Structural Parameters in Combinatorial Objects
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Combinatorial Design University of Southern California Non-Parametric Computational Design Strategies
Jose Sanchez Combinatorial design University of Southern California Non-parametric computational design strategies 1 ABSTRACT This paper outlines a framework and conceptualization of combinatorial design. Combinatorial 1 Design – Year 3 – Pattern of one design is a term coined to describe non-parametric design strategies that focus on the permutation, unit in two scales. Developed by combinatorial design within a game combination and patterning of discrete units. These design strategies differ substantially from para- engine. metric design strategies as they do not operate under continuous numerical evaluations, intervals or ratios but rather finite discrete sets. The conceptualization of this term and the differences with other design strategies are portrayed by the work done in the last 3 years of research at University of Southern California under the Polyomino agenda. The work, conducted together with students, has studied the use of discrete sets and combinatorial strategies within virtual reality environments to allow for an enhanced decision making process, one in which human intuition is coupled to algo- rithmic intelligence. The work of the research unit has been sponsored and tested by the company Stratays for ongoing research on crowd-sourced design. 44 INTRODUCTION—OUTSIDE THE PARAMETRIC UMBRELLA To start, it is important that we understand that the use of the terms parametric and combinatorial that will be used in this paper will come from an architectural and design background, as the association of these terms in mathematics and statistics might have a different connotation. There are certainly lessons and a direct relation between the term ‘combinatorial’ as used in this paper and the field of combinatorics and permutations in mathematics. -
Spectral Gap Bounds for the Simplicial Laplacian and an Application To
Spectral gap bounds for the simplicial Laplacian and an application to random complexes Samir Shukla,∗ D. Yogeshwaran† Abstract In this article, we derive two spectral gap bounds for the reduced Laplacian of a general simplicial complex. Our two bounds are proven by comparing a simplicial complex in two different ways with a larger complex and with the corresponding clique complex respectively. Both of these bounds generalize the result of Aharoni et al. (2005) [1] which is valid only for clique complexes. As an application, we investigate the thresholds for vanishing of cohomology of the neighborhood complex of the Erdös- Rényi random graph. We improve the upper bound derived in Kahle (2007) [15] by a logarithmic factor using our spectral gap bounds and we also improve the lower bound via finer probabilistic estimates than those in Kahle (2007) [15]. Keywords: spectral bounds, Laplacian, cohomology, neighborhood complex, Erdös-Rényi random graphs. AMS MSC 2010: 05C80. 05C50. 05E45. 55U10. 1 Introduction Let G be a graph with vertex set V (G) (often to be abbreviated as V ) and let L(G) denote the (unnormalized) Laplacian of G. Let 0 = λ1(G) ≤ λ2(G) ≤ . ≤ λ|V |(G) denote the eigenvalues of L(G) in ascending order. Here, the second smallest eigenvalue λ2(G) is called the spectral gap. The clique complex of a graph G is the simplicial complex whose simplices are all subsets σ ⊂ V which spans a complete subgraph of G. We shall denote the kth k arXiv:1810.10934v2 [math.CO] 9 Oct 2019 reduced cohomology of a simplicial complex X by H (X). -
Message Passing Simplicial Networks
Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks Cristian Bodnar * 1 Fabrizio Frasca * 2 3 Yu Guang Wang * 4 5 6 Nina Otter 7 Guido Montufar´ * 4 7 Pietro Lio` 1 Michael M. Bronstein 2 3 v0 Abstract v6 v10 v9 The pairwise interaction paradigm of graph ma- v chine learning has predominantly governed the 8 v v modelling of relational systems. However, graphs 1 5 v3 alone cannot capture the multi-level interactions v v present in many complex systems and the expres- 4 7 v2 sive power of such schemes was proven to be lim- ited. To overcome these limitations, we propose Figure 1. Message Passing with upper and boundary adjacencies Message Passing Simplicial Networks (MPSNs), illustrated for vertex v2 and edge (v5; v7) in the complex. a class of models that perform message passing on simplicial complexes (SCs). To theoretically anal- 1. Introduction yse the expressivity of our model we introduce a Simplicial Weisfeiler-Lehman (SWL) colouring Graphs are among the most common abstractions for com- procedure for distinguishing non-isomorphic SCs. plex systems of relations and interactions, arising in a broad We relate the power of SWL to the problem of range of fields from social science to high energy physics. distinguishing non-isomorphic graphs and show Graph neural networks (GNNs), which trace their origins that SWL and MPSNs are strictly more power- to the 1990s (Sperduti, 1994; Goller & Kuchler, 1996; Gori ful than the WL test and not less powerful than et al., 2005; Scarselli et al., 2009; Bruna et al., 2014; Li et al., the 3-WL test. -
Some Properties of Glued Graphs at Complete Clone
Sci.Int.(Lahore),27(1),39-47,2014 ISSN 1013-5316; CODEN: SINTE 8 39 SOME PROPERTIES OF GLUED GRAPHS AT COMPLETE CLONE IN THE VIEW OF ALGEBRAIC COMBINATORICS Seyyede Masoome Seyyedi , Farhad Rahmati Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran. Corresponding Author: Email: [email protected] ABSTRACT: A glued graph at complete clone is obtained from combining two graphs by identifying edges of of each original graph. We investigate how to change some properties such as height, big height, Krull dimension, Betti numbers by gluing of two graphs at complete clone. We give a sufficient and necessary condition so that the glued graph of two Cohen-Macaulay chordal graphs at complete clone is a Cohen-Macaulay graph. Moreover, we present the conditions that the edge ideal of gluing of two graphs at complete clone has linear resolution whenever the edge ideals of original graphs have linear resolution. We show when gluing of two independence complexes, line graphs, complement graphs can be expressed as independence complex, line graph and complement of gluing of two graphs. Key Words: Glued graph, Height, Big height, Krull dimension, Projective dimension, Linear resolution, Betti number, Cohen-Macaulay. INTRODUCTION The concept edge ideal was first introduced by Villarreal in Our paper is organized as follows. In section 2, we give an [23], that is, let be a simple (no loops or multiple edges) explicit formula for computing the height of glued graph at graph on the vertex set and the edge set complete clone. Also, we present a lower bound for the big . -
The Book Review Column1 by William Gasarch Department of Computer Science University of Maryland at College Park College Park, MD, 20742 Email: [email protected]
The Book Review Column1 by William Gasarch Department of Computer Science University of Maryland at College Park College Park, MD, 20742 email: [email protected] In this column we review the following books. 1. Combinatorial Designs: Constructions and Analysis by Douglas R. Stinson. Review by Gregory Taylor. A combinatorial design is a set of sets of (say) {1, . , n} that have various properties, such as that no two of them have a large intersection. For what parameters do such designs exist? This is an interesting question that touches on many branches of math for both origin and application. 2. Combinatorics of Permutations by Mikl´osB´ona. Review by Gregory Taylor. Usually permutations are viewed as a tool in combinatorics. In this book they are considered as objects worthy of study in and of themselves. 3. Enumerative Combinatorics by Charalambos A. Charalambides. Review by Sergey Ki- taev, 2008. Enumerative combinatorics is a branch of combinatorics concerned with counting objects satisfying certain criteria. This is a far reaching and deep question. 4. Geometric Algebra for Computer Science by L. Dorst, D. Fontijne, and S. Mann. Review by B. Fasy and D. Millman. How can we view Geometry in terms that a computer can understand and deal with? This book helps answer that question. 5. Privacy on the Line: The Politics of Wiretapping and Encryption by Whitfield Diffie and Susan Landau. Review by Richard Jankowski. What is the status of your privacy given current technology and law? Read this book and find out! Books I want Reviewed If you want a FREE copy of one of these books in exchange for a review, then email me at gasarchcs.umd.edu Reviews need to be in LaTeX, LaTeX2e, or Plaintext. -
