Social Networks 11 (1989) 387-401

North-Holland 387

DEMARCATING THE BOUNDARIES BETWEEN SELF AND THE SOCIAL: THE ANATOMY OF CENTRALITY IN SOCIAL NETWORKS

Tony TAM *

University of Chicago

The conceptualization and quantification of centrality has been a major focus in modern network

analysis. Mizruchi et al. (1986) have recently pointed out that a conceptual and formal distinction

between reflected and derived centrality is germane to the understanding of network organization

in many substantive contexts, e.g. business power structure and the emergence of social move-

ment. The present paper offers a critique of Mizruchi et d’s (1986) solution to the problem and

provides a novel alternative that takes all possible reflection and derivation of centrality into

account. The problem of interpretation is extensively discussed.

1. The problem

When discussing the fundamentals of economic theory, Gary S. Becker made the following remarks:

it has been said that in economics everything depends upon every- thing else. Critics have even accused economists of circular reasoning when describing the operation of the interdependent pricing mecha- nism.

The French economist Walras analyzed this problem of interde- pendence and showed that there is no circular reasoning, just mutual determination or general equilibrium. Anyone who has studied high school algebra knows that each of the unknowns in a system of simultaneous equations can be determined provided a sufficient number of independent relations are available. Walras similarly demonstrated that all prices and quantities in the economic system can be simultaneously determined because there are a sufficient

* Department of Sociology. University of Chicago, Chicago, IL 60637, U.S.A

037%8733/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

388 T. Tam / Anmmy of centrul;ry

number of independent supply and demand equations. Walras’ dem- onstration was a major achievement with far-reaching implications, perhaps the greatest achievement of nineteenth-century economics - simple as it appears to a modern student. (1971: 4-5)

The coordination of seemingly atomistic choices by market prices is indeed the heart of economic theory. Without Walras’ conceptual breakthrough, it is hard to imagine that the power of economic theory could have unleashed.

In modern network analysis, a similar idea arises in the conceptuali- zation of centrality. Sociologists have long recognized the relevance of a conception of centrality that builds upon mutual determination (Katz 1953; Hubbell 1965; Bonacich 1972). The work of Walras has the same bearing on the sociological problem as it does on economic problems. Interdependence is not a circular idea. A further problem has to be addressed here, however. If the centrality of the units in a social network depends upon each other, can we attribute any part of the centrality of a unit to be self-originated? Unique in the sociological context is, therefore, a problem of demarcation - identifying the boundaries between self and the social. While this is largely irrelevant to an economist, it is often important for a sociologist to specify the sources of the centrality of a sociological unit.

Mizruchi, Mariolis, Schwartz and Mintz (1986) (I shall hereafter abbreviate the paper by these four authors as MMSM) have correctly pointed out that a conceptual and formal distinction between reflected (self-originated) and derived (alter-originated) centrality is germane to the understanding of network organization in many substantive con- texts, e.g. business power structure, social movements, power and leadership in small groups, community structure and elites, social organization of key actors in national policy domains, interorganiza- tional relations and communication networks. The main conceptual and methodological issue underlying the distinction is: if we conceive the centrality of the position of an ego as a function of those of the alters to whom the ego is related, can we meaningfully separate the component of centrality due to oneself from that due to alters? This is the focus of the present paper. I shall confine the following discussion to the theoretical aspects of the problem. In a sequel to this paper, I shall deal with practical aspects of the problem and apply the result of this paper to empirical problems.

T Tam / Anutomy of centrality 389

2. Background

2.1. Conceptual ideas

At the most elementary level, network analysis may involve two tasks: first, the quantification (or measurement) of network positions in terms of some meaningful dimensions (e.g. centrality); second, the classifica- tion (or aggregation) of positions into subgroups or clusters. In this paper, I shall restrict my attention to the development of the first task. Besides MMSM, the most distinctive contributions are Freeman (1979) Knoke and Burt (1983) and Bonacich (1987). Among these, MMSM is particularly interesting because it stresses more than any other papers the relevance of disaggregating an actor or a position’s centrality - what I would call the anatomy of centrality.

The foundations of MMSM are twofold. First, it adopts a concep- tion of centrality in which the centrality of an ego (viz. an actor, or his/her network position) is a function of (a) the strength of its relations to alters (other actors or positions in a specific network) and (b) the levels of centrality of alters. This is what I call endogenous centrality and the concept has a positive feedback structure. Endoge- nous centrality is distinct from the three families of centrality discussed by Freeman (1979). This endogenous conception of centrality has spurred many empirical applications in the study of power (e.g. Mars- den and Laumann 1977; Burt 1982; Knoke and Burt 1983; Mizruchi et al. 1986 and the substantive literature cited there).

