Exact Bounds for Degree Centralization

Transcription

1 Exact Bounds for Degree Carter T. Butts 5/1/04 Abstract Degree centralization is a simple and widely used index of degree distribution concentration in social networks. Conventionally, the centralization score is normalized by the maximum value attainable at the observed order; this results in a measure on the unit interval. Here, exact bounds are provided for degree centralization given density as well as order. It is shown that roughly half of the region of conceivable degree centralization scores is actually feasible, and that the geometry of the feasible region alters with graph size. Concentration of the distribution of graphs within the feasible region is shown, and a renormalized family of degree centralization indices which adjusts for both density and order is provided. Keywords: degree centralization, density, graph order, graph level indices 1 Introduction A prominent line of inquiry since the early decades of network analysis concerns the causes and consequences of centralization in network structure (e.g., Bavelas, 1950; Marwell et al., 1988; Buskens, 1998; Topper and Carley, 1999); i.e., the extent to which certain vertices are far more central than others within the network in question. Freeman s (1979) definition of centralization in terms of the family of indices C (G) ( max c ( v v ) ) c (v), (1) V v V This material is based upon work supported by the National Science Foundation under award NSF ITR # Department of Sociology and Institute for Mathematical Behavioral Sciences, University of California, Irvine; SSPA 2145, Irvine, CA 92697; buttsc@uci.edu 1

2 for arbitrary centrality index c and graph G (V, E), continues to serve as the standard formal notion of this concept. While Freeman s centralization family has been used profitably for a quarter-century, many questions still remain regarding the properties of these indices. Here, we focus on one of the simplest and most widely used centralization indices degree centralization deriving exact bounds for attainable values as a function of order (network size) and density. As we shall show, a substantial fraction of the space of conceivable degree centralization scores is not actually feasible, and (consistent with Anderson et al., 1999) the overwhelming majority of graphs is concentrated on a small section of this feasible region. Using the centralization bounds, we propose renormalized versions of the standard degree centralization measures which account for limits on possible centralization scores due to order and density. 2 Bounding Degree The raw or unnormalized degree centralization of graph G (V, E) under degree measure d is defined as C d (G) v V ( d (v)), (2) where max v V d i (Freeman, 1979). While degree centralization is conventionally normalized so as to be restricted to the interval from 0 (for a complete or null graph) to 1 (for the appropriate maximal star configuration 1 ), it does not follow that all values within this range are feasible. In addition to restrictions arising from the granularity of the degree distribution, graphs with a given density are often heavily constrained as to the centralization scores they can exhibit. Indeed, this is immediately apparent from the above: densities of 0 or 1 must be associated with centralization scores of 0 (under any measure), while maximum centralization is only possible for the appropriate star density ( 2 1 V or V, depending on the graph type and degree measure employed). Here, we follow this intuition further, obtaining exact bounds for degree centralization given order and density (in the spirit of Snijders, 1981). We begin with the undirected case, proceeding next to centralization of indegree, outdegree, and total degree for directed graphs. 1 The degree centralization is maximized by a maximal star, instar, outstar, or reciprocated (bidirectional) star, with respect to undirected degree, indegree, outdegree, and total degree (respectively). 2

3 2.1 Undirected Case We begin our derivation of bounds on the degree centralization, C d, by demonstrating that C d can be expressed as a function of size, density, and maximum degree. Let G be a simple graph of order N with mean degree d and maximum degree, and let d(v ) be the degree of vertex v V. By definition, the normalized degree centralization of G is given by v V ( d (v)) C d (G) (N 1)(N 2). (3) Rewriting the sum in the numerator gives us v V v V d (v), (4) (N 1)(N 2) which, after substituting N d for the degree sum, becomes N ( d ) (N 1)(N 2). (5) Now, let d G be the density of G. Since d d G (N 1), substitution into the above yields N( d G (N 1) ) (6) (N 1)(N 2) N ( ) N 2 N 1 d G. (7) Thus, the degree centralization of a simple graph can be shown to be a fairly straightforward function of size, density, and maximum degree. Interestingly, we can also see from Equation 7 that (ceteris paribus) C d falls linearly in density; since d G C d (G) N N 2, density and degree centralization exhibit an approximate 1:1 tradeoff for large N. By turns, scaling of C d for large graphs will be roughly proportional to 1 N given fixed density and maximum degree. It follows, then, that large and/or dense graphs cannot be highly centralized. Note that the relationship of Equation 7 still depends upon the maximum degree of G; however, maximum degree is itself related to both size and density. For instance, a lower bound on the maximum degree is obtained 3

4 when edges are distributed as evenly as possible through the graph. In this case, we have d (8) d G (N 1). (9) For the corresponding upper bound, we observe that the maximum degree in any order-n graph is trivially N 1, provided that sufficient edges are available. Adding this second constraint leads to the bound min [ N 1, d G N(N 1) 2 ]. (10) Combining the maximum degree bounds with Equation 7 gives us the lower bound C d (G) N ( ) dg (N 1) d G (11) N 2 N 1 and the upper bound ( C d (G) N ( min N 1, dg N(N 1) ) ) d G (12) N 2 N 1 [ ( )] N min N 2 (1 d N N G), d g N (13) To gain some intuition regarding the behavior of these bounds, it is useful to plot the range of potential degree centralization scores by density, for graphs of varying sizes. Such a plot is shown in Figure 1. Interestingly, the region of potential scores is roughly triangular, with an upper limit which rises linearly from a value of 0 at the extreme densities to a value of 1 at density 2 N. Hence, the peak of the triangle recedes towards the left axis as N. The lower edge of the viable region has a distinctive sawtooth structure, owing to the fact that ties cannot be distributed evenly, in general, among vertices thus, the period and amplitude of the oscillation are approximately 1 N (with some distortion, in certain cases, due to granularity effects). As this implies, the lower bound gradually approaches a constant (0) as N, although (as in the leftward drift of the maximum value) this process slows markedly as size increases. For very large N, then, we can roughly characterize the set of possible centralization scores as occupying the right triangle formed by the lower left half of the unit square: values for which C d (G) > 1 d G are generally impossible. 4

5 Degree by, Graph Order 5 Degree by, Graph Order 10 5 Degree by, Graph Order 20 Degree by, Graph Order 50 Figure 1: Degree Bounds and Distribution, by Order and, Undirected Case

6 In addition to tracing the boundary of the set of possible centralization scores, Figure 1 also depicts the approximate distribution of graphs over density/centralization space, by order. (All plots based on samples of 100,000 graphs, sampled uniformly conditional on order.) At each unique density/centralization value observed, a circle is placed; the area and shading of each circle is proportional to the fraction of graphs residing in the associated cell. As Figure 1 shows, the distribution of centrality scores (like densities) becomes heavily concentrated for graphs of even modest size. This is consistent with past simulation studies (e.g. Anderson et al., 1999) which have suggested that the interaction of size and density with graph-level indices is often substantial and complex. Viewing the centralization distribution within the bounds of Equations 11 and 13 provides additional context for these results. 2.2 Directed Case Turning to the directed case, we first note that there are three distinct notions of degree with respect to which centralization may be defined. These are, respectively, outdegree (d +, the number of edges leaving a given vertex), indegree (d, the number of edges incident upon a given vertex), and total degree (d t, the number of edges having a given vertex as an endpoint). Clearly, d t (v) d + (v) + d (v). Perhaps somewhat less obviously, v V d+ (v) v V d (v) E, establishing that the mean indegree (d ) must be equal to the mean outdegree (d + ). For this reason, we shall simplify our notation by simply using d to refer generically to d + or d. This notation also preserves the connection with the undirected case with respect to density, since d d + d d G (N 1). Given the above, we now proceed to derive bounds for indegree/outdegree centralization. (Note that, by symmetry, we may treat the two cases equivalently.) Without loss of generality, let us focus on outdegree. Since the maximum attainable raw outdegree centralization for a digraph G of order N (obtained in the case of a maximum outstar) is (N 1) 2, the normalized outdegree centrality of G is defined as v V G d +(G) ( + d + (v)) (N 1) 2, (14) 6

7 where + max v V d + (v). Since, using the above notation, v V d+ (v) Nd, we can rewrite the numerator as which, substituting d d G (N 1), becomes N ( + d ) (N 1) 2, (15) N ( + d G (N 1)) (N 1) 2 (16) N ( ) + N 1 N 1 d G. (17) As indicated above, we may employ an analogous argument for indegree centrality, giving us the parallel result, C d (G) N ( ) N 1 N 1 d G, (18) where max v V d (v). Given Equations 17 and 18, the derivation of bounds on C d +, C d now depends only on finding bounds for + and. Here, as before, the argument runs parallel to the undirected case. For both indegree and outdegree, the maximum value will be minimized by a distribution which divides degrees as evenly as possible. Hence, we have +, d (19) d G (N 1). (20) Likewise, the maximum indegrees/outdegrees are obtained for instars/outstars (respectively), both of which involve N 1 edges. This gives us +, min [N 1, d G N(N 1)]. (21) To find the C d +, C d bounds, we now simply substitute Equations 20 and 21 into Equations 17 and 18. The combined bounds can be summarized by the inequality ( ) [ ] N dg (N 1) N d G C N 1 N 1 d +(G), C d (G) min N 1 (1 d G), d G N. (22) 7

8 Having accounted for indegree and outdegree centralization, we are now left with the case of total degree. Since the maximum raw total degree centralization occurs for a maximum reciprocated star (i.e., a maximum star in which all edges are reciprocated), we can discern that the normalizing factor for the order-adjusted centralization score will be (N 1)(2N 2 2) 2(N 1)(N 2). Given this, the total degree centralization can be written ( t d t (v) ) C d t(g) v V 2(N 1)(N 2), (23) with t max v V d t (v); this may, in turn, be re-expressed as ( v V t d + (v) d (v) ) 2(N 1)(N 2) Rewriting in terms of density gives us v V t v V d+ (v) v V d (v) 2(N 1)(N 2) (24) (25) N ( t 2d ) 2(N 1)(N 2). (26) N ( t 2d G (N 1) ) (27) 2(N 1)(N 2) N ( t ) N 2 2(N 1) d G. (28) To obtain the centralization bounds, we turn once more to the bounds implied by N and d G for maximum degree. In the case of the minimum maximum total degree, the even allocation argument must take both indegrees and outdegrees into account; a maximally even division, then, must satisfy t 2d (29) 2d G (N 1). (30) An upper bound on the maximum total degree will be obtained in the reciprocal star case, in which the central vertex will be an endpoint of 2(N 1) edges. Taking into account the number of edges available, this leads to the bound t min [2(N 1), d G N(N 1)]. (31) 8

