Section 7.13: Homophily (or Assortativity) By: Ralucca Gera, NPS

Transcription

1 Section 7.13: Homophily (or Assortativity) By: Ralucca Gera, NPS

2 Are hubs adjacent to hubs? How does a node s degree relate to its neighbors degree? Real networks usually show a non-zero degree correlation (defined later compared to random). If it is positive, the network is said to have assortatively mixed degrees (assortativity based other attributes can also be considered). If it is negative, it is disassortatively mixed. According to Newman, social networks tend to be assortatively mixed, while other kinds of networks are generally disassortatively mixed. 2

3 Partitioning the network Sociologists have observed network partitioning based on the following characteristics: Friendships, aquintances, business relationships Relationships based on certain characteristics: Age Nationality Language Education Income level Homophily: It is the tendency of individuals to choose friends with similar characteristic. Like links with like.

4 Example of homophily 4

5 Assortativity by race James Moody 5

6 Friendship networks at a US highschool Homophily or Assortative mixing Homophily (or assortative mixing) is the strong tendency of people to associate with others whom they perceive are similar to themselves in some way Disassortative mixing is the tendency for people to associate with others who are unlike them (such as gender in sexual contact networks, where the majority of partnerships are between opposite sex individuals) 6

7 Assortative mixing Titter data: political retweet network Red = Republicans Blue = Democrats Note that they mostly tweet and re-tweet to each other Conover et at.,

8 Homophily (or Assortative mixing) Homophily or assortativity is a common property of social networks (but not necessary): Papers in citation networks tend to cite papers in the same field Websites tend to point to websites in the same language Political views Race Obesity 8

9 Homophily in Gephi here 9

10 Homophily in Python In NetworkX, to check degree Assortativity: r = nx.attribute_assortativity_coefficient(g) To check an attribute s assortativity: assortivity_val=nx.attribute_assortativity_coefficient(g, "color ) The attribute color can be replaced by other attributes that your data was tagged with. 10

11 Disassortative Disassortative mixing: like links with dislike. Dissasortative networks are the ones in which adjacent nodes tend to be dissimilar: Dating network (females/males) Food web (predator/prey) Economic networks (producers/consumers) 11

12 Assortative mixing (homophily) We will study two types of assortative mixing: 1. Based on enumerative characteristics (the characteristics don t fall in any particular order), such as: 1. Nationality 2. Race 3. Gender 2. Based on scalar characteristics, such as: 1. Age 2. Income 3. By degree: high degree connect to high degree 12

13 Based on enumerative characteristics (characteristics that don t fall in any particular order), such as: 1. Nationality 2. Race 3. Gender 13

14 Possible defn assortativity A network is assortative if there is a significant fraction of the edges between same-type vertices How to quantify how assortative the network is? Let be the assortative class of vertex i, S = # Then: What is the assortativity if we consider V(G) as the only class? Does it make sense? 14

15 Alternative definitions Alternate defintions, as before, will allow to compare the Assortativity of the current network to the one of a random graph: Compute the fraction of edges in in the given network Compute the fraction of edges in in a random graph Consider their difference 15

16 Compute the fraction of edges in in the given network Let be the class of vertex i Let be the total number of classes Let ) = be the Kronecker that accounts for vertices in the same class. Then the number of edges of the same type is: Checks if vertices are in the same class 16

17 Compute the fraction of edges in in the random network Construct a random graph with the same degree distrib. Let be the class of vertex i Let be the total number of classes Let ( ) = be the Kronecker Pick an arbitrary edge in the random graph: If pick a vertex i there are deg edges incident with it, so deg choices for i to be the 1st end vertex of our arbitrary edge and then there are deg choices for j to be the other end edges are placed at random, the expected number of edges between i and j is # of vertices at the end of the E(G) edges Checks if vertices are in the same class 17

18 Let be the class of vertex i Let be the total number of classes Compute the fraction of edges in in the random network Let ( ) = be the Kronecker Checks if vertices are in the same class The expected number of edges between i and j if m edges are placed at random is Then the number of edges of the same type is: duplications: as you choose vertex j above, the edge ji will be counted after edge ij was counted 18

19 Consider their difference - = Checks if vertices are in the same class And now normalize by m = E(G) : Modularity = Which is the same defn as the modularity in community detection (assortativity based on deg). 19