The conception can be formally stated. Consider a network of n units, defined by an n X n relational matrix R. The centrality scores of the units in the network, formally expressed as an n X 1 vector C, have the following structure:

C = RC/A 0)

where X is the largest eigenvalue of R, introduced to ensure the existence of a feasible solution for the vector C under usual conditions (MMSM, pp. 28-29). This conception assumes that R denotes a digraph. Each relation must be directed from an ego i to an alter j and in a sense transmits the centrality of j to i. Since the scale of the vector C is indeterminate, many normalizations have been considered in the

390 T. Tam / Anatomy of centrality

literature (e.g. Bonacich 1972, 1987; MMSM). For the present discus- sion, it suffices to allow for the following general constraint:

Cc,=t

with the summation taken over i from 1 to n. Even though the centrality of an ego is a function of its relations to

alters and the centrality of alters, the formal definition also reveals that all the information is contained in the matrix R. It thus suggests that the endogenous centrality of an ego may be expressed as strictly a function of the pattern of relations embedded in R. This observation has fundamental importance and I shall use it in section 3.

Second, MMSM makes a conceptual distinction between the re- flected and the derived components of centrality. By definition, each ego’s centrality is a function of alters’ centrality. This leads to the question whether it is possible to separate, for a pair of units i and ,j, the component of i’s centrality that is “reflective” of i ‘s own centrality from the component of i’s centrality that is “derived” from j’s centrality. MMSM suggests that we focus on a two-step process:

Although part of the centrality that i acquires from j is based on j’s centrality, j’s centrality is also based on i’s centrality. That is, unit i sends some of its own centrality to unit j at step 1 and then receives some of it back at step 2. We call this component of centrality reflected centrality. The remainder of the centrality that unit i receives from unit j is purely a result of j’s centrality. We call this component derived centrality. (1986: 32)

In the literature, an actor or position connected to many peripheral actors is called a hub, whereas an actor or position connected to a few central actors is a bridge. Hence a hub should have high proportion of reflected centrality, whereas a bridge should have high proportion of derived centrality. In other words, we may conceive reflected centrality as hub centrality and derived centrality as bridge centrality. Thus, we have in effect extended the conventional discrete typology of hubs and bridges into two distinct continuous dimensions. Since MMSM has elaborated on the substantive meanings of these ideas, I shall not repeat them here.

T Tam /Anatomy of centrality 391

2.2. Formalization

Before assessing the adequacy of MMSM’s proposal of decomposing centrality, I shall introduce the main mathematical procedure that MMSM utilizes in operationalizing the distinction of the two compo- nents of centrality. The procedure is based on a simple fact. Given that C satisfies (1) then substituting (1) into the right-hand side of (1) itself gives

C= R(RC/A),‘h (3)

which implies

(X’)C= (R*)C. (4)

This substitution can be iterated indefinitely. The interpretation of this fact is that, if C is the eigenvector of R, then it is also the eigenvector of R2 and any other power of R. In network theoretic terms, centrality defined with the endogenous feedback structure of (1) can be seen as the result of a reflection and derivation process of one-step, two-step, or, to be sure, any number of steps.

MMSM capitalizes on this fact and defines the two-step reflected centrality and derived centrality of ego i from alter j as, respectively,

C;j.2r= R,j(RjiCz)/(X2) (5)

C ‘].2d= RijCj/X - ‘i/.2r (6)

Notice that, using (1) the first term of (6) can be decomposed into

C lJ.2 = CRij/‘)(CRjkck/‘)l (7) ’ k /

which defines the derived centrality of i from j in two steps. That is, various k’s centrality first inputs to j and then j transmits it to i.

Equation (6) can therefore be simplified as

C rJ.2d = C R jkCk/h k

= Rij c RjkCk i

/(A’). k#i

(8)

392 T. Tam / Anatom, of centrali<b

Comparing (5). (7) and (8) the logic of the decomposition is very clear. The centrality of i acquired from j is purtitioned according to how j

generates its own centrality in one step. The component of j’s centrality generated from i contributes to the reflected component of i’s centrality acquired from j. The rest of j’s centrality contributes to the derived component of i’s centrality acquired from j.

