Uncovering disassortativity in large scale-free networks Nelly Litvak University of Twente, Stochastic Operations Research group Joint work with Remco van der Hofstad Supported by EC FET Open project NADINE Trento, Italy, 23-07-2012
Power laws ◮ degree of the node = # links, [fraction nodes degree k ] = p k , [ N. Litvak, SOR group ] 2/30
Power laws ◮ degree of the node = # links, [fraction nodes degree k ] = p k , ◮ Power law: p k ≈ const · k − α , α > 1. [ N. Litvak, SOR group ] 2/30
Power laws ◮ degree of the node = # links, [fraction nodes degree k ] = p k , ◮ Power law: p k ≈ const · k − α , α > 1. ◮ Power laws: Internet, WWW, social networks, biological networks, etc... [ N. Litvak, SOR group ] 2/30
Power laws ◮ degree of the node = # links, [fraction nodes degree k ] = p k , ◮ Power law: p k ≈ const · k − α , α > 1. ◮ Power laws: Internet, WWW, social networks, biological networks, etc... ◮ Model for high variability, scale-free graph [ N. Litvak, SOR group ] 2/30
Power laws ◮ degree of the node = # links, [fraction nodes degree k ] = p k , ◮ Power law: p k ≈ const · k − α , α > 1. ◮ Power laws: Internet, WWW, social networks, biological networks, etc... ◮ Model for high variability, scale-free graph ◮ signature log-log plot: log p k = log ( const ) − α log k [ N. Litvak, SOR group ] 2/30
Power laws ◮ degree of the node = # links, [fraction nodes degree k ] = p k , ◮ Power law: p k ≈ const · k − α , α > 1. ◮ Power laws: Internet, WWW, social networks, biological networks, etc... ◮ Model for high variability, scale-free graph ◮ signature log-log plot: log p k = log ( const ) − α log k ◮ Faloutsos, Faloutsos, Faloutsos (1999): power laws in Internet [ N. Litvak, SOR group ] 2/30
But Power Law is not everything! Example: Robustness of the Internet. ◮ Albert, Jeong and Barabasi (2000): Achille’s heel of Internet: Internet is sensitive to targeted attack [ N. Litvak, SOR group ] 3/30
But Power Law is not everything! Example: Robustness of the Internet. ◮ Albert, Jeong and Barabasi (2000): Achille’s heel of Internet: Internet is sensitive to targeted attack ◮ Doyle et al. (2005): Robust yet fragile nature of Internet: Internet is not a random graph, it is designed to be robust [ N. Litvak, SOR group ] 3/30
But Power Law is not everything! (cont.) Example: Spread of infections ◮ Classical epidemiology, e.g. Adnerson and May (1991): epidemic only if infection rate exceeds a critical value [ N. Litvak, SOR group ] 4/30
But Power Law is not everything! (cont.) Example: Spread of infections ◮ Classical epidemiology, e.g. Adnerson and May (1991): epidemic only if infection rate exceeds a critical value ◮ Vespignani et al. (2001): power law networks have a zero critical infection rate! [ N. Litvak, SOR group ] 4/30
But Power Law is not everything! (cont.) Example: Spread of infections ◮ Classical epidemiology, e.g. Adnerson and May (1991): epidemic only if infection rate exceeds a critical value ◮ Vespignani et al. (2001): power law networks have a zero critical infection rate! ◮ Eguiluz et al. (2002): a specially wired highly clustered network is resistant up to a certain critical infection rate. [ N. Litvak, SOR group ] 4/30
But Power Law is not everything! (cont.) Example: Spread of infections ◮ Classical epidemiology, e.g. Adnerson and May (1991): epidemic only if infection rate exceeds a critical value ◮ Vespignani et al. (2001): power law networks have a zero critical infection rate! ◮ Eguiluz et al. (2002): a specially wired highly clustered network is resistant up to a certain critical infection rate. Example: Technological versus economical networks [ N. Litvak, SOR group ] 4/30
Degree-degree correlations ◮ It is clearly important how the network is wired [ N. Litvak, SOR group ] 5/30
Degree-degree correlations ◮ It is clearly important how the network is wired ◮ To start with: do hubs connect to each other? [ N. Litvak, SOR group ] 5/30
Degree-degree correlations ◮ It is clearly important how the network is wired ◮ To start with: do hubs connect to each other? YES for banks, NO for Internet [ N. Litvak, SOR group ] 5/30
Degree-degree correlations ◮ It is clearly important how the network is wired ◮ To start with: do hubs connect to each other? YES for banks, NO for Internet ◮ Assortative networks: nodes with similar degree connect to each other. ◮ Disassortative networks: nodes with large degrees tend to connect to nodes with small degrees. [ N. Litvak, SOR group ] 5/30
Assortativity coefficient ◮ G = ( V , E ) undirected graph of n nodes ◮ d i degree of node i = 1, 2, . . . , n [ N. Litvak, SOR group ] 6/30
Assortativity coefficient ◮ G = ( V , E ) undirected graph of n nodes ◮ d i degree of node i = 1, 2, . . . , n ◮ We are interested in correlations between degrees of neighboring nodes [ N. Litvak, SOR group ] 6/30
Assortativity coefficient ◮ G = ( V , E ) undirected graph of n nodes ◮ d i degree of node i = 1, 2, . . . , n ◮ We are interested in correlations between degrees of neighboring nodes ◮ Newman (2002): assortativity measure ρ n � 2 � � � 1 1 1 ij ∈ E d i d j − 2 ( d i + d j ) ij ∈ E | E | | E | ρ n = � 2 � � � 1 1 1 1 2 ( d 2 i + d 2 2 ( d i + d j ) j ) − ij ∈ E ij ∈ E | E | | E | ◮ Statistical estimation of the correlation coefficient between degrees on two ends of a random edge [ N. Litvak, SOR group ] 6/30
Assortativity coefficient ◮ G = ( V , E ) undirected graph of n nodes ◮ d i degree of node i = 1, 2, . . . , n ◮ We are interested in correlations between degrees of neighboring nodes ◮ Newman (2002): assortativity measure ρ n � 2 � � � 1 1 1 ij ∈ E d i d j − 2 ( d i + d j ) ij ∈ E | E | | E | ρ n = � 2 � � � 1 1 1 1 2 ( d 2 i + d 2 2 ( d i + d j ) j ) − ij ∈ E ij ∈ E | E | | E | ◮ Statistical estimation of the correlation coefficient between degrees on two ends of a random edge ◮ Very popular measure of assortativity! [ N. Litvak, SOR group ] 6/30
Is there something wrong with ρ n ? ◮ Preferential Attachment graph appears to be assortatively neutral (Newman 2003, Dorogovtsev et al. 2010) ◮ Recent criticism: ρ n depends on the size of the networks (Raschke et al. 2010; Dorogovtsev et al. 2010) [ N. Litvak, SOR group ] 7/30
What IS assortativity measure? ◮ ρ n is a statistical estimation for the coefficient of variation ρ = E ( XY ) − [ E ( X )] 2 , Var ( X ) ◮ X and Y are the degrees of the nodes on the two ends of a randomly chosen edge [ N. Litvak, SOR group ] 8/30
What IS assortativity measure? ◮ ρ n is a statistical estimation for the coefficient of variation ρ = E ( XY ) − [ E ( X )] 2 , Var ( X ) ◮ X and Y are the degrees of the nodes on the two ends of a randomly chosen edge ◮ Problems? [ N. Litvak, SOR group ] 8/30
What IS assortativity measure? ◮ ρ n is a statistical estimation for the coefficient of variation ρ = E ( XY ) − [ E ( X )] 2 , Var ( X ) ◮ X and Y are the degrees of the nodes on the two ends of a randomly chosen edge ◮ Problems? YES!!! [ N. Litvak, SOR group ] 8/30
What IS assortativity measure? ◮ ρ n is a statistical estimation for the coefficient of variation ρ = E ( XY ) − [ E ( X )] 2 , Var ( X ) ◮ X and Y are the degrees of the nodes on the two ends of a randomly chosen edge ◮ Problems? YES!!! ◮ X and Y are power law r.v.’s, exponent α − 1 P ( X = k ) = kp k / E ( degree ) . ◮ In real networks (WWW) we often have 2 < α < 3, so kp k � E ( X ) = k E ( degree ) = ∞ k [ N. Litvak, SOR group ] 8/30
What IS assortativity measure? ◮ ρ n is a statistical estimation for the coefficient of variation ρ = E ( XY ) − [ E ( X )] 2 , Var ( X ) ◮ X and Y are the degrees of the nodes on the two ends of a randomly chosen edge ◮ Problems? YES!!! ◮ X and Y are power law r.v.’s, exponent α − 1 P ( X = k ) = kp k / E ( degree ) . ◮ In real networks (WWW) we often have 2 < α < 3, so kp k � E ( X ) = k E ( degree ) = ∞ k ◮ ρ is not defined in the power law model! Then: what are we measuring? [ N. Litvak, SOR group ] 8/30
Assortative and disassortative graphs ◮ Newman(2003) [ N. Litvak, SOR group ] 9/30
Assortative and disassortative graphs ◮ Newman(2003) ◮ Technological and biological networks are disassortative, ρ n < 0 ◮ Social networks are assortative, ρ n > 0 [ N. Litvak, SOR group ] 9/30
Assortative and disassortative graphs ◮ Newman(2003) ◮ Technological and biological networks are disassortative, ρ n < 0 ◮ Social networks are assortative, ρ n > 0 ◮ Note: large networks are never strongly disassortative... [ N. Litvak, SOR group ] 9/30
ρ n in terms of moments of the degrees ◮ Write � � � � 1 d 2 2 ( d 2 1 i + d 2 d 3 2 ( d i + d j ) = i , j ) = i ij ∈ E i ∈ V ij ∈ E i ∈ V [ N. Litvak, SOR group ] 10/30
ρ n in terms of moments of the degrees ◮ Write � � � � 1 d 2 1 2 ( d 2 i + d 2 d 3 2 ( d i + d j ) = i , j ) = i ij ∈ E i ∈ V ij ∈ E i ∈ V ◮ Then � � � 2 � 1 i ∈ V d 2 ij ∈ E d i d j − i | E | ρ n = � 2 . � � � 1 i ∈ V d 3 i ∈ V d 2 i − | E | i [ N. Litvak, SOR group ] 10/30
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