Title
Simplicial complexes and complex networks
Creator
Maletić, Slobodan V.
Copyright date
2013
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Select license
Autorstvo-Nekomercijalno-Bez prerade 3.0 Srbija (CC BY-NC-ND 3.0)
License description
Dozvoljavate samo preuzimanje i distribuciju dela, ako/dok se pravilno naznačava ime autora, bez ikakvih promena dela i bez prava komercijalnog korišćenja dela. Ova licenca je najstroža CC licenca. Osnovni opis Licence: http://creativecommons.org/licenses/by-nc-nd/3.0/rs/deed.sr_LATN. Sadržaj ugovora u celini: http://creativecommons.org/licenses/by-nc-nd/3.0/rs/legalcode.sr-Latn
Language
English
Cobiss-ID
Theses Type
Doktorska disertacija
description
Datum odbrane: 11.10.2013.
Other responsibilities
mentor
Rajković, Milan
član komisije
Knežević, Milan, 1952-
član komisije
Elezović-Hadžić, Sunčica, 1961-
član komisije
Miljković, Vladimir, 1967-
University
Univerzitet u Beogradu
Faculty
Fizički fakultet
Alternative title
Simplicijalni kompleksi i kompleksne mreže-uticaj (pod)struktura višeg reda na karakteristike mreže
Publisher
[S. V. Maletić]
Format
VIII, 66 str.
description
physics-statistical physics/fizika-statistička fizika
Abstract (en)
In modern theoretical physics (quantum gravity, computational electromagnetism,
gauge theories, elasticity, to name a few) simplicial complexes have become an important
objects due to their computational convenience and power of algebraic topological
concepts. On the other hand, physics (and mathematics) of complex systems
formed by the large number of elements interacting through pairwise interactions in
highly irregular manner, is the most commonly restricted to concepts and methods
of the graph theory. Such systems are called complex networks and notions of graph
and complex network are used interchangeably. The achievements of the complex
networks research are important for modern world and largely reshape our notion
of a large class of complex phenomena, primarily because seemingly random and
disorganized phenomena display meaningful structure and organization. The same
stands also for the aggregations of complex network’s elements into communities
(modules or clusters), which as a major drawback has that they are restricted to
the collections of pairwise interactions.
In this thesis to the notions of structure and substructure of complex systems,
exemplified by complex networks, are given a new meaning through the changing the
notion of community, by defining a simplicial community. Unlike the common notion
of community, simplicial community is characterized by higher-order aggregations
of complex network’s elements. Namely, starting from typical properties of complex
systems it was shown that the natural substructure of complex networks emerges
like the aggregations of a multidimensional simplices. It was further shown that
simplicial complexes may be constructed from complex networks in several different
ways, indicating the possible different hidden organizational patterns leading to
the final structure of complex network and which are responsible for the network
properties. In this thesis two simplicial complexes obtained from complex networks
are studied: the neighborhood and the clique complex.
Relying on the combinatorial algebraic topology concepts a unified mathematical
framework for the study of their properties is proposed. The topological quantities,
like structure vectors, Betti numbers, combinatorial Laplacian operator are calculated
for diverse models real-world networks. Properties of spectra of combinatorial
iii
Laplacian operator of simplicial complexes are explored, and the necessity of higher
order spectral analysis is discussed and compared with results for ordinary graphs...
Abstract (sr)
U savremenoj teorijskoj fizici (na primer, kvantnoj gravitaciji, raˇcunskom elektromagnetizmu,
gejdˇz teoriji, elastiˇcnosti) simplicijalni kompleksi su postali vaˇzni
objekti zbog njihove raˇcunske pogodnosti i mo´ci koncepata algebarske topologije.
Sa druge strane, fizika (i matematika) kompleksnih sistema formiranih od velikog
broja elemenata koji interaguju parnim interakcijama na izrazito neregularan naˇcin,
najˇceˇs´ce je ograniˇcena na koncepte i metode teorije grafova. Takvi sistemi se nazivaju
kompleksne mreˇze i pojmovi graf i kompleksna mreˇza se poistove´cuju. Doprinosi
istraˇzivanja kompleksnih mreˇza su vaˇzni za savremeni svet i umnogome preoblikuju
naˇse poimanje velike klase kompleksnih fenomena, pre svega zbog toga
ˇsto naizgled sluˇcajni i neured-eni fenomeni pokazuju smislenu strukturu i organizaciju.
