Labor market implications of weak ties.

1. Introduction

Many of us have heard the phrase "it's not what you know but who you know." While this may not be strictly true, approximately one half of workers in the United States do find employment through friends, relatives, and other social contacts (Granovetter 1995). Further, in a seminal paper, Granovetter (1973) argues that, for the purpose of finding jobs, it is acquaintances, not close friends, who are most important in helping individuals find employment. He claims that friends whom one knows well are likely to be close in social space. Close friends know one's other close friends and tend to have social networks similar to each other. Thus, any information held by a given close friend may be obtained from other close friends as well; the job information of close friends is often redundant. In contrast, acquaintances tend to have less similar social networks and know different people from one's close friends. Granovetter argued that acquaintances also are more likely to have different information about available jobs, which makes them valuable job contacts. Granovetter labeled acquaintances as weak ties as opposed to close friends, who are known as strong ties.

In this article, I am defining weak and strong ties as Granovetter (1983) did. Weak ties are acquaintances and strong ties are close friends. The important point of these definitions is the idea that, on average, acquaintances are less likely to know one's other friends than strong ties. On average, the social network of strong ties has more overlap than the social network of weak ties. As I will explain in more detail below, the lack of overlap of acquaintances implies that a social network with more weak ties will be larger (will have more range) than a social network with fewer weak ties.

Since Granovetter's article, much work has been done by other scholars to validate the importance of weak ties in labor markets by answering two questions: First, researchers asked whether weak ties are a common source of finding employment. These studies have been successful in demonstrating that weak ties are indeed an important source of finding employment. For instance, in Granovetter (1973), of the people finding jobs through referral sources (such as friends, family, and other personal contacts), 27.8% use weak ties, 16.7% use strong ties, and the majority, 55.6%, use ties of moderate strength in between weak and strong. Others have replicated the finding that weak ties are a common means of finding employment. (1) In summary, this line of research has demonstrated that, as Granovetter claimed, weak ties do play an important role in matching workers and jobs. Second, because weak ties appear to be important in locating jobs, researchers have asked if the use of weak ties as a job-finding method increases income (Bridges and Villemez 1986; Marsden and Hurlbert 1988; Wegener 1991). The findings of these studies suggest that there is little evidence of a relationship between weak ties and income. (Note that the hypothesized link between weak ties and income was not a part of Granovetter's hypothesis, as he makes clear in Granovetter [1983].)

The present article can be seen as an extension to this second line of questioning. In this article, I use an example to show that previous methods used to measure the effect of weak ties on income will underestimate the effect, if it exists. I then provide a new method of estimating the effect of weak ties on income that directly considers the social network of individuals. Granovetter's idea that weak ties are more likely to provide novel information rests on the idea that weak ties are less likely to overlap than strong ties, on average. In other words, weak ties know a smaller fraction of one's other friends compared with strong ties, on average. I use this idea to show that less overlap in friendships increases the range of an individual's social network. Individuals with less overlap have more individuals in their social network. Thus, they have more opportunities to acquire job information. I find that the range of an individual's social network has a positive and meaningful effect on income. Because weak ties increase the range of an individual's nework (Granovetter 1973), I find strong empirical support for the notion that having weak ties in one's social network increases income.

In the next section, I provide a short overview of the previous findings with regard to weak ties and income. I then provide an example in Section 3 that demonstrates why previous studies underestimate the effect of weak ties on income. Section 3 also describes the important features of the potential relationship between weak ties and income that will be used in Section 4. In Section 4, I describe an explicit model of social networks that can be used to estimate the size of an agent's social network relative to other members in the population. I then use this estimation to measure the effect of weak ties on income using data from the General Social Survey (Davis and Smith 1985). In Section 5, I briefly review the main findings of the article and discuss implications for future research.

2. An Overview of Weak Ties and Labor Markets

Establishing and measuring the effect of weak ties on income is the focus of this article. Because weak ties appear to be especially helpful in finding jobs, it has been argued that individuals with many weak ties in their social network may have higher income than those with few weak ties. In order to test this claim, researchers (Bridges and Villemez 1986; Marsden and Hurlbert 1988; Wegener 1991; Smith 2000; Mouw 2002, 2003) performed studies that compare the income of workers who found jobs using various search methods. These studies used survey information from respondents who were questioned about their current (and sometimes past) jobs, how each job was found, and income or wages in addition to other relevant variables, such as education, work experience, etc. The studies statistically analyzed how income is affected by the methods used to find jobs.

The workers in Granovetter's original study (Granovetter 1973) who found their job using weak ties earned higher income, on average, than those who found their job by other means. However, the purpose of Granovetter's study was to show that weak ties were a common means of finding employment. Again, as he makes clear in Granovetter (1983), he was not testing the hypothesis that weak ties increase income. Thus, he did not attempt to control for items such as level of education, age, etc. Although the later findings of Bridges and Villemez (1986) and Marsden and Hurlbert (1988) substantiate that individuals who find their jobs through weak ties earn higher income on average, they also find that the effect is statistically insignificant when control variables, such as level of education, are introduced. Wegener (1991) provides an exception. He finds a positive effect on income of having found one's job using weak ties but only for high-status individuals. He argues that low-status individuals with weak ties still only connect to other low-status individuals. Thus, Wegener's findings suggest that the job information available through weak ties is not better than the job information they receive from strong ties for low-status individuals. In contrast, Wegener argues that high-status individuals using weak ties more easily bridge their social networks to other high-status individuals with useful job information.