20 Q =, Modularity Checks if vertices are in the same class Measure used to quantify the like vertices being connected to like vertices -1 < Q < 0 means there are fewer edges between like vertices in a class compared to a random network i.e. disassortative network 0 < Q < 1 means there are more edges between like vertices in a class compared to a random network i.e. assortative network Q = 0 means it behaves like a random network. 20

21 Ranges of the modularity values Range: -1 < Q < 1 However, it never reaches 1 (we ll see shortly), even in a perfectly assortative network (where all like nodes are adjacent) The modularity is generally far from 1 So what is a good modularity value (Q >?) that says the network is assortative? Should we normalize Q again? What are some choices? 21

22 Enumerative characteristics We normalize the modularity value Q, by the maximum value that it can get Perfect mixing is when all edges fall between vertices of the same type 1 Q max 1 2m (2m k k i j ij 2m (c i,c j )) Then, the assortativity coefficient is given by: Q ( Aij kk i j /2 m) ( ci, c ) ij j 1 1 Q 2 m ( kk /2 m) ( c, c ) max ij i j i j 22

23 Alternative form for edge list An alternative faster to compute formula, if the network is given by a list of edges rather than adjacency matrix: Let be the fraction of 2m A ij(c i,r)(c j, s) ij edges that join vertices of type to vertices of type Let a r 1 be the fraction of ends of 2m k i(c i, r) ij edges attached to vertices of type r We further have that the edges between group and is (c i,c j ) (c i, r)(c j,r) e rs 1 r 23

24 Q 1 2m r r ij A ij k i k j 2m r Alternative form for edge list (c i,r)(c j,r) 1 2m A ij(c i,r)(c j,r) 1 ij 2m (e rr a r 2 ) i k 1 i(c i,r) 2m k i(c j, r) j Where we defined a r 1 is the fraction of edges 2m k i(c i, r) ij between nodes of type and e rs 1 is the fraction of edges between and, 2m A ij(c i,r)(c j, s) ij and deg 24

25 Based on scalar characteristics, such as: 1. Age 2. Income 25

26 Scalar characteristics Scalar characteristics take values that come in a particular order (enumerative) and take numerical values Such as age, income Age: In this case two people can be considered similar if they are born the same day or within a year or within x years etc. If people are friends with others of the same age, we consider the network assortatively mixed by age (or stratified by age) 26

27 Assortativity by race James Moody 27

28 Assortativity by grade/age James Moody 28

29 Scalar characteristics When we consider scalar characteristics we basically have an approximate notion of similarity between adjacent vertices (i.e. how far/close the values are) There is no approximate similarity that can be measured this way when we talk about enumerative characteristics; rather present/absent 29

30 Friendship networks at a US highschool Recall Homophily (or assortative mixing) is the strong tendency of people to associate with others whom they perceive are similar to themselves in some way Disassortative mixing is the tendency for people to associate with others who are unlike them (such as gender in sexual contact networks, where the majority of partnerships are between opposite sex individuals) 30

31 Scalar characteristics Friendships at the same US high school: each dot represents a friendship (an edge from the network) Denser along the y = x line (because of the way data is displayed) Sparser as the difference in grades increases 31

32 Strongly assortative Top: scatter plot of 1141 married couples Bottom: same data showing a histogram of the age difference Data: 1995 US National Survey of Family Growth Newman, Phys Rev E. 67, (2003) 32

33 Scalar characteristics How do we measure this assortative mixing? Would the idea of the enumartive assortative mixing work? That is to place vertices in bins based on scalar values: Treat vertices that fall in the same bin (such as age) as like vertices or identical Apply modularity metric for enumerative characteristics 33

34 Scalar characteristics This approach misses much of the point of scalar part of the scalar characteristics Why are two vertices identical if they are in the same bin? Should they only be approximately the same type? Why two vertices that fall in different bins are different? That is, if the bins are close enough (1 grade apart), could the vertices be more related than these that fall farther (like 4 grades apart)? We don t have just a binary choice as we did for enumerative characteristics 34

35 Scalar characteristics We will use a covariance measure (next slide) Let x i (or x j ) be the value of the scalar characteristic of vertex i (j), such as the age Then the expected mean of at the end of an edge over all edges of the graph (not just the mean value of ) is: And so ij ij Ax ij i i kx i i 1 A 2m 2m ij A vertex of degree appears at the end of edges i kx i i 35

36 Scalar characteristics What is the covariance of two variables i and j (based on the scalar characteristic x i and x j ) over all edges of the graph? cov(x i, x j ) Which is similar to the enumerative modularity Q = ij A ij (x i )(x j ) ij A ij m ij (A ij k i k j 2m )x i x j Either 0 or 1 Product of values (a function of two variables) 36