Geometry, Combinatorial Designs and Cryptology Fourth Pythagorean Conference
Geometry, Combinatorial Designs and Cryptology Fourth Pythagorean Conference Sunday 30 May to Friday 4 June 2010 Index of Talks and Abstracts Main talks 1. Simeon Ball, On subsets of a finite vector space in which every subset of basis size is a basis 2. Simon Blackburn, Honeycomb arrays 3. G`abor Korchm`aros, Curves over finite fields, an approach from finite geometry 4. Cheryl Praeger, Basic pregeometries 5. Bernhard Schmidt, Finiteness of circulant weighing matrices of fixed weight 6. Douglas Stinson, Multicollision attacks on iterated hash functions Short talks 1. Mari´en Abreu, Adjacency matrices of polarity graphs and of other C4–free graphs of large size 2. Marco Buratti, Combinatorial designs via factorization of a group into subsets 3. Mike Burmester, Lightweight cryptographic mechanisms based on pseudorandom number generators 4. Philippe Cara, Loops, neardomains, nearfields and sets of permutations 5. Ilaria Cardinali, On the structure of Weyl modules for the symplectic group 6. Bill Cherowitzo, Parallelisms of quadrics 7. Jan De Beule, Large maximal partial ovoids of Q−(5, q) 8. Bart De Bruyn, On extensions of hyperplanes of dual polar spaces 1 9. Frank De Clerck, Intriguing sets of partial quadrangles 10. Alice Devillers, Symmetry properties of subdivision graphs 11. Dalibor Froncek, Decompositions of complete bipartite graphs into generalized prisms 12. Stelios Georgiou, Self-dual codes from circulant matrices 13. Robert Gilman, Cryptology of infinite groups 14. Otokar Groˇsek, The number of associative triples in a quasigroup 15. Christoph Hering, Latin squares, homologies and Euler’s conjecture 16. Leanne Holder, Bilinear star flocks of arbitrary cones 17. Robert Jajcay, On the geometry of cages 18. -
Topological Graph Persistence
CAIM ISSN 2038-0909 Commun. Appl. Ind. Math. 11 (1), 2020, 72–87 DOI: 10.2478/caim-2020-0005 Research Article Topological graph persistence Mattia G. Bergomi1*, Massimo Ferri2*, Lorenzo Zuffi2 1Veos Digital, Milan, Italy 2ARCES and Dept. of Mathematics, Univ. of Bologna, Italy *Email address for correspondence: [email protected] Communicated by Alessandro Marani Received on 04 28, 2020. Accepted on 11 09, 2020. Abstract Graphs are a basic tool in modern data representation. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological constructions can be used to gain information otherwise concealed by the low-dimensional nature of graphs. We do this by extending previous work in homological persistence, and proposing novel graph-theoretical constructions. Beyond cliques, we use independent sets, neighborhoods, enclaveless sets and a Ramsey-inspired extended persistence. Keywords: Clique, independent set, neighborhood, enclaveless set, Ramsey AMS subject classification: 68R10, 05C10 1. Introduction Currently data are produced massively and rapidly. A large part of these data is either naturally organized, or can be represented, as graphs or networks. Recently, topological persistence has proved to be an invaluable tool for the exploration and understanding of these data types. Albeit graphs can be considered as topological objects per se, given their low dimensionality, only limited information can be obtained by studying their topology. It is possible to grasp more information by superposing higher-dimensional topological structures to a given graph. Many research lines, for ex- ample, focused on building n-dimensional complexes from graphs by considering their (n+1)-cliques (see, e.g., section 1.1). -
Persistent Homology of Unweighted Complex Networks Via Discrete