More recently, Smith (2000) studied the wage effects of using personal contacts to find a job conditional on race and gender. She finds that the effect varies. For instance, white men and Latinos earn 9% and 10% less than workers finding jobs through formal channels. But white women earn an average of 14% more if using personal contacts. Further, the white male to female wage gap shrinks for personal-contact job finders versus formal method job finders. Additionally, she finds that the wage differential between white males and Latinos using personal contacts is greater than the difference when more formal job-finding methods are used. In summary, her results suggest that the effect of job-finding method on wages is race and gender dependent. Mouw also has recently studied the effect of personal-contact use in job finding for black workers (Mouw 2002) and a broad sample of workers from the National Longitudinal Study of Youth (NLSY) and the Multi City Study of Urban Inequality (MCSUI) (Mouw 2003). He does not find a strong effect of personal-contact use increasing income in either case. In summary, despite the intuitive appeal of the notion that using weak ties to find a job may increase income, for the most part, past efforts to show a clear empirical link between weak ties and income have failed (Mouw 2003).

All of the previously mentioned studies measure individual income as a function of how the individual found a particular job. More specifically, they estimate the effect of job-finding method (found job by direct application, found by newspaper advertisement, found by weak tie, found by strong tie, etc.) on income while controlling for other relevant labor-market variables, such as education, experience, gender, etc. Essentially, these articles test the hypothesis that weak ties provide superior job information that leads to better jobs or higher paying jobs. As Granovetter (1995) has argued, however, there is no empirical evidence that weak ties do provide superior information. Recall that Granovetter's weak ties hypothesis claims that having many weak ties allows one to find more information both about jobs and other socially transmitted information because weak ties increase the range of one's social network. In effect, weak ties increase the amount of nonredundant information a job seeker receives but may not increase the quality of information.

For there to be an effect of weak ties on income, it is not necessary that weak ties specifically provide superior information (Granovetter 1995). I will show that, because weak ties increase the range of a social network (as originally argued by Granovetter), individuals with a larger proportion of weak ties in their social network should expect to learn of more job information compared with individuals with a smaller proportion of weak ties in their social network. My results show that, if there is an effect of weak ties on income, it is more likely to occur because having more weak ties in one's social network allows one to learn of more job information, not better job information.

As an example, suppose that everyone in a population can find one job on their own, say through a newspaper advertisement. But having many weak ties allows one person to find one additional job through her social contacts. Another person only finds the one job from a newspaper. Having a second job offer may allow the first person to gain higher income from being in a better bargaining position even if she accepts the job found from the newspaper advertisement and not the job found from the weak tie. Equally compelling, an individual with many weak ties may have better knowledge about offers made to other individuals in the population that aids in bargaining or job acquisition.

With this in mind, the previous lack of empirical support for an effect of weak ties on income becomes easier to understand. The link between weak ties and income is not directly tied to how one finds the particular job she holds. Instead, weak ties allow one to gather more (and novel) information (again as Granovetter originally suggested). Other researchers have recognized this: Montgomery (1992) argues that the importance of weak ties can only be understood if one considers the entire network structure. Obviously, this is a daunting task for researchers. The difficulties of constructing such a data set for a large population sample clearly leads one to understand why researchers attempted to measure the effect of weak ties using job-search method as a proxy for network structure.

However, I will apply recent advances in our understanding of how network structure impacts information flows to estimate the effect of weak ties in a labor market. (2) "Network structure generally refers to the range of one's network of ties as indicated by the number of ties one has, the number of different status groups in one's network (network diversity), the proportion of contacts who are intimately tied to each other (redundancy), and the proportion of ties who are weakly tied to an individual (density). It is generally argued that the greater an individual's range, the greater his or her likelihood to receive information for status and income advancement" (Smith 2000). In this article, I will estimate the network structure of respondents in the General Social Survey (using number of contacts, redundancy, and density) and then estimate the effect of network structure on income.

More specifically, I characterize the network structure of individuals by looking at the social structure of their close friends. While far from an exhaustive description of the social network of an individual, this method provides a way to estimate the range of an individual's network through the amount of overlap among the individual's close friends. Weak ties greatly increase the range of an individual's social network and thus allow the individual to have more information sources for jobs. I then find that the effect of the estimated range of an individual's social network on income is positive and relatively large. In other words, I find strong empirical support for the argument that weak ties have a positive effect on income.

3. Estimating the Effect of Weak Ties

Strong ties are defined to be close friends. Weak ties are defined as acquaintances. Again, the intuition of Granovetter's original article is that weak ties provide novel information sources because it is less common for weak ties to overlap with one's other friends. Strong ties are more likely to know one's other friends. Thus, strong ties are more likely to provide redundant information sources. (3) The intuition of this article follows Granovetter's original intuition about strong and weak ties and assumes that, on average, strong ties are more likely to know one's other friends than weak ties.

I begin this section with an example where one's group of friends consists of a core group, where everyone in the core group knows everyone else in the core group. In addition, everyone has a set of random friends who do not know anyone in the core group. This example moves us toward the modeling to be employed in this article. What I define as a core friend may be thought to introduce information effects similar to Granovetter's strong ties. (One would expect there to be redundancy of information in the core group because everyone knows everyone else.) What I define as random friends may be thought to introduce information effect similar to Granovetter's weak ties. (One would expect there to be little redundancy of information among random friends because they are less likely to know one's other friends.) (4)

Why Random Friends Matter Most

To begin, let us consider a simple example. The example will demonstrate two things: First the number of random friends (weak ties) in an individual's social network grows exponentially as a function of network distance. One has far more random friends than might be thought. Thus, the fact that many referral job offers come through weak ties instead of strong ties is less surprising than one may think. Second, estimating the effect of weak ties by coding the method by which someone finds a job underestimates the effect of weak ties on income. This second item helps to explain why the previously mentioned articles failed to find an effect of weak ties on income.

