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Assortativity and Mixing Assortativity and Mixing Definition General mixing Complex Networks, Course 303A, Spring, 2009 Assortativity by degree Contagion Prof. Peter Dodds References Department of Mathematics & Statistics University of Vermont Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License . Frame 1/26 Outline Assortativity and Mixing Definition General mixing Definition Assortativity by degree General mixing References Contagion Assortativity by degree Contagion References Frame 2/26 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. Assortativity and Mixing Definition General mixing Assortativity by degree Contagion References 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.: 1. degree 2. demographics (age, gender, etc.) 3. group affiliation I We speak of mixing patterns, correlations, biases... I Networks are still random at base but now have more global structure. I Build on work by Newman [3, 4] . Frame 3/26 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 Frame 4/26 Connection to degree distribution: Assortativity and Mixing Definition General mixing Assortativity by degree Contagion References Frame 5/26 Notes: 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 and Mixing Definition General mixing Assortativity by degree Contagion References Frame 6/26 Correlation coefficient: Assortativity and Mixing Definition I Quantify the level of assortativity with the following assortativity coefficient [4] : P P aµ bµ Tr E − ||E 2 ||1 µ eµµ − P µ r= = 1 − µ aµ bµ 1 − ||E 2 ||1 General mixing 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 Frame 7/26 Assortativity and Mixing Correlation coefficient: Definition Notes: I General mixing r = −1 is inaccessible if three or more types are presents. 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 = Assortativity by degree Contagion References −||E 2 ||1 1 − ||E 2 ||1 where −1 ≤ rmin < 0. Frame 8/26 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 Frame 9/26 Assortativity and Mixing Degree-degree correlations I I Natural correlation is between the degrees of connected nodes. Definition General mixing Assortativity by degree Now define ejk with a slight twist: Contagion 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 . References Frame 10/26 Assortativity and Mixing Degree-degree correlations Definition I Notation reconciliation for undirected networks: P j k j k (ejk − Rj Rk ) r= σR2 General mixing 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 Frame 11/26 Degree-degree correlations Assortativity and Mixing Definition General mixing Assortativity by degree Error estimate for r : I I Remove edge i and recompute r to obtain ri . Contagion References Repeat for all edges and compute using the jackknife method () [1] σr2 = X (ri − r )2 . i I Mildly sneaky as variables need to be independent for us to be truly happy and edges are correlated... Frame 12/26 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$. Measurements of degree-degree correlations Type Size n Assortativity r Error & r a a b c d e f Physics coauthorship Biology coauthorship Mathematics coauthorship Film actor collaborations Company directors Student relationships Email address books undirected undirected undirected undirected undirected undirected directed 52 909 1 520 251 253 339 449 913 7 673 573 16 881 0.363 0.127 0.120 0.208 0.276 !0.029 0.092 0.002 0.0004 0.002 0.0002 0.004 0.037 0.004 g h i j Power grid Internet World Wide Web Software dependencies undirected undirected directed directed 4 941 10 697 269 504 3 162 !0.003 !0.189 !0.067 !0.016 0.013 0.002 0.0002 0.020 k l m n o Protein interactions Metabolic network Neural network Marine food web Freshwater food web undirected undirected directed directed directed 2 115 765 307 134 92 !0.156 !0.240 !0.226 !0.263 !0.326 0.010 0.007 0.016 0.037 0.031 Group Social Technological Biological Network Assortativity and Mixing Definition General mixing Assortativity by degree Contagion References !58,59$. 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 Frame 13/26 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" 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 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 βk 1 , we can handle various problems: Assortativity by degree 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. Frame 14/26 Spreading on degree-correlated networks Assortativity and Mixing Definition General mixing I I I I Goal: Find fn,j = Pr an edge emanating from a degree j + 1 node leads to a finite active subcomponent of size n. Assortativity by degree Contagion References Repeat: a node of degree k is in the game with probability βk 1 . ~1 = [βk 1 ]. Define β Plan: Find the generating function ~1 ) = P∞ fn,j x n . Fj (x; β n=0 Frame 15/26 Spreading on degree-correlated networks I Assortativity and Mixing Definition Recursive relationship: General mixing ~1 ) = x 0 Fj (x; β +x ∞ X ejk k =0 ∞ X k =0 Rj (1 − βk +1,1 ) h ik ejk ~1 ) . βk +1,1 