www.nature.com/scientificreports OPEN Persistent homology of unweighted complex networks via discrete Morse theory Received: 17 January 2019 Harish Kannan1, Emil Saucan2,3, Indrava Roy1 & Areejit Samal 1,4 Accepted: 6 September 2019 Topological data analysis can reveal higher-order structure beyond pairwise connections between Published: xx xx xxxx vertices in complex networks. We present a new method based on discrete Morse theory to study topological properties of unweighted and undirected networks using persistent homology. Leveraging on the features of discrete Morse theory, our method not only captures the topology of the clique complex of such graphs via the concept of critical simplices, but also achieves close to the theoretical minimum number of critical simplices in several analyzed model and real networks. This leads to a reduced fltration scheme based on the subsequence of the corresponding critical weights, thereby leading to a signifcant increase in computational efciency. We have employed our fltration scheme to explore the persistent homology of several model and real-world networks. In particular, we show that our method can detect diferences in the higher-order structure of networks, and the corresponding persistence diagrams can be used to distinguish between diferent model networks. In summary, our method based on discrete Morse theory further increases the applicability of persistent homology to investigate the global topology of complex networks. In recent years, the feld of topological data analysis (TDA) has rapidly grown to provide a set of powerful tools to analyze various important features of data1. In this context, persistent homology has played a key role in bring- ing TDA to the fore of modern data analysis. -
Arxiv:2006.02870V1 [Cs.SI] 4 Jun 2020
The why, how, and when of representations for complex systems Leo Torres Ann S. Blevins [email protected] [email protected] Network Science Institute, Department of Bioengineering, Northeastern University University of Pennsylvania Danielle S. Bassett Tina Eliassi-Rad [email protected] [email protected] Department of Bioengineering, Network Science Institute and University of Pennsylvania Khoury College of Computer Sciences, Northeastern University June 5, 2020 arXiv:2006.02870v1 [cs.SI] 4 Jun 2020 1 Contents 1 Introduction 4 1.1 Definitions . .5 2 Dependencies by the system, for the system 6 2.1 Subset dependencies . .7 2.2 Temporal dependencies . .8 2.3 Spatial dependencies . 10 2.4 External sources of dependencies . 11 3 Formal representations of complex systems 12 3.1 Graphs . 13 3.2 Simplicial Complexes . 13 3.3 Hypergraphs . 15 3.4 Variations . 15 3.5 Encoding system dependencies . 18 4 Mathematical relationships between formalisms 21 5 Methods suitable for each representation 24 5.1 Methods for graphs . 24 5.2 Methods for simplicial complexes . 25 5.3 Methods for hypergraphs . 27 5.4 Methods and dependencies . 28 6 Examples 29 6.1 Coauthorship . 29 6.2 Email communications . 32 7 Applications 35 8 Discussion and Conclusion 36 9 Acknowledgments 38 10 Citation diversity statement 38 2 Abstract Complex systems thinking is applied to a wide variety of domains, from neuroscience to computer science and economics. The wide variety of implementations has resulted in two key challenges: the progenation of many domain-specific strategies that are seldom revisited or questioned, and the siloing of ideas within a domain due to inconsistency of complex systems language. -
Modules Over Semisymmetric Quasigroups
Modules over semisymmetric quasigroups Alex W. Nowak Abstract. The class of semisymmetric quasigroups is determined by the iden- tity (yx)y = x. We prove that the universal multiplication group of a semisym- metric quasigroup Q is free over its underlying set and then specify the point- stabilizers of an action of this free group on Q. A theorem of Smith indicates that Beck modules over semisymmetric quasigroups are equivalent to modules over a quotient of the integral group algebra of this stabilizer. Implementing our description of the quotient ring, we provide some examples of semisym- metric quasigroup extensions. Along the way, we provide an exposition of the quasigroup module theory in more general settings. 