Suppose that the friends of individuals can be divided into two groups: core and random friends. First, each person in the example has one or more friends from a core group, where each friend in the core group knows each other. Thus, the core group is fully connected (everyone knows everyone else). Second, in addition to having friends from one's core group, each individual has a set of acquaintances or "random" friends who do not belong to their core group and thus are not connected to the individual's other friends. The friends in the core group will introduce effects similar to Granovetter's strong ties and the random friends will introduce effects similar to his weak ties.

I want to show how the mix of core and random friends affects the size of an individual's social network. I want to measure the size of a social network as distance from the individual increases. I define a distance-one network for individual j as the number of friends of j. I define a distance-two network as the number of friends of j plus all of the friends of the friends of j. A distance-three network includes the distance two network plus friends of friends. And so on.... I define [F.sub.i] as the size of a social network within distance i. In the example, suppose that each individual in a population has [F.sub.1] friends within distance one of the individual. Of these friends, suppose that C are core friends and R are random friends, where core and random friends are as defined above. [F.sub.1] = C + R.

Now calculate the size of the distance-two network of the individual, [F.sub.2]. To do this, one must count the number of new friends that each friend in the distance-one network adds to the social network. Note, however, that random friends always add their full complement of friends as new friends. Their core friends are different from the original core and random friends are new by definition. Each random friend adds C + R new friends to the network. However, core friends only add their random friends because the core friends of core friends have already been counted in the previous step. Each core friend only adds R new friends to the network. (5) Thus, the total new friends added to your network are R(C + R)+CR. Or [F.sub.2]=[F.sub.1]+R(C + R)+CR = C + R + 2CR+[R.sup.2]. Through similar calculations, one can find that [F.sub.3] = [F.sub.2] + [C.sup.2]R + 3[R.sup.2]C + [R.sup.3] = C + R + 2CR + [R.sup.2] + [C.sup.2]R + 3[R.sup.2]C + [R.sup.3].

To see how the network size changes as a function of the fraction of core and random friends in a social network, suppose that, in one network, everyone has 20 core friends and 10 random friends (for a total of 30) and, in another network, everyone also has 30 total friends but only 10 of them are core friends and 20 are random friends. The network sizes are shown in Table 1.

From this example, a second reason becomes clear why weak ties are such a good information source: there are exponentially more of them than strong ties. (6) In addition, note that the network with more random friends expands substantially faster than the network with fewer. This increased range provided by having a larger proportion of random friends in one's social network may be one reason why job information appears to come from weak as opposed to strong ties. Note that this idea is not in conflict with Granovetter's original notion that weak ties are important because they provide novel information sources. Not only can weak ties provide novel information, they also provide novel information from a very large number of sources.

You may ask, if individuals have so many weak ties, why don't almost all referrals come from them? There are at least two reasons: The first is part of Granovetter's original conception of the weak ties hypothesis. Because strong ties are closer to the individual, these friends are more motivated to provide job-finding assistance than weak ties. Second, by definition, individuals spend more time with their strong ties than their weak ties. Strong ties are more likely to know a given friend is looking for a job, and that friend may come to mind more often when a strong tie learns of employment opportunities. There is a balance between having few strong ties with whom you are in frequent contact and who have a specific motivation to help and having may weak ties with whom you are in less frequent contact and who are less specifically motivated to help.

In order to simplify the analysis in this article, I assume that the amount of information one receives from friends is proportional to one's network size. Thus, if individual i has twice as many individuals in his or her social network as individual j, then i receives twice as much job information from friends. This assumption simplifies the analysis on at least three levels: First, it ignores the fact that strong ties may be more willing to help in a job search than weak ties. Second, because strong ties are close friends they may have more knowledge about your job needs and may be more likely to work in your field. Thus, strong ties may be more likely to provide better information. Third, as mentioned above, the information of strong ties may be more redundant than the information of weak ties. My model takes account of effect three but not effects one and two. (7)

Initially, the first two effects may appear to bias my results in favor of finding an effect of weak ties on income, but the opposite is actually true. Suppose that individual A has two strong ties and one weak tie (and thus a network size of 3), and individual B has one strong tie and three weak ties (and thus a network size of 4). Suppose that strong ties are twice as valuable as weak ties (each strong tie increases income by $2 and each weak tie increases income by $1). If this is the case, individual A will have income of $5 and individual B will have income of $4. Individual A has smaller network size than individual B but a higher income than individual B. If I assume all friends are equal in terms of the information they provide, and run a regression of network size on income, I will find that network size does not increase income. In fact, I will find a negative relationship between network size and income because individual B has a larger network but smaller income compared with individual A. If strong ties are more important than weak ties for job-finding purposes and I assume that all friends are equal, this will lead to understating the effect of network size in the estimation results. Assuming all ties provide equivalent information biases my results against finding an effect of weak ties if strong ties are more valuable.

Sorting by Job Method Underestimates the Effect of Weak Ties

I now turn to the question of estimating the effect of weak ties by extending the intuition of the core and random friends example above. Suppose there are two groups in the population; one with many random friends (weak ties) but few core friends (strong ties), group W, and one with many core friends (strong ties) but few random friends (weak ties), group S. I develop the example such that group W individuals, who have many weak ties, have an advantage in the labor market. Then I show that estimating the effect of weak ties by using job-finding method will only reveal a fraction of the true advantage. Suppose it is such that the group S individuals have 500 friends within distance two of themselves and the group W individuals have 1000. For the moment, I am not going to make a distinction between weak and strong ties among these numbers and assume that they are all weak ties. Let each individual search for jobs both through formal means (newspaper advertisements, direct application, etc.) and through friends. Suppose that each individual finds 10 jobs through formal means and 1 of every 100 friends tells the individual about a job. Thus, group S individuals find 5 jobs through friends and 10 through formal means, for a total of 15. Group W individuals find 10 jobs through friends and 10 through formal means, for total of 20. Further, for simplicity, suppose that each job found is equally likely to result in an accepted offer. (8) Thus, group S individuals get one third of their jobs through friends and two thirds of their jobs through formal means. Group W individuals get one half of their jobs through friends and one half of their jobs through formal means. Suppose there are an equal number of individuals in each group.

