text: abstract—this paper examines the dependence of performance measures on network size with a focus on large networks. we develop a framework where it is shown that poor
  performance can be attributed to dimension–dependent scaling
  of network energy. drawing from previous work, we show
  that such scaling in undirected networks can be attributed
  to the proximity of network spectrum to unity, or distance
  to instability. however, such a simple characterization does
  not exist for the case of directed networks. in fact, we show
  that any arbitrary scaling can be achieved for a fixed distance
  to instability. the results here demonstrate that it is always
  favorable, in terms of performance scaling, to balance a large
  network. this justifies a popular observation that undirected
  networks generally perform better. we show the relationship
  between network energy and performance measures, such as
  output shortfall probability or centrality, that are used in
  economic or financial networks. the strong connection between
  them explains why a network topology behaves qualitatively
  similar under different performance measures. our results
  suggest that there is a need to study performance degradation
  in large networks that focus on topological dependence and
  transcend application specific analyses.