Distances in power-law random graphs

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1 Distances in power-law random graphs Sander Dommers Supervisor: Remco van der Hofstad February 2, 2009 Where innovation starts

2 Introduction There are many complex real-world networks, e.g. Social networks (friendships, business relationships, sexual contacts,... ) Information networks (World Wide Web, citations,... ) Technological networks (Internet, airline routes,... ) Biological networks (protein interactions, neural networks,... ) 2/16 Sexual network Colorado Springs, USA (Potterat, et al., 2002)

3 Introduction There are many complex real-world networks, e.g. Social networks (friendships, business relationships, sexual contacts,... ) Information networks (World Wide Web, citations,... ) Technological networks (Internet, airline routes,... ) Biological networks (protein interactions, neural networks,... ) 2/16 Small part of the Internet steve/stuff/ipmap/

Introduction 2/16 There are many complex real-world networks, e.g. I Social networks (friendships, business relationships, sexual contacts,... ) I Information networks (World Wide Web, citations,.

4 Introduction 2/16 There are many complex real-world networks, e.g. I Social networks (friendships, business relationships, sexual contacts,... ) I Information networks (World Wide Web, citations,... ) I Technological networks (Internet, airline routes,... ) I Biological networks (protein interactions, neural networks,... ) Yeast protein interaction network (Jeong, et al., 2001) / department of mathematics and computer science

5 Properties of complex networks Power law behavior Number of vertices with degree k is proportional to k τ 3/16 Small worlds Distances in the network are small

6 Random graph models Erdős-Rényi random graph Start with n vertices Draw edges between each pair of vertices with probability p Exponential tails, so no power-law distribution 4/16

7 Random graph models Erdős-Rényi random graph Start with n vertices Draw edges between each pair of vertices with probability p Exponential tails, so no power-law distribution 4/16 Configuration model Start with n vertices Let vertex i have Di half-edges D i has power-law distribution with exponent τ Connect the half-edges uniformly at random Power-law because of Strong Law of Large Numbers

8 Random graph models (continued) Preferential attachment (PA) Networks are growing Popular vertices are more likely to get more edges than unpopular vertices 5/16

9 Random graph models (continued) Preferential attachment (PA) Networks are growing Popular vertices are more likely to get more edges than unpopular vertices 5/16 PA model Choose integer m, and parameter δ > m Start at time t = 2 with 2 vertices connected with 2m edges At each time step, say t + 1, add a new vertex with m edges Connect its m edges independently according to P [ edge of t + 1 i PA m,δ (t) ] = D i (t) + δ t(2m + δ), for i [t].

10 Properties of PA model 6/16 Power law with exponent So any exponent τ > 2 possible τ = 3 + δ m

11 Properties of PA model 6/16 Power law with exponent So any exponent τ > 2 possible τ = 3 + δ m The diameter of a graph at time t, diam(t), is the largest distance present in the graph at time t

12 Properties of PA model Power law with exponent 6/16 So any exponent τ > 2 possible τ = 3 + δ m The diameter of a graph at time t, diam(t), is the largest distance present in the graph at time t τ = 3 (Bollobás and Riordan, 2004) For m 2 and δ = 0, with high probability, log t (1 ε) log log t log t diam(t) (1 + ε) log log t for all ε > 0

13 Properties of PA model (continued) 7/16 τ > 3 (Van der Hofstad and Hooghiemstra, 2008) For m 2 and δ > 0, with high probability, for some c 1, c 2 > 0 c 1 log t diam(t) c 2 log t τ (2, 3) (Van der Hofstad and Hooghiemstra, 2008) For m 2 and δ ( m, 0), with high probability, diam(t) c log log t for some c > 0

14 log log lowerbound for τ > 2 8/16 Theorem Consider preferential attachment with m 2 and δ > m (i.e. τ > 2) Let k = ε log log t, with 0 < ε < 1 log m Then, with high probability, diam(2t) k

15 Proper k-exploration trees Construction of k-exploration tree of a vertex i Start from vertex i Connect its m edges vertices at distance 1 from vertex i Connect the m edges of vertices at distance 1,... Continue in same fashion, up to distance k 9/16 Example for m = 2, k = 3 and i = 30

16 Proper k-exploration trees Construction of k-exploration tree of a vertex i Start from vertex i Connect its m edges vertices at distance 1 from vertex i Connect the m edges of vertices at distance 1,... Continue in same fashion, up to distance k 9/16 Example for m = 2, k = 3 and i = 30

17 Proper k-exploration trees Construction of k-exploration tree of a vertex i Start from vertex i Connect its m edges vertices at distance 1 from vertex i Connect the m edges of vertices at distance 1,... Continue in same fashion, up to distance k 9/16 Example for m = 2, k = 3 and i = 30

18 Proper k-exploration trees Construction of k-exploration tree of a vertex i Start from vertex i Connect its m edges vertices at distance 1 from vertex i Connect the m edges of vertices at distance 1,... Continue in same fashion, up to distance k 9/16 Example for m = 2, k = 3 and i = 30

19 Proper k-exploration trees Construction of k-exploration tree of a vertex i Start from vertex i Connect its m edges vertices at distance 1 from vertex i Connect the m edges of vertices at distance 1,... Continue in same fashion, up to distance k 9/16 Example for m = 2, k = 3 and i = 30

20 Proper k-exploration trees Construction of k-exploration tree of a vertex i Start from vertex i Connect its m edges vertices at distance 1 from vertex i Connect the m edges of vertices at distance 1,... Continue in same fashion, up to distance k 9/16 Example for m = 2, k = 3 and i = 30

21 Proper k-exploration trees (continued) 10/16 A k-exploration tree is proper when The k-exploration tree has no collisions No other vertex connects to a vertex in the tree All vertices of the tree are in [2t]\[t] Example for m = 2, k = 3 and i = 30

22 Proper k-exploration trees (continued) 10/16 A k-exploration tree is proper when The k-exploration tree has no collisions No other vertex connects to a vertex in the tree All vertices of the tree are in [2t]\[t] Example for m = 2, k = 3 and i = 30

23 Proper k-exploration trees (continued) 10/16 A k-exploration tree is proper when The k-exploration tree has no collisions No other vertex connects to a vertex in the tree All vertices of the tree are in [2t]\[t] Example for m = 2, k = 3 and i = 30

24 Proper k-exploration trees (continued) 10/16 A k-exploration tree is proper when The k-exploration tree has no collisions No other vertex connects to a vertex in the tree All vertices of the tree are in [2t]\[t] Example for m = 2, k = 3 and i = 30

25 Number of proper k-exploration trees Let Z k (2t) = # proper k-exploration trees at time 2t 11/16

26 Number of proper k-exploration trees Let Z k (2t) = # proper k-exploration trees at time 2t 11/16 Then P [diam(2t) < k] P [Z k (2t) = 0]

27 Number of proper k-exploration trees Let Z k (2t) = # proper k-exploration trees at time 2t 11/16 Then P [diam(2t) < k] P [Z k (2t) = 0] Using Chebychev inequality gives P [Z k (2t) = 0] Var [Z k (2t)] E [Z k (2t)] 2? 0

28 Number of proper k-exploration trees Let Z k (2t) = # proper k-exploration trees at time 2t 11/16 Then P [diam(2t) < k] P [Z k (2t) = 0] Using Chebychev inequality gives We need P [Z k (2t) = 0] Var [Z k (2t)] E [Z k (2t)] 2? 0 E [Z k (2t)] and Var [Z k (2t)] E [Z k (2t)] 2

29 E [Z k (2t)] 12/16 Let B T = no other vertex connects to a vertex in the tree T Then E [Z k (2t)] = E = T possible T possible = T possible I{T PA m,δ (2t) and T is proper} P [T PA m,δ (2t) and T is proper] P [T PA m,δ (2t)] P [B T T PA m,δ (2t)]

30 E [Z k (2t)] (continued) 13/16 E [Z k (2t)] = P [T PA m,δ (2t)] P [B T T PA m,δ (2t)] T possible

31 E [Z k (2t)] (continued) 13/16 E [Z k (2t)] = P [T PA m,δ (2t)] P [B T T PA m,δ (2t)] T possible ( ) 1 T 1 t

32 E [Z k (2t)] (continued) 13/16 E [Z k (2t)] = P [T PA m,δ (2t)] P [B T T PA m,δ (2t)] T possible ( ) 1 T 1 ( 1 T ) mt t t

33 E [Z k (2t)] (continued) 13/16 E [Z k (2t)] = P [T PA m,δ (2t)] P [B T T PA m,δ (2t)] T possible ( ) t T ( ) 1 T 1 ( 1 T ) mt T t t

34 E [Z k (2t)] (continued) 13/16 E [Z k (2t)] = P [T PA m,δ (2t)] P [B T T PA m,δ (2t)] T possible ( ) t T ( ) 1 T 1 ( 1 T ) mt T t t ( 1 t (log t) ε ) (log t) ε m(log t)ε e

35 Var [Z k (2t)] E [Z k (2t)] 2 14/16 Study covariance of two different possible trees being formed in the graph and being proper Take exact look at probabilities involved Conclusion Var [Z k (2t)] c (log t)2 E [Z k (2t)] 2 + E [Z k (2t)] E [Z k (2t)] 2 t

36 log log lowerbound for τ > 2 15/16 Theorem Consider preferential attachment with m 2 and δ > m (i.e. τ > 2) Let k = ε log log t, with 0 < ε < 1 log m Then, with high probability, diam(2t) k Proof P[diam(2t) k] 1 P [Z k (2t) = 0] 1 Var [Z k (2t)] E [Z k (2t)] 2 1

37 Conclusion Small world property holds in preferential attachment model 16/16 Future research Better bounds on distances Distances in other extended PA models Processes on random graphs

Random graphs and complex networks

Random graphs and complex networks Julia Komjathy, Remco van der Hofstad Random Graphs and Complex Networks (2WS12) course Complex networks Figure 2 Ye a s t p ro te in in te ra c tio n n e tw o rk. A