This is where the problem with estimating the effect of weak ties by job-finding methods begins: Even if some individuals have fewer weak ties (group S) than others (group W), these individuals still find some jobs through friends. Individuals with more weak ties still find some jobs through formal means. If a researcher sorts a population sample by job-finding method, the found-by-friend group will include individuals of both types. Of course, on average, there will be more individuals with lots of weak ties in the found-by-friend group. But, as I will show below, even a relatively small amount of mixing can cause the effect of weak ties to be underestimated by a large amount.

As the final step in the example, let us suppose that there truly is an effect of weak ties on wages through novel knowledge of job information, bargaining, acquiring more job offers, or any other means one may suggest. Suppose that having more weak ties allows the group W individuals to earn more than the group S individuals. If a researcher is estimating income by running a regression of job-finding method used on income, what would be the amount of the effect estimated?

For simplicity, assume that there are equally many group W and group S individuals. Let [R.sub.w] be the percentage of jobs found by referral of friends for the group W individuals and define [R.sub.s] similarly. Let b be the base income earned by the group S individuals and let d be the income difference between the group W and group S individuals; group W individuals earn b + d. The following equation states the estimated income difference, [E.sub.d], between job-finding method a researcher would find as a function of [R.sub.w], [R.sub.s], b, d:

[E.sub.d] = [[R.sub.s]b +[R.sub.w](b + d)/[R.sub.s] + [R.sub.w]] - [(1 - [R.sub.s])b + (1 - [R.sub.w](b + d))/ (1 - [R.sub.s]) + (1-[R.sub.w])], (1)

which simplifies to d([R.sub.w] - [R.sub.s]). If we divide by d, this yields the difference found as a percent of the true income difference, [R.sub.w] - [R.sub.s]. Thus, the percent of the true difference found is simply the difference in the percentage of jobs that the group W individuals find by referral of friends less the percentage of jobs that the group S individuals find by referral. From this, one can see that the amount of the bias decreases as the groups sort themselves more strongly by how a job is found. As the correlation increases between number of weak ties and how a job is found, the underestimation decreases. In the example that began this subsection, group W individuals found one half of their jobs through friends and group S individuals found one third of their jobs through friends. Thus, a researcher would find 1/ 2 - 1/3 = 1/6 of the true difference, if estimating by job-search method.

Because individuals find jobs through both formal and informal means, regardless of the amount of weak ties, estimating the effect of weak ties through the type of method used to find a job is strongly biased against the hypothesis that having more weak ties increases income. The mixing of people in the how-a-job-was-found categories places people from both types in each category. The true effect of weak ties is not accurately measured. Weak ties will appear to have a smaller effect than they truly do. The example above shows that the measured amount of the effect can be so small as to find virtually no effect. This is essentially what the data have produced. In most instances, researchers who have used the method described above have found that individuals finding jobs through weak ties tend to earn higher income, but the differences are not statistically significant when control variables are introduced. (9)

By how much may the effect of weak ties be underestimated? The amount of underestimation depends primarily on two things: the correlation between type and job method success and the preference for accepting jobs found from referral versus formal sources. Depending on changes in these items, the amount of underestimation of the effect of weak ties may be larger or smaller than that described in the example above.

First, let us consider how the underestimation changes as a function of the correlation between job-finding method and type. For instance, suppose the mix of friends implies that group S individuals get 25% of jobs through friends and 75% through formal means ([R.sub.s] = 0.25) and the mix is 75% friends and 25% formal for group W individuals ([R.sub.w] = 0.75). If a researcher estimates the difference in wages using how a job was found in this scenario, she would find only 50% ([R.sub.w] - [R.sub.s] = 0.50) of the true difference in income even when the group W individuals get three times as many jobs through friends compared with the group S individuals! One would have to have [R.sub.w] = 1.0 and [R.sub.s] = 0.0 to find the true income difference between the groups when a researcher sorts by job-finding method.

Conversely, the underestimation may actually be worse than the original example above. Holzer (1988) has found that jobs found through friends are more likely to be accepted than jobs found through formal means. If individuals prefer to accept jobs found through friends, then the correlation between type and job-finding method decreases. If this bias was included in the example above, it would be even more difficult to identify the difference in income using the method of job finding as the primary independent variable. Differences in the networks would be masked by an individual bias to accepting jobs found through friends.

In any case, given perfect data, a researcher would still find an effect of weak ties on income, if one exists, even if estimating by the type of job-search method used. (The effect will be smaller than the true effect, as described above.) However, income data tends to be far from perfect. Income data tends to be noisy for a variety of reasons, such as compensating differentials, biases in self-reporting, etc. Because of the noise in income data, the standard errors will be larger than the true standard errors associated with the relevant variables. Using job-finding method as the independent variable of interest leads to underestimating the magnitude of the effect of weak ties in data that will have large standard errors. Thus, even if having more weak ties leads to an increase in income, it may be very difficult when using job-finding method to identify an economically important difference.

Given the problems associated with estimating the effect of weak ties through job-finding method described above, I estimate the effect of weak ties on income in a new, more direct way; I estimate the size of an agent's network using information on the close friends of an individual. In the next section, I describe a method for formalizing the important points of the example above. I then use this model to estimate the effect of weak ties on income.