We are now in a position to assess the adequacy of MMSM’s decomposition. By the definition of C in (1) the centrality of each unit is a function of the centrality of other units in the network. There is a vital implication: as long as a unit i is reachable by another unit j in some finite number of steps, i will be reachable by j via infinitely many different numbers of steps. In my interpretation of (4), I have already hinted that the decomposition of reflected and derived centrality is not unique. Depending on the number of steps of the feedback process one looks at, one will obtain different results.

Thus, MMSM’s proposed decomposition and the conceptual defini- tions of endogenous centrality have a major inconsistency: the quota- tion from MMSM (p. 32) shows that the concepts are intended to capture pure components while the formal definitions only manage to tease out the two-step reflected component, leaving infinitely many other ways of reflection out of the picture. As a result, MMSM’s proposed decomposition will approximate the true decomposition only if the two-step process dominates all other processes. Fortunately, despite the existence of infinitely many routes of reflection, the net result is in general a convergent series. We can, therefore, assign a finite numerical value to each component of centrality.

Another inadequacy of MMSM’s procedure is apparent even in its definition of two-step centrality in (5). Notice that C, in the right-hand side of (5) is not a pure function of reflection. According to the definition of endogenous centrality in (l), C, is a function of i’s relations to alters and the centrality of alters. The form of the defini- tion in (5) thus inherently confounds reflection with derivation. A viable decomposition should not have the centrality of the ego or alters appearing explicitly in the expansion. In the next section, I shall derive a definition of endogeneous centrality that is equivalent to (1) but appears only as a function of the patterns of relations in the network.

T. Tam / Anatomy of centrality 393

3. An alternative decomposition

While MMSM’s proposed decomposition is conceptually appealing, the formal procedure they propose is logically unsatisfactory and could produce misleading results. My goal here is to present an alternative procedure that starts with the definition of endogeneous centrality in (1) and avoids the two pitfalls of MMSM. My method exploits two mathematical ideas: a closed-form expression of eigenvector via matrix inversion, and an infinite series expansion of an inverse matrix. ’

3. I. Reduced-form expression for endogenous centrality

The first idea is involved in redefining C with a closed-form or reduced-form equation. This is useful because it will allow us to view C as a function of exogenous quantities in the system, hence facilitate the interpretation of this endogenously defined concept. We, may proceed as follows.

To begin with, we need a reformulation of (l), the formal definition of centrality. Helpful to the reformulation is a new reference for the relational matrix that defines the network under study. We shall define the new reference as:

Z=R/X. (9)

Clearly the largest eigenvalue of Z is 1 and the corresponding eigenvec- tor is C. This transformation of R amounts to controlling for the effect of the scale of R on A. Notice that, for a given configuration of R,

changing the scale of R by any constant factor p will only change the eigenvalue A by a factor of p while the corresponding eigenvector C remains intact. That is, Z is invariant to the arbitrary choice of scale for R. Consequently, this transformation allows us to replace (1) with a formal definition that directly operationalizes the conceptual definition of endogenous centrality, i.e.

c=zc. 00)

’ To my best knowledge, Coleman (forthcoming) is the only place where the two ideas are

combined to address a sociological problem - the problem of restricted exchange. I have extended Coleman’s method of solving eigenvectors and applied a somewhat different decomposition and a

novel interpretation for endogenous centrality.

394 T. Tam / Anatomy of centraliry

As mentioned above, given an arbitrary non-negative R, a non-trivial solution for C = RC usually does not exist. That is, the equation is usually satisfied only when C is a vector of zeros. The use of Z is, therefore, not only meaningful as just stated but also eliminates the problem of non-existence of C.

Now consider the problem of expressing C in a reduced form. Suppose the system consists of n units, and we define

e = CZ,,/n’.

The constant e is the average strength of relation in the system. It is also the expected level of strength for each Z,, if there is no variation in the strength of relations in this system. Further, define E as an n x n matrix with entries e. It is easy to derive the following equation from (10):

(I-Z)C=O (12)

where 0 is a vector of 0’s. Next, add the term EC to both sides of (12). We have

(I-Z)C-tEC=EC (13)

which, by factorization and the assumption of invertability, gives

c= [(I- Z) + E]-‘EC

= (en)(t/n)[I- (Z- E)]-‘1

= s(t/n)Gl (14)

where s = en, a scale factor, and 1 is a vector of 1’s. Note that the second step of (14) is due to the constraint of (2), that is,

EC= et1 = en(t/n)l. (15)

Hence C can now be expressed in a reduced-form equation: the product of a scale factor, average centrality (i.e. t/n) and a vector of adjustment factors determined by the vector Gl. This method will provide identical solutions of C for any matrix E with constant

T Tam / Anatomy of centrality 395

elements. We select e to be the average relational strength here because, as we shall see shortly, it yields a uniquely interpretable infinite series decomposition of centrality.

