DOCSLIB.ORG

Modularity Analysis by the Laplacian Matrix

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Modularity Analysis by the Laplacian Matrix Linear Software Models: Modularity Analysis by the Laplacian Matrix Iaakov Exman and Rawi Sakhnini Software Engineering Dept., The Jerusalem College of Engineering – JCE-Azrieli, Jerusalem, Israel Keywords: Linear Software Models, Modularity Matrix, Laplacian Matrix, Connected Components, Eigenvectors, Zero-valued Eigenvalues, Bipartite Graph, Software Redesign, Coupling Resolution, Outliers. Abstract: We have recently shown that one can obtain the number and sizes of modules of a software system from the eigenvectors of the Modularity Matrix weighted by an affinity matrix. However such a weighting still demands a suitable definition of an affinity. This paper obtains the same results by means of a Laplacian Matrix, directly based upon the Modularity Matrix without the need of weighting. These formalizations are different alternatives leading to the same outcomes based upon a central idea: modules are connected components. The important point is that, independently of specific advantages of given techniques, there is just one single unified algebraic theory of software composition – the Linear Software Models – behind the different approaches. The specifics of the Laplacian Matrix technique, after its formal enunciation, are illustrated by calculations made for case studies. 1 INTRODUCTION 1.1 Modularity Matrix Concepts Software engineering is an exercise in composition Modularity is the antithesis of coupling between (Fall, 2016). A software system is composed from software components. Thus, a theory of modular subsystems, which are made of sub-subsystems and composition must clearly define coupling. An so on for each hierarchy level of the system, down to intuitive notion of coupling is given in the Design indivisible components. The problem to be solved is Patterns GoF book Glossary (Gamma et al., 1995): which components are needed to satisfy the system coupling is the degree to which software required functionalities. It is widely accepted that components depend on each other. In our Linear modularity is essential to solve this problem. Software Models, this is translated to "coupling is Our Linear Software Models approach uses linear dependence" among software components. Modularity Matrices linking component structures to To apply the theory, a Modularity Matrix their functionalities, see (Exman, 2012 and 2013). (Exman, 2014) is constructed for each software The outcome is a set of modules in the respective system. The matrix displays relations between the hierarchical level of the software system. software system architectural entities: column Recently we have shown that numbers and sizes structors, and row functionals. Structors generalize of the referred modules can be precisely obtained UML classes and functionals generalize methods. A from eigenvectors of the Modularity Matrix suitably 1-valued matrix element means that the column weighted by an affinity matrix (Exman, 2015). structor provides a functional in the respective row. In this paper we show that the same numbers and To avoid couplings, a standard Modularity sizes of modules are obtained from the eigenvectors Matrix must have all its structors and respectively all of a Laplacian Matrix, directly derived from the its functionals linearly independent. From linear Modularity Matrix, without the need of weighting. algebra it follows that the standard Modularity In this Introduction we review properties of the Matrix is square. If certain structor sub-sets provide Modularity Matrix and the Laplacian Matrix. sub-sets of functionals disjoint to other sub-sets, the matrix is strictly block-diagonal. These blocks are the desired independent modules. 