Fk (x; β 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; β Assortativity by degree Contagion References Frame 16/26 Spreading on degree-correlated networks Assortativity and Mixing Definition I I ~1 ), set x = 1, and rearrange. Differentiate Fj (x; β ~1 ) = 1 which is true when no giant We use Fk (1; β component exists. We find: General mixing Assortativity by degree Contagion References ~1 ) = Rj Fj0 (1; β ∞ X k =0 I ejk βk +1,1 + ∞ X ~1 ). kejk βk +1,1 Fk0 (1; β k =0 Rearranging and introducing a sneaky δjk : ∞ X k =0 ∞ X ~1 ) = ejk βk +1,1 . δjk Rk − k βk +1,1 ejk Fk0 (1; β k =0 Frame 17/26 Spreading on degree-correlated networks Assortativity and Mixing Definition General mixing I In matrix form, we have Assortativity by degree ~ 0 (1; β ~1 ) = Eβ ~1 AE,β~ F 1 Contagion References where h i AE,β~ = δjk Rk − k βk +1,1 ejk , 1 j+1,k +1 h i ~ 0 (1; β ~1 ) ~1 ), F = Fk0 (1; β k +1 h i ~1 [E]j+1,k +1 = ejk , and β = βk +1,1 . k +1 Frame 18/26 Spreading on degree-correlated networks I So, in principle at least: Assortativity and Mixing Definition ~ 0 (1; β ~1 ) = A−1 Eβ ~1 . F ~ E,β1 I ~ 0 (1; β ~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: General mixing Assortativity by degree Contagion References det AE,β~ = 0 1 . Frame 19/26 Spreading on degree-correlated networks I General condition details: det AE,β~ = det δjk Rk −1 − (k − 1)βk ,1 ej−1,k −1 = 0. 1 I I The above collapses to our standard contagion condition when ejk = Rj Rk . ~1 = β~1, we have the condition for a simple When β Assortativity and Mixing Definition General mixing Assortativity by degree Contagion References disease model’s successful spread det δjk Rk −1 − β(k − 1)ej−1,k −1 = 0. I ~1 = ~1, we have the condition for the existence When β of a giant component: det δjk Rk −1 − (k − 1)ej−1,k −1 = 0. I Bonusville: We’ll find another (possibly better) version of this set of conditions later... Frame 20/26 Spreading on degree-correlated networks 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. Assortativity and Mixing Definition General mixing Assortativity by degree Contagion References Triggering probability: I Generating function: ~1 ) = x H(x; β ∞ X h ik ~1 ) . Pk Fk −1 (x; β k =0 I Generating function for vulnerable component size is more complicated. Frame 21/26 Spreading on degree-correlated networks Assortativity and Mixing Definition I Want probability of not reaching a finite component. Ptrig = Strig ~1 ) =1 − H(1; β ∞ h ik X ~1 ) . Pk Fk −1 (1; β =1 − General mixing Assortativity by degree Contagion References k =0 I ~1 ). Last piece: we have to compute Fk −1 (1; β I Nastier (nonlinear)—we have to solve the recursive expression P we started with when x = 1: ~1 ) = ∞ ejk (1 − βk +1,1 )+ Fj (1; β k =0 Rj h ik P∞ ejk ~1 ) . β F (1; β k +1,1 k k =0 Rj I Iterative methods should work here. Frame 22/26 Spreading on degree-correlated networks I I I Truly final piece: Find final size using approach of Gleeson [2] , a generalization of that used for uncorrelated random networks. Need to compute θj,t , the probability that an edge leading to a degree j node is infected at time t. Assortativity and Mixing Definition General mixing Assortativity by degree Contagion References Evolution of edge activity probability: 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 βki . 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 βki . Frame 23/26 Spreading on degree-correlated networks I I I 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( ~ ~ ~ Linearize G around θ0 = 0. ∞ ∞ X X ∂Gj (~0) ∂ 2 Gj (~0) 2 ~ θ +... θj,t+1 = Gj (0) + θk ,t + ∂θk ,t ∂θk2,t k ,t k =1 k =1 Assortativity and Mixing Definition General mixing Assortativity by degree Contagion References If Gj (~0) 6= 0 for at least one j, always have some infection. ∂Gj (~0) ~ If Gj (0) = 0 ∀ j, largest eigenvalue of must ∂θk ,t exceed 1. Condition for spreading is therefore dependent on eigenvalues of this matrix: ∂Gj (~0) ej−1,k −1 = (k − 1)βk 1 ∂θk ,t Rj−1 Insert question from assignment 5 () Frame 24/26 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 Definition General mixing Assortativity by degree 1.0 I giant component S 0.8 0.6 0.4 assortative neutral disassortative 0.2 0.0 Assortativity and Mixing 1 10 I 100 exponential parameter κ FIG. 1. Size of the [3] 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). More assortative networks percolate for lower average degrees Contagion References But disassortative networks end up with higher extents of spreading. Frame 25/26 References I [1] B. Efron and C. Stein. The jackknife estimate of variance. The Annals of Statistics, 9:586–596, 1981. pdf () Assortativity and Mixing Definition General mixing Assortativity by degree Contagion [2] J. P. Gleeson. Cascades on correlated and modular random networks. Phys. Rev. E, 77:046117, 2008. pdf () References [3] M. Newman. Assortative mixing in networks. Phys. Rev. Lett., 89:208701, 2002. pdf () [4] M. E. J. Newman. Mixing patterns in networks. Phys. Rev. E, 67:026126, 2003. pdf () Frame 26/26