37 Scalar characteristics If the covariance is positive, we have assortativity mixing, If the covariance is negative, we have disassortativity As with the modularity Q we need to normalize the above covariance value to be closer to 1 for a perfectly mixed network 37

38 Scalar characteristics Recall 1 kk i j cov( x, x ) ( A ) xx 2m 2m i j ij i j ij Setting x i =x j for all the existing edges (i,j) we get the maximum possible value (which is the variance of the variable) for perfectly assortative network: 1 kk 1 kk 2 i j i j max ( Ax ij i xx i j) ( kiij ) xx i j 2m ij 2m 2m ij 2m This can then be used for comparison 38

39 Scalar characteristics Then the assortativity coefficient r is defined as: r ij ij (A ij k i k j /2m)x i x j (k i ij k i k j /2m)x i x j Similar to the enumerative one again r=1 Perfectly assortative network r=-1 Perfectly disassortative network r=0 no correlation Similar to the Pearson correlation coefficient 39

40 Simpler reformulation for Corr. Coeff. =.621 for the network below (strongly assort.) is the fraction of edges in a network that connect a vertex of type i to one of type j, form the matrix above. Ref: Newman, Phys Rev E. 67, (2003) 40

41 Computer Science faculty 88 Computer Science faculty: vertices are PhD granting institutions in North America Edge (i,j) means that PhD at i and now faculty at j labels are US census regions + Canada Five Lectures on Networks, by Aaron Clauset 41

42 Assortative community structure Communities are tight and dense groups that connect to other dense groups Five Lectures on Networks, by Aaron Clauset 42

43 Community structure Five Lectures on Networks, by Aaron Clauset 43

44 By degree: high degree connect to high degree 44

45 Assortative mixing by degree A special case is when the characteristic of interest is the degree of the node Commonly used in social networks (the most used one of the scalar characteristics) More interesting since degree is a topologial property of the network (not just a value like in the previously seen scalars) In this case studying the assortative mixing will show if high/low-degree nodes connect with other high/low-degree nodes or not

46 Assortative mixing by degree Assortative network by degree core of high degrees and a periphery of low degrees (Figure (a) below) Disassortative network by degree uniform (low degree adjacent to high degree) Either 0 or 1 (a) (b)

47 Assortative mixing by degree

48 Assortative mixing by degree measured like the other scalar assortative requires the same adjacency information of the network cov(k i, k j ) 1 2m ij (A ij k i k j 2m )k i k j The correlation coefficient is the same normalize value: degcorr_ coeff ij ij ( A i ij ( k ij k k i k k i j / 2m) k k j i i j / 2m) k k The scalar is the degree j 48

49 Bottom line Same formula: Q ( Aij kk i j /2 m) ( ci, c ) ij j 1 1 Q 2 m ( kk /2 m) ( c, c ) max i j i j If enumerative: either the property is or is not present (the delta Kronecker function is used) If scalar: use the value of the scalar instead of the delta Kronecker function In particular one could use the degree ij 49

50 Examples Newman, Phys. Rev. E. 67, (2003) 50

51 Same slide with more details Newman, Phys. Rev. E. 67, (2003) 51

52 Assortative mixing by degree The assortative mixing by degree coefficient,, measured how connected are the vertices of the same degree (compared to random graphs) In general the absolute values of the assortativity coefficient r is not large in observed networks (the correlation between the degrees is not especially strong) 52

53 Table 8.1 page 237 r = assortativity coefficient 53

54 Assortative mixing by degree Technological/biological/information networks tend to have negative (disassortative mixing based on degree), while social networks tend to have positive (assortative) Good time to share your thoughts on why. Why? One possible reason is that technological, biological and information networks are simple graphs (no edge multiplicity) and Maslov et al. showed these networks tend to be disassortative since the number of vertices between high degree vertices is limited by the number of vertices of high degree 54

55 Assortative mixing by degree The multiplicity of edges in social networks tend to drive the value of up There might be other biases in place in social networks that cause a slight assortative mixing: Communities: low degree nodes adjacent to low degree nodes within the same community, and the high degree ones connecting the communities might have better chances of being adjacent 55

56 References Besides the references at the bottom left of different slides, this is a good overall reference: Newman, Mark EJ. "Mixing patterns in networks." Physical Review E 67.2 (2003):