100 Exman, I. and Sakhnini, R. Linear Software Models: Modularity Analysis by the Laplacian Matrix. DOI: 10.5220/0005985601000108 In Proceedings of the 11th International Joint Conference on Software Technologies (ICSOFT 2016) - Volume 2: ICSOFT-PT, pages 100-108 ISBN: 978-989-758-194-6 Copyright c 2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved Linear Software Models: Modularity Analysis by the Laplacian Matrix In a standard Modularity Matrix modules are strictly block-diagonal Modularity Matrix – the irreducible – cannot be broken into smaller modules. same results are obtained by means of the All the matrix elements outside modules are zero- eigenvectors of the Laplacian Matrix; valued, the modules are orthogonal thus mutually case of outliers – a neat procedure is formulated linearly independent. One easily shows that to highlight residual coupling. eigenvectors of a suitably weighted Modularity The argument centrepiece is that modules are Matrix precisely reflect module sizes. This spectral connected components. This is true for blocks of the approach (Exman, 2015) is illustrated in Fig. 1. block-diagonal Modularity Matrix (Exman, 2014), sub-lattices of a Modularity Lattice (Exman and Speicher, 2015) or sub-graphs obtaining a Laplacian Matrix in this paper. One follows the next steps: a) Generate a Bipartite Graph – from the Modularity Matrix; b) Obtain the Laplacian Matrix – from the generated graph; c) Software Modules are Connected Components – seen in the graph and in the corresponding Laplacian; d) Number of Modules – is set by the number of zero-valued Laplacian eigenvalues; e) Sizes of Modules – are given by the non-zero elements of Laplacian eigenvectors; Figure 1: Schematic Modularity Matrix with a fitting eigenvector – The square block-diagonal matrix has 4 f) Modules Sparsity – may demand module numbered modules (blue background). The eigenvector at splitting to assure sparsity below a threshold, the right-hand-side of the matrix precisely reflects the size thereby highlighting outlier coupling in need of of module #2 (with hatched background). redesign. The above steps will be first intuitively illustrated by 1.2 Laplacian Matrix Concepts an introductory example. Then, formal proofs will be provided in a subsequent section. A generic Laplacian matrix is defined upon a graph. For software systems the graph is undirected, and 1.4 Organization of the Paper edges link the referred architectural units – structors and functionals – where each unit is a vertex in the The remaining of this paper is organized as follows. graph. Formally the Laplacian L is given by: Section 2 mentions related work. Section 3 displays an introductory example. Section 4 formulates LDA (1) theoretical considerations. Section 5 presents case- where D is the Degree Matrix – showing the degree studies. A discussion concludes the paper. deg(vi ) of vertex vi in its diagonal element Dii – and A is the Adjacency Matrix – showing for each ij, pair of vertices whether they are adjacent in the 2 RELATED WORK graph. Adjacent vertices have a 1-valued Aij matrix 2.1 Modularity Analysis element and 0-valued otherwise. Laplacian properties of interest are: Various techniques are suitable for modularity L is symmetric; analysis. Here is a short sample of approaches. The number of zero-valued eigenvalues of the Baldwin and Clark, in their “Design Rules” book Laplacian Matrix is the number of graph (Baldwin and Clark, 2000), describe a Design connected components. Structure Matrix (DSM) upon which one adds an economic design model. DSM has also been applied 1.3 Modules as Connected Components to software engineering (Cai and Sullivan, 2006). Conceptual lattices were introduced in (Wille, This paper’s purpose is to show that for: 1982) within Formal Concept Analysis (FCA). This 101 ICSOFT-PT 2016 - 11th International Conference on Software Paradigm Trends technique has been used for modularization and software system design, e.g. (Siff and Reps, 1999). In previous spectral work we have used eigenvectors of the Modularity Matrix symmetrized and weighted by an Affinity (Exman, 2015) to extract software system modules' numbers and sizes. 2.2 Laplacian Matrix Techniques Spectral techniques based upon the Laplacian Matrix Figure 2: Prototype Design Pattern Modularity Matrix – have been used in various contexts, including There are 4 structors and functionals in this matrix. These software engineering. A good survey on Laplacian form 3 modules (blue background): a- upper-left shape matrices of graphs is (Merris, 1994). role; b- middle shapes-cache role; c- lower-right the prototype client. Zero-valued matrix elements outside the A tutorial on spectral clustering, with emphasis modules are omitted for clarity in all matrix figures. on Laplacian techniques is (von Luxburg, 2007). A more recent review on clustering methodologies for As an illustration, each 1-valued element in the software engineering, of spectral methods in general Matrix originates one edge, e.g. a 1-valued element and in particular the Laplacian is (Shtern and in column S3 