4. Modeling Network Structure

The example above highlights two important items in characterizing the size of the social networks of individuals: The number of connections held by each individual and the amount of random friends or weak ties in the social network of the individual. (10) I use these two features of social networks to develop a model that will estimate the range of an individual's network relative to other members of the population.

One may represent the social network of individuals as a graph, where individuals are nodes and friendships are edges on the graph. Thus, the graph contains N nodes, one for each member of the population. For each pair of friends i and j in the population create an undirected edge (i, j) between the nodes. The resulting social graph represents the social network of the population.

Define the number of people to whom agent i is connected, the degree of i, as [k.sub.i]. We can formalize the mix of what I call core and random friends by looking at the overlap of friends in an individual's social network. If agent i has [k.sub.i] adjacent neighbors in the social graph, then this set of neighbors defines a subgraph with at most [k.sub.i]([k.sub.i] - 1)/2 edges. Define the clustering coefficient of i, [C.sub.i], as the fraction of these edges that exist (Watts and Strogatz 1998). (11) [C.sub.i] measures the fraction of the friends of i that are friends with each other. A relatively larger (smaller) [C.sub.i] implies i has more friends similar to core (random) friends than random (core) friends. If all of the friends of i are friends with each other (all friends are strictly core friends), then [C.sub.i] = 1. If none of the friends of i are friends with each other (all friends are strictly random friends), then [C.sub.i] = 0. [C.sub.i] between 0 and 1 implies that there is some but not complete overlap in the connections of friends. For instance, if [C.sub.i] = 0.5, then one half of the potential friendships between i's friends exist. We also can define the average clustering for the population,

[bar]C = [SIGMA].sup.n.sub.i] [C.sub.i] / N. (2)

And average degree in the population is

[bar.K] - [SIGMA].sup.n.sub.i] [k.sub.i] / N. (3)

I can use these items to estimate the number of friends an agent has within a given distance on the social graph. By definition, agent i has [k.sub.i] agents within distance one. For distance two, each of these [k.sub.i] connections has [bar]k connections on average. But, we must consider two items: First, one of the [bar.k] connections is agent i. Thus, on average, there are only [bar]K - 1 potential new friends. Second, there likely will be some overlap in the connections of each of these [bar]K - 1 friends. Some of these [bar]K - 1 friends will be members of the distance-one network of agent i. We can estimate this overlap by the clustering coefficient of agent i. Suppose that [k.sub.i] = 4, [bar]K = 4, and [C.sub.i] = 1/2. Agent i has four friends within distance one. How many within distance 2? At most, each of the four friends within distance one have three new friends because one of the 4 is i. If this was the case, there would be 4 + 4 X (4 - l) = 16 friends within distance 2. But we know some overlap is likely to exist among the 12 additional people added who are distance two from i. Because [C.sub.i] measures how many of i's friends are connected to each other, it may be used to estimate the clustering near i in the social graph. Thus, in the example, there is expected to be 4 + 4 x (4 - 1) x 1/2 = 10 friends within distance two. More generally, there will be

friends2 = [k.sub.i] + [k.sub.i] x ([bar]K - 1)(1 - [C.sub.i]) (4)

friends within distance 2, where friends2 is the expected number of friends within distance two on the social graph of i. So, friends2 is the [k.sub.i] friends within distance one. Plus, each of those [k.sub.i] friends has [bar]K - 1 friends (not including i), on average. But a fraction [C.sub.i] of those friends are members of the distance-one network of i. So, only fraction 1 - [C.sub.i] of these are additional friends of i.

The analysis can be carried out to further distances: However, additional technical restrictions result: First, as one moves further from the individual of interest, the clustering of the social network begins to resemble C, given in Equation 2, and not [C.sub.i]. Second, additional overlaps not accounted for by [C.sub.i] may occur. For instance, a friend in distance three, call her m, may be friends with two individuals i and j in the distance-two network but i and j are not connected. For a discussion of this issue as well as other similar difficulties, see Newman (2003a). Third, the distribution of friends (the distribution of [k.sub.i]) becomes increasingly important. If there are individuals with large numbers of friends, this can have a large effect on the expected number of people to which one connects in a given distance. Again, see Newman (2003a) or Albert and Barabasi (2001) for a discussion of this topic.

Because of these difficulties, and more importantly the idea that closer ties are more likely to be motivated to help an individual than distant ties, I do not extend the analysis beyond looking at networks of distance two in the data. (12)

Data

Until this point in the article, I only have argued in words that the increase in number of contacts provided by weak ties may increase income. I test this hypothesis specifically using data from the General Social Survey to analyze the effect of network structure on income. This is the first attempt to measure the effect of weak ties on income that does not use job-search method employed as the primary independent variable of social networks. The results are encouraging. Even though the data is far from optimal, I find that the additional range provided by weak ties has a measurable and positive effect on income.

The data used for this analysis comes from the 1985 General Social Survey Social Network Module (GSS). As part of the social-network module, the survey asked respondents to identify individuals with whom they have "discussed important matters" over the last 6 months. Further, each of the respondents was asked additional information on up to five of those named (fewer if the individual named fewer than five friends). Respondents were asked if and how well each of the friends named knew each other. From this information, I am able to estimate the clustering coefficient of i, [C.sub.i], in the social network of the respondent. The data also contains standard information on the education, age, gender, work status, and race of the individual. Income information is self-reported over 13 ranges. In the analysis below, I include all respondents from the social-network module who report positive individual income. In total, there are 958 individuals in the sample used. Summary statistics are given in Table 2.

Approximately 72.5% of the workers in the sample work full time, 46.5% are female, 87.3% are white, and 9.3% are black. The average worker has 21.8 years of work experience and 13.1 years of education. Regarding the social networks of the respondents, the average level of clustering ([bar.C]) for the sampled friends was approximately 0.628; 62.8% of the potential friendships between respondent's friends existed. For this study, the most important variable is friends2. This variable is an estimate of the range of the social network of the respondent as calculated from Equation 4 above. The average value of the friends2 variable is 5.79 and the maximum is 22.08.

