Complex Networks, Course 303A, Spring, Prof. Peter Dodds

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1 Complex Networks, Course 303A, Spring, 2009 Prof. Peter Dodds Department of Mathematics & Statistics University of Vermont Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. Frame 1/26

2 Outline Frame 2/26

3 Basic idea: Random networks with arbitrary 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.: demographics (age, gender, etc.) 3. group affiliation We speak of mixing patterns, correlations, biases... Networks are still random at base but now have more global structure. Build on work by Newman [3, 4]. Frame 3/26

4 between node categories 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 e µν = Pr to a node of type ν ) a µ = Pr(an edge comes from a node of type µ) b ν = Pr(an edge leads to a node of type ν) Write E = [e µν ], a = [a µ ], and b = [b ν ]. Requirements: e µν = 1, e µν = a µ, and e µν = b ν. ν µ µ ν Frame 4/26

5 Connection to distribution: Frame 5/26

6 Notes: Varying e µν allows us to move between the following: 1. Perfectly assortative networks where nodes only connect to like nodes, and the network breaks into subnetworks. Requires e µν = 0 if µ ν and µ e µµ = Uncorrelated networks (as we have studied so far) For these we must have independence: e µν = a µ b ν. 3. Disassortative networks where nodes connect to nodes distinct from themselves. Disassortative networks can be hard to build and may require constraints on the e µν. Basic story: level of assortativity reflects the to which nodes are connected to nodes within their group. Frame 6/26

7 Correlation coefficient: Quantify the level of assortativity with the following assortativity coefficient [4] : µ r = e µµ µ a µb µ 1 µ a = Tr E E 2 1 µb µ 1 E 2 1 where 1 is the 1-norm = sum of a matrix s entries. Tr E is the fraction of edges that are within groups. E 2 1 is the fraction of edges that would be within groups if connections were random. 1 E 2 1 is a normalization factor so r max = 1. When Tr e µµ = 1, we have r = 1. When e µµ = a µ b µ, we have r = 0. Frame 7/26

8 Correlation coefficient: Notes: r = 1 is inaccessible if three or more types are presents. Disassortative networks simply have nodes connected to unlike nodes no measure of how unlike nodes are. Minimum value of r occurs when all links between non-like nodes: Tr e µµ = 0. where 1 r min < 0. r min = E E 2 1 Frame 8/26

9 Scalar quantities Now consider nodes defined by a scalar integer quantity. Examples: age in years, height in inches, number of friends,... e jk = Pr a randomly chosen edge connects a node with value j to a node with value k. a j and b k are defined as before. Can now measure correlations between nodes based on this scalar quantity using standard Pearson correlation coefficient ( ): r = j k j k(e jk a j b k ) σ a σ b = j 2 a j 2 a jk j a k b k 2 b k 2 b This is the observed normalized deviation from randomness in the product jk. Frame 9/26

10 Degree- correlations Natural correlation is between the s of connected nodes. Now define e jk with a slight twist: ( an edge connects a j + 1 node e jk = Pr to a k + 1 node ) ( an edge runs between a node of in- j = Pr and a node of out- k ) Useful for calculations (as per R k ) Important: Must separately define P 0 as the {e jk } contain no information about isolated nodes. Directed networks still fine but we will assume from here on that e jk = e kj. Frame 10/26

11 Degree- correlations Notation reconciliation for undirected networks: j k r = j k(e jk R j R k ) σ 2 R where, as before, R k is the probability that a randomly chosen edge leads to a node of k + 1, and σ 2 R = j j 2 R j j 2 jr j. Frame 11/26

12 Degree- correlations Error estimate for r: Remove edge i and recompute r to obtain r i. Repeat for all edges and compute using the jackknife method ( ) [1] σ 2 r = i (r i r) 2. Mildly sneaky as variables need to be independent for us to be truly happy and edges are correlated... Frame 12/26

