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Deterministic Scale-Free Networks
Authors
Albert-Laszlo Barabasi,Erzsebet Ravasz
Tamas Vicsek,Katrine Søborg Spang,Maria Czaja,MASSIMO MAZZER,Ruey-Shin Juang,David Suarez,Frederic Villieras,Christine Gaylarde,Boaventura F. Reis,Masabumi Minami,Andrea Spagnoli,Geoffrey Grime,Amina Suleymanova,Jorge Casaus,Paul Ornstein,Javier Berdugo,Ivan Aleksic,Tomoki Fukai,Raz Yirmiya,Wesley Metzger,Bruna Bertucci,Noriaki Takagi,Joachim Mnich,Giovanni Ambrosi,Pablo García Abia,Marco Pieri,Lucas Taylor,John Vienna,Kirill Kolosov,Irina Novikova,Olga Lozovskaya,Nina Kulchenko,Irina Snegireva,Vladimir Makarov,Donal Bradley,Andrzej Stokłosa,JUAN PEÑA LOPEZ,Pavlos Vranas,Evandro Salvador Alves de Oliveira,Niels Grønbech-Jensen,Hooman Momen,Patrice Bourgeois,María Dolores Estrada de Oya,Carsten Benndorf,Xiao-Ying Huang,Maria Jesus Dianez,S.A. Bedina,Silvia PONDRELLI,Julienne Grant,Jaël Muls,Ana Pintar,Tahir Bachar Issa,Michael Milona,Cristin Chall,Wendy Flores-Fuentes,Roman Drnovsek,Konrad Reszka,Akimichi Takemura,Christophe Dutang,Galina Kutekhova,Mengqi Li,Timo Baumann,Roderick Hooker,Natalia Volodina,Silvio Alejandro Alcivar Molina,Riccardo Ferretti,Mikhail Zhilin,Janet Scull,Xueping Liu,Ricardo Silva,Inesa Sheremet,Kara Wunderlich,Jouni Tuisku,Xing-Li Zhao,valentina gonzalez jaramillo,Grigory Vasiliev,Kelsey Alland,Alexander Bukreyev,Rajmund Porada,Jehad Azar,Takao Kasuga,Uta Sailer,Jinshan Wen,Jonas Rieger,Fredrik Eika,Hamed Heydari,Eusebio Sanchez,Thomas Ferguson,Pier Giorgio Rancoita,Leonello Servoli,Jasper Kirkby,Gianni Pastore,Gonzalo Gómez Marqués,Guido Castellini,Tatiana Iukhta,Maya Kashuba,Alla Omelchenko,Diane Smith,Anna Lepekhina,Сирануш Нигарян,Svitlana Stavytska,Tim Muris,Inga Mühlenpfordt,Olga Lytvynova,Zhi-wen Chen,Shigeyuki KON,Viet Khoi Nguyen,Goyitom Gebremedhn,Tian Long,George Kiroff,Gabriela Vlase,Kirsten Thonicke,Ignasi Mundet Riera,Sileshi Degu,Wenlong Ma,José Germán Serrano-Gamboa,Naoto Tominaga,Amire Seyedfaraji,Yinghui Peng,Augusto Tamayo,Pierre Vanhove,Pupa Gilbert,Christopher Fuchs,Laura Kaplan-Weisman,Robert Nistico,Antonio Bennasar,Roger Karlsson,Francisco Salvà-Serra,Sergey Volkov,Rochelle Basden,Jingxiu Xie,WENDY MILAGROS TERCERO MARTINEZ,José Augusto Afonso,Charlie Siu,Григорий Колесников,Werner Grass,Jeanne Kagle,Cecilia Bonolo de Campos,红 张,Grzegorz Moroz,Itzhak Barkana,Hemalatha Bhagavan,Néstor Antonio Luque Vergara,Ahmet Nusret Bulgurcuoğlu,Reade Roberts,Mohammad Ali Mahjoub,Татьяна Димитриади,Majid Abbasi,Ravindra Belur,shekhar gogna,William Matern,Dennis Barrett,Stuart Rowlands,Soichiro Watanabe,Rafael Peñafiel,Xavier SANTARELLI,giuseppe munda,Thomas Mason,Gregory Broderick,Lonnie Zeltzer,Ana M Ibarra,Hiroshi Okuyama,James Adair,Dr. Alexis Diaz-Torres,Wen-Chen Chang,Steven Overbury,Soledad Anguiano Igea,Paolo Bartalini,Vuko Brigljević,Jürgen Kesselmeier,Adriaan König,Dimitri Bourilkov,Helmut Vogel,Arno Straessner,Isabel Josa Mutuberria,Francesca Cavallari,Franck cleymand,Shailesh Chandrasekharan,Hamid Kachkachi,Vincent Perreten,Francisco E Olucha-Bordonau,Jennifer Scott,N.F. 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arXiv:cond-mat/0107419v3 [cond-mat.stat-mech] 6 Feb 2002 Deterministic Scale-Free Networks
Albert-L´aszl´o Barab´asia, Erzs´ebet Ravasza, Tam´as Vicsekb
aDepartment of Physics, University of Notre Dame,
Notre Dame, IN 46556-5670, USA
bDepartment of Biological Physics, E¨otv¨os University,
P´azm´any P´eter S´etany 1-A, Budapest, Hungary, H-1117
Abstract
Scale-free networks are abundant in nature, describing such diverse systems as
the world wide web, the web of human sexual contacts, or the chemical network of