9 To obtain the total degree centralization bounds, we now substitute the bounds for t into Equation 28, yielding ( ) [ ( )] N 2dG (N 1) N d G C N 2 2(N 1) d t(g) min N 2 (1 d N N G), d G N (32) Clearly, the bounds shown in Equations 22 and 32 are quite similar in form to those of Equations As is shown in Figures 2 and 3, the centralization bounds in the directed case do indeed form the same sawtoothed triangle observed previously in the undirected case. Although some details differ (e.g., the 1 N peak for in/outdegree centralization versus the 2 N peak for undirected and total degree centralization), the size and structure of the feasible region is essentially the same. In particular, it should be noted that for all forms of degree centralization, the feasible region approaches the lower half-triangle of the unit square thus, the result that approximately one-half of all conceivable degree centralization values are actually feasible holds for both directed and undirected graphs. Beyond the bounds themselves, we may also consider the distribution of centralization scores by order for indegree, outdegree, and total degree centralization. Estimated densities based on simulated networks are shown for each respective case in Figures 2 and 3; due to the fact that the distributions for indegree and outdegree centralization are identical, they are jointly represented by Figure 2. (As with Figure 1, each plot is based on 100,000 uniform draws, with each unique density/centralization pair represented by a circle whose area and shading are proportional to its estimated frequency.) The estimated centralization distributions reveal patterns of both similarity and difference vis a vis each other and the undirected case. All four centralization measures are seen to be alike in converging to a fairly small, ovoid region as N becomes large, indicating that the overwhelming majority of graphs are of moderately low centralization. The four measures are also alike in exhibiting clear banding behavior, a visual representation of the fact that small changes to density cannot be made without simultaneously changing centralization. Since this is a type of granularity effect, it is perhaps unsurprising to find bands to be most widely spaced in the undirected case, followed by in/outdegree, and finally total degree (which can take the widest range of values). Another subtle difference is the nature of concentration in the centralization distribution for N small. Simple degree and total degree centralization indices initially concentrate more mass in the central 9

10 In/Outdegree by, Graph Order 5 In/Outdegree by, Graph Order In/Outdegree by, Graph Order 20 In/Outdegree by, Graph Order 50 Figure 2: In/Outdegree Bounds and Distribution, by Order and, Directed Case

11 region than do in/outdegree centralization indices, the latter giving rise to somewhat more elongated densities. On the whole, however, the form of the centralization distribution appears fairly similar across notions of degree, particularly for large N. 3 -Normalized Scores Having derived bounds for centralization scores given both size and density, it is now natural to ask whether one might construct a renormalized set of scores which assess the extent to which a graph is centralized relative to values which are attainable given these other structural properties. In particular, one might seek an index which, with respect to the raw centralization score of Equation 2, shows the fraction of possible degree centralization which is actually realized, over and above that which is necessary given size and density. Such a measure may serve as a more reasonable indicator of actual centralizing/equalizing processes than raw or order-normalized centralization scores, particularly where size and density are the result of exogenous factors. To implement this notion with respect to centralization measure C, we employ the affine transformation ( ) C (G) min C (G ) C (G) {G : V N,δ(G)d G } ( ) ( ), (33) max C (G ) {G : V N,δ(G)d G } min C (G ) {G : V N,δ(G)d G } where δ(g) is the density of G. In the event that the minimum and maximum centralization scores are the same, Equation 33 is poorly defined; since, for the measures considered here, this affects a small region of length 3 2N occupying the density extremes, it is unlikely to be a problem in most settings. Consistent with the above motivation, however, it is suggested that C be defined equal to 0 in such cases. Likewise, we assume that the expressions below treat only the nondegenerate case in which N 2, since centralization is necessarily 0 for N 1. We begin our derivation of the density-normalized degree centralization index with the undirected case. Substituting the results of Equations 7, 11, 11

12 Total Degree by, Graph Order 5 Total Degree by, Graph Order Total Degree by, Graph Order 20 Total Degree by, Graph Order 50 Figure 3: Total Degree Bounds and Distribution, by Order and, Directed Case

13 and 13 into Equation 33 gives us N N 2 ( N 1 d G C d (G) [ min N N 2 (1 d G), d g N N 2 ) ( N dg (N 1) N 2 ( N 2 1)] N N 2 N 1 d G ) ( dg (N 1) N 1 d G ); (34) note that, under the assumption that N 2, we can pull the N 2 from the minimum term in the denominator and cancel it, giving us the simplification N 1 d G(N 1) N 1 min [ ( 1 d G, d N G 2 1)] d G(N 1) N 1 + d G (35) d G (N 1) (N 1) ( d G + min [ ( 1 d G, d N G 2 1)]) d G (N 1). (36) Intuitively, then, C d reduces to the excess maximum degree (i.e., over and above the minimum ), scaled by the maximum possible surplus (given N and d G ). We now repeat this process for the directed case. For outdegree centralization, we substitute Equations 17 and 22 into the general form of Equation 33 to obtain ( ) ( ) N + C d N 1 N 1 d G N dg (N 1) N 1 N 1 d G (G) [ ] ( ) (37) + min N N 1 (1 d G), d G N N dg (N 1) N 1 N 1 d G + N 1 d G(N 1) N 1 min [1 d G, d G (N 1)] d G(N 1) N 1 + d G (38) + d G (N 1) (N 1) (d G + min [1 d G, d G (N 1)]) d G (N 1). (39) Note that this form is essentially similar to that of Equation 36. Since the indegree centralization of Equation 18 is identical to that used here save in the substitution of for +, and since it obeys the same bounds (given in Equation 22), the degree-normalized indegree follows immediately from Equation 39: C d (G) d G (N 1) (N 1) (d G + min [1 d G, d G (N 1)]) d G (N 1). (40) N 13

14 Finally, for total degree centralization, substitution of the results of Equations 28 and 32 into Equation 33 yields ( ) ( ) N t C d t(g) N 2 2(N 1) d G N 2dG (N 1) N 2 2(N 1) d G [ min N N 2 (1 d ( ( ) N G), d N G N 2 2 1)] N 2dG (N 1) N 2 2(N 1) d G (41) t 2(N 1) 2d G(N 1) 2(N 1) min [ ( 1 d G, d N G 2 1)] 2d (42) G(N 1) 2(N 1) + d G t 2d G (N 1) 2 (N 1) ( d G + min [ ( 1 d G, d N G 2 1)]) 2d G (N 1). (43) Once again, we obtain a form which reduces to a rescaled excess maximum degree. As this suggests, this behavior is a property of degree centralization generally, rather than the specific notion of degree being employed. With the above equations in hand, it seems reasonable to inquire as to the distribution of the renormalized centralization measure over the space of graphs. To that end, we employ the same simulation strategy used earlier, i.e., taking 100,000 uniform draws from each of four distinct orders (5, 10, 20, and 50) and plotting each unique C value with a circle whose area is proportional to the observed frequency. The results for the simulated draws are shown in Figure 4. (Note that, since the distributions of C d and C d are + identical, only the latter is depicted here.) While the affine transformation of Equation 33 cannot change the essentially discrete nature of the centralization score (nor remove all aspects of the order/density/centralization relationship), the distributions shown in Figure 4 have some attractive properties relative to those of Figures 1 3. By design, the renormalized distributions use a larger portion of the [0,1] interval, even for N large; because the density bound has been removed, a lack of coverage over the full range is due to frequency rather than necessity. Another interesting side effect of the renormalization is a change in the banded structure of the distribution. While small changes to density generally necessitated changes in centralization for the standard measures (as witnessed by the diagonal bands of possible values), this is not generally true in the renormalized case. Although one encounters threshold densities beyond which further increases (or decreases) result in a centralization change, small changes in density do not typically require associated changes in the renormalized centralization score. This may make renormalized centralization scores somewhat easier 14