Because each respondent is allowed to provide information on at most five friends in the survey, the social-network data is far from complete; friends2 does not count the total number of friends in an individual's social network. In using friends2 as my variable of network size, I am using a lower bound on the distance-two network of respondents because the GSS places an upper limit on the number of friends one can name. Because the network is limited, respondents are more likely to name closer friends in the survey. Because close friends are more likely to know each other than nonclose friends, I am biasing my sample against finding individuals with weak ties. (If close friends do not know each other, it is unlikely that nonclose friends do.) However, because we have information on the amount of overlap of the named friends, we can use friends2 as an estimate of the size of the social network of an individual relative to others in the population. Thus, the friends2 variable is a relative estimate of the social networks of individuals in the sample. I use this comparison of relative network structure to estimate the importance of weak ties in the next subsection.

Estimation

I use two methods to estimate the effect of network structure on income. First, because the income data in the GSS are sorted into bins, I estimate the effect of the variables mentioned above using an ordered logit model. Second, I estimate a linear model where the log of the midpoint of the reported income bin is used as the dependent variable. (For example, suppose individual i reports income within bin j and bin j contains income levels between x and y. I use log ([x + y]/2) as the income of respondent i.) The results of the models are provided in Table 3. The results presented in the table indicate that education, experience, experience squared, and the dummy variables for gender and full-time workers are significant at the 99% confidence level. Note that these variables all have signs consistent with expectations. Neither of the race dummy variables (the omitted group was other races) is statistically significant. This may be due to there being very few nonwhite individuals in the sample. Of the 958 individuals in the sample, 836 were classified as white. In terms of the signs of the coefficients and statistical significance, the two models report very similar results. Interpreting the size of the coefficients will be discussed below.

I am primarily interested in the effect of social networks on individual income; however, I need to control for other relevant income influences, such as education and work experience of the respondents, in order to isolate the effects of social networks. Because the control variables discussed above match with expectations, I concentrate discussion on the effect of the social-network variables in the remainder of this section. Most important for this article, the coefficient on number of friends within distance two is positive, significant, and of a meaningful size.

As a reminder, the network variables are defined as [C.sub.i] = the fraction of i's friends that are friends with each other; friends1 = [k.sub.i] = the number of friends i reported in the survey;friends2 = [k.sub.i] + [k.sub.i] X ([bar]K - 1)(1 - [Csub.i]), where [bar]K is the average number of friends reported in the sample population.

Estimating the magnitude of the effect of friends2 in the ordered logit model is not straightforward. The coefficients of an ordered logit do not directly lead to an estimate of magnitude. One can only calculate the change in likelihood of moving between income bins for a marginal change in friends2. Because the coefficient on friends2 is positive, one knows that if friends2 increases, the likelihood of being in the highest income bin increases and the likelihood of being in the lowest income bin decreases. But the change in the likelihood of being in one of the interior bins cannot be interpreted directly from the coefficients. Because the sum of the probabilities must equal one, the sum of the marginal changes must be zero; some of the marginal changes are positive and others are negative. (13) But because the coefficient is positive, we know that increasing the friends2 variable for a respondent increases the likelihood of the respondent being in a higher income bin. (14) Thus, it is not straightforward to put a dollar amount on the effect that friends have from an ordered logit model.

The log linear model is much more direct and useful if we are interested in the effect of changes in the network variables for the average respondent in the sample. Using the average income as the base point, increasing friends2 by one friend increases income by approximately 3.2%. If one earns $30,000 a year, this translates to $960, a sizeable but not huge amount. The friends2 variable is a function of both clustering, [C.sub.i], and friends1. Thus, it is also informative to know how a change in these variables impacts income through the friends2 variable. For instance, if friends1 increases by 10%, the average respondent with a clustering coefficient of 0.628 will see an increase in friends2 of 0.1 + 0.1 ([bar]K - 1)(1 - 0.628) = 0.1 + 0.1 (2.22)(0.372) = 0.18. Friends2 will increase by 0.18 for a 10% increase in friends1. This translates into an income increase of (0.18)(0.032) = 0.005 or 0.5%. This is $150 for someone earning $30,000. Similarly, if clustering were to increase by 0.10 for the average respondent (from 0.628 to 0.728), this would decrease friends2 to [bar]K + [bar]K([bar]K - 1)(1 - 0.728) = 3.22 + (3.22)(2.22)(0.272) = 5.12 from 5.79. Or friends2 decreases by 0.67. This would be about a 2% ((0.67)(0.032)) decrease in income ($600 for someone with income of $30,000). As one can see from these estimates, the effect of network structure on income is of an economically meaningful magnitude, especially if one considers that these values accrue on a yearly basis.

One potential concern with the results presented above is an issue of the endogeneity of the social-network variables (specifically the friends2 variable) and income. Others have found that individuals with higher socioeconomic status tend to have larger social networks (Homans 1950; Campbell, Marsden, and Hurlbert 1986). Moreover, other research has found that network size may be positively correlated with education and family income (Smith 2000). This suggests a concern that friends2 and income may have a simultaneous relationship, In other words, the friends2 variable may be correlated with the error term, in my estimation.

One method to test for endogeneity is the Hausman specification test. To perform the test, one estimates a less efficient but consistent model and the fully efficient model. (15) Call these models A and B. One then compares the resulting covariance matrices,

H = ([[??].sub.A] - [[??].sub.B])' [(var[A] - var[B]).sup.-1] ([[??].sub.A] - [[??].sub.B]),

which follows a chi-squared distribution. The null hypothesis is that there is no systematic difference between the models; in other words, there is no endogeneity. Unfortunately, the Hausman test is not applicable to an ordered logit model because the variance estimates for second-stage maximum likelihood are not consistent. Thus, one cannot formally carry out the Hausman test in the ordered logit framework.

