Outline Introduction The Barab´ asi-Albert model The copying model The fitness model Optimization models Introduction to network dynamics Ramon Ferrer-i-Cancho & Argimiro Arratia Universitat Polit` ecnica de Catalunya Version 0.4 Complex and Social Networks (20 20 -20 21 ) Master in Innovation and Research in Informatics (MIRI) Ramon Ferrer-i-Cancho & Argimiro Arratia Introduction to network dynamics
Outline Introduction The Barab´ asi-Albert model The copying model The fitness model Optimization models Official website: www.cs.upc.edu/~csn/ Contact: ◮ Ramon Ferrer-i-Cancho, rferrericancho@cs.upc.edu, http://www.cs.upc.edu/~rferrericancho/ ◮ Argimiro Arratia, argimiro@cs.upc.edu, http://www.cs.upc.edu/~argimiro/ Ramon Ferrer-i-Cancho & Argimiro Arratia Introduction to network dynamics
Outline Introduction The Barab´ asi-Albert model The copying model The fitness model Optimization models Introduction The Barab´ asi-Albert model The effect of replacing preferential by random attachment The copying model The fitness model Zipf’s law Optimization models Ramon Ferrer-i-Cancho & Argimiro Arratia Introduction to network dynamics
Outline Introduction The Barab´ asi-Albert model The copying model The fitness model Optimization models Models that generate networks [Caldarelli, 2007] ◮ The Barab´ asi-Albert model (growth and preferential attachment). ◮ Copying models ◮ Fitness based model ◮ Optimization models Each model produces a network through different dynamical principles/rules. Ramon Ferrer-i-Cancho & Argimiro Arratia Introduction to network dynamics
Outline Introduction The Barab´ asi-Albert model The effect of replacing preferential by random attachment The copying model The fitness model Optimization models The Barab´ asi-Albert model Example from citation networks, where p ( k ) ∼ k − 3 [Redner, 1998]. The evolution of an undirected network over time t . 1. t = 0, a disconnected set of n 0 vertices (no edges). 2. At time t > 0, add a new vertex with m 0 edges: ◮ The new vertex connects to the i -th vertex with probability k i π ( k i ) = � j k j Thus n = n 0 + t n m = 1 � k i = m 0 t 2 j =1 Ramon Ferrer-i-Cancho & Argimiro Arratia Introduction to network dynamics
Outline Introduction The Barab´ asi-Albert model The effect of replacing preferential by random attachment The copying model The fitness model Optimization models The growth of a vertex degree over time I The dependence of k i on time ◮ Treat k i as a continuous variable (although it is not). ◮ The variation of degree over time (on average) is ∂ k i 2 m 0 t = k i k i ∂ t = m 0 π ( k i ) = m 0 2 t ◮ t i is the time at which the i -th vertex was introduced. ◮ m 0 is the degree of the i -th vertex at time t i . ◮ Integrate on both sides of � k i � t ∂ k i = ∂ t ∂ k i = 1 ∂ t 2 t → 2 k i k i t m 0 t i Ramon Ferrer-i-Cancho & Argimiro Arratia Introduction to network dynamics
Outline Introduction The Barab´ asi-Albert model The effect of replacing preferential by random attachment The copying model The fitness model Optimization models The growth of a vertex degree over time II Finally, � t � 1 / 2 k i ( t ) ≈ m 0 t i Ramon Ferrer-i-Cancho & Argimiro Arratia Introduction to network dynamics
Outline Introduction The Barab´ asi-Albert model The effect of replacing preferential by random attachment The copying model The fitness model Optimization models A non-rigorous proof that p ( k ) ≈ k − 3 I Sketch of the proof [Barab´ asi et al., 1999] � 1 / 2 � ◮ Starting point: k i ( t ) = m 0 t t i ◮ Final goal: obtain p ( k ) through p ( k ) ≈ ∂ p ( k i < k ) ∂ k ◮ Intermediate goal: calculate p ( k i < k ) A rigorous proof is available [Bollob´ as et al., 2001] Ramon Ferrer-i-Cancho & Argimiro Arratia Introduction to network dynamics
Outline Introduction The Barab´ asi-Albert model The effect of replacing preferential by random attachment The copying model The fitness model Optimization models A non-rigorous proof that p ( k ) ≈ k − 3 II ◮ p ( k i < k ): the probability that the i -th vertex has degree lower than k . ◮ � t � � � 1 / 2 t i > m 2 � 0 t � p ( k i < k ) = p < k = p m 0 k 2 t i ◮ p ( t i = τ ) = 1 / ( n 0 + t ) for n 0 = 1 (for t i ≤ τ ). ◮ p ( t i = τ ) ≈ 1 / ( n 0 + t ) for n 0 > 1 but small. m 2 0 t k 2 t i > m 2 t i ≤ m 2 � 0 t � � 0 t � � = 1 − p = 1 − p ( t i = τ ) p k 2 k 2 τ =0 Ramon Ferrer-i-Cancho & Argimiro Arratia Introduction to network dynamics