a cell. All models used to generate a scale-free topology are stochastic, that is they
create networks in which the nodes appear to be randomly connected to each other.
Here we propose a simple model that generates scale-free networks in a deterministic
fashion. We solve exactly the model, showing that the tail of the degree distribution
follows a power law.
Key words: Disordered Systems; Networks; Scale-free networks; Scaling
PACS:
Networks are ubiquitous in nature and society [1–3], describing various com-
plex systems, such as the society, a network of individuals linked by various
social links [4–7]; the Internet, a network of routers connected by various
physical connections [8,9]; the world wide web, a virtual web of documents
connected by uniform resource connectors [10] or the cell, a network of sub-
strates connected by chemical reactions [11,12]. Despite their diversity, most
networks appearing in nature follow universal organizing principles. In partic-
ular, it has been found that many networks of scientific interest are scale-free
[13,14], that is, the probability that a randomly selected node has exactly k
links decays as a power law, following P (k) ∼ k−γ , where γ is the degree
exponent. The list of documented scale-free networks now include the world
wide web [10,15], the Internet [8], the cell [11,12], the web of human sexual
contacts [16], the language [17], or the web of actors in Hollywood [18], most
of which appear to have degree exponents between two and three.
The high interest in understanding the topology of complex networks has
resulted in the development of a considerable number of network models
Preprint submitted to Elsevier Preprint 22 October 2018
Albert-L´aszl´o Barab´asia, Erzs´ebet Ravasza, Tam´as Vicsekb
aDepartment of Physics, University of Notre Dame,
Notre Dame, IN 46556-5670, USA
bDepartment of Biological Physics, E¨otv¨os University,
P´azm´any P´eter S´etany 1-A, Budapest, Hungary, H-1117
Abstract
Scale-free networks are abundant in nature, describing such diverse systems as
the world wide web, the web of human sexual contacts, or the chemical network of
a cell. All models used to generate a scale-free topology are stochastic, that is they
create networks in which the nodes appear to be randomly connected to each other.
Here we propose a simple model that generates scale-free networks in a deterministic
fashion. We solve exactly the model, showing that the tail of the degree distribution
follows a power law.
Key words: Disordered Systems; Networks; Scale-free networks; Scaling
PACS:
Networks are ubiquitous in nature and society [1–3], describing various com-
plex systems, such as the society, a network of individuals linked by various
social links [4–7]; the Internet, a network of routers connected by various
physical connections [8,9]; the world wide web, a virtual web of documents
connected by uniform resource connectors [10] or the cell, a network of sub-
strates connected by chemical reactions [11,12]. Despite their diversity, most
networks appearing in nature follow universal organizing principles. In partic-
ular, it has been found that many networks of scientific interest are scale-free
[13,14], that is, the probability that a randomly selected node has exactly k
links decays as a power law, following P (k) ∼ k−γ , where γ is the degree
exponent. The list of documented scale-free networks now include the world
wide web [10,15], the Internet [8], the cell [11,12], the web of human sexual
contacts [16], the language [17], or the web of actors in Hollywood [18], most
of which appear to have degree exponents between two and three.