Isto vaˇzi i za agregacije elemenata kompleksne mreˇze u zajednice (module ili
klastere), koje kao najve´ci nedostatak imaju osobinu da su ograniˇcene na kolekcije
parnih interakcija.
U ovoj tezi pojmovima strukture i podstrukture kompleksnog sistema, kroz
primer kompleksne mreˇze, dato je novo znaˇcenje menjanjem pojma zajednice, definisanjem
simplicijalne zajednice. Za razliku od uobiˇcajenog pojma zajednice, simplicijalna
zajednica je karakterisana sa agregacijama viˇseg reda elemenata mreˇze.
Naime, poˇsavˇsi od tipiˇcnih osobina kompleksnih sistema pokazano je da se kao
prirodna podstruktura kompleksne mreˇze pojavljuju agregacije multidimenzionalnih
simpleksa. Pokazano je, dalje, da se simplicijalni kompleksi mogu iz kompleksnih
mreˇza konstruisati na nekoliko razliˇcitih naˇcina, ukazuju´ci na postojanje razliˇcitih
skrivenih organizacionih obrazaca koji vode do konaˇcne strukture kompleksne mreˇze
i koji su odgovorni za osobine mreˇze. U ovoj tezi su razmatrana dva simplicijalna
kompleksa dobijena iz kompleksne mreˇze: kompleks susedstva i klika kompleks.
Oslanjaju´ci se na koncepte kombinatorijalne algebarske topologije predloˇzen je
objedinjeni matematiˇcki okvir za prouˇcavanje njihovih osobina. Topoloˇske veliˇcine,
kao ˇsto su strukturni vektori, Betti brojevi, operator kombinatorni laplasijan, raˇcunate
su za razliˇcite modele realnih mreˇza. Ispitivane su osobine spektra operatora kombinatorni
laplasijan simplicijalnog kompleksa, i razmatrana je neophodnost spektralne
analize viˇseg reda koja je pored-ena sa rezultatima za obiˇcne grafove...
Authors Key words
statistical mechanics, complex systems, graph, complex networks, combinatorial algebraic
topology, simplicial complexes, topological invariant, combinatorial Laplacian,
entropy
Authors Key words
statistička mehanika, kompleksni sistemi, graf, kompleksne mreže, kombinatorna
algebarska topologija, simplicijalni kompleksi, topološka invarijanta, kombinatorni
laplasijan, entropija
Classification
533.9(043.3)
Type
Tekst
Abstract (en)
In modern theoretical physics (quantum gravity, computational electromagnetism,
gauge theories, elasticity, to name a few) simplicial complexes have become an important
objects due to their computational convenience and power of algebraic topological
concepts. On the other hand, physics (and mathematics) of complex systems
formed by the large number of elements interacting through pairwise interactions in
highly irregular manner, is the most commonly restricted to concepts and methods
of the graph theory. Such systems are called complex networks and notions of graph
and complex network are used interchangeably. The achievements of the complex
networks research are important for modern world and largely reshape our notion
of a large class of complex phenomena, primarily because seemingly random and
disorganized phenomena display meaningful structure and organization. The same
stands also for the aggregations of complex network’s elements into communities
(modules or clusters), which as a major drawback has that they are restricted to
the collections of pairwise interactions.
In this thesis to the notions of structure and substructure of complex systems,
exemplified by complex networks, are given a new meaning through the changing the
notion of community, by defining a simplicial community. Unlike the common notion
of community, simplicial community is characterized by higher-order aggregations
of complex network’s elements. Namely, starting from typical properties of complex
systems it was shown that the natural substructure of complex networks emerges
like the aggregations of a multidimensional simplices. It was further shown that
simplicial complexes may be constructed from complex networks in several different
ways, indicating the possible different hidden organizational patterns leading to
the final structure of complex network and which are responsible for the network
properties. In this thesis two simplicial complexes obtained from complex networks
are studied: the neighborhood and the clique complex.
Relying on the combinatorial algebraic topology concepts a unified mathematical
framework for the study of their properties is proposed. The topological quantities,
like structure vectors, Betti numbers, combinatorial Laplacian operator are calculated
for diverse models real-world networks. Properties of spectra of combinatorial
iii
Laplacian operator of simplicial complexes are explored, and the necessity of higher
order spectral analysis is discussed and compared with results for ordinary graphs...
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