Limits and trade-offs of topological network robustness

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Abstract

We investigate the trade-off between the robustness against random and targeted removal of nodes from a network. To this end we utilize the stochastic block model to study ensembles of infinitely large networks with arbitrary large-scale structures. We present results from numerical two-objective optimization simulations for networks with various fixed mean degree and number of blocks. The results provide strong evidence that three different blocks are sufficient to realize the best trade-off between the two measures of robustness, i.e. to obtain the complete front of Pareto-optimal networks. For all values of the mean degree, a characteristic three block structure emerges over large parts of the Pareto-optimal front. This structure can be often characterized as a core-periphery structure, composed of a group of core nodes with high degree connected among themselves and to a periphery of low-degree nodes, in addition to a third group of nodes which is disconnected from the periphery, and weakly connected to the core. Only at both extremes of the Pareto-optimal front, corresponding to maximal robustness against random and targeted node removal, a two-block core-periphery structure or a one-block fully random network are found, respectively.

Figures

  • Figure 1. Giant connected component of a Erdős-Rényi random network with a Poisson (dashed green) and modified Poisson (solid red) degree distribution (excluding nodes with k~0) as function the average mean degree of the network. The inset shows the regular and modified Poisson distribution for a mean degree of SkT~1:2 with the same color coding. doi:10.1371/journal.pone.0108215.g001
  • Figure 2. Pareto-optimal fonts of robustness against targeted and random removal of nodes for mean degree SkT~2:5 and various number of blocks. For three, four and five blocks the curves match exactly which implies that no more than three blocks are necessary to achieve the best robustness values. Since at the and of the curves all of them match only two or even one block is enough to achieve best robustness. doi:10.1371/journal.pone.0108215.g002
  • Figure 3. Pareto-optimal fronts of RRandom versus RTargeted for optimal block model networks with B~3 and for various mean degrees (colored symbols). For SkTw2, the smaller gray symbols to the left of each Pareto front indicate solutions which maximize RRandom for fixed RTargeted but which are not Pareto-optimal (see main text). doi:10.1371/journal.pone.0108215.g003
  • Figure 4. Parameters of the optimized networks as a function of RTargeted obtained from a three-block optimization with SkT~2:5. The upper row shows the elements of the edge matrix ers, where the areas of the squares is proportional to the logarithm of the element. The positions for which these Hinton plots are shown are marked with dashed lines in the other panels. The second row shows the trade-off curve already displayed in Fig. 2, while the third and fourth display the mean degree of the blocks, on a logarithmic and a linear scale, respectively. The last row shows the relative sizes of the blocks. The coloring of the blocks and their index is determined by their mean degree, where the block with the highest mean degree is shown in blue and always has index r~0, followed by green with r~1 (second highest) and red (r~2, lowest degree). doi:10.1371/journal.pone.0108215.g004
  • Figure 5. Parameters of the networks along the trade-off curve for the five block optimization with SkT~2:0. The rows show the elements of the edge matrix ers , the trade-off curve already displayed in Fig. 2, the mean degree of the blocks on a logarithmic and on a linear scale, as well as the relative sizes of the blocks, from top to bottom, respectively. See caption of Fig. 4 for more details on the coloring and box sizes. doi:10.1371/journal.pone.0108215.g005
  • Figure 6. Parameters of the networks along the trade-off curve for the five block optimization with SkT~3:5. The rows show the elements of the edge matrix ers , the trade-off curve already displayed in Fig. 2, the mean degree of the blocks on a logarithmic and on a linear scale, as well as the relative sizes of the blocks, from top to bottom, respectively. See caption of Fig. 4 for more details on the coloring and box sizes. doi:10.1371/journal.pone.0108215.g006

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CITATION STYLE

APA

Priester, C., Schmitt, S., & Peixoto, T. P. (2014). Limits and trade-offs of topological network robustness. PLoS ONE, 9(9). https://doi.org/10.1371/journal.pone.0108215

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