DOCSLIB.ORG

Linear Algebraic Louvain Method in Python

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Linear Algebraic Louvain Method in Python c c 2020 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. 1 Linear Algebraic Louvain Method in Python Tze Meng Low, Daniele G. Spampinato Scott McMillan Michel Pelletier Electrical and Computer Engineering Department Software Engineering Institute Graphegon, Inc. Carnegie Mellon University Carnegie Mellon University Portland, OR, USA Pittsburgh, PA, USA Pittsburgh, PA, USA [email protected] [email protected], [email protected] [email protected] Abstract—We show that a linear algebraic formulation of Step 2) A new graph is created from the newly par- • the Louvain method for community detection can be derived titioned graph. Identified communities form the set of systematically from the linear algebraic definition of modularity. new vertices, and edges between vertices in different Using the pygraphblas interface, a high-level Python wrapper for the GraphBLAS C Application Programming Interface (API), we communities are aggregated into a new set of edges. Step demonstrate that the linear algebraic formulation of the Louvain 1 is then applied to this newly created graph. method can be rapidly implemented. The two steps above are repeated until no further improvement Index Terms—Louvain Method, Community Detection, Graph in modularity is obtained. Algorithms, GraphBLAS A. Definition of Modularity I. INTRODUCTION Modularity is a scalar measure used to quantify the strengh Community detection is a critical component in the anal- of community structures within a graph [3]. It is computed as isys of real-world network systems with important appli- 1 k k Q = A i j δ(c , c ), cations in social media analytics, biology, recommendation 2m ij − 2m i j ij systems, and telecommunications among others [1]. The Lou- X vain method [2] based on modularity optimization [3] is a where Aij is the edge weight between vertices vi, vj V , m ∈ commonly adopted technique to identify communities. is the sum of all edge weights, ki is the sum of the weights Current implementations of the Louvain method adopt a of all edges incident on vertex vi, ci C is the community ∈ traditional vertex-based approach in their algorithmic for- to which vi is currently assigned, and δ(ci, cj) = 1 if vertices mulation despite there being a linear algebraic formulation vi and vj are in the same community (i.e., ci = cj) and 0 for the computation of modularity [4]. In this paper, we otherwise. show that by starting with the linear algebraic formulation We will build on the matrix formulation of Q from [4]: of modularity, a linear algebraic algorithm for computing 1 Q = Γ(ST BS), (1) the Louvain method can be determined. This formulation 2m can leverage recent community-driven efforts behind linear where Γ( ) computes the trace of a matrix, S is a V C algebraic graph interfaces, such as the GraphBLAS [5], to · | | × | | matrix that captures the community membership of each rapidly develop an implementation of the Louvain method. vertex, and B is the V V modularity matrix where element Productivity is enhanced through the use of pygraphblas [6], | |×| | Bij is defined as a python interface for the GraphBLAS effort. k k B = A i j . ij ij − 2m II. BACKGROUND Each column si of S represents a community of vertices, th We assume the goal is to identify communities within a where a one in the j position of si indicates that vertex vj graph graph = (V, E) where V is the set of vertices G is in community ci. We denote vertex vj as a column basis and E is the set of weighted edges connecting them. The vector e of length V , and s is the sum of all vertices that j | | i Louvain method is an iterative algorithm where each iteration belong to the same community ci, i.e., is composed of two major steps: Step 1) Vertices are moved between communities such si = ej. • j:v c that each move increases the overall modularity of G. Xj ∈ i Step 1 is initialized by creating V communities where ~ | | An empty community is simply the zero vector, i.e. si = 0. each vertex is in its own community. Vertices are se- quentially reassigned to new communities as long as III. LINEAR ALGEBRAIC LOUVAIN METHOD modularity can be increased. The choice of communities In this section we develop a fully linear algebraic formula- for a vertex is based on the community that yields the tion of the Louvain algorithm starting from the linear algebraic highest modularity increase. definition of modularity. A. Modularity of a Community The maximum increase in modularity is computed as κ = Starting with Eqn. (1) and expanding on the definition of max(q1), and the community c` associated with κ can be the trace of a matrix, we express the modularity Q as identified by t = (q1 == κ), 1 T T T Q = s0 Bs0 + s1 Bs1 + ... + s C 1Bs C 1 2m | |− where t is a boolean vector of length V indicating the | |− | | C 1 C 1 v 1 | |− 1 | |− new community for vertex j. The original community = sT Bs = Q , c for vertex v can be computed by extracting the jth 2m i i 2m i i j i=0 i=0 S X X row of . Using the original community and the vector where Qi is the modularity of community ci. t, the community matrix S can be updated as Consider an arbitrary vertex vj where vj is a vertex T T S = S + ejt ejej S. (possibly, the only one) in community ci. We can define − a community vector si in terms of the vertex vj and the Step 1 is repeated for all vertices until no increase in remaining set of vertices si j as follows modularity can be achieved. \{ } Step 2) A new graph 0 = (V 0,E0) is constructed, where si = si j + ej. • G \{ } the set of vertices V are the communities identified in Using the above definition, we can derive the following defini- 0 Step 1. The new adjacency matrix A is computed as tion of modularity for community i in terms of the contribution 0 T to modularity by introducing vertex vj into community i: A0 = S AS, T Qi = si Bsi where the weight at location Aij0 is the sum of edge = (e + (s e ))T B(e + (s e )) weights in A between vertices in community ci and j i − j j i − j T T T T vertices in community cj. The diagonal elements Aii0 are = ej Bej + si j Bej + ej Bsi j + si j Bsi j \{ } \{ } \{ } \{ } the sum of edge weights in A belonging to the same T T T = Bjj + si j Bej + ej Bsi j +si j Bsi j communities ci. \{ } \{ } \{ } \{ } Contribution by vertex vj IV. IMPLEMENTATION CONSIDERATIONS B. Linear| Algebraic Louvain{z Method } In this section, we focus on Step 1 of the previously We address the two steps of Louvain separately. presented algorithm and discuss several optimizations. Step 1) Initially, vertices are assigned to their own • 1) Removing redundant computation. Recall the objective community and the community matrix S is the identity in Step 1 is to find the maximum increase in modularity. matrix, I. Since the expression eT Bs +sT Be is removed Each vertex, v is evaluated for the change in modularity j i j i j j j from all modularity computation,\{ } it\{ does} not contribute when reassigned from community ci to community c`. to the decision of which community should vertex vj Mathematically, the change in modularity due to such T be reassigned to. Thus, the expression ej Bsi j + movement (∆Q j , where i = `) can be computed as: \{ } i ` 6 sT Be does not need to be computed. Hence, the −→ i j j T T following\{ } computation can be performed instead: ∆Q j = (Bjj + ej Bs` j + s` j Bej) i ` \{ } \{ } −→ T T ˜ T T (Bjj + ej Bsi j + si j Bej) ∆Q j = ej Bs` j + s` j Bej. − \{ } \{ } i ` \{ } \{ } T T −→ T T = ej Bs` j + s` j Bej = e (B + B )s (3) \{ } \{ } j ` j T T \{ } (ej Bsi j + si j Bej). − \{ } \{ } 2) Bulk computation of q. Notice that Eqn. (3) can be Essentially, ∆Q j is computed by subtracting the con- computed at once for all values of ` as: i ` tribution of vertex−→vj to community ci from the contri- T T T ¯ q = ej (B + B )S. bution of vertex vj to community c`. Let q be a vector of length V defined as follows: where | | ¯ T S = S ejej S T q = ∆Q j ... ∆Q j . − i 0 i V 1 = (I e eT )S, (4) −→ −→| |− − j j As the Louvain algorithm only considers moving a vertex which is the community matrix after removing vertex v to a community that includes at least one neighbor, we j from its original community. can restrict the modularity computation to the commu- 3) Eliminating modularity matrix B. Modularity matrix B nities containing neighbors of v via a mask in the j is a dense V V matrix. In order to reduce the following manner: | | × | | memory requirements, recall that qT = qT ((eT (A + AT ))S = 0). (2) 1 ◦ j 6 1 B = A kkT , tq − 2m | {z } where k is a vector where the ith value is the node B. Discussion degree of vertex vi. Therefore, In terms of productivity, our pygraphblas implementation 1 1 can be mapped nearly one-to-one with the linear algebraic B + BT = (A kkT ) + (A kkT )T − 2m − 2m formulation of the algorithm, and requires less than 40 lines 1 of code (LOC).
Load more

Recommended publications
  • Scikit-Network Documentation Release 0.24.0 Bertrand Charpentier
    scikit-network Documentation Release 0.24.0 Bertrand Charpentier Jul 27, 2021 GETTING STARTED 1 Resources 3 2 Quick Start 5 3 Citing 7 Index 205 i ii scikit-network Documentation, Release 0.24.0 Python package for the analysis of large graphs: • Memory-efficient representation as sparse matrices in the CSR formatof scipy • Fast algorithms • Simple API inspired by scikit-learn GETTING STARTED 1 scikit-network Documentation, Release 0.24.0 2 GETTING STARTED CHAPTER ONE RESOURCES • Free software: BSD license • GitHub: https://github.com/sknetwork-team/scikit-network • Documentation: https://scikit-network.readthedocs.io 3 scikit-network Documentation, Release 0.24.0 4 Chapter 1. Resources CHAPTER TWO QUICK START Install scikit-network: $ pip install scikit-network Import scikit-network in a Python project: import sknetwork as skn See examples in the tutorials; the notebooks are available here. 5 scikit-network Documentation, Release 0.24.0 6 Chapter 2. Quick Start CHAPTER THREE CITING If you want to cite scikit-network, please refer to the publication in the Journal of Machine Learning Research: @article{JMLR:v21:20-412, author= {Thomas Bonald and Nathan de Lara and Quentin Lutz and Bertrand Charpentier}, title= {Scikit-network: Graph Analysis in Python}, journal= {Journal of Machine Learning Research}, year={2020}, volume={21}, number={185}, pages={1-6}, url= {http://jmlr.org/papers/v21/20-412.html} } scikit-network is an open-source python package for the analysis of large graphs. 3.1 Installation To install scikit-network, run this command in your terminal: $ pip install scikit-network If you don’t have pip installed, this Python installation guide can guide you through the process.
    [Show full text]
  • A Survey of Results on Mobile Phone Datasets Analysis
    Blondel et al. RESEARCH A survey of results on mobile phone datasets analysis Vincent D Blondel1*, Adeline Decuyper1 and Gautier Krings1,2 *Correspondence: [email protected] Abstract 1Department of Applied Mathematics, Universit´e In this paper, we review some advances made recently in the study of mobile catholique de Louvain, Avenue phone datasets. This area of research has emerged a decade ago, with the Georges Lemaitre, 4, 1348 increasing availability of large-scale anonymized datasets, and has grown into a Louvain-La-Neuve, Belgium Full list of author information is stand-alone topic. We will survey the contributions made so far on the social available at the end of the article networks that can be constructed with such data, the study of personal mobility, geographical partitioning, urban planning, and help towards development as well as security and privacy issues. Keywords: mobile phone datasets; big data analysis 1 Introduction As the Internet has been the technological breakthrough of the ’90s, mobile phones have changed our communication habits in the first decade of the twenty-first cen- tury. In a few years, the world coverage of mobile phone subscriptions has raised from 12% of the world population in 2000 up to 96% in 2014 – 6.8 billion subscribers – corresponding to a penetration of 128% in the developed world and 90% in de- veloping countries [1]. Mobile communication has initiated the decline of landline use – decreasing both in developing and developed world since 2005 – and allows people to be connected even in the most remote places of the world. In short, mobile phones are ubiquitous.
    [Show full text]
  • Multilevel Hierarchical Kernel Spectral Clustering for Real-Life Large
    Multilevel Hierarchical Kernel Spectral Clustering for Real-Life Large Scale Complex Networks Raghvendra Mall, Rocco Langone and Johan A.K. Suykens Department of Electrical Engineering, KU Leuven, ESAT-STADIUS, Kasteelpark Arenberg,10 B-3001 Leuven, Belgium {raghvendra.mall,rocco.langone,johan.suykens}@esat.kuleuven.be Abstract Kernel spectral clustering corresponds to a weighted kernel principal compo- nent analysis problem in a constrained optimization framework. The primal formulation leads to an eigen-decomposition of a centered Laplacian matrix at the dual level. The dual formulation allows to build a model on a repre- sentative subgraph of the large scale network in the training phase and the model parameters are estimated in the validation stage. The KSC model has a powerful out-of-sample extension property which allows cluster affiliation for the unseen nodes of the big data network. In this paper we exploit the structure of the projections in the eigenspace during the validation stage to automatically determine a set of increasing distance thresholds. We use these distance thresholds in the test phase to obtain multiple levels of hierarchy for the large scale network. The hierarchical structure in the network is de- termined in a bottom-up fashion. We empirically showcase that real-world networks have multilevel hierarchical organization which cannot be detected efficiently by several state-of-the-art large scale hierarchical community de- arXiv:1411.7640v2 [cs.SI] 2 Dec 2014 tection techniques like the Louvain, OSLOM and Infomap methods. We show a major advantage our proposed approach i.e. the ability to locate good quality clusters at both the coarser and finer levels of hierarchy using internal cluster quality metrics on 7 real-life networks.
    [Show full text]
  • Xerox University Microfilms
    INFORMATION TO USERS This material was produced from a microfilm copy of the original document. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the original submitted. The following explanation of techniques is provided to help you understand markings or patterns which may appear on this reproduction. I.The sign or "target" for pages apparently lacking from the document photographed is "Missing Page(s)". If it was possible to obtain the missing page(s) or section, they are spliced into the film along with adjacent pages. This may have necessitated cutting thru an image and duplicating adjacent pages to insure you complete continuity. 2. When an image on the film is obliterated with a large round black mark, it is an indication that the photographer suspected that the copy may have moved during exposure and thus caijse a blurred image. You will find a good image of die page in the adjacent frame. 3. When a map, drawing or chart, etif., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet ang to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below tf e first row and continuing on until complete. 4. The majority of users indicate that thetextual content is of greatest value, however, a somewhat higher quality reproduction could be made from "photographs" if essential to the understanding of the dissertation.
    [Show full text]
  • Fast Community Detection Using Local Neighbourhood Search
    Fast community detection using local neighbourhood search Arnaud Browet,∗ P.-A. Absil, and Paul Van Dooren Universit´ecatholique de Louvain Department of Mathematical Engineering, Institute of Information and Communication Technologies, Electronics and Applied Mathematics (ICTEAM) Av. Georges Lema^ıtre 4-6, B-1348 Louvain-la-Neuve, Belgium. (Dated: August 30, 2013) Communities play a crucial role to describe and analyse modern networks. However, the size of those networks has grown tremendously with the increase of computational power and data storage. While various methods have been developed to extract community structures, their computational cost or the difficulty to parallelize existing algorithms make partitioning real networks into commu- nities a challenging problem. In this paper, we propose to alter an efficient algorithm, the Louvain method, such that communities are defined as the connected components of a tree-like assignment graph. Within this framework, we precisely describe the different steps of our algorithm and demon- strate its highly parallelizable nature. We then show that despite its simplicity, our algorithm has a partitioning quality similar to the original method on benchmark graphs and even outperforms other algorithms. We also show that, even on a single processor, our method is much faster and allows the analysis of very large networks. PACS numbers: 89.75.Fb, 05.10.-a, 89.65.-s INTRODUCTION ity is encoded in the edge weight, this task is known in graph theory as community detection and has been introduced by Newman [9]. Community detection has Over the years, the evolution of information and com- already proven to be of great interest in many different munication technology in science and industry has had research areas such as epidemiology [10], influence and a significant impact on how collected data are being spread of information over social networks [11, 12], anal- used to understand and analyse complex phenomena [1].
    [Show full text]
  • A Survey of Results on Mobile Phone Datasets Analysis
    Blondel et al. EPJ Data Science (2015)4:10 DOI 10.1140/epjds/s13688-015-0046-0 R E V I E W Open Access A survey of results on mobile phone datasets analysis Vincent D Blondel1*, Adeline Decuyper1 and Gautier Krings1,2 *Correspondence: [email protected] Abstract 1Department of Applied Mathematics, Université catholique In this paper, we review some advances made recently in the study of mobile phone de Louvain, Avenue Georges datasets. This area of research has emerged a decade ago, with the increasing Lemaitre, 4, Louvain-La-Neuve, availability of large-scale anonymized datasets, and has grown into a stand-alone 1348, Belgium Full list of author information is topic. We survey the contributions made so far on the social networks that can be available at the end of the article constructed with such data, the study of personal mobility, geographical partitioning, urban planning,andhelp towards development as well as security and privacy issues. Keywords: mobile phone datasets; big data analysis; data mining; social networks; temporal networks; geographical networks 1 Introduction As the Internet has been the technological breakthrough of the ’s, mobile phones have changed our communication habits in the first decade of the twenty-first century. In a few years, the world coverage of mobile phone subscriptions has raised from % of the world population in up to % in - . billion subscribers - corresponding to a penetration of % in the developed world and % in developing countries []. Mobile communication has initiated the decline of landline use - decreasing both in developing and developed world since - and allows people to be connected even in the most remote places of the world.
    [Show full text]
  • Generalized Louvain Method for Community Detection in Large Networks
    Generalized Louvain method for community detection in large networks Pasquale De Meo∗, Emilio Ferrarax, Giacomo Fiumara∗, Alessandro Provetti∗,∗∗ ∗Dept. of Physics, Informatics Section. xDept. of Mathematics. University of Messina, Italy. ∗∗Oxford-Man Institute, University of Oxford, UK. fpdemeo, eferrara, gfiumara, [email protected] Abstract—In this paper we present a novel strategy to a novel measure of edge centrality, in order to rank all discover the community structure of (possibly, large) networks. the edges of the network with respect to their proclivity This approach is based on the well-know concept of network to propagate information through the network itself; iii) modularity optimization. To do so, our algorithm exploits a novel measure of edge centrality, based on the κ-paths. its computational cost is low, making it feasible even for This technique allows to efficiently compute a edge ranking large network analysis; iv) it is able to produce reliable in large networks in near linear time. Once the centrality results, even if compared with those calculated by using ranking is calculated, the algorithm computes the pairwise more complex techniques, when this is possible; in fact, proximity between nodes of the network. Finally, it discovers because of the computational constraints, the adoption of the community structure adopting a strategy inspired by the well-known state-of-the-art Louvain method (henceforth, LM), some existing techniques is not viable when considering efficiently maximizing the network modularity. The experi- large networks, and their application is only limited to small ments we carried out show that our algorithm outperforms case-studies. other techniques and slightly improves results of the original This paper is organized as follows: in the next Section we LM, providing reliable results.
    [Show full text]
  • Principles of Tissue Fractionation
    1. Theoret. Biol. (1964) 6, 33-59 Principles of Tissue Fractionation CHRISTIAN DE DUVE Laboratory of Physiological Chemistry University of Louvain, Belgium and The Rockefeiler Institute, New York, N. Y., U.S.A. (Received 15 April 1963) This paper representsan attempt to develop logically the basic premises that tissuefractionation is : (a) a chemical method to be conducted according to the ground rules which govern chemical fractionation in general; (b) potentially applicable to the separationand characterization of all elements of cellular organization, whether known or unknown, which are not irretrievably lost in the initial grinding of the cells. It is shownthat the approach which best answersthese prerequisites is a purely analytical one, untrammeled by any preconceivedidea of the cyto- logical composition of the isolated fractions, and allowing their bio- chemicalproperties to be expressedas continuous functions of the physical parameterwhich determinesthe behaviour of subcellularcomponents in the fractionation systemchosen, Examples are given which illustrate the appli- cation of density gradient centrifugation in this type of approach, as well as the advantageswhich can be derived from the useof mediaof different composition. The resultsof suchexperiments are expressedin the form of distribution curves of biochemical constituents, as with other fractionation methods such as chromatography or electrophoresis,but with the differencethat the independent variable is related to a property, not of the constituent but of its host-particles. These curves can be taken to represent the mass distribution of the particles themselvesif the constituent is assumedto be homogeneouslydistributed amongstthem (postulateof biochemicalhomo- geneity). This assumptionhas been verified for a number of enzymes,which provide valuable markers to fix the position of their host-particleson the distribution diagrams.
    [Show full text]
  • A Weight-Based Information Filtration Algorithm for Stock-Correlation Networks
    A Weight-based Information Filtration Algorithm for Stock-Correlation Networks Seyed Soheil Hosseini, Nick Wormald ∗, and Tianhai Tian School of Mathematics, Monash University Abstract Several algorithms have been proposed to filter information on a complete graph of corre- lations across stocks to build a stock-correlation network. Among them the planar maximally filtered graph (PMFG) algorithm uses 3n − 6 edges to build a graph whose features include a high frequency of small cliques and a good clustering of stocks. We propose a new algo- rithm which we call proportional degree (PD) to filter information on the complete graph of normalised mutual information (NMI) across stocks. Our results show that the PD algorithm produces a network showing better homogeneity with respect to cliques, as compared to eco- nomic sectoral classification than its PMFG counterpart. We also show that the partition of the PD network obtained through normalised spectral clustering (NSC) agrees better with the NSC of the complete graph than the corresponding one obtained from PMFG. Finally, we show that the clusters in the PD network are more robust with respect to the removal of random sets of edges than those in the PMFG network. Key words— stock-correlation network, PD network, PMFG network, normalised mutual in- formation arXiv:1904.06007v1 [q-fin.ST] 12 Apr 2019 1 Introduction ‘Complex systems’ is the term referring to the study of systems with a significant number of com- ponents in which we want to find out how the relationships between those components affect the behaviour of the system. The study of complex systems includes concepts from various disciplines such as mathematics, statistics, and computer science.
    [Show full text]
  • Community Detection Problem Based on Polarization Measures: an Application to Twitter: the COVID-19 Case in Spain
    mathematics Article Community Detection Problem Based on Polarization Measures: An Application to Twitter: The COVID-19 Case in Spain Inmaculada Gutiérrez 1,* , Juan Antonio Guevara 1 , Daniel Gómez 1,2 , Javier Castro 1,2 and Rosa Espínola 1,2 1 Faculty of Statistics, Complutense University Puerta de Hierro, 28040 Madrid, Spain; [email protected] (J.A.G.); [email protected] (D.G.); [email protected] (J.C.); [email protected] (R.E.) 2 Instituto de Evaluación Sanitaria, Complutense University, 28040 Madrid, Spain * Correspondence: [email protected] Abstract: In this paper, we address one of the most important topics in the field of Social Networks Analysis: the community detection problem with additional information. That additional information is modeled by a fuzzy measure that represents the risk of polarization. Particularly, we are interested in dealing with the problem of taking into account the polarization of nodes in the community detection problem. Adding this type of information to the community detection problem makes it more realistic, as a community is more likely to be defined if the corresponding elements are willing to maintain a peaceful dialogue. The polarization capacity is modeled by a fuzzy measure based on the JDJpol measure of polarization related to two poles. We also present an efficient algorithm for finding groups whose elements are no polarized. Hereafter, we work in a real case. It is a network obtained from Twitter, concerning the political position against the Spanish government taken by Citation: Gutiérrez, I.; Guevara, J.A.; several influential users. We analyze how the partitions obtained change when some additional Gómez, D.; Castro, J.; Espínola, R.
    [Show full text]
  • Towards More Efficient Parallel SAT Solving
    Towards more efficient parallel SAT solving Ludovic Le Frioux To cite this version: Ludovic Le Frioux. Towards more efficient parallel SAT solving. Distributed, Parallel, and Cluster Computing [cs.DC]. Sorbonne Université, 2019. English. NNT : 2019SORUS209. tel-03030122 HAL Id: tel-03030122 https://tel.archives-ouvertes.fr/tel-03030122 Submitted on 29 Nov 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. i Thesis submitted to obtain the degree of doctor of philosophy from Sorbonne Université École doctorale EDITE de Paris (ED130) Informatique, Télécommunication et Électronique Laboratoire de Recherche et Développement d’EPITA (LRDE) Laboratoire d’Informatique de Paris 6 (LIP6) Towards more efficient parallel SAT solving – Vers une parallélisation efficace de la résolution du problème de satisfaisabilité Ludovic LE FRIOUX Presented on July 3, 2019 in front of a jury composed of: Reviewers: - Michaël KRAJECKI, Professor at Université de Reims, CReSTIC - Laurent SIMON, Professor at Université de Bordeaux, LaBRI, CNRS Examiners: - Daniel LE BERRE, Professor
    [Show full text]
  • The Resurgence of Integrism the Action of the Holy Office Against René Draguet
    EphemeridesTheologicaeLovanienses 91/2 (2015) 295-309. doi: 10.2143/ETL.91.2.3085095 © 2015 by Ephemerides Theologicae Lovanienses. All rights reserved. The Resurgence of Integrism The Action of the Holy Office against René Draguet Ward DE PRIL KULeuven I. INTRODUCTION On July 4, 1942, the Belgian Archbishop Ernest Van Roey received a letter from the Holy Office, instructing him – without any further explana- tion – to prudently remove René Draguet (1896-1980) from his teaching duties at the Louvain Faculty of Theology1. Draguet, a secular priest, theo- logian and orientalist, had been teaching fundamental theology – at that time called dogmaticageneralis– since 1927. One day earlier, Van Roey had received the instruction from the Holy Office that the Catholic Univer- sity of Leuven had to deprive Vincenze Buffon, a former student of the Faculty of Theology, of his doctor’s title2. Buffon’s dissertation on Paolo Sarpi – written under the direction of Draguet – had been “condemned” by the Holy Office two weeks earlier (June 17, 1942)3. These decisions hit the University of Louvain very hard: one of its professors and one of its doctors were denounced by the Holy Office. Some months earlier, Louvain theologians had been startled when Louis Charlier’s Essaisurleproblèmethéologique4 had been put on the Index, together with Marie-Dominique Chenu’s UneÉcoledethéologie:Le Saulchoir5. Charlier was a professor at the Dominican studiumin Louvain. It was common knowledge that he shared the outspoken views of Draguet on the nature and method of theology. Ever since the Essaisurlepro- blèmethéologiquehad come out in late 1938, rumours were circulating that Draguet was its real author.
    [Show full text]