3.2. Decomposing endogenous centrality

The second mathematical idea, applied in this paper to expand the inverse matrix in (14), is a well-known result in matrix algebra (e.g. Strang 1980: 204):

(I-A)-‘=z+A +A2+A3+ . . . (16)

where A is a square matrix with the same dimensionality as that of I. This infinite series can be applied to (14) and gives

c=s(t/n)[r+(z-E.)+(Z-E)2+...]1. (17)

Each element of the matrix G in [14] turned out to be an infinite series. Alternatively, we may look at an individual element of C:

c, = s(t/n,( 1+ C(z;;-e) i

+~~(Z,k-e)(Zki-e)+...}. .i k

(18)

The summations over j are due to the product of G and 1 whereas the other summations are due to the expansion of powers of the difference matrix (2 - E). Note that each factor of the terms in (18) comes from the difference matrix (Z - E) and indicates a deviation of Z from e, i.e. deviation from the average strength of relation. The heart of the decomposition of each C, therefore lies in the ith row of matrix G in [14], i.e. [I - (Z - E)]-‘. It should be clear that each element of G is an infinite series in itself. For j # i:

Gjj= (z;j-e) + ~(zlk-e)(zk,-e) k

+CC(Zlk-e)(Zk,-e)(Z,j-e)+.... k I

(19)

396 i? Tam / Anatomy of centrality

For j = i:

G,, = 1 + (z,, - e) + c(G - e)(-G, - e)

+xCC(Zi,-e)(Z,,-e)(Z,,-e)+ . . . . k I

(20)

The key feature to take note of is that each element G,, is the total input from j to i through all possible paths of any number of steps. G,, is, similarly, the total input from i to i itself through all possible alters and all

possible paths of any number of steps. From (18) (19) and (20), we can accordingly formulate the alternative decomposition promised earlier:

c, = 4Vn)CG,,

= s(t/n){l + (6 - 1) + c cl,) j#f = {scale factor}

* {average centrality}

* { 1 + [net adjustment due to total reflected centrality]

+ [net adjustment due to total derived centrality] } . (21)

The term net is used to indicate the fact that some adjustments are positive while some are negative. The term total refers to the considera- tion of all possible paths. This decomposition is clearly different from that of MMSM. It is, to be sure, more consistent with the conceptual definitions of reflected and derived centrality than MMSM’s disaggre- gation.

4. Interpretation

As mentioned earlier, different choices of the constant matrix E will not affect the solution for C. The important implication here is that the different choices of E mean differences in (a) the reference point to

T Tam / Anatomy of centrahiy 397

assess the deviation of Z and (b) the scale factor s. I have also asserted that choosing e to be the average strength of relation has unique interpretive advantage. What are the precise meanings of the terms in the infinite series expansion suggested above? Crucial to the interpreta- tion of individual terms of the series are three assumptions and two keys.

First, we presume that R or Z, which generates endogenous central- ity C, denotes directed relations from rows (egos) to columns (alters). No matter if the relations are empirically symmetric (viz. fully recipro- cated) or not, each relation also connotates a directed feedback from alter to ego. This is what equations (1) and (10) signify. I have, therefore, departed from Knoke and Burt’s (1983) stipulation that centrality be restricted to intrinsically symmetric relations.

Second, each alter j may have infinitely many effects on the central- ity of an ego i; the precise effect depends on the paths we consider. A pair of units, i and j, may be linked in many different steps and through different intermediaries. When looking at the direct relation of i to j, my proposed decomposition compares Z,j with the global average strength, e. The assumption is that, if the relation of i to j is stronger than average, i gains above-average centrality from j. When extended to all alters, if Cj( Z,, - e) is positive, then i’s centrality is augmented above the average centrality t/n by virtue of its direct relations to alters (including absence of ties to others in the system). This second assumption thus specifies how each unit derives its central- ity from direct relations.

