Section 7.13: Homophily (or Assortativity) By: Ralucca Gera, NPS 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 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.” Example of homophily 4 http://www.slideshare.net/NicolaBarbieri/homophily-and-influence-in-social-networks Assortativity by race James Moody 5 Homophily or Assortative mixing Friendship networks at a US highschool • 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 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., 2011 7 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 Homophily in Gephi here https://gephi.org/tutorials/gephi-tutorial-layouts.pdf 9 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 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 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 Based on enumerative characteristics (characteristics that don’t fall in any particular order), such as: 1. Nationality 2. Race 3. Gender 13 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 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 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 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 Checks if vertices are in the same class • Let ( ) = be the Kronecker • Pick an arbitrary edge in the random graph: – 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 • If 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 17 Compute the fraction of edges in in the random network • Let be the class of vertex i • Let be the total number of classes Checks if vertices are in the same class • Let ( ) = be the Kronecker • 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 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 Modularity •Q = ∑ ∙ , 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 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 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 1 kik j Qmax (2m (ci,cj )) 2m ij 2m • Then, the assortativity coefficient is given by: (/2)(,)Akkmcc Q ij ij i j i j 11 Qmkkmcc2(/2)(,) max ij ij i j 22 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: 1 • Let e rs A ij ( c i , r ) ( c j , s ) be the fraction of 2m edges that joinij vertices of type to vertices of type 1 • Letar ki(ci,r) be the fraction of ends of 2m edges attachedij to vertices of type r • We further have that the edges between group and is (ci,cj ) (ci,r)(cj,r) r 23 Alternative form for edge list 1 kik j Q Aij (ci,r)(c j,r) 2m ij 2m r 1 1 1 Aij(ci,r)(cj,r) ki(ci,r) ki(c j,r) r 2m ij 2m i 2m j 2 (err ar ) r 1 Where we defined a r k i ( c i, r ) is the fraction of edges 2m ij 1 between nodes of type and e rs A ij ( c i , r ) ( c j, s ) is the fraction 2m of edges between and , ij 24 and deg Based on scalar characteristics, such as: 1. Age 2. Income 25 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 Assortativity by race James Moody 27 Assortativity by grade/age James Moody 28 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 Recall Friendship networks at a US highschool • 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 Scalar characteristics Denser along the Friendships at the y = x line same US high school: (because of the each dot represents a way data is friendship (an edge displayed) from the network) Sparser as the difference in grades increases 31 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, 026126 (2003) 32 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 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