Figure 2. Distributions of empirical networks according to three sets of control measures.

Each set includes three measures that sum to 1 for a given network. The distributions are shown on ternary plots, where a network at the center corresponds to a set of values (1/3, 1/3, 1/3) and a network at a corner corresponds to e.g. (1, 0, 0). Networks are represented with colored according to their maximal parameter. Interior lines indicate the regions where each parameter is largest. (a) The control profile of Ruths & Ruths²³. Directly-controlled nodes are either source nodes (η_s), arise due to internal dilations (η_i), or arise due to external dilations (η_e). (b) The fraction of nodes that are always (ν_a), sometimes (ν_s), or never (ν_n) directly controlled, when considering all control schemes that minimize the number of controls. (c) The fraction of edges that are always (ε_a), sometimes (ε_s), or never (ε_n) on a control signal path, when considering all control schemes that minimize the number of controls. (d–f) The degeneracy measures applied independently to the cases where η_s, η_e, and η_i are the dominant parameter in the control profile. Each plot is uniformly colored according to the corresponding dominant control profile parameter (as labeled on the left of the panel). The formatting of each plot otherwise follows (a–c).