When extended to indirect relations, the situation is complicated. While there are only (n - 1) direct relations for each ego in a system of n units, there are infinitely many indirect relations connecting an ego to alters. The effects of a given alter on the centrality of a given ego are manifold and are contingent on the paths under examination. Consider a two-step indirect relation in which ego i is linked to alter j via k. With respect to this relation, k may be conceived as having a contribu- tion to the centrality of i by mediating i’s indirect relation to j. The third assumption pertains to the optimal allocation of relational strength that takes the contribution of k into account. If the contribution of k is positive, then i’s centrality will be increased above average by an above-average strength of relation to k, and decreased below average by a below-average strength of relation to k. If the contribution of k is negative, however, i’s centrality will be augmented above average by a below-average strength of relation to k, depreciated below average by

398 T. Tam / Anatomy of centrality

an above-average strength of relation to k. This final assumption can be readily generalized to deal with any m-step indirect relation and it is critical to the interpretation of the decomposition in this paper. 2

Consider now the building blocks of (19) (and the same principle applies to (20)). Notice that each term of (19) is a summation of multiplicative terms. We need two more keys to understand the precise interpretation of these individual terms. The first key is to see that this summation of products is equivalent to a product of summations. In general, we have

(22)

This property will enable us to view (19) and (20) in a new light, as I will show shortly.

The second key is to start the interpretation with the last factor of each term and move forward. For instance, (Z,, - e) is the last factor for the third (i.e. three-step) term of G,,. The meaning of this factor is straightforward: it denotes the deviation of l’s strength of relation to j from the global average. This one-step value of I due to l’s one-step relation to j may be called, in short, the one-step value of 1 due to j. Then, if we couple the last two factors and apply result (22),

C(z,,-e)P,,-4 (23)

can be interpreted as the net two-step value of k due to j. This summation is the net two-step value of k generated through all possible two-step relations to j.

The nice thing about this perspective is that an mth term of (19) has a simple structure:

C(Z,k-4{(m-1>- step value of k due to j } , (24)

which is simply the m-step value of i due to j. Together with the third assumption above, this structure enables us to see each term of (19)

’ As repeated stressed above, the contribution of an alter depends much on the path in question. That is why it is so important to take every possibility into account in order to assess whether the

relation of i to j augments or depreciates the centrality of i.

T Tam / Anatomy of centralify 399

simply as the net centrality of i generated from i’s m-step value due to j. There are different routes to go from i to i for different numbers of steps; the net value of i generated will vary accordingly. When all these variations are added together, we obtain the net centrality of i due to j through all possible routes. Armed with this interpretation and the decomposition in (21), we see that G,i is the multiplicative adjustment factor of average centrality and it is an additive function of all types of values of i generated from j through all possible paths. If the m-step value of i due to j is positive, j augments i’s centrality above average when all possible m-step indirect relations of i to j are considered. If the net value of i due to j, i.e. G,j, is positive, j augments i’s centrality above average when all possible indirect relations of i to j are consid- ered.

The foregoing development has proceeded at a relatively high level of generality. The conditions on C (2) and R are very general indeed. Given the prominence of Coleman’s system of action approach to network analysis in general (1973, 1977, forthcoming), and the study of power in particular, let me briefly consider the special case of column stochastic R because this is a vital feature of Coleman’s models. In this case, all columns sum to 1, i.e.

CR,, = 1. (25)

With this condition, 1 is necessarily the largest eigenvalue of R (e.g. Strang 1980: 200-201). Besides, the average strength of relation is, by definition of e, l/n. Thus the scale factor s becomes n/n = 1 and one can substitute l/n for e in all the expansions.

5. Conclusion

To recapitulate, I have presented a new method to implement the theoretically appealing distinction between reflected and derived centrality. This distinction extends the traditional typology of hub and bridge centrality in network analysis. The proposed method is a correc- tion to an analogous procedure of disaggregation suggested by MMSM. My procedure draws upon two simple mathematical ideas to accom- plish the objective under a wide range of data conditions. The decom-

400 T. Tam / Anatomy of centrolit~

position has two levels: (1) the centrality of a unit is first decomposed into a multiplicative function of an overall scale factor, the average centrality in the system, and an adjustment factor when all possible feedbacks are considered; (2) the adjustment factor is decomposed into an additive function of the reflected and derived effects on the central- ity of the unit expected under a null model of random or unbiased network. The interpretation of the matrix G appears to be new. The use of Z = R/X facilitates a formal definition of endogenous centrality that is simpler than the one used by MMSM and, at the same time, avoids the problem of non-existence of the centrality vector (even when R is not column stochastic).

All in all, even if we espouse a world view in which everybody is dependent on everybody else, we may still meaningfully separate the reflected (self-originated) from the derived (alter-originated) compo- nent. In order words, the boundaries between an ego and its social context can be precisely demarcated even in a fully endogenous system. The fundamental premise of sociology does not imply that self is lost in the mist of social interdependence. I have demonstrated this in the specific context of centrality in social networks. Further implications for empirical work will be dealt with in a separate paper.

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