1. Introduction A quasigroup (Q, ·, /, \) is a set Q equipped with three binary operations: · denoting multiplication, / right division, and \ left division; these operations satisfy the identities (IL) y\(y · x)= x; (IR) x = (x · y)/y; (SL) y · (y\x)= x; (SR) x = (x/y) · y. Whenever possible, we convey multiplication of quasigroup elements by concatena- tion. As a class of algebras defined by identities, quasigroups constitute a variety (in the universal-algebraic sense), denoted by Q. Moreover, we have a category of quasigroups (also denoted Q), the morphisms of which are quasigroup homomor- phisms respecting the three operations. arXiv:1908.06364v1 [math.RA] 18 Aug 2019 If (G, ·) is a group, then defining x/y := x · y−1 and x\y := x−1 · y furnishes a quasigroup (G, ·, /, \). That is, all groups are quasigroups, and because of this fact, constructions in the representation theory of quasigroups often take their cue from familiar counterparts in group theory. -
ON the POLYHEDRAL GEOMETRY of T–DESIGNS
ON THE POLYHEDRAL GEOMETRY OF t{DESIGNS A thesis presented to the faculty of San Francisco State University In partial fulfilment of The Requirements for The Degree Master of Arts In Mathematics by Steven Collazos San Francisco, California August 2013 Copyright by Steven Collazos 2013 CERTIFICATION OF APPROVAL I certify that I have read ON THE POLYHEDRAL GEOMETRY OF t{DESIGNS by Steven Collazos and that in my opinion this work meets the criteria for approving a thesis submitted in partial fulfillment of the requirements for the degree: Master of Arts in Mathematics at San Francisco State University. Matthias Beck Associate Professor of Mathematics Felix Breuer Federico Ardila Associate Professor of Mathematics ON THE POLYHEDRAL GEOMETRY OF t{DESIGNS Steven Collazos San Francisco State University 2013 Lisonek (2007) proved that the number of isomorphism types of t−(v; k; λ) designs, for fixed t, v, and k, is quasi{polynomial in λ. We attempt to describe a region in connection with this result. Specifically, we attempt to find a region F of Rd with d the following property: For every x 2 R , we have that jF \ Gxj = 1, where Gx denotes the G{orbit of x under the action of G. As an application, we argue that our construction could help lead to a new combinatorial reciprocity theorem for the quasi{polynomial counting isomorphism types of t − (v; k; λ) designs. I certify that the Abstract is a correct representation of the content of this thesis. Chair, Thesis Committee Date ACKNOWLEDGMENTS I thank my advisors, Dr. Matthias Beck and Dr. -
FACE VECTORS of FLAG COMPLEXES 1. Introduction We
FACE VECTORS OF FLAG COMPLEXES ANDY FROHMADER Abstract. A conjecture of Kalai and Eckhoff that the face vector of an arbi- trary flag complex is also the face vector of some particular balanced complex is verified. 1. Introduction We begin by introducing the main result. Precise definitions and statements of some related theorems are deferred to later sections. The main object of our study is the class of flag complexes. A simplicial complex is a flag complex if all of its minimal non-faces are two element sets. Equivalently, if all of the edges of a potential face of a flag complex are in the complex, then that face must also be in the complex. Flag complexes are closely related to graphs. Given a graph G, define its clique complex C = C(G) as the simplicial complex whose vertex set is the vertex set of G, and whose faces are the cliques of G. The clique complex of any graph is itself a flag complex, as for a subset of vertices of a graph to not form a clique, two of them must not form an edge. Conversely, any flag complex is the clique complex of its 1-skeleton. The Kruskal-Katona theorem [6, 5] characterizes the face vectors of simplicial complexes as being precisely the integer vectors whose coordinates satisfy some par- ticular bounds. The graphs of the \rev-lex" complexes which attain these bounds invariably have a clique on all but one of the vertices of the complex, and sometimes even on all of the vertices. Since the bounds of the Kruskal-Katona theorem hold for all simplicial com- plexes, they must in particular hold for flag complexes.