Outline Introduction The Barab´ asi-Albert model The effect of replacing preferential by random attachment The copying model The fitness model Optimization models A non-rigorous proof that p ( k ) ≈ k − 3 III ◮ t i > m 2 ≈ 1 − m 2 � � 0 t 0 t n 0 + t k − 2 p k 2 ◮ ≈ 2 m 2 p ( k ) ≈ ∂ p ( k i < k ) 0 t n 0 + t k − 3 ∂ k ◮ p ( k ) ≈ ck − γ with γ = 3 and c = 2 m 2 0 t n 0 + t . More rigorous proofs are available [Newman, 2010]. Exercise: a more precise calculation for p ( t i = τ ). Ramon Ferrer-i-Cancho & Argimiro Arratia Introduction to network dynamics
Outline Introduction The Barab´ asi-Albert model The effect of replacing preferential by random attachment The copying model The fitness model Optimization models Deeper thinking ◮ m 0 ≤ n 0 is needed. ◮ Initial conditions: if there are n 0 disconnected vertices, then π ( k i ) is undefined initially. Solutions: ◮ Another initial condition, e.g., a complete graph of n 0 nodes. ◮ Same initial condition but different preferential attachment rule, e.g., k i + 1 π ( k i ) = � j ( k j + 1) ◮ Some limitations: ◮ Global knowledge is required by π . ◮ p ( k ) ∼ k − γ with γ = 3 is suitable for article citation networks [Redner, 1998] but γ < 3 in many real networks, e.g., global syntactic dependency networks (lab session and [Ferrer-i-Cancho et al., 2004]). Ramon Ferrer-i-Cancho & Argimiro Arratia Introduction to network dynamics
Outline Introduction The Barab´ asi-Albert model The effect of replacing preferential by random attachment The copying model The fitness model Optimization models The origins of the power-law in Barab´ asi-Albert model I Controlling for the role of growth and preferential attachment [Barab´ asi et al., 1999] ◮ Hypothesis: preferential attachment is vital for obtaining a power-law (in that model) ◮ Test: Replacing the preferential attachment by uniform attachment (all vertices are equally likely) → p ( k ) = ae − ck . ◮ Hypothesis: growth is vital for obtaining a power-law (in that model) ◮ Test: suppressing growth: fixed number vertices → k follows a ”Gausian” distribution. Ramon Ferrer-i-Cancho & Argimiro Arratia Introduction to network dynamics
Outline Introduction The Barab´ asi-Albert model The effect of replacing preferential by random attachment The copying model The fitness model Optimization models The origins of the power-law in Barab´ asi-Albert model II Controlling for the hidden assumptions of the preferential attachment rule ◮ Generalizing the preferential attachment [Krapivsky et al., 2000] k δ i π ( k i ) = j k δ � j ◮ δ = 1 → original B.A. model. ◮ δ > 1 → one node dominates (very pronounced effect for δ > 2). ◮ δ < 1 → combination of power-law with stretched exponential. Ramon Ferrer-i-Cancho & Argimiro Arratia Introduction to network dynamics
Outline Introduction The Barab´ asi-Albert model The effect of replacing preferential by random attachment The copying model The fitness model Optimization models The effect of replacing preferential attachment by random attachment The growth of a vertex degree over time ◮ Recall n ( t ) = n 0 + t . ◮ The variation of degree over time (on average) is ∂ k i m 0 ∂ t = n ( t − 1) ◮ Integrate on both sides of � k i � t ∂ t ∂ t ∂ k i = m 0 n ( t − 1) → ∂ k i = m 0 n ( t − 1) m 0 t i Ramon Ferrer-i-Cancho & Argimiro Arratia Introduction to network dynamics
Outline Introduction The Barab´ asi-Albert model The effect of replacing preferential by random attachment The copying model The fitness model Optimization models The effect of replacing preferential by random attachment Finally, � � log n ( t − 1) k i ( t ) ≈ m 0 n ( t i − 1) + 1 � � log n 0 + t − 1 = m 0 n 0 + t i − 1 + 1 Ramon Ferrer-i-Cancho & Argimiro Arratia Introduction to network dynamics
Outline Introduction The Barab´ asi-Albert model The effect of replacing preferential by random attachment The copying model The fitness model Optimization models A non-rigorous proof that p ( k ) ∼ e k / m 0 I Sketch of the proof [Barab´ asi et al., 1999] � � log n 0 + t − 1 ◮ Starting point: k i ( t ) = m 0 n 0 + t i − 1 + 1 ◮ Final goal: obtain p ( k ) through p ( k ) ≈ ∂ p ( k i < k ) ∂ k ◮ Intermediate goal: calculate p ( k i < k ) Ramon Ferrer-i-Cancho & Argimiro Arratia Introduction to network dynamics
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