The high interest in understanding the topology of complex networks has
resulted in the development of a considerable number of network models
Preprint submitted to Elsevier Preprint 22 October 2018
[13,14,19–27]. Most of these are based on two mechanisms: incremental growth
and preferential attachment [13,14]. Incremental growth captures the fact that
networks are assembled through the addition of new nodes to the system,
while preferential attachment encodes the hypothesis that new nodes connect
with higher probability to more connected nodes. Both of these mechanisms
have been supported by extensive empirical measurements [6,28,29], indicating
that they are simultaneously present in many systems with scale-free network
topology.
Stochasticity is a common feature of all network models that generate scale-
free topologies. That is, new nodes connect using a probabilistic rule to the
nodes already present in the system. This randomness present in the models,
while in line with the major features of networks seen in nature, makes it
harder to gain a visual understanding of what makes them scale-free, and
how do different nodes relate to each other. It would therefore be of major
theoretical interest to construct models that lead to scale-free networks in
a deterministic fashion. Here we present such a simple model, generating a
deterministic scale-free network using a hierarchical construction.
1 Model description
The construction of the model, that follows a hierarchical rule commonly used
in deterministic fractals [30,31], is shown in Figure 1. The network is built in an
iterative fashion, each iteration repeating and reusing the elements generated
in the previous steps as follows:
Step 0: We start from a single node, that we designate as the root of the graph.
Step 1: We add two more nodes, and connect each of them to the root.
Step 2: We add two units of three nodes, each unit identical to the network
created in the previous iteration (step 1), and we connect each of the bottom
nodes (see Figure 1) of these two units to the root. That is, the root will gain
four more new links.
Step 3: We add two units of nine nodes each, identical to the units generated
in the previous iteration, and connect all eight bottom nodes of the two new
units to the root.
These rules can be easily generalized. Indeed, step n would involve the follow-
ing operation:
Step n: Add two units of 3n−1 nodes each, identical to the network created in
the previous iteration (step n − 1), and connect each of the 2n bottom nodes
2
and preferential attachment [13,14]. Incremental growth captures the fact that
networks are assembled through the addition of new nodes to the system,
while preferential attachment encodes the hypothesis that new nodes connect
with higher probability to more connected nodes. Both of these mechanisms
have been supported by extensive empirical measurements [6,28,29], indicating
that they are simultaneously present in many systems with scale-free network
topology.
Stochasticity is a common feature of all network models that generate scale-
free topologies. That is, new nodes connect using a probabilistic rule to the
nodes already present in the system. This randomness present in the models,
while in line with the major features of networks seen in nature, makes it
harder to gain a visual understanding of what makes them scale-free, and
how do different nodes relate to each other. It would therefore be of major
theoretical interest to construct models that lead to scale-free networks in
a deterministic fashion. Here we present such a simple model, generating a
deterministic scale-free network using a hierarchical construction.
1 Model description
The construction of the model, that follows a hierarchical rule commonly used
in deterministic fractals [30,31], is shown in Figure 1. The network is built in an
iterative fashion, each iteration repeating and reusing the elements generated
in the previous steps as follows:
Step 0: We start from a single node, that we designate as the root of the graph.
Step 1: We add two more nodes, and connect each of them to the root.
Step 2: We add two units of three nodes, each unit identical to the network
created in the previous iteration (step 1), and we connect each of the bottom
nodes (see Figure 1) of these two units to the root. That is, the root will gain
four more new links.
Step 3: We add two units of nine nodes each, identical to the units generated
in the previous iteration, and connect all eight bottom nodes of the two new
units to the root.
These rules can be easily generalized. Indeed, step n would involve the follow-
ing operation:
Step n: Add two units of 3n−1 nodes each, identical to the network created in
the previous iteration (step n − 1), and connect each of the 2n bottom nodes
2
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