Optimization of the robustness of multimodal networks

We investigate the robustness against both random and targeted node removal of networks in which $P\left(k\right)$, the distribution of nodes with degree $k$, is a multimodal distribution, $P\left(k\right)\propto {\sum }_{i=1}^m{a}^{-(i-1)}\delta (k-k_i)$ with $k_i\propto b^{-(i-1)}$ and Dirac’s delta function $\delta \left(x\right)$. We refer to this type of network as a scale-free multimodal network. For $m=2$, the network is a bimodal network; in the limit $m$ approaches infinity, the network models a scale-free network. We calculate and optimize the robustness for given values of the number of modes $m$, the total number of nodes $N$, and the average degree $⟨k⟩$, using analytