Exact Bounds for Degree Centralization

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) =
1
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.
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 ) Cd falls ∂ Cd (G) = − NN linearly in density; since ∂d −2 , density and degree centralizaG tion exhibit an approximate 1:1 tradeoff for large N . By turns, scaling of C d 1 for large graphs will be roughly proportional to 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
2
Bounding Degree Centralization
The “raw” or unnormalized degree centralization of graph G = (V, E ) under degree measure d∗ is defined as Cd∗ (G) =
v ∈V
v ∈V
∗
max c∗ v − c∗ (v ) ,
v ∈V
(1)
This material is based upon work supported by the National Science Foundation under award NSF ITR #0331707. † Department of Sociology and Institute for Mathematical Behavioral Sciences, University of California, Irvine; SSPA 2145, Irvine, CA 92697; buttsc@
v ∈V
(3)
Rewriting the sum in the numerator gives us =
v ∈V
∆ − v∈V d (v ) , (N − 1)(N − 2)
(4)
whichdegree sum, becomes = N ∆−d . (N − 1)(N − 2) (5)
(∆∗ − d∗ (v )) ,
(2)
where ∆∗ = maxv∈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 configuration1 ), 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 pos2 1 sible for the appropriate star density ( |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.
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).
1
2
2.1
Undirected Case
We begin our derivation of bounds on the degree centralization, Cd , by demonstrating that Cd 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 Cd (G) = (∆ − d (v )) . (N − 1)(N − 2)