Albert-Lszl Barabsi With Emma K. Towlson, Sebastian Ruf, Michael - PowerPoint PPT Presentation

Network Science Class 6: Evolving Networks Albert-László Barabási With Emma K. Towlson, Sebastian Ruf, Michael Danziger, and Louis Shekhtman www.BarabasiLab.com

Questions 1. Bianconi-Barabasi Model 2. Bose-Einstein Condensation 3. Initial attractiveness 4. Role of internal links. 5. Node deletion. 6. Accelerated growth.

Section 1 Introduction

Section 1

EVOLVING NETWORK MODELS The BA model is only a minimal model. Makes the simplest assumptions: k  • linear growth 2 m   • linear preferential attachment ( k ) k i i Does not capture variations in the shape of the degree distribution variations in the degree exponent the size-independent clustering coefficient Hypothesis : The BA model can be adapted to describe most features of real networks. We need to incorporate mechanisms that are known to take place in real networks: addition of links without new nodes, link rewiring, link removal; node removal, constraints or optimization Network Science: Evolving Network Models

Section 6.2 Bianconi-Barabasi model

Can Latecomers Make It? SF model: k(t)~t ½ (first mover advantage) h i k i Fitness model: fitness ( η ) )  ( k i ) @ k(η,t)~t β ( η ) å h j k j j β η =η / C ( ) Degree (k) time Bianconi & Barabási, Physical Review Letters 2001; Europhys. Lett. 2001.

Section 6.2 Bianconi-Barabasi Model (definition) • Growth In each timestep a new node j with m links and fjtness j is added to the network, where j is a random number chosen from a fitness dis- tribution . Once assigned, a node’s fjtness does not change. • Preferential Attac hment The probability that a link of a new node connects to node i is propor- tional to the product of node i ’s degree k i and its fjtness i , h k .   (6.1) i i å i h k j j j

Section 2 Fitness Model

Section 6.2 Bianconi-Barabasi Model (Analytical) k i i k i . t  m j k j k ( i ) i ( t , t i )  m t k . t i

Section 6.2 Bianconi-Barabasi Model (Analytical) k i i k i . t  m j k j k ( i ) i ( t , t i )  m t k . t i Cmt = C ⩽ 2 η max

Section 2 Fitness Model BA model: k(t)~t ½ (first mover advantage) BB model: k(η,t)~t β ( η ) (fit-gets-richer) β η =η / C ( )

Section 2 Fitness Model-Degree distribution C m + 1 p k ∼ C ∫ d η ρ ( η ) η ( k ) η Uniform fitness distribution: fitness uniformly distributed in the [0,1] interval. C* = 1.255

Section 6.2 Bianconi-Barabasi Model (Analytical) ( i ) i ( t , t i )  m t C m + 1 p k ∼ C ∫ d η ρ ( η ) η ( k . k ) η t i

Section 6.2 Same Fitness C m + 1 p k ∼ C ∫ d η ρ ( η ) η ( k ) η

Section 6.2 Uniform Fitnesses C m + 1 p k ∼ C ∫ d η ρ ( η ) η ( k ) η

Section 6.2 Uniform Fitnesses

Section 6.3 Measuring Fitness

Section 6.3 Measuring Fitness: Web documents

Section 3 The Fitness of a scientific publication Π i ∼ η i k i P i ( t ) ln ( t )− μ i β η i A Φ ( σ i ) − 1 ) k i ( t )= m ( e x 1 2 / 2 dy − y Φ( x )= √ 2 π ∫ e −∞

Section 3 The Fitness of a scientific publication ln ( t )− μ i β η i A Φ ( σ i ) − 1 ) k i ( t )= m ( e x 1 2 / 2 dy − y Φ( x )= √ 2 π ∫ e −∞ Ultimate Impact: t  ∞ β η i A − 1 ) k i (∞)= m ( e