Figure 1.
An adjustable hierarchical network with the different edge types.
The blue edges belong to the original arborescence graph that is used as the backbone of the adjustable hierarchical (AH) network. There are three type of possible edges added to the graph: down edges (green), horizontal edges (orange) and up edges (red). They have different effects on the hierarchical structure of the directed tree. Down edges conserve the hierarchy, horizontal edges has a slight influence and up edges make strong changes in the structure.
Table 1.
Heterogeneity of the distribution of the local reaching centrality for different network types.
Figure 2.
Distribution of the local reaching centrality for the adjustable hierarchical network.
Distribution of the local reaching centrality in the adjustable hierarchical (AH) network model at different parameter values. Each distribution is averaged over 1000 AH networks with
and
. The standard deviations of the distributions are comparable to the averages only for relative frequencies less than 0.002. Note that from the
(highly random) to the
(fully hierarchical) state the distribution changes continuously and monotonously with
.
Figure 3.
The global reaching centrality at different p values in the adjustable hierarchical model.
All curves show averages over an ensemble of 1000 networks with and different average degrees. Standard deviations grow with
, but they are clearly below the average values of the GRC. Note that for larger density, it is less likely to obtain the same level of hierarchy.
Figure 4.
The global reaching centrality versus average degree in the Erdös–Rényi and scale-free networks.
Dots show averages for 1000 graphs with nodes. In the Erdös–Rényi and scale-free networks, standard deviations of the GRC are comparable with its averages only for
and
, respectively.
Table 2.
Hierarchical properties of real networks.
Table 3.
The Pearson correlation of the GRC and defined by Liu et al.
Table 4.
Pearson correlation of the GRC and in the switchboard dynamics.
Figure 5.
Visualization of three network types based on the local reaching centrality.
Visualization of (A) an Erdös–Rényi (ER) network, (B) a scale-free (SF) network and (C) a directed tree with random branching number between 1 and 5. All three graphs have nodes and the ER and SF graphs have
. In each network
was set to
.
Figure 6.
Diagram illustrating the process of visualizing an ensemble of networks.
First, we compute the layout based on the selected local quantity for each graph in the ensemble (top right). Next, we separate the levels logarithmically and scale each layout into the unit square (bottom left). Last, we overlay all rescaled layouts and plot the obtained density of nodes in the unit square (bottom right, see color scale also). In the heat maps, the color scale shows
, where
is the average density of the ensemble.
Figure 7.
Visualization of network ensembles.
Visualizations of the (A) Erdös–Rényi, (B) scale-free, (C) directed tree and (D)–(L) AH network ensembles (subfigures (D)–(L) are for different values of the model parameter: ). In each case the color scale shows
where
is the density averaged over 1000 graphs.
and
were set. In every network,
was set to
. The corresponding GRC values are: 0.997 (A), 0.058 (B), 0.127 (C), 0.135 (D), 0.161 (E), 0.194 (F), 0.238 (G), 0.290 (H), 0.361 (I), 0.452 (J), 0.581 (K) and 0.775 (L).
Figure 8.
Visualization of real networks.
The hierarchy-based visualization of (A) the GrassLand food web, (B) the electrical circuit benchmark s9234, (C) the transcriptional regulatory network of yeast and (D) the core of the Enron network. In every network was set to
.
Figure 9.
Distributions of the local reaching centrality for different network types.
For each network type and for the Erdös–Rényi (ER) and scale-free (SF) networks
. All curves show averages of the distributions over an ensemble of 1000 graphs. Standard deviations are comparable with the averages only near the peaks in the ER and SF models. Although the standard deviations at the peaks are large, they do not change the positions of the peaks, and thus, do not affect the distributions.