 2020 ISSN: 2579-0625 (Online), 2579-0617 (Paper) Performance Evaluation of Genetic Algorithm and Simulated Annealing in solving Kirkman Schoolgirl Problem *1Christopher A. Oyeleye, 2Victoria O. Dayo-Ajayi, 3Emmanuel Abiodun and 4Alabi O. Bello 1Department of Information Systems, Ladoke Akintola University of Technology, Ogbomoso, Nigeria 2Department of Computer Science, Ladoke Akintola University of Technology, Ogbomoso, Nigeria 3Department of Computer Science, Kwara State Polytechnic, Ilorin, Nigeria 4Department of Mathematical and Physical Sciences, Afe Babalola University, Ado-Ekiti, Nigeria [email protected]|[email protected]|[email protected]|[email protected] Received: 15-FEB-2020; Reviewed: 12-MAR-2020; Accepted: 28-MAY-2020 http://dx.doi.org/10.46792/fuoyejet.v5i2.477 Abstract- This paper provides performance evaluation of Genetic Algorithm and Simulated Annealing in view of their software complexity and simulation runtime. Kirkman Schoolgirl is about arranging fifteen schoolgirls into five triplets in a week with a distinct constraint of no two schoolgirl must walk together in a week. The developed model was simulated using MATLAB version R2015a. The performance evaluation of both Genetic Algorithm (GA) and Simulated Annealing (SA) was carried out in terms of program size, program volume, program effort and the intelligent content of the program. The results obtained show that the runtime for GA and SA are 11.23sec and 6.20sec respectively. The program size for GA and SA are 2.01kb and 2.21kb, respectively. The lines of code for GA and SA are 324 and 404, respectively. The program volume for GA and SA are 1121.58 and 3127.92, respectively. -
Percolation Thresholds for Robust Network Connectivity
Percolation Thresholds for Robust Network Connectivity Arman Mohseni-Kabir1, Mihir Pant2, Don Towsley3, Saikat Guha4, Ananthram Swami5 1. Physics Department, University of Massachusetts Amherst, [email protected] 2. Massachusetts Institute of Technology 3. College of Information and Computer Sciences, University of Massachusetts Amherst 4. College of Optical Sciences, University of Arizona 5. Army Research Laboratory Abstract Communication networks, power grids, and transportation networks are all examples of networks whose performance depends on reliable connectivity of their underlying network components even in the presence of usual network dynamics due to mobility, node or edge failures, and varying traffic loads. Percolation theory quantifies the threshold value of a local control parameter such as a node occupation (resp., deletion) probability or an edge activation (resp., removal) probability above (resp., below) which there exists a giant connected component (GCC), a connected component comprising of a number of occupied nodes and active edges whose size is proportional to the size of the network itself. Any pair of occupied nodes in the GCC is connected via at least one path comprised of active edges and occupied nodes. The mere existence of the GCC itself does not guarantee that the long-range connectivity would be robust, e.g., to random link or node failures due to network dynamics. In this paper, we explore new percolation thresholds that guarantee not only spanning network connectivity, but also robustness. We define and analyze four measures of robust network connectivity, explore their interrelationships, and numerically evaluate the respective robust percolation thresholds arXiv:2006.14496v1 [physics.soc-ph] 25 Jun 2020 for the 2D square lattice. -
Blocking Set Free Configurations and Their Relations to Digraphs and Hypergraphs
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector DISCRETE MATHEMATICS ELSIZI’IER Discrete Mathematics 1651166 (1997) 359.-370 Blocking set free configurations and their relations to digraphs and hypergraphs Harald Gropp* Mihlingstrasse 19. D-69121 Heidelberg, German> Abstract The current state of knowledge concerning the existence of blocking set free configurations is given together with a short history of this problem which has also been dealt with in terms of digraphs without even dicycles or 3-chromatic hypergraphs. The question is extended to the case of nonsymmetric configurations (u,, b3). It is proved that for each value of I > 3 there are only finitely many values of u for which the existence of a blocking set free configuration is still unknown. 1. Introduction In the language of configurations the existence of blocking sets was investigated in [3, 93 for the first time. Further papers [4, 201 yielded the result that the existence problem for blocking set free configurations v3 has been nearly solved. There are only 8 values of v for which it is unsettled whether there is a blocking set free configuration ~7~:u = 15,16,17,18,20,23,24,26. The existence of blocking set free configurations is equivalent to the existence of certain hypergraphs and digraphs. In these two languages results have been obtained much earlier. In Section 2 the relations between configurations, hypergraphs, and digraphs are exhibited. Furthermore, a nearly forgotten result of Steinitz is described In his dissertation of 1894 Steinitz proved the existence of a l-factor in a regular bipartite graph 20 years earlier then Kiinig.