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Assortativity and Mixing Assortativity and Mixing Definition Complex Networks CSYS/MATH 303, Spring, 2011 Assortativity by degree General mixing Contagion References Prof. Peter Dodds Department of Mathematics & Statistics Center for Complex Systems Vermont Advanced Computing Center University of Vermont Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. 1 of 26 Outline Assortativity and Mixing Definition General mixing Definition Assortativity by degree Contagion References General mixing Assortativity by degree Contagion References 2 of 26 Basic idea: I I I I Random networks with arbitrary degree distributions cover much territory but do not represent all networks. Moving away from pure random networks was a key first step. We can extend in many other directions and a natural one is to introduce correlations between different kinds of nodes. Node attributes may be anything, e.g.: Assortativity and Mixing Definition General mixing Assortativity by degree Contagion References 1. degree 2. demographics (age, gender, etc.) 3. group affiliation I I I We speak of mixing patterns, correlations, biases... Networks are still random at base but now have more global structure. Build on work by Newman [4, 5] , and Boguñá and Serano. [1] . 3 of 26 General mixing between node categories I I Assume types of nodes are countable, and are assigned numbers 1, 2, 3, . . . . Consider networks with directed edges. an edge connects a node of type µ eµν = Pr to a node of type ν Assortativity and Mixing Definition General mixing Assortativity by degree Contagion References aµ = Pr(an edge comes from a node of type µ) bν = Pr(an edge leads to a node of type ν) I I Write E = [eµν ], ~a = [aµ ], and ~b = [bν ]. Requirements: X X X eµν = 1, eµν = aµ , and eµν = bν . µν ν µ 4 of 26 Notes: Assortativity and Mixing Definition General mixing I Varying eµν allows us to move between the following: 1. Perfectly assortative networks where nodes only connect to like nodes, and the network breaks into subnetworks. P Requires eµν = 0 if µ 6= ν and µ eµµ = 1. 2. Uncorrelated networks (as we have studied so far) For these we must have independence: eµν = aµ bν . 3. Disassortative networks where nodes connect to nodes distinct from themselves. I Disassortative networks can be hard to build and may require constraints on the eµν . I Basic story: level of assortativity reflects the degree to which nodes are connected to nodes within their group. Assortativity by degree Contagion References 5 of 26 Correlation coefficient: Assortativity and Mixing Definition General mixing I Quantify the level of assortativity with the following assortativity coefficient [5] : P P aµ bµ Tr E − ||E 2 ||1 µ eµµ − P µ r= = 1 − µ aµ bµ 1 − ||E 2 ||1 Assortativity by degree Contagion References where || · ||1 is the 1-norm = sum of a matrix’s entries. I Tr E is the fraction of edges that are within groups. I ||E 2 ||1 is the fraction of edges that would be within groups if connections were random. I 1 − ||E 2 ||1 is a normalization factor so rmax = 1. I When Tr eµµ = 1, we have r = 1. X I When eµµ = aµ bµ , we have r = 0. X 6 of 26 Assortativity and Mixing Correlation coefficient: Definition General mixing Notes: I Assortativity by degree r = −1 is inaccessible if three or more types are present. I Disassortative networks simply have nodes connected to unlike nodes—no measure of how unlike nodes are. I Minimum value of r occurs when all links between non-like nodes: Tr eµµ = 0. I rmin = Contagion References −||E 2 ||1 1 − ||E 2 ||1 where −1 ≤ rmin < 0. 7 of 26 Assortativity and Mixing Scalar quantities I I I I I Now consider nodes defined by a scalar integer quantity. Examples: age in years, height in inches, number of friends, ... ejk = Pr (a randomly chosen edge connects a node with value j to a node with value k ). aj and bk are defined as before. Can now measure correlations between nodes based on this scalar quantity using standard Pearson correlation coefficient (): P hjk i − hjia hk ib j k j k (ejk − aj bk ) r= =q q σa σb hj 2 i − hji2 hk 2 i − hk i2 a I a b This is the observed normalized deviation from randomness in the product jk . Definition General mixing Assortativity by degree Contagion References b 8 of 26 Assortativity and Mixing Degree-degree correlations Definition I I Natural correlation is between the degrees of connected nodes. General mixing Assortativity by degree Now define ejk with a slight twist: Contagion References ejk = Pr = Pr an edge connects a degree j + 1 node to a degree k + 1 node an edge runs between a node of in-degree j and a node of out-degree k I Useful for calculations (as per Rk ) I Important: Must separately define P0 as the {ejk } contain no information about isolated nodes. I Directed networks still fine but we will assume from here on that ejk = ekj . 9 of 26 Assortativity and Mixing Degree-degree correlations Definition General mixing I Notation reconciliation for undirected networks: P j k j k (ejk − Rj Rk ) r= σR2 Assortativity by degree Contagion References where, as before, Rk is the probability that a randomly chosen edge leads to a node of degree k + 1, and σR2 = X j 2 X j 2 Rj − jRj . j 10 of 26 Degree-degree correlations Assortativity and Mixing Definition General mixing Assortativity by degree Error estimate for r : Contagion I Remove edge i and recompute r to obtain ri . I Repeat for all edges and compute using the jackknife method () [2] X σr2 = (ri − r )2 . References i I Mildly sneaky as variables need to be independent for us to be truly happy and edges are correlated... 11 of 26 age A to package B indicates that A relies on components of B for its operation. Biological networks: "k# protein-protein interaction network in the yeast S. Cerevisiae !53$; "l# metabolic network of the bacterium E. Coli !54$; "m# neural network of the nematode worm C. Elegans !5,55$; tropic interactions between species in the food webs of "n# Ythan Estuary, Scotland !56$ and "o# Little Rock Lake, Wisconsin !57$. Measurements of degree-degree correlations Group Social Technological Biological Network Assortativity and Mixing Definition Type Size n Assortativity r Error & r General mixing Assortativity by degree a a b c d e f Physics coauthorship Biology coauthorship Mathematics coauthorship Film actor collaborations Company directors Student relationships Email address books undirected undirected undirected undirected undirected undirected directed 52 909 1 520 251 253 339 449 913 7 673 573 16 881 0.363 0.127 0.120 0.208 0.276 !0.029 0.092 0.002 0.0004 0.002 0.0002 0.004 0.037 0.004 g h i j Power grid Internet World Wide Web Software dependencies undirected undirected directed directed 4 941 10 697 269 504 3 162 !0.003 !0.189 !0.067 !0.016 0.013 0.002 0.0002 0.020 k l m n o Protein interactions Metabolic network Neural network Marine food web Freshwater food web undirected undirected directed directed directed 2 115 765 307 134 92 !0.156 !0.240 !0.226 !0.263 !0.326 0.010 0.007 0.016 0.037 0.031 Contagion References !58,59$. The algorithm is as follows. lmost never is. In this paper, therefore, we take an I Social "1# Given the desired edge distribution e jk , we first caltend to be assortative (homophily) ive approach, making usenetworks of computer simulation. culate the corresponding distribution of excess degrees q k would like to generate on a computer a random netfrom networks Eq. "23#, and then invertto Eq.be "22# to find the degree Technological tend aving, forIinstance, a particular valueand of thebiological matrix distribution: his also fixes the degree distribution, via Eq. "23#.$ In disassortative 2$ we discussed one possible way of doing this using q k!1 /k rithm similar to that of Sec. II C. One would draw . "27# p k" 12 of 26 rom the desired distribution e jk and then join the deq j!1 / j % Spreading on degree-correlated networks Assortativity and Mixing Definition General mixing Assortativity by degree I I Next: Generalize our work for random networks to degree-correlated networks. As before, by allowing that a node of degree k is activated by one neighbor with probability Bk 1 , we can handle various problems: Contagion References 1. find the giant component size. 2. find the probability and extent of spread for simple disease models. 3. find the probability of spreading for simple threshold models. 13 of 26 Spreading on degree-correlated networks Assortativity and Mixing Definition General mixing Assortativity by degree Contagion I Goal: Find fn,j = Pr an edge emanating from a degree j + 1 node leads to a finite active subcomponent of size n. I Repeat: a node of degree k is in the game with probability Bk 1 . ~ 1 = [Bk 1 ]. Define B I I References Plan: Find the generating function ~ 1 ) = P∞ fn,j x n . Fj (x; B n=0 14 of 26 Spreading on degree-correlated networks Assortativity and Mixing Definition I General mixing Recursive relationship: ~ 1) = x 0 Fj (x; B +x ∞ X ejk k =0 ∞ X k =0 Rj Assortativity by degree (1 − Bk +1,1 ) Contagion References h ik ejk ~ 1) . Bk +1,1 Fk (x; B Rj I First term = Pr that the first node we reach is not in the game. I Second term involves Pr we hit an active node which has k outgoing edges. I Next: find average size of active components reached by following a link from a degree j + 1 node ~ 1 ). = Fj0 (1; B 15 of 26 Spreading on degree-correlated networks Assortativity and Mixing Definition General mixing I I ~ 1 ), set x = 1, and rearrange. Differentiate Fj (x; B ~ 1 ) = 1 which is true when no giant We use Fk (1; B Assortativity by degree component exists. We find: References ~ 1) = Rj Fj0 (1; B ∞ X k =0 I ejk Bk +1,1 + ∞ X Contagion ~ 1 ). kejk Bk +1,1 Fk0 (1; B k =0 Rearranging and introducing a sneaky δjk : ∞ X k =0 ∞ X ~ 1) = ejk Bk +1,1 . δjk Rk − kBk +1,1 ejk Fk0 (1; B k =0 16 of 26 Spreading on degree-correlated networks Assortativity and Mixing Definition General mixing I Assortativity by degree In matrix form, we have Contagion ~ 0 (1; B ~ 1 ) = EB ~1 AE,B~ F References 1 where h i AE,B~ = δjk Rk − kBk +1,1 ejk , 1 j+1,k +1 h i ~ 0 (1; B ~ 1) ~ 1 ), F = Fk0 (1; B k +1 h i ~1 [E]j+1,k +1 = ejk , and B = Bk +1,1 . k +1 17 of 26 Spreading on degree-correlated networks Assortativity and Mixing Definition I So, in principle at least: General mixing ~ 0 (1; B ~ 1 ) = A−1 EB ~ 1. F ~ E,B1 Assortativity by degree Contagion References I ~ 0 (1; B ~ 1 ), the average size of an active Now: as F component reached along an edge, increases, we move towards a transition to a giant component. I Right at the transition, the average component size explodes. I Exploding inverses of matrices occur when their determinants are 0. I The condition is therefore: det AE,B~ = 0 1 . 18 of 26 Spreading on degree-correlated networks I General condition details: det AE,B~ = det δjk Rk −1 − (k − 1)Bk ,1 ej−1,k −1 = 0. 1 I I The above collapses to our standard contagion condition when ejk = Rj Rk . ~ 1 = B~1, we have the condition for a simple When B Assortativity and Mixing Definition General mixing Assortativity by degree Contagion References disease model’s successful spread det δjk Rk −1 − B(k − 1)ej−1,k −1 = 0. I ~ 1 = ~1, we have the condition for the When B existence of a giant component: det δjk Rk −1 − (k − 1)ej−1,k −1 = 0. I Bonusville: We’ll find a much better version of this set of conditions later... 19 of 26 Spreading on degree-correlated networks Assortativity and Mixing Definition We’ll next find two more pieces: 1. Ptrig , the probability of starting a cascade 2. S, the expected extent of activation given a small seed. General mixing Assortativity by degree Contagion References Triggering probability: I Generating function: ~ 1) = x H(x; B ∞ X h ik ~ 1) . Pk Fk −1 (x; B k =0 I Generating function for vulnerable component size is more complicated. 20 of 26 Spreading on degree-correlated networks Assortativity and Mixing Definition General mixing I Want probability of not reaching a finite component. ~ 1) Ptrig = Strig =1 − H(1; B ∞ h ik X ~ 1) . Pk Fk −1 (1; B =1 − Assortativity by degree Contagion References k =0 I ~ 1 ). Last piece: we have to compute Fk −1 (1; B I Nastier (nonlinear)—we have to solve the recursive expression we started with when x = 1: ~ 1 ) = P∞ ejk (1 − Bk +1,1 )+ Fj (1; B k =0 Rj h ik P∞ ejk ~ B F (1; B ) . k 1 k =0 Rj k +1,1 I Iterative methods should work here. 21 of 26 Spreading on degree-correlated networks I Truly final piece: Find final size using approach of Gleeson [3] , a generalization of that used for uncorrelated random networks. I Need to compute θj,t , the probability that an edge leading to a degree j node is infected at time t. I Evolution of edge activity probability: Assortativity and Mixing Definition General mixing Assortativity by degree Contagion References θj,t+1 = Gj (θ~t ) = φ0 + (1 − φ0 )× ∞ k −1 X ej−1,k −1 X k − 1 i θk ,t (1 − θk ,t )k −1−i Bki . Rj−1 i k =1 I i=0 Overall active fraction’s evolution: φt+1 = φ0 +(1−φ0 ) ∞ X k =0 Pk k X k i=0 i θki ,t (1−θk ,t )k −i Bki . 22 of 26 Spreading on degree-correlated networks I I I As before, these equations give the actual evolution of φt for synchronous updates. ~ θ~t ). Contagion condition follows from θ~t+1 = G( ~ around θ~0 = ~0. Expand G θj,t+1 = Gj (~0)+ ∞ X ∂Gj (~0) k =1 I I I ∂θk ,t ∞ 1 X ∂ 2 Gj (~0) 2 θk ,t + θ +. . . 2! ∂θk2,t k ,t Assortativity and Mixing Definition General mixing Assortativity by degree Contagion References k =1 If Gj (~0) 6= 0 for at least one j, always have some infection. ∂Gj (~0) ~ If Gj (0) = 0 ∀ j, want largest eigenvalue > 1. ∂θk ,t Condition for spreading is therefore dependent on eigenvalues of this matrix: ∂Gj (~0) ej−1,k −1 = (k − 1)Bk 1 ∂θk ,t Rj−1 Insert question from assignment 9 () 23 of 26 tively mixed network. These findings are intuitively reasonable. If the network mixes assortatively, then the high-degree vertices will tend to stick together in a subnetwork or core group of higher mean degree than the network as a whole. It is reasonable to suppose that percolation would occur earlier within such a subnetwork. Conversely, since percolation will be restricted to this subnetwork, it is not surprising How the giant component changes with assortativity: General mixing Contagion 0.8 giant component S Definition Assortativity by degree 1.0 I More assortative networks percolate for lower average degrees I But disassortative networks end up with higher extents of spreading. 0.6 0.4 assortative neutral disassortative 0.2 0.0 Assortativity and Mixing 1 10 100 exponential parameter κ FIG. 1. Size of the [4] giant component as a fraction of graph from Newman, 2002 size for graphs with the edge distribution given in Eq. (9). The points are simulation results for graphs of N ! 100 000 vertices, while the solid lines are the numerical solution of Eq. (8). Each point is an average over ten graphs; the resulting statistical errors are smaller than the symbols. The values of p are 0.5 (circles), p0 ! 0:146 . . . (squares), and 0:05 (triangles). References 24 of 26 References I Assortativity and Mixing Definition General mixing [1] M. Boguñá and M. Ángeles Serrano. Generalized percolation in random directed networks. Phys. Rev. E, 72:016106, 2005. pdf () Assortativity by degree Contagion References [2] B. Efron and C. Stein. The jackknife estimate of variance. The Annals of Statistics, 9:586–596, 1981. pdf () [3] J. P. Gleeson. Cascades on correlated and modular random networks. Phys. Rev. E, 77:046117, 2008. pdf () [4] M. Newman. Assortative mixing in networks. Phys. Rev. Lett., 89:208701, 2002. pdf () 25 of 26 References II Assortativity and Mixing Definition General mixing Assortativity by degree Contagion References [5] M. E. J. Newman. Mixing patterns in networks. Phys. Rev. E, 67:026126, 2003. pdf () 26 of 26