13 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. Measurements Coli 54 ; m neural network of theof nematode - worm C. Elegans 5,55 ; tropic interactions correlations between species in the food webs of n Ythan Estuary, Scotland 56 and o Little Rock Lake, Wisconsin 57. Group Network Type Size n Assortativity r Error r a Physics coauthorship undirected a Biology coauthorship undirected b Mathematics coauthorship undirected Social c Film actor collaborations undirected d Company directors undirected e Student relationships undirected f address books directed g Power grid undirected Technological h Internet undirected i World Wide Web directed j Software dependencies directed k Protein interactions undirected l Metabolic network undirected Biological m Neural network directed n Marine food web directed o Freshwater food web directed most never is. In this paper, therefore, we take an ve approach, making use of computer simulation. ould like to generate on a computer a random netving, for instance, a particular value of the matrix is also fixes the distribution, via Eq. 23. In we discussed disassortative one possible way of doing this using ithm similar to that of Sec. II C. One would draw om the desired distribution e jk and then join the de- 58,59. The algorithm is as follows. 1 Given the desired edge distribution e jk, we first calculate the corresponding distribution of excess s q k from Eq. 23, and then invert Eq. 22 to find the distribution: Frame 13/26 p k q k 1 /k. 27 q j 1 / j Social networks tend to be assortative (homophily) Technological and biological networks tend to be

14 Spreading on -correlated networks Next: Generalize our work for random networks to -correlated networks. As before, by allowing that a node of k is activated by one neighbor with probability β k1, we can handle various problems: 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

15 Spreading on -correlated networks Goal: Find f n,j = Pr an edge emanating from a j + 1 node leads to a finite active subcomponent of size n. Repeat: a node of k is in the game with probability β k1. Define β 1 = [β k1 ]. Plan: Find the generating function F j (x; β 1 ) = n=0 f n,jx n. Frame 15/26

16 Spreading on -correlated networks Recursive relationship: F j (x; β 1 ) = x 0 + x k=0 k=0 e jk R j (1 β k+1,1 ) e [ jk β k+1,1 F k (x; R k β 1 )]. j First term = Pr that the first node we reach is not in the game. Second term involves Pr we hit an active node which has k outgoing edges. Next: find average size of active components reached by following a link from a j + 1 node = F j (1; β 1 ). Frame 16/26

17 Spreading on -correlated networks Differentiate F j (x; β 1 ), set x = 1, and rearrange. We use F k (1; β 1 ) = 1 which is true when no giant component exists. We find: R j F j (1; β 1 ) = e jk β k+1,1 + ke jk β k+1,1 F k (1; β 1 ). k=0 k=0 Rearranging and introducing a sneaky δ jk : ( ) δjk R k kβ k+1,1 e jk F k (1; β 1 ) = e jk β k+1,1. k=0 k=0 Frame 17/26

18 Spreading on -correlated networks In matrix form, we have where [A E, β1 ] A E, β1 F (1; β 1 ) = E β 1 = δ jkr k kβ k+1,1 e jk, j+1,k+1 [ F (1; β 1 )] = F k (1; β 1 ), k+1 [ ] [E] j+1,k+1 = e jk, and β1 = β k+1,1. k+1 Frame 18/26

19 Spreading on -correlated networks So, in principle at least: F (1; β 1 ) = A 1 E, β 1 E β 1. Now: as F (1; β 1 ), the average size of an active component reached along an edge, increases, we move towards a transition to a giant component. Right at the transition, the average component size explodes. Exploding inverses of matrices occur when their determinants are 0. The condition is therefore: det A E, β1 = 0. Frame 19/26

20 Spreading on -correlated networks General condition details: det A E, β1 = det [ δ jk R k 1 (k 1)β k,1 e j 1,k 1 ] = 0. The above collapses to our standard contagion condition when e jk = R j R k. When β 1 = β 1, we have the condition for a simple disease model s successful spread det [ δ jk R k 1 β(k 1)e j 1,k 1 ] = 0. When β 1 = 1, we have the condition for the existence of a giant component: det [ δ jk R k 1 (k 1)e j 1,k 1 ] = 0. Bonusville: We ll find another (possibly better) version of this set of conditions later... Frame 20/26