However, it is straightforward to implement the test in the log-linear model. As noted above, there is a close link between family income and network size, which suggests that family income may be a useful proxy for my friends2 variable. In order to create the less efficient but consistent model needed for comparison, I instrument for friends2 by using the respondent's family income when they were 16. I then use the predicted values of friends2 from this estimation in place of friends2 in the original model and compare the results of the two models using the statistic above. For my data and the loglinear model, I find [chi square](8) = 5.02. Therefore, I fail to reject the null with a high degree of confidence. The log-linear model does not exhibit endogeneity as determined by the Hausman test. Again, it would be best if I also could perform this test for the ordered logit model. But because this is not possible, I remind the reader that the econometric results of the log-linear model are statistically and qualitatively similar to those of the ordered logit model and the two models use the same data. Thus, given endogeneity is not a problem in the log-linear model, it is likely not a problem in the ordered logit model either.

Now, recall the earlier discussion about measuring the effect of weak ties through method of finding a job. Using the examples above as a baseline, assume that 20% of the true difference in types is measurable if estimating the effect of weak ties by job-finding method. Then the measured effect would only be a little under $200 (20% of $960 from the first marginal effect discussed in the proceeding paragraph). Given the inherent noise in most income data sets due to compensating differentials, preferences for status attainment, as opposed to income, etc., finding a statistically significant difference of $200 would be unlikely except in large data sets. Thus, the estimates here lead one to believe that previous attempts to measure the effect of weak ties by job-finding method were unlikely to find statistically significant differences. Given the size of the effects estimated in this article, the previous studies were doomed to fail.

Recall that the data on the friendship networks in this study are brief and noncomplete and that the binning of income makes the dependent variable nonexact. With this in mind, I am encouraged that the estimate for number of friends appears to affect the income of individuals by a relatively large and statistically significant amount even in this nonideal data set. Given more exact income data and more complete social-network data, better estimates are possible.

Also, as Granovetter's original hypothesis suggested, having weak ties helps one to find jobs because they provide novel information. If this is the only effect of weak ties, then the analysis of friends2 above may capture Granovetter's notion within the friends2 variable because friends2 is increasing in the number of weak ties. In order to account for this, I also include the individual clustering coefficient, [C.sub.i], as a measure of the number of weak ties and the number of friends of each agent, friends1, in the estimation. If the only effect of the weak ties hypothesis is novel information, then that should be directly picked up through [C.sub.i]. However, I find [C.sub.i] and friends1 to be statistically insignificant at the most common confidence levels in both models. Now, recall from Equation 4 that both clustering and the number of friends within distance one enter the measurement of friends2. That friends2 is statistically significant and of a substantial size indicates that the interaction of weak ties (clustering) and the size of an individual's network (friends1) jointly determine the importance of social networks in the labor market. One needs a large base of strong ties, which weak ties then expand. (16)

As a final note on the empirical results of the article, I mention differences between the social networks of the black and white respondents of the survey. Consistent with previous work on the GSS (Patterson 1998), white respondents in my subsample of workers have significantly broader social networks than black respondents. The average value of friends2 for white respondents is 5.92, compared with 4.69 for black respondents. Given that wages are positively correlated with friends2, the results of this article support the belief that observed levels of black-white income differentials may be due in part to differences in their social networks.(17) Of course, the results of this article are far from conclusive in regard to the effects of social networks on differences in group level income inequality but, instead, suggest that further investigation in future data sets is warranted.

5. Conclusion

The findings of this article examine the microfoundations of the weak ties hypothesis in labor markets and suggest that, along with weak ties, an individual also needs a sufficiently large network of strong and weak ties, which one can then expand through additional weak ties. This finding is in agreement with Granovetter's original claim that weak ties are an important information source. But, the importance of having novel information only appears meaningful if one has a broad enough network such that many information sources are reached. The primary importance of having weak ties in one's social network appears to be in allowing one's social network to expand more quickly.

This study suggests that more completely considering the structure of social networks is important in understanding the full impact of Granovetter's strength of weak ties hypothesis. Given the intuitive appeal of the hypothesis, it is somewhat surprising that larger effects have not been found previously. But given the empirical results and examples in this article, one begins to understand why. Previous empirical studies have used methods that underestimate, perhaps greatly, the effects of weak ties on income. Yet few data sets exist that allow better analysis.

To fully understand the impact of weak ties on labor-market outcomes, substantially better and more complete data are needed. Even the data used here are far from ideal. The binning procedure makes the income data nonexact and the information on social networks is limited. That the estimate of the number of friends within distance two is both of an economically meaningful size and statistically significant makes a strong case that, with better data, the effect of the weak ties hypothesis on income may be measurable and important. It is my hope that the optimistic results in this article will lead to further efforts to collect data that allow for better measurement of social networks, combined with more exact income data.

More generally, the primary importance of understanding the effects of social networks on income concern issues of income inequality. Many believe that social networks in the labor market are a primary cause of the persistence of income inequality for some minority groups. Many theoretical models, such as those mentioned above, continue to be produced that document plausible effects of social networks on income. Some even extend to the education choices of individuals (Mailath, Samuelson, and Shaked 2000; Calvo-Armengol and Jackson 2002). If social networks and education are both important in acquiring jobs, and a given individual does not have the required social networks to enter a chosen field, she may forgo investing time and money in education. Because her employment prospects diminish, she becomes a less valuable connection for the purpose of acquiring jobs. In turn, this may affect the education choices and employment prospects of her friends, and so on. From this, one can understand that the reliance on social networks in searching for employment may generate externalities in both employment and education choices. Thus, implications for the effect of social networks on income spill over into many aspects of social inequality more broadly. I hope the results of this study encourage the further collection of data that will help to make these potentially important economic implications of social networks more clear.

I thank Mary Beth Combs, Forrest Nelson, George Neumann, Scott E. Page, and two anonymous referees for many helpful suggestions and comments.

Received June 2004; accepted May 2005

References

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Troy Tassier, Department of Economics, E528 Dealy Hall, Fordham University, Bronx, NY 10458, USA; E-mail [e-mail].

(1) See the following studies for details: Langlois (1977), Lin, Ensel, and Vaughn (1981), Bridges and Villemez (1986), Marsden and Hurlbert (1988), Smith (2000), Mouw (2002), and Mouw (2003). There is a discussion of some of these in Granovetter (1983). According to these studies, between 13% and 76% of jobs were found by using information obtained through a weak tie (Granovetter 1983).

(2) For a general overview of some of the recent research on networks, see Barabasi (2002) or Newman (2003b).

(3) Of course, some of one's weak ties may be well acquainted with one's other friends and some may have a best friend (a strong tie) who happens to live in another country and, thus, does not know any of one's other friends.

(4) Most friends fall somewhere between core and random friends and, thus, know a percentage of one's other friends. I will account for this in the formal modeling contained in section 4.

(5) Note that this analysis ignores some subtleties, which will be considered later in the actual estimation. For instance, when counting the friends of the friends of individual i. one needs to note that i is one of the friends.

(6) This fact was originally brought to my attention by Scott E. Page. Donald Light apparently mentioned a similar point to Granovetter, as noted in footnote 18 of his original paper.

(7) I thank a reviewer for pointing out this fact.

(8) I discuss biases in accepting job offers found through friends versus formal offers below. If workers are biased toward accepting offers found through friends, it is even more difficult to estimate the effect of weak ties using job-finding method.

(9) See Granovetter (1995) or Granovetter (1983) for a discussion.

(10) The distribution of number of connections is important as well. but the data discussed below do not allow for its inclusion. See Newman (2003a) for a discussion of this topic.

(11) Clustering is also referred to as density (Wasserman and Faust 1994).

(12) Granovetter (1995) found that, of individuals using referrals, 38% knew the contact directly, 45% knew the contact through one friend, and 13% knew the contact through the friend of a friend. Thus, 96% of the referrals came through referrals of distance three or less and 83% came through a referral of distance two or less. Thus, only including referrals of distance two seems reasonable given our knowledge of referral generation.

(13) See Greene (1993) for a more complete discussion of ordered logit models and Crawford, Pollak, and Vella (1998) for a discussion on the interpretation of coefficients in ordered logit models.

(14) Because the coefficient is positive, there is a bin m such that [partial derivative][P.sub.j]/[partial derivative][friend2].sub.i] > 0 for all j [greater than or equal to] m and [partial derivative][P.sub.j]/[partial derivative][friend2.sub.i] < 0 for all j < m, where [P.sub.j] is the probability of being in the jth income bin. See Crawford, Pollak, and Vella (1998) for more details on interpretation of ordered logit coefficients.

(15) See Greene (1993) for details.

(16) If friends2 is dropped from the estimation, then clustering has the expected negative relationship with income. But clustering still remains statistically insignificant from zero in both the ordered logit model and the log-linear model. The friends1 variable has the expected positive relationship with income and is statistically significant from zero at the 90% confidence level in the ordered logit model and is statistically insignificant at any reasonable level in the log-linear model.

(17) See Granovetter (1995) for a review of earlier work on the effect of referral hiring and differences in wage. More recent work discussing this topic includes Mouw (1999), Arrow and Borzekowski (2000), Calvo-Armengol and Jackson (2002), and Calvo-Armengol (2004).

Table 1. Network Size by Distance for a
Fixed Set of 30 Friends

            C = 20,   C = 10,
            R = 10    R = 20

[F.sup.1]        30        30
[F.sup.2]       530       830
[F.sup.3]    11,530    22,830

Table 2. Summary Statistics

Variable                   Average             Standard
                             Value             Deviation

Full time                    0.725               0.45
Experience                   21.9                14.5
Education                    13.1                2.97
Race = White                 0.873               0.33
Race = black                 0.093               0.29
Female                       0.465               0.499
Clustering (C)               .628                0.41
Friends1 ([k.sup.i])         3.22                1.67
Friends2                     5.79                4.11

Table 3. Estimation Results

Independent             Ordered Logit         Log-linear
Variable                Coef (Std Err)      Coef (Std Err)

Constant                      --           6.927 ** (0.212)
Full time              1.989 ** (0.150)    0.889 ** (0.063)
Experience             0.137 ** (0.015)    0.059 ** (0.006)
Experience             -0.002 ** (0.000)   -.0001 ** (0.000)
Education              0.272 ** (0.025)    0.102 ** (0.010)
Race = white            -0.314 (0.334)      -0.075 (0.145)
Race = black             -0.317 (0.378      -0.097 (0.167)
Female                 -1.563 ** (0.126)   -0.600 ** (0.054)
Clustering (C.sub.i])    0.348 (0.265)       0.117 (0.125)
Friends1 ([k.sub.i])    -0.093 (0.081)      -0.050 (0.038)
Friends2                0.077 * (0.031)     0.032 * (0.014)

The pseudo-[R.sub.2] for the ordered logit is 0.143 and
the [R.sup.2] for the log-linear model is 0.437.

* Significant at the 0.025 level.

** Significant at the 0.01 level.