Chapter 4 Networks in Their Surrounding Contexts - Cornell University

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88 CHAPTER 4. NETWORKS IN THEIR SURROUNDING CONTEXTS Figure 4.2: Using a numerical measure, one can determine whether small networks such as this one (with nodes divided into two types) exhibit homophily. network, it inevitably becomes difficult to attribute any individual link to a single factor. And ultimately, one expects most links to in fact arise from a combination of several factors — partly due to the effect of other nodes in the network, and partly due to the surrounding contexts. Measuring Homophily. When we see striking divisions within a network like the one in Figure 4.1, it is important to ask whether they are “genuinely” present in the network itself, and not simply an artifact of how it is drawn. To make this question concrete, we need to formulate it more precisely: given a particular characteristic of interest (like race, or age), is there a simple test we can apply to a network in order to estimate whether it exhibits homophily according to this characteristic? Since the example in Figure 4.1 is too large to inspect by hand, let’s consider this question on a smaller example where we can develop some intuition. Let’s suppose in particular that we have the friendship network of an elementary-school classroom, and we suspect that it exhibits homophily by gender: boys tend to be friends with boys, and girls tend to be friends with girls. For example, the graph in Figure 4.2 shows the friendship network of a (small) hypothetical classroom in which the three shaded nodes are girls and the six unshaded nodes are boys. If there were no cross-gender edges at all, then the question of homophily would be easy to resolve: it would be present in an extreme sense. But we expect that homophily should be a more subtle effect that is visible mainly in aggregate — as it is, for example, in the real data from Figure 4.1. Is the picture in Figure 4.2 consistent with homophily? There is a natural numerical measure of homophily that we can use to address questions
4.1. HOMOPHILY 89 like this [202, 319]. To motivate the measure (using the example of gender as in Figure 4.2), we first ask the following question: what would it mean for a network not to exhibit homophily by gender? It would mean that the proportion of male and female friends a person has looks like the background male/female distribution in the full population. Here’s a closely related formulation of this “no-homophily” definition that is a bit easier to analyze: if we were to randomly assign each node a gender according to the gender balance in the real network, then the number of cross-gender edges should not change significantly relative to what we see in the real network. That is, in a network with no homophily, friendships are being formed as though there were random mixing across the given characteristic. Thus, suppose we have a network in which a p fraction of all individuals are male, and a q fraction of all individuals are female. Consider a given edge in this network. If we independently assign each node the gender male with probability p and the gender female with probability q, then both ends of the edge will be male with probability p 2 , and both ends will be female with probability q 2 . On the other hand, if the first end of the edge is male and the second end is female, or vice versa, then we have a cross-gender edge, so this happens with probability 2pq. So we can summarize the test for homophily according to gender as follows: Homophily Test: If the fraction of cross-gender edges is significantly less than 2pq, then there is evidence for homophily. In Figure 4.2, for example, 5 of the 18 edges in the graph are cross-gender. Since p =2/3 and q =1/3 in this example, we should be comparing the fraction of cross-gender edges to the quantity 2pq =4/9 = 8/18. In other words, with no homophily, one should expect to see 8 cross-gender edges rather than than 5, and so this example shows some evidence of homophily. There are a few points to note here. First, the number of cross-gender edges in a random assignment of genders will deviate some amount from its expected value of 2pq, and so to perform the test in practice one needs a working definition of “significantly less than.” Standard measures of statistical significance (quantifying the significance of a deviation below a mean) can be used for this purpose. Second, it’s also easily possible for a network to have a fraction of cross-gender edges that is significantly more than 2pq. In such a case, we say that the network exhibits inverse homophily. The network of romantic relationships in Figure 2.7 from Chapter 2 is a clear example of this; almost all the relationships reported by the highschool students in the study involved opposite-sex partners, rather than same-sex partners, so almost all the edges are cross-gender. Finally, it’s easy to extend our homophily test to any underlying characteristic (race, ethnicity, age, native language, political orientation, and so forth). When the characteristic can only take two possible values (say, one’s voting preference in a two-candidate election), then we can draw a direct analogy to the case of two genders, and use the same formula
Page 1 and 2: From the book Networks, Crowds, and
Page 3: 4.1. HOMOPHILY 87 Figure 4.1: Homop
Page 7 and 8: 4.2. MECHANISMS UNDERLYING HOMOPHIL
Page 9 and 10: 4.3. AFFILIATION 93 Anna Daniel Lit
Page 11 and 12: 4.3. AFFILIATION 95 graphs are ofte
Page 13 and 14: 4.4. TRACKING LINK FORMATION IN ON-
Page 23 and 24: 4.5. A SPATIAL MODEL OF SEGREGATION
Page 33 and 34: 4.6. EXERCISES 117 seven people in

Chapter 4 Networks in Their Surrounding Contexts - Cornell University

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