21 Spreading on -correlated networks We ll next find two more pieces: 1. P trig, the probability of starting a cascade 2. S, the expected extent of activation given a small seed. Triggering probability: Generating function: H(x; β 1 ) = x P k [F k 1 (x; β k 1 )]. k=0 Generating function for vulnerable component size is more complicated. Frame 21/26

22 Spreading on -correlated networks Want probability of not reaching a finite component. P trig = S trig =1 H(1; β 1 ) =1 P k [F k 1 (1; β k 1 )]. k=0 Last piece: we have to compute F k 1 (1; β 1 ). Nastier (nonlinear) we have to solve the recursive expression we started with when x = 1: F j (1; β 1 ) = e jk k=0 R j (1 β k+1,1 )+ k=0 Iterative methods should work here. e jk R j β k+1,1 [F k (1; β 1 )] k. Frame 22/26

23 Spreading on -correlated networks Truly final piece: Find final size using approach of Gleeson [2], a generalization of that used for uncorrelated random networks. Need to compute θ j,t, the probability that an edge leading to a j node is infected at time t. Evolution of edge activity probability: θ j,t+1 = G j ( θ t ) = φ 0 + (1 φ 0 ) k=1 e j 1,k 1 R j 1 k 1 ( ) k 1 θk,t i i (1 θ k,t) k 1 i β ki. i=0 Overall active fraction s evolution: φ t+1 = φ 0 +(1 φ 0 ) k P k k=0 i=0 ( ) k θk,t i i (1 θ k,t) k i β ki. Frame 23/26

24 Spreading on -correlated networks As before, these equations give the actual evolution of φ t for synchronous updates. condition follows from θ t+1 = G( θ t ). Linearize G around θ 0 = 0. θ j,t+1 = G j ( 0) + k=1 G j ( 0) θ k,t θ k,t + 2 G j ( 0) θk,t k=1 θ 2 k,t If G j ( 0) 0 for at least one j, always have some infection. [ ] If G j ( G 0) = 0 j, largest eigenvalue of j ( 0) θ k,t must exceed 1. Condition for spreading is therefore dependent on eigenvalues of this matrix: G j ( 0) θ k,t = e j 1,k 1 (k 1)β k1 R j 1 Insert question from assignment 5 ( ) Frame 24/26

25 tively mixed network. These findings are intuitively reasonable. If the net- How the giant component changes with assortativity work mixes assortatively, then the high- vertices will tend to stick together in a subnetwork or core group of higher mean 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 giant component S exponential parameter κ assortative neutral disassortative FIG. 1. Size of the giant component as a fraction of graph from Newman, 2002 [3] size for graphs with the edge distribution given in Eq. (9). The points are simulation results for graphs of N 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), p 0 0: (squares), and 0:05 (triangles). More assortative networks percolate for lower average s But disassortative networks end up with higher extents of spreading. Frame 25/26

26 I [1] B. Efron and C. Stein. The jackknife estimate of variance. The Annals of Statistics, 9: , pdf ( ) [2] J. P. Gleeson. Cascades on correlated and modular random networks. Phys. Rev. E, 77:046117, pdf ( ) [3] M. Newman. Assortative mixing in networks. Phys. Rev. Lett., 89:208701, pdf ( ) [4] M. E. J. Newman. patterns in networks. Phys. Rev. E, 67:026126, pdf ( ) Frame 26/26

Assortativity and Mixing. Outline. Definition. General mixing. Definition. Assortativity by degree. Contagion. References. Contagion.

Assortativity and Mixing. Outline. Definition. General mixing. Definition. Assortativity by degree. Contagion. References. Contagion. Outline Complex Networks, Course 303A, Spring, 2009 Prof Peter Dodds Department of Mathematics & Statistics University of Vermont Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike