Chapter 9:  Ego networks (2024)

9. Ego networks

• Introduction
• Ego network data
• Ego network density
• Structural holes
• Brokerage
• Summary

In the previous chapter we looked at the idea of the amount of"embedding" in whole networks -- loosely: the extent to which actorsfind themselves in social structures characterized by dense, reciprocal, transitive,strong ties. The main theme was to understand and index the extent andnature of the pattern of "constraint" on actors that results from theway that they are connected to others. These approaches may tell us someinteresting things about the entire population and its sub-populations; but,they don't tell us very much about the opportunities and constraints facingindividuals.

If we want to understand variation in the behavior of individuals, we need totake a closer look at their local circ*mstances. Describing and indexingthe variation across individuals in the way they are embedded in"local" social structures is the goal of the analysis of ego networks.

We need some definitions.

"Ego" is an individual "focal" node. A network hasas many egos as it has nodes. Egos can be persons, groups, organizations,or whole societies.

"Neighborhood" is the collection of ego and all nodes to whom egohas a connection at some path length. In social network analysis, the"neighborhood" is almost always one-step; that is, it includes onlyego and actors that are directly adjacent. The neighborhood alsoincludes all of the ties among all of the actors to whom ego has a directconnection. The boundaries of ego networks are defined in terms ofneighborhoods.

"N-step neighborhood" expands the definition of the size of ego'sneighborhood by including all nodes to whom ego has a connection at a pathlength of N, and all the connections among all of these actors.Neighborhoods of greater path length than 1 (i.e. egos adjacent nodes) arerarely used in social network analysis. When we use the term neighborhoodhere, we mean the one-step neighborhood.

"In" and "Out" and other kinds of neighborhoods.Most of the analysis of ego networks uses simple graphs (i.e. graphs that aresymmetric, and show only connection/not, not direction). If we are workingwith a directed graph, it is possible to define different kinds ofego-neighborhoods. An "out" neighborhood would include all theactors to whom ties are directed from ego. An "in" neighborhoodwould include all the actors who sent ties directly to ego. We might wantto define a neighborhood of only those actors to whom ego had reciprocatedties. There isn't a single "right" way to define an egoneighborhood for every research question.

"Strong and weak tie neighborhoods." Most analysis of egonetworks uses binary data -- two actors are connected or they aren't, and thisdefines the ego neighborhood. But if we have measured the strength of therelation between two actors, and even its valence (positive or negative), weneed to make choices about when we are going to decide that another actor isego's neighbor. With ties that are measured as strengths or probabilities,a reasonable approach is to define a cut-off value (or, better, explore severalreasonable alternatives). Where the information about ties includesinformation about positive/negative, the most common approach is to analyze thepositive tie neighborhood and the negative tie neighborhood separately.

Ego network data commonly arise in two ways:

Surveys may be used to collect information on ego networks. We can askeach research subject to identify all of the actors to whom they have aconnection, and to report to us (as an informant) what the ties are among theseother actors. Alternatively, we could use a two-stage snowball method;first ask ego to identify others to whom ego has a tie, then ask each of thoseidentified about their ties to each of the others identified.

Data collected in this way cannot directly inform us about the overallembeddedness of the networks in a population, but it can give us information onthe prevalence of various kinds of ego networks in even very largepopulations. When data are collected this way, we essentially have a datastructure that is composed of a collection of networks. As the actors ineach network are likely to be different people, the networks need to be treatedas separate actor-by-actor matrices stored as different data sets (i.e. it isn'ta good idea to "stack" the multiple networks in the same data file,because the multiple matrices do not represent multiple relations among the sameset of actors).

A modification of the survey method can give rise to a multi-plex datastructure (that is, a "stack" of actor-by-actor matrices of equaldimension). If we ask each ego to characterize their relation with theoccupants of social roles (or a particular occupant of a role), and to alsoreport on the relations among occupants of those roles, we can build"conformable" matrices for each ego. For example, suppose thatwe asked a number of egos: "do you have a male friend or friends inyour classroom?" "Do you have a female friend or friends in yourclassroom?" and "Are your male friends, friends of your femalefriends?" The resulting data for each ego would have three nodes(ego, "male friends," "female friends") and the ties amongthem. Since each ego's matrix would have the same nodes (in the sense ofsocial roles, but not individuals) they could be treated as a type of multi-plexdata that we will discuss more later on.

The second major way in which ego network data arise is by"extracting" them from regular complete network data. The Data>Extractapproach can be used to select a single actor and their ties, but would notinclude the ties among the "alters." The Data>Subgraphsfrom partitions approach could be used if we had previouslyidentified the members of a particular ego neighborhood, and stored this as anattribute vector.

More commonly, though, we would want to extract multiple, or even all of theego networks from a full network to be stored as separate files. For thistask, the Data>Egonet tool isideal. Here is an example of the dialog for using the tool:

Figure 9.1. Dialog for Data>Egonet

Here we are focusing on ballot proposition campaigns in California that areconnected by having donors in common (i.e. CA_Props is aproposition-by-proposition valued matrix). We've said that we want toextract a network that includes the 3rd, 11th, 17th, and 19th rows/columns, andall the nodes that are connected to any of these actors. More commonly, wemight select a single "ego." The list of focal nodes can beprovided either as an attribute file, by typing in the list of row numbers, orby selecting the node labels of the desired actors.

A picture of part of the resulting data, stored as a new file called "Neighbor_example"is shown in figure 9.2.

Figure 9.2. (Partial) output of Data>Egonet

Extracting sub-graphs, based on a focal actor or set of actors (e.g."elites") can be a very useful way of looking at a part of a wholenetwork, or the condition of an individual actor. The Data>Egonettool is helpful for creating data sets that are good for graphing and separateanalysis -- particularly when the networks in which the focal actor/actors areembedded are quite large.

It is not necessary, however, to create separate ego-network datasets foreach actor to be analyzed. The approaches to analysis that we'll reviewbelow generate output for the first-order ego network of every node in adataset. For small datasets, there is often no need to extract separateego networks.

There are quite a few characteristics of the ego-neighborhoods of actors thatmay be of interest. The Network>Ego networks>Density tools in UCINETcalculate a substantial number of indexes that describe aspects of theneighborhood of each ego in a data set. Here is an example of the dialog,applied to the Knoke information exchange data (these are binary, directedconnections).

Figure 9.3. Dialog for Network>Ego networks>Density

In this example, we've decided to examine "out neighborhoods" (inneighborhoods or undirected neighborhoods can also be selected). We've elected not tosave the output as a dataset (if you wanted to do further analysis, or treat egonetwork descriptive statistics as node attributes, you might want to save theresults as a file for use in other routines or Netdraw). Here are theresults:

Figure 9.4 Ego network density output for Knoke informationout-neighborhoods

There's a lot of information here, and we should make a few comments.

Note that there is a line of data for each of the 10 organizations in thedata set. Each line describes the one-step ego neighborhood of aparticular actor. Of course, many of the actors are members of many of theneighborhoods -- so each actor may be involved in many lines of data.

Size of ego network is the number of nodes that one-step out neighborsof ego, plus ego itself. Actor 5 has the largest ego network, actors 6, 7,and 9 have the smallest networks.

Number of directed ties is the number of connections among all thenodes in the ego network. Among the four actors in ego 1's network, thereare 11 ties.

Number of ordered pairs is the number of possible directed ties ineach ego network. In node 1's network there are four actors, so there are4*3 possible directed ties.

Density is, as the output says, the number of ties divided by thenumber of pairs. That is, what percentage of all possible ties in each egonetwork are actually present? Note that actor 7 and 9 live inneighborhoods where all actors send information to all other actors; they areembedded in very dense local structures. The welfare rights organization(node 6) lives in a small world where the members are not tightlyconnected. This kind of difference in the constraints and opportunitiesfacing actors in their local neighborhoods may be very consequential.

Average geodesic distance is the mean of the shortest path lengthsamong all connected pairs in the ego network. Where everyone is directlyconnected to everyone (e.g. node 7 and 9) this distance is one. In ourexample, the largest average path length for connected neighbors is for actor 5(average distances among members of the neighborhood is 1.57).

Diameter of an ego network is the length of the longest path betweenconnected actors (just as it is for any network). The idea of a networkdiameter, is to index the span or extensiveness of the network -- how far apartare the two furthest actors. In the current example, they are not very farapart in the ego networks of most actors.

In addition to these fairly basic and reasonably straight-forward measures,the output provides some more exotic measures that get at some quite interestingideas about ego neighborhoods that have been developed by a number of socialnetwork researchers.

Number of weak components. A weak component is the largestnumber of actors who are connected, disregarding the direction of the ties (astrong component pays attention to the direction of the ties for directeddata). If ego was connected to A and B (who are connected to one another),and ego is connected to C and D (who are connected to one another), but A and Bare not connected in any way to C and D (except by way of everyone beingconnected to ego) then there would be two "weak components" in ego'sneighborhood. In our example, there are no such cases -- each ego isembedded in a single component neighborhood. That is, there are no caseswhere ego is the only connection between otherwise dis-joint sets of actors.

Number of weak components divided by size. The likelihood thatthere would be more than one weak components in ego's neighborhood would be afunction of neighborhood size if connections were random. So, to get asense of whether ego's role in connecting components is "unexpected"given the size of their network, it is useful to normalize the count ofcomponents by size. In our example, since there are no cases of multiplecomponents, this is a pretty meaningless exercise.

Two-step reach goes beyond ego's one-step neighborhood to report thepercentage of all actors in the whole network that are within two directed stepsof ego. In our example, only node 7 cannot get a message to all otheractors within "friend-of-a-friend" distance.

Reach efficiency (two-step reach divided by size) norms the two-stepreach by dividing it by size. The idea here is: how much (non-redundant)secondary contact to I get for each unit of primary contact? If reachefficiency is high, then I am getting a lot of "bang for my buck" inreaching a wider network for each unit of effort invested in maintaining aprimary contact. If my neighbors, on the average, have few contactsthat I don't have, I have low efficiency.

Brokerage (number of pairs not directly connected). The idea ofbrokerage (more on this, below) is that ego is the "go-between" forpairs of other actors. In an ego network, ego is connected to every otheractor (by definition). If these others are not connected directly to oneanother, ego may be a "broker" ego falls on a the paths between theothers. One item of interest is simply how much potential for brokeragethere is for each actor (how many times pairs of neighbors in ego's network arenot directly connected). In our example, actor number 5, who is connectedto almost everyone, is in a position to broker many connections.

Normalized brokerage (brokerage divided by number of pairs) assessesthe extent to which ego's role is that of broker. One can be in abrokering position a number of times, but this is a small percentage of thetotal possible connections in a network (e.g. the network is large). Giventhe large size of actor 5's network, the relative frequency with which actor 5plays the broker role is not so exceptional.

Betweenness is an aspect of the larger concept of"centrality." A later chapter provides a more in-depth treatmentof the concept and its application to whole networks. For the moment,though, it' pretty easy to get the basic idea. Ego is "between"two other actors if ego lies on the shortest directed path from one to theother. The ego betweenness measure indexes the percentage of all geodesicpaths from neighbor to neighbor that pass through ego.

Normalized Betweenness compares the actual betweenness of ego to themaximum possible betweenness in neighborhood of the size and connectivity ofego's. The "maximum" value for betweenness would be achievedwhere ego is the center of a "star" network; that is, no neighborscommunicate directly with one another, and all directed communications betweenpairs of neighbors go through ego.

In several important works, Ronald Burt coined and popularized the term"structural holes" to refer to some very important aspects ofpositional advantage/disadvantage of individuals that result from how they areembedded in neighborhoods. Burt's formalization of these ideas, and hisdevelopment of a number of measures (including the computer program Structure, thatprovides these measures and other tools) has facilitated a great deal of furtherthinking about how and why the ways that an actor is connected affect theirconstraints and opportunities, and hence their behavior.

The basic idea is simple, as good ideas often are.

Imagine a network of three actors (A, B, and C), in which each is connectedto each of the others as in figure 9.5.

Figure 9.5. Three actor network with no structural holes

Let's focus on actor A (of course, in this case, the situations of B and Care identical in this particular network). Suppose that actor A wanted toinfluence or exchange with another actor. Assume that both B and C mayhave some interest in interacting or exchanging, as well. Actor A will notbe in a strong bargaining position in this network, because both of A'spotential exchange partners (B and C) have alternatives to treating withA; they could isolate A, and exchange with one another.

Now imagine that we open a "structural hole" between actors B andC, as in figure 9.6. That is, a relation or tie is "absent" suchthat B and C cannot exchange (perhaps they are not aware of one another, orthere are very high transaction costs involved in forming a tie).

Figure 9.6. Three actor network with a structural hole

In this situation, actor A has an advantaged position as a direct result ofthe "structural hole" between actors B and C. Actor A has twoalternative exchange partners; actors B and C have only one choice, if theychoose to (or must) enter into an exchange.

Real networks, of course, usually have more actors. But, as networksgrow in size, they tend to become less dense (how many relations can each actorsupport?). As density decreases, more "structural holes" arelikely to open in the "social fabric." These holes, and how andwhere they are distributed can be a source of inequality (in both the strictmathematical sense and the sociological sense) among actors embedded innetworks.

Network>Ego Networks>Structural Holesexamines the position of each actor in their neighborhood for the presence ofstructural holes. A number of measures (most proposed by Burt) thatdescribe various aspects of the advantage or disadvantage of the actor are alsocomputed. Figure 9.7 shows a typical dialog box; we're looking at theKnoke information network again.

Figure 9.7. Network>Ego Networks>Structural Holes dialog

Measures related to structural holes can be computed on both valued andbinary data. The normal practice in sociological research has been to usebinary (a relation is present or not). Interpretation of the measuresbecomes quite difficult with valued data (at least I find it difficult).As an alternative to losing the information that valued data may provide, theinput data could be dichotomized (Transform>Dichotomize)at various levels of strength. The structural holes measures may becomputed for either directed or undirected data -- and the interpretation, ofcourse, depends on which is used. Here, we've used the directed binarydata. Three output arrays are produced, and can be saved as separate files(or not, as the output reports all three).

The results are shown in figure 9.8, and need a bit of explanation.

Figure 9.8. Structural holes results for the Knoke information exchangenetwork

Dyadic redundancy means that ego's tie to alter is"redundant." If A is tied to both B and C, and B is tied to C(as in figure 9.5) A's tie to B is redundant, because A can influence B by wayof C. The dyadic redundancy measure calculates, for each actor in ego'sneighborhood, how many of the other actors in the neighborhood are also tied tothe other. The larger the proportion of others in the neighborhood who aretied to a given "alter," the more "redundant" is ego'sdirect tie. In the example, we see that actor 1's (COUN) tie to actor 2 (COMM)is largely redundant, as 72% of ego's other neighbors also have ties withCOMM. Actors that display high dyadic redundancy are actors who areembedded in local neighborhoods where there are few structural holes.

Dyadic constraint is an measure that indexes the extent to which therelationship between ego and each of the alters in ego's neighborhood"constrains" ego. A full description is given in Burt's 1992monograph, and the construction of the measure is somewhat complex. At thecore though, A is constrained by its relationship with B to the extent that Adoes not have many alternatives (has few other ties except that to B), and A'sother alternatives are also tied to B. If A has few alternatives toexchanging with B, and if those alternative exchange partners are also tied toB, then B is likely to constrain A's behavior. In our example constraintmeasures are not very large, as most actors have several ties. COMM andMAYR are, however, exerting constraint over a number of others, and are not veryconstrained by them. This situation arises because COMM and MAYR haveconsiderable numbers of ties, and many of the actors to whom they are tied donot have many independent sources of information.

Effective size of the network (EffSize) is the number of alters thatego has, minus the average number of ties that each alter has to otheralters. Suppose that A has ties to three other actors. Suppose thatnone of these three has ties to any of the others. The effective size ofego's network is three. Alternatively, suppose that A has ties to threeothers, and that all of the others are tied to one another. A's networksize is three, but the ties are "redundant" because A can reach allthree neighbors by reaching any one of them. The average degree of theothers in this case is 2 (each alter is tied to two other alters). So, theeffective size of the network is its actual size (3), reduced by its redundancy (2), to yield an efficient size of 1.

Efficiency (Efficie) norms the effective size of ego's network by its actual size. That is, what proportion of ego's ties to its neighborhoodare "non-redundant." The effective size of ego's network maytell us something about ego's total impact; efficiency tells us how much impactego is getting for each unit invested in using ties. An actor can beeffective without being efficient; and and actor can be efficient without beingeffective.

Constraint (Constra) is a summary measure that taps the extent towhich ego's connections are to others who are connected to one another. Ifego's potential trading partners all have one another as potential tradingpartners, ego is highly constrained. If ego's partners do not have otheralternatives in the neighborhood, they cannot constrain ego's behavior.The logic is pretty simple, but the measure itself is not. It would begood to take a look at Burt's 1992 Structural Holes. The idea ofconstraint is an important one because it points out that actors who have manyties to others may actually lose freedom of action rather than gain it --depending on the relationships among the other actors.

Burt's approach to understanding how the way that an actor is embedded in itsneighborhood is very useful in understanding power, influence, and dependencyeffects. We'll examine some similar ideas in the chapter oncentrality. Burt's underlying approach is that of the rational individualactor who may be attempting to maximize profit or advantage by modifying the wayin which they are embedded. The perspective is decidedly"neo-classical."

Fernandez and Gould also examined the ways in which actor's embedding mightconstrain their behavior. These authors though, took a quite different approach; they focus on the roles that ego plays in connectinggroups. That is, Fernandez and Gould's "brokerage" notionsexamine ego's relations with its neighborhood from the perspective of ego actingas an agent in relations among groups (though, as a practical matter, the groupsin brokerage analysis can be individuals).

To examine the brokerage roles played by a given actor, we find everyinstance where that actor lies on the directed path between two others.So, each actor may have many opportunities to act as a "broker."For each one of the instances where ego is a "broker," we examinewhich kinds of actors are involved. That is, what are the groupmemberships of each of the three actors? There are five possiblecombinations.

In figure 9.9, the ego who is "brokering" (node B), and both thesource and destination nodes (A and C) are all members of the same group.In this case, B is acting as a "coordinator" of actors within the samegroup as itself.

Figure 9.9. Ego B as "coordinator"

In figure 9.10, ego B is brokering a relation between two members of the samegroup, but is not itself a member of that group. This is called a"consulting" brokerage role.

Figure 9.10. Ego B as "consultant"

In figure 9.11, ego B is acting as a gatekeeper. B is a member of agroup who is at its boundary, and controls access of outsiders (A) to thegroup.

Figure 9.11. Ego B as "gatekeeper"

In figure 9.12, ego B is in the same group as A, and acts as the contactpoint or representative of the red group to the blue.

Figure 9.12. Ego B as "representative"

Lastly, in figure 9.13, ego B is brokering a relation between two groups, andis not part of either. This relation is called acting as a"liaison."

Figure 9.13. Ego B as "liaison"

To examine brokerage, you need to create an attribute file that identifieswhich actor is part of which group. You can select one of the attributesfrom a user-created attribute file, or use output files from other UCINETroutines that store descriptors of nodes as attributes. As an example,we've taken the Knoke information exchange network, and classified each of theorganizations as either a general government organization (coded 1), a privatenon-welfare organization (coded 2), or an organizational specialist (coded3). Figure 9.14 shows the attribute (or partition) as we created it usingthe UCINET spreadsheet editor.

Figure 9.14. Partition vector for Knoke information exchange

Using the network data set and the attribute vector we just created, we canrun Network>Ego Networks>Brokerage, asshown in figure 9.15.

The option "unweighted" needs a little explanation. Supposethat actor B was brokering a relation between actors A and C, and was acting asa "liaison." In the unweighted approach, this would count as onesuch relation for actor B. But, suppose that there was some other actor Dwho also was acting as a liaison between A and C. In the"weighted" approach, both B and D would get 1/2 of the credit for thisrole; in the unweighted approach, both B and D would get full credit.Generally, if we are interested in ego's relations, the unweighted approachwould be used. If we were more interested in group relations, a weightedapproach might be a better choice.

The output produced by Network>Ego Networks>Brokerage is quiteextensive. We'll break it up into a few parts and discuss themseparately. The first piece of the output (figure 9.16) is a census of thenumber of times that each actor serves in each of the five roles.

Figure 9.16. Unnormalized brokerage scores for Knoke information network

The actors have been grouped together into "partitions" forpresentation; actors 1, 3, and 5, for example, form the first type oforganization. Each row counts the raw number of times that each actorplays each of the five roles in the whole graph. Two actors (5 and 2) arethe main sources of inter-connection among the three organizationalpopulations. Organizations in the third population (6, 8, 9, 10), thewelfare specialists, have overall low rates of brokerage. Organizations inthe first population (1, 3, 5), the government organizations seem to be moreheavily involved in liaison than other roles. Organizations in the secondpopulation (2, 4, 7), non-governmental generalists play more diverseroles. Overall, there is very little coordination within each of thepopulations.

We might also be interested in how frequently each actor is involved inrelations among and within each of the groups. Figure 9.17 shows theseresults for the first two nodes.

Figure 9.17. Group-to-group brokerage map

We see that actor 1 (who is in group 1) plays no role in connections fromgroup 1 to itself or the other groups (i.e. the zero entries in the first row ofthe matrix). Actor 1 does, however, act as a "liaison" in makinga connection from group 2 to group 3. Actor 1 also acts as a"consultant" in connecting a member of group 3 to another member ofgroup 3. The very active actor 2 does not broker relations within group 2, butis heavily involved in ties in both directions of all three groups to oneanother, and relations among members of groups 1 and 3.

These two descriptive maps can be quite useful in characterizing the"role" that each ego is playing in the relations among groups by wayof their inclusion in its local neighborhood. These roles may help us tounderstand how each ego may have opportunities and constraints in access to theresources of the social capital of groups, as well as individuals. Theoverall maps also inform us about the degree and form of cohesion within andbetween the groups.

There may be some danger of "over interpreting" the informationabout individuals brokerage roles as representing meaningful acts of"agency." In any population in which there are connections,partitioning will produce brokerage -- even if the partitions are notmeaningful, or even completely random. Can we have any confidence that thepatterns we are seeing in real data are actually different from a random result?

In Figure 9.18, we see the number of relations of each type that would beexpected by pure random processes. We ask: what if actors wereassigned to groups as we specify, and each actor has the same number of ties toother actors that we actually observe; but, the ties are distributed at randomacross the available actors? What if the pattern of roles was generatedentirely by the number of groups of various sizes, rather than representingefforts by the actors to deliberately construct their neighborhoods to deal withthe constraints and opportunities of group relations?

Figure 9.18. Expected values under random assignment

If we examine the actual brokerage relative to this random expectation, wecan get a better sense of which parts of which actors roles are"significant." That is, occur much more frequently than we wouldexpect in a world characterized by groups, but random relations among them.

Figure 9.19. Normalized brokerage scores

The normalized brokerage scores in this example need to be treated with alittle caution. As with most "statistical" approaches, largersamples (more actors) produce more stable and meaningful results. Sinceour network does not contain large numbers of relations, and does not have highdensity, there are many cases where the expected number of relations is small,and finding no such relations empirically is not surprising. Both actor 2and actor 5, who do broker many relations, do not have profiles that differgreatly from what we would expect by chance. The lack of large deviationsfrom expected values suggests that we might want to have a good bit of cautionin interpreting our seemingly interesting descriptive data as being highly"significant."

In this chapter we've taken another look at the notion of embedding; thistime, our focus has been on the individual actor, rather than the network as awhole.

The fundamental idea here is that the ways in which individuals are attachedto macro-structures is often by way of their local connections. It is thelocal connections that most directly constrain actors, and provide them withaccess to opportunities. Examining the ego-networks of individuals canprovide insight into why one individual's perceptions, identity, and behaviordiffer from another's. Looking at the demography of ego networks in awhole population can tell us a good bit about its differentiation and cohesion- from a micro point of view.

In the next several chapters we will examine additional concepts andalgorithms that have been developed in social network analysis to describeimportant dimensions of the ways in which individuals and structuresinteract. We'll start with one of the most important, but also mosttroublesome, concepts: power.