and row F3 of the matrix originates an Tzerpos, 2012). edge between the S3 and F3 vertices in Fig. 3. Ng et al. (Ng et al., 2001) deal with spectral clustering, analysing an algorithm, explicitly referring to the Laplacian. Shokoufandeh et al. (Shokoufandeh, 2005) extract a Module Dependency Graph (MDG) from software source code for clustering into MDG partitions. Their Modularization Quality criterion is reformulated as a Laplacian eigenvalue problem. Figure 3: Prototype Bipartite undirected Graph – It is obtained from the Modularity Matrix in Fig. 2. A structor 3 INTRODUCTORY EXAMPLE: Sj is linked to a functional Fi by a graph edge if the respective (i, j) Matrix element is 1-valued. Separate graph PROTOTYPE PATTERN modules Mk, having no common edges, are highlighted by light
Load more

Recommended publications
  • Arxiv:1711.06300V1
    EXPLICIT BLOCK-STRUCTURES FOR BLOCK-SYMMETRIC FIEDLER-LIKE PENCILS∗ M. I. BUENO†, M. MARTIN ‡, J. PEREZ´ §, A. SONG ¶, AND I. VIVIANO k Abstract. In the last decade, there has been a continued effort to produce families of strong linearizations of a matrix polynomial P (λ), regular and singular, with good properties, such as, being companion forms, allowing the recovery of eigen- vectors of a regular P (λ) in an easy way, allowing the computation of the minimal indices of a singular P (λ) in an easy way, etc. As a consequence of this research, families such as the family of Fiedler pencils, the family of generalized Fiedler pencils (GFP), the family of Fiedler pencils with repetition, and the family of generalized Fiedler pencils with repetition (GFPR) were con- structed. In particular, one of the goals was to find in these families structured linearizations of structured matrix polynomials. For example, if a matrix polynomial P (λ) is symmetric (Hermitian), it is convenient to use linearizations of P (λ) that are also symmetric (Hermitian). Both the family of GFP and the family of GFPR contain block-symmetric linearizations of P (λ), which are symmetric (Hermitian) when P (λ) is. Now the objective is to determine which of those structured linearizations have the best numerical properties. The main obstacle for this study is the fact that these pencils are defined implicitly as products of so-called elementary matrices. Recent papers in the literature had as a goal to provide an explicit block-structure for the pencils belonging to the family of Fiedler pencils and any of its further generalizations to solve this problem.
    [Show full text]
  • "Distance Measures for Graph Theory"
    Distance measures for graph theory : Comparisons and analyzes of different methods Dissertation presented by Maxime DUYCK for obtaining the Master’s degree in Mathematical Engineering Supervisor(s) Marco SAERENS Reader(s) Guillaume GUEX, Bertrand LEBICHOT Academic year 2016-2017 Acknowledgments First, I would like to thank my supervisor Pr. Marco Saerens for his presence, his advice and his precious help throughout the realization of this thesis. Second, I would also like to thank Bertrand Lebichot and Guillaume Guex for agreeing to read this work. Next, I would like to thank my parents, all my family and my friends to have accompanied and encouraged me during all my studies. Finally, I would thank Malian De Ron for creating this template [65] and making it available to me. This helped me a lot during “le jour et la nuit”. Contents 1. Introduction 1 1.1. Context presentation .................................. 1 1.2. Contents .......................................... 2 2. Theoretical part 3 2.1. Preliminaries ....................................... 4 2.1.1. Networks and graphs .............................. 4 2.1.2. Useful matrices and tools ........................... 4 2.2. Distances and kernels on a graph ........................... 7 2.2.1. Notion of (dis)similarity measures ...................... 7 2.2.2. Kernel on a graph ................................ 8 2.2.3. The shortest-path distance .......................... 9 2.3. Kernels from distances ................................. 9 2.3.1. Multidimensional scaling ............................ 9 2.3.2. Gaussian mapping ............................... 9 2.4. Similarity measures between nodes .......................... 9 2.4.1. Katz index and its Leicht’s extension .................... 10 2.4.2. Commute-time distance and Euclidean commute-time distance .... 10 2.4.3. SimRank similarity measure .........................
    [Show full text]
  • Chapter 4 Introduction to Spectral Graph Theory
    Chapter 4 Introduction to Spectral Graph Theory Spectral graph theory is the study of a graph through the properties of the eigenvalues and eigenvectors of its associated Laplacian matrix. In the following, we use G = (V; E) to represent an undirected n-vertex graph with no self-loops, and write V = f1; : : : ; ng, with the degree of vertex i denoted di. For undirected graphs our convention will be that if there P is an edge then both (i; j) 2 E and (j; i) 2 E. Thus (i;j)2E 1 = 2jEj. If we wish to sum P over edges only once, we will write fi; jg 2 E for the unordered pair. Thus fi;jg2E 1 = jEj. 4.1 Matrices associated to a graph Given an undirected graph G, the most natural matrix associated to it is its adjacency matrix: Definition 4.1 (Adjacency matrix). The adjacency matrix A 2 f0; 1gn×n is defined as ( 1 if fi; jg 2 E; Aij = 0 otherwise. Note that A is always a symmetric matrix with exactly di ones in the i-th row and the i-th column. While A is a natural representation of G when we think of a matrix as a table of numbers used to store information, it is less natural if we think of a matrix as an operator, a linear transformation which acts on vectors. The most natural operator associated with a graph is the diffusion operator, which spreads a quantity supported on any vertex equally onto its neighbors. To introduce the diffusion operator, first consider the degree matrix: Definition 4.2 (Degree matrix).
    [Show full text]
  • MINIMUM DEGREE ENERGY of GRAPHS Dedicated to the Platinum
    Electronic Journal of Mathematical Analysis and Applications Vol. 7(2) July 2019, pp. 230-243. ISSN: 2090-729X(online) http://math-frac.org/Journals/EJMAA/ |||||||||||||||||||||||||||||||| MINIMUM DEGREE ENERGY OF GRAPHS B. BASAVANAGOUD AND PRAVEEN JAKKANNAVAR Dedicated to the Platinum Jubilee year of Dr. V. R. Kulli Abstract. Let G be a graph of order n. Then an n × n symmetric matrix is called the minimum degree matrix MD(G) of a graph G, if its (i; j)th entry is minfdi; dj g whenever i 6= j, and zero otherwise, where di and dj are the degrees of ith and jth vertices of G, respectively. In the present work, we obtain the characteristic polynomial of the minimum degree matrix of graphs obtained by some graph operations. In addition, bounds for the largest minimum degree eigenvalue and minimum degree energy of graphs are obtained. 1. Introduction Throughout this paper by a graph G = (V; E) we mean a finite undirected graph without loops and multiple edges of order n and size m. Let V = V (G) and E = E(G) be the vertex set and edge set of G, respectively. The degree dG(v) of a vertex v 2 V (G) is the number of edges incident to it in G. The graph G is r-regular if and only if the degree of each vertex in G is r. Let fv1; v2; :::; vng be the vertices of G and let di = dG(vi). Basic notations and terminologies can be found in [8, 12, 14]. In literature, there are several graph polynomials defined on different graph matrices such as adjacency matrix [8, 12, 14], Laplacian matrix [15], signless Laplacian matrix [9, 18], seidel matrix [5], degree sum matrix [13, 19], distance matrix [1] etc.
    [Show full text]
  • MATH7502 Topic 6 - Graphs and Networks
    MATH7502 Topic 6 - Graphs and Networks October 18, 2019 Group ID fbb0bfdc-79d6-44ad-8100-05067b9a0cf9 Chris Lam 41735613 Anthony North 46139896 Stuart Norvill 42938019 Lee Phillips 43908587 Minh Tram Julien Tran 44536389 Tutor Chris Raymond (P03 Tuesday 4pm) 1 [ ]: using Pkg; Pkg.add(["LightGraphs", "GraphPlot", "Laplacians","Colors"]); using LinearAlgebra; using LightGraphs, GraphPlot, Laplacians,Colors; 0.1 Graphs and Networks Take for example, a sample directed graph: [2]: # creating the above directed graph let edges = [(1, 2), (1, 3), (2,4), (3, 2), (3, 5), (4, 5), (4, 6), (5, 2), (5, 6)] global graph = DiGraph(Edge.(edges)) end [2]: {6, 9} directed simple Int64 graph 2 0.2 Incidence matrix • shows the relationship between nodes (columns) via edges (rows) • edges are connected by exactly two nodes (duh), with direction indicated by the sign of each row in the edge column – a value of 1 at A12 indicates that edge 1 is directed towards node 1, or node 2 is the destination node for edge 1 • edge rows sum to 0 and constant column vectors c(1, ... , 1) are in the nullspace • cannot represent self-loops (nodes connected to themselves) Using Strang’s definition in the LALFD book, a graph consists of nodes defined as columns n and edges m as rows between the nodes. An Incidence Matrix A is m × n. For the above sample directed graph, we can generate its incidence matrix. [3]: # create an incidence matrix from a directed graph function create_incidence(graph::DiGraph) M = zeros(Int, ne(graph), nv(graph)) # each edge maps to a row in the incidence
    [Show full text]
  • Two One-Parameter Special Geometries
    Two One-Parameter Special Geometries Volker Braun1, Philip Candelas2 and Xenia de la Ossa2 1Elsenstrasse 35 12435 Berlin Germany 2Mathematical Institute University of Oxford Radcliffe Observatory Quarter Woodstock Road Oxford OX2 6GG, UK arXiv:1512.08367v1 [hep-th] 28 Dec 2015 Abstract The special geometries of two recently discovered Calabi-Yau threefolds with h11 = 1 are analyzed in detail. These correspond to the 'minimal three-generation' manifolds with h21 = 4 and the `24-cell' threefolds with h21 = 1. It turns out that the one- dimensional complex structure moduli spaces for these manifolds are both very similar and surprisingly complicated. Both have 6 hyperconifold points and, in addition, there are singularities of the Picard-Fuchs equation where the threefold is smooth but the Yukawa coupling vanishes. Their fundamental periods are the generating functions of lattice walks, and we use this fact to explain why the singularities are all at real values of the complex structure. Contents 1 Introduction 1 2 The Special Geometry of the (4,1)-Manifold 6 2.1 Fundamental Period . .6 2.2 Picard-Fuchs Differential Equation . .8 2.3 Integral Homology Basis and the Prepotential . 11 2.4 The Yukawa coupling . 16 3 The Special Geometry of the (1,1)-Manifold(s) 17 3.1 The Picard-Fuchs Operator . 17 3.2 Monodromy . 19 3.3 Prepotential . 22 3.4 Integral Homology Basis . 23 3.5 Instanton Numbers . 24 3.6 Cross Ratios . 25 4 Lattice Walks 27 4.1 Spectral Considerations . 27 4.2 Square Lattice . 28 1. Introduction Detailed descriptions of special geometries are known only in relatively few cases.
    [Show full text]
  • Seidel Signless Laplacian Energy of Graphs 1. Introduction
    Mathematics Interdisciplinary Research 2 (2017) 181 − 191 Seidel Signless Laplacian Energy of Graphs Harishchandra S. Ramane?, Ivan Gutman, Jayashri B. Patil and Raju B. Jummannaver Abstract Let S(G) be the Seidel matrix of a graph G of order n and let DS (G) = diag(n − 1 − 2d1; n − 1 − 2d2; : : : ; n − 1 − 2dn) be the diagonal matrix with di denoting the degree of a vertex vi in G. The Seidel Laplacian matrix of G is defined as SL(G) = DS (G) − S(G) and the Seidel signless Laplacian + matrix as SL (G) = DS (G) + S(G). The Seidel signless Laplacian energy ESL+ (G) is defined as the sum of the absolute deviations of the eigenvalues of SL+(G) from their mean. In this paper, we establish the main properties + of the eigenvalues of SL (G) and of ESL+ (G). Keywords: Seidel Laplacian eigenvalues, Seidel Laplacian energy, Seidel sign- less Laplacian matrix, Seidel signless Laplacian eigenvalues, Seidel signless Laplacian energy. 2010 Mathematics Subject Classification: 05C50. 1. Introduction Let G be a simple, undirected graph with n vertices and m edges. We say that G is an (n; m)-graph. Let v1; v2; : : : ; vn be the vertices of G. The degree of a vertex vi is the number of edges incident to it and is denoted by di. If di = r for all i = 1; 2; : : : ; n, then G is said to be an r-regular graph. By G will be denoted the complement of the graph G. The adjacency matrix of a graph G is the square matrix A(G) = (aij) , in which aij = 1 if vi is adjacent to vj and aij = 0 otherwise.
    [Show full text]
  • A Short Tutorial on Graph Laplacians, Laplacian Embedding, and Spectral Clustering
    A Short Tutorial on Graph Laplacians, Laplacian Embedding, and Spectral Clustering Radu Horaud INRIA Grenoble Rhone-Alpes, France [email protected] http://perception.inrialpes.fr/ Radu Horaud Graph Laplacian Tutorial Introduction The spectral graph theory studies the properties of graphs via the eigenvalues and eigenvectors of their associated graph matrices: the adjacency matrix and the graph Laplacian and its variants. Both matrices have been extremely well studied from an algebraic point of view. The Laplacian allows a natural link between discrete representations, such as graphs, and continuous representations, such as vector spaces and manifolds. The most important application of the Laplacian is spectral clustering that corresponds to a computationally tractable solution to the graph partitionning problem. Another application is spectral matching that solves for graph matching. Radu Horaud Graph Laplacian Tutorial Applications of spectral graph theory Spectral partitioning: automatic circuit placement for VLSI (Alpert et al 1999), image segmentation (Shi & Malik 2000), Text mining and web applications: document classification based on semantic association of words (Lafon & Lee 2006), collaborative recommendation (Fouss et al. 2007), text categorization based on reader similarity (Kamvar et al. 2003). Manifold analysis: Manifold embedding, manifold learning, mesh segmentation, etc. Radu Horaud Graph Laplacian Tutorial Basic graph notations and definitions We consider simple graphs (no multiple edges or loops), G = fV; Eg: V(G) = fv1; : : : ; vng is called the vertex set with n = jVj; E(G) = feijg is called the edge set with m = jEj; An edge eij connects vertices vi and vj if they are adjacent or neighbors. One possible notation for adjacency is vi ∼ vj; The number of neighbors of a node v is called the degree of v and is denoted by d(v), d(v ) = P e .
    [Show full text]
  • The Distance Matrix and Its Variants for Graphs and Digraphs by Carolyn
    The distance matrix and its variants for graphs and digraphs by Carolyn Reinhart A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Program of Study Committee: Leslie Hogben, Major Professor Steve Butler Jonas Hartwig Sung-Yell Song Michael Young The student author, whose presentation of the scholarship herein was approved by the program of study committee, is solely responsible for the content of this dissertation. The Graduate College will ensure this dissertation is globally accessible and will not permit alterations after a degree is conferred. Iowa State University Ames, Iowa 2021 Copyright © Carolyn Reinhart, 2021. All rights reserved. ii TABLE OF CONTENTS Page ACKNOWLEDGMENTS . iii ABSTRACT . iv CHAPTER 1. GENERAL INTRODUCTION . .1 1.1 Definitions for graphs . .3 1.2 Digraphs . .4 1.3 References . .5 CHAPTER 2. THE NORMALIZED DISTANCE LAPLACIAN . .7 2.1 Introduction . .7 2.2 The normalized distance Laplacian . 11 2.3 Using twin vertices to determine eigenvalues . 18 2.4 Characteristic polynomials . 21 2.5 Cospectral graphs . 25 2.5.1 Cospectral pairs on 10 or fewer vertices . 26 2.5.2 The number of graphs with a cospectral mate . 32 2.6 Concluding remarks . 33 2.7 References . 34 CHAPTER 3. A NOTE ON THE PRESERVATION OF GRAPH PARAMETERS BY COSPECTRALITY FOR THE DISTANCE MATRIX AND ITS VARIANTS . 37 3.1 Introduction . 37 3.2 Preservation of parameters . 39 3.3 Concluding remarks . 44 3.4 References . 44 CHAPTER 4. DISTANCE COSPECTRALITY IN DIGRAPHS . 46 4.1 Introduction . 46 4.2 The number of digraphs with distance cospectral mates .