15 to interpret in cases for which density cannot be specified exactly. It should be emphasized that these properties to not make the renormalized centralization indices more correct than the conventional versions one should use the index which accords with the properties of greatest substantive interest but they do suggest that renormalized centralization indices may be especially useful in certain types of settings. 4 Discussion and Conclusions In the foregoing, exact bounds were derived for degree centralization in both the simple and directed case. These bounds, in and of themselves, provide us with some general intuitions regarding the behavior of degree centralization in social networks, particularly with respect to the interaction of centralization with size and density. As we have seen, the convex hull of degree centralization statistics is triangular, with a peak at density 2 N (simple graphs or digraphic total degree) or 1 N (digraphic indegree/outdegree). The upper bound on degree centralization falls away linearly in density from this point, approaching 0 at the density extremes of the null and complete graphs. Since the density of maximum potential centralization (which corresponds to the density of the appropriate star form) approaches 0 in N, we observed that the region of feasible centralization scores was confined to the lower left half of the unit square in the large-n limit. Thus, large graphs cannot be more centralized than (approximately) one minus their density, regardless of other factors. Most graphs of moderate to large size do not approach this constraint, however, as (for reasons of combinatorics) the overwhelming majority are concentrated in a small region of moderate density and low centralization. This fact, however, should not preclude the constraint s being of consequence for empirically derived networks. Turning to the lower bound on degree centralization, we find that it arises from a fundamental limit on how equitably edges can be allocated through a graph. This is not a major consideration in large networks (since the magnitude of the effect relative to the maximum obtainable centralization is on the order of 1 N ), but the effect can be substantial in small groups. Depending on density, a sizeable fraction (e.g., 10 20%) of the degree centralization in groups of size 5 10 will result from irreducible edge allocation effects. This implies, on the one hand, that some nontrivial degree of centralization is inevitable in small groups (again, depending upon network density); on the other, this underscores the notion that one must be cautious in seeking 15

16 C d G, V 5 C d G, V 10 C d G, V 20 C d G, V 50 C d + G, V 5 C d + G, V 10 C d + G, V 20 C d + G, V 50 C d t G, V 5 C d t G, V 10 C d t G, V 20 C d t G, V 50 Figure 4: Estimated and Order Normalized Degree Score Distributions, by Order and Degree Type 16

17 complex explanations for modest tendencies towards centralization in such settings. In either case, we must acknowledge the importance of density as a network property with far-reaching consequences: a social process producing slight changes in density can engender reasonably large changes in centralization, as a side effect. In addition to showing bounds for degree centralization, we have also used those bounds to derive a family of renormalized measures which take into account both minimum and maximum centralization constraints due to size and density. Although these expressions may appear complex at first blush, all of them admit a very simple substantive interpretation: the relative degree centralization score is merely the extent to which the maximum degree exceeds its minimum possible value, divided by the maximum possible excess. This result serves to emphasize the extent to which degree centralization is fundamentally centered on the maximum degree statistic, to the exclusion (for good or ill) of many other properties of the degree distribution. Since extreme order statistics are inherently unstable, such a property might give us pause; degree centralization is unlikely to be robust to measurement error, for instance. On the other hand, the underlying simplicity of the index is a substantive virtue, and the maximum order statistic provides a fairly direct means of assessing the upper tail of the degree distribution. As this general issue affects all measures of the form shown in Equation 1, its significance extends beyond the scope of the present work. On a slightly more programmatic note, the presence of interesting and subtle behavior for a simple index such as centralization underscores the continuing need for (and value of) work which explores the connections among structural properties at the graph level. Even for the three classic centrality measures defined by Freeman (1979) (and their corresponding centralization indices), there is much which remains to be known. It is hoped that the present work will aid in this ongoing effort. 5 References Anderson, B. S., Butts, C. T., and Carley, K. M. (1999). The interaction of size and density with graph-level indices. Social Networks, 21(3): Bavelas, A. (1950). Communication patterns in task oriented groups. Journal of the Acoustical Society of America, 22: Buskens, V. (1998). The social structure of trust. Social Networks, 20(3):

18 Freeman, L. C. (1979). Centrality in social networks: Conceptual clarification. Social Networks, 6: Marwell, G., Oliver, P. E., and Prahl, R. (1988). Social networks and collective action: A theory of the critical mass. III. American Journal of Sociology, 94(3): Snijders, T. A. B. (1981). The degree variance: an index of graph heterogeneity. Social Networks, 3(3): Topper, C. M. and Carley, K. (1999). A structural perspective on the emergence of network organizations. Journal of Mathematical Sociology, 24(1):