Mathematics of networks

Author(s):  
Mark Newman

An introduction to the mathematical tools used in the study of networks. Topics discussed include: the adjacency matrix; weighted, directed, acyclic, and bipartite networks; multilayer and dynamic networks; trees; planar networks. Some basic properties of networks are then discussed, including degrees, density and sparsity, paths on networks, component structure, and connectivity and cut sets. The final part of the chapter focuses on the graph Laplacian and its applications to network visualization, graph partitioning, the theory of random walks, and other problems.

2009 ◽  
Vol 3 (4) ◽  
pp. 266-278 ◽  
Author(s):  
G.S. Thakur ◽  
A.W.M. Dress ◽  
R. Tiwari ◽  
S.-S. Chen ◽  
M.T. Thai

2013 ◽  
Vol 58 (2) ◽  
pp. 119-142
Author(s):  
Елена Коницкая

Фразеологизмы служат действенным средством лаконизации речи и, следовательно, экономии языковых средств. Роль языковой экономии во фразеологии проявляется как на синхроническом уровне (возникновение модификаций фразеологизмов, не нарушающих цельности фразеологизма и норм его использования), так и на диахроническом уровне (закрепляясь в языке, модификации могут превращаться в новые устойчивые выражения, включающиеся во фразеологический фонд). Эти процессы, хорошо изученные в русском языке, недостаточно исследованы в словенском. В статье рассматривается роль языковой экономии в образовании словенских фразеологизмов на фоне русской фразеологии; сопоставляются результаты действия принципа языковой экономии, отмечаются как сходства, так и различия в двух славянских языках.Ключевые слова: словенская фразеология, русская фразеология, происхождение фразеологизмов, языковая экономия, эллипсис, имплицитность....Jelena KonitskayaRole of the economy principle in language in the Composition of Slovenian phraseologisms (compared with those in Russian) Phraseologisms serve as a telling resource of economy of language features. The role of the economy principle in language, in its phraseology, appears at the synchronic level (derivation of modifications of phraseological units) and at the diachronic level (derivation of new stable expressions incorporated into the stock of phraseology). The paper deals with the diachronic aspect of the problem. The results of compression (ellipsis, implication) caused by the economy principle in language (in phraseology) have been studied rather well in the Russian language (e.g., works of V. Mokiyenko and others), however, insufficiently in the Slovenian language. The paper comprises an analysis of the role of the economy principle in the composition of Slovenian phraseologisms against the background of the Russian phraseology; a comparison of the effect of the economy principle in language, both the similarities and differences in these two Slavic languages are pointed out. The similarities appear in the derivation of phraseologisms through elimination of the final part of expression(proverbs or phraseologisms of a broader component structure), e.g., Kovačeva kobila je vedno bosa > kovačeva kobila, in comparison with the Russian кто во что горазд, тот тем и промышляет (тот в то и играет)> кто во что горазд. Different constituents of a stable expression may turn elliptic in both languages, e.g., the collocation posipati si glavo s pepelom > posipati se s pepelom, speljati vodo na svoj mlin > voda na mlin, рус. дорого не по карману > не по карману, разбить в пух и прах > разбить в пух etc. In both languages, elimination (ellipsis) of a constituent may cause fogging of the inner form of phraseologisms; e.g., the collocation gledati kot čuk na palici > gledati kot čuk; ne priplаvati po prežgani juhi > ne priplаvati po juhi. Implication of additions in phraseologisms, leading to the derivation of structures with a broken syntactic relationship, is noted in both languages; e.g., Russian безо всяких, наша взяла, и никаких, прокатить на вороных, etc., the collocation ne reči ne bele ne črne, ne biti pri čisti, huda / tenka / slaba / tesna prede (komu), tristo kosmatih medvedov / hudičev etc. Forward pronominalisation may be consideredas a separate case of the manifestation of the economy principle in a language, noted in expressions such as ima ga pod kapo (‘быть пьяным’, literatim “иметь его под шапкой”), pobrisati jo (‘убежать’, literatim “почистить ее”), dobiti jih (‘быть избитым’, literatim “получить их”). The paper covers the sight of different possibilities of explaining of the derivation of Slavic expressions taking into account the role of the economy principle in language.


1999 ◽  
Vol 10 (02) ◽  
pp. 225-246 ◽  
Author(s):  
MICHAEL HOLZRICHTER ◽  
SUELY OLIVEIRA

The problem of partitioning a graph such that the number of edges incident to vertices in different partitions is minimized, arises in many contexts. Some examples include its recursive application for minimizing fill-in in matrix factorizations and load-balancing for parallel algorithms. Spectral graph partitioning algorithms partition a graph using the eigenvector associated with the second smallest eigenvalue of a matrix called the graph Laplacian. The focus of this paper is the use graph theory to compute this eigenvector more quickly.


2016 ◽  
Vol 103 ◽  
pp. 165-180 ◽  
Author(s):  
Víctor M. López Millán ◽  
Vicent Cholvi ◽  
Antonio Fernández Anta ◽  
Luis López

2013 ◽  
Vol 23 (1) ◽  
pp. 013104 ◽  
Author(s):  
Yang Wang ◽  
An Zeng ◽  
Zengru Di ◽  
Ying Fan

2015 ◽  
Vol 581 ◽  
pp. 45-66 ◽  
Author(s):  
Atish Das Sarma ◽  
Anisur Rahaman Molla ◽  
Gopal Pandurangan

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