    [Show full text]
  • Eigenvector Localization
    First, parse the title ... Eigenvector localization: • Eigenvectors are “usually” global entities • But they can be localized in extremely sparse/noisy graphs/matrices Implicit regularization: • Usually “exactly” optimize f+λg, for some λ and g • Regularization often a side effect of approximations to f Algorithmic anti-differentiation: • What is the objective that approximate computation exactly optimizes Large-scale graphs and network data: • Small versus medium versus large versus big • Social/information networks versus “constructed” graphs Outline Motivation: large informatics graphs • Downward-sloping, flat, and upward-sloping NCPs (i.e., not “nice” at large size scales, but instead expander-like/tree-like) • Implicit regularization in graph approximation algorithms Eigenvector localization & semi-supervised eigenvectors • Strongly and weakly local diffusions • Extension to semi-supervised eigenvectors Implicit regularization & algorithmic anti-differentiation • Early stopping in iterative diffusion algorithms • Truncation in diffusion algorithms Outline Motivation: large informatics graphs • Downward-sloping, flat, and upward-sloping NCPs (i.e., not “nice” at large size scales, but instead expander-like/tree-like) • Implicit regularization in graph approximation algorithms Eigenvector localization & semi-supervised eigenvectors • Strongly and weakly local diffusions • Extension to semi-supervised eigenvectors Implicit regularization & algorithmic anti-differentiation • Early stopping in iterative diffusion algorithms • Truncation
    [Show full text]
  • 35. Finding Clusters in Graphs
    MITOCW | 35. Finding Clusters in Graphs The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. GILBERT OK. Now, clustering for graphs. So this is a topic-- this is one of the important things you can STRANG: try to do with a graph. So you have a large graph. Let me kind of divide it into two clusters. So you've got a giant graph. And then the job is to make some sense out of it. And one possible step is to be able to subdivide it, if, as I see here, there's a cut between two reasonably equal parts of the graph-- reasonable-- reasonably same size. And therefore, that graph could be studied in two pieces. So the question is, how do you find such a cut by a algorithm? What's an algorithm that would find that cut? So that's a problem. Let's say we're looking for two clusters. We could look for more clusters, but let's say we want to look for two clusters. So what are we trying to do? We're trying to minimize. So this is the problem, then. So we look for-- find positions x and y, let's say. Two which will be the centers, so to speak, of the-- and really, it's just these points that-- so the data is the points and the edges, as always-- the nodes and the edges.
    [Show full text]
  • Laplacian Matrix and Applications
    LAPLACIAN MATRIX AND APPLICATIONS Alice Nanyanzi Supervisors: Dr. Franck Kalala Mutombo & Dr. Simukai Utete [email protected] August 24, 2017 1 Complex systems & Complex Networks 2 Networks Overview 3 Laplacian Matrix Laplacian Centrality Diffusion on networks Alice Nanyanzi (AIMS-SU) Laplacian Matrix August 24, 2017 1 / 22 Complex Systems; Complex Network/Large graph Approach Alice Nanyanzi (AIMS-SU) Laplacian Matrix August 24, 2017 2 / 22 Figure Introduction to Networks Intuition of Networks Whenever one mentions the word 'network', one normally thinks of an interconnection of items or things. Alice Nanyanzi (AIMS-SU) Laplacian Matrix August 24, 2017 3 / 22 Introduction to Networks Intuition of Networks Whenever one mentions the word 'network', one normally thinks of an interconnection of items or things. Formal Definition A network, G, is a pair (V ; E). Where V is the set of vertices (nodes) of G and E is the set of edges (links) of G. (Estrada & Knight). Categories of networks include simple networks, directed networks, undirected networks, weighted networks, etc (Estrada, 2015) Alice Nanyanzi (AIMS-SU) Laplacian Matrix August 24, 2017 3 / 22 Real-world Networks (a) Internet (b) Protein-Protein (c) Food web (d) Citation network Source: www.wikipedia.com Alice Nanyanzi (AIMS-SU) Laplacian Matrix August 24, 2017 4 / 22 Laplacian Matrix Definition Consider a simple undirected network, the Laplacian matrix L is the difference between the Degree matrix D and Adjacency matrix A i.e L = D − A. The entries of L are given as 8 k if i = j <> i Li;j = −1 if i 6= j and i is adjacent to j :>0 otherwise; where ki denotes the degree of node i (Estrada, 2011).
    [Show full text]