Diagram illustrating the process of visualizing an ensemble of networks.

<p>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.</p

An adjustable hierarchical network with the different edge types.

<p>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.</p

The global reaching centrality at different p values in the adjustable hierarchical model.

<p>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.</p

Pearson correlation of the GRC and in the switchboard dynamics.

<p>The correlations are all negative (except for the Internet networks) and most of them are very close to −1. Thus, under the switchboard dynamics the GRC (strength of hierarchy) and are strongly negatively correlated.</p

Visualization of network ensembles.

<p>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).</p

Visualization of three network types based on the local reaching centrality.

<p>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 .</p

Heterogeneity of the distribution of the local reaching centrality for different network types.

<p>The two measures of heterogeneity presented here are the global reaching centrality () and (standard deviation of ). Means and variances are shown for an ensemble of 1000 networks.</p

Distribution of the local reaching centrality for the adjustable hierarchical network.

<p>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 .</p

Hierarchical properties of real networks.

<p>We show the order (), average degree (), and global reaching centrality for the original () and for the randomized networks (). References to data sources are included. Suits next to the GRC values show comparison to the randomized networks: whether the original networks are more hierarchical than their randomization (club suit) or they are more egalitarian (diamond suit) with a 98% confidence level. The meaning of edges is also indicated.</p

The Pearson correlation of the GRC and defined by Liu et al.

<p>With only one exception, all correlations are positive and many of them are above 0.6, i.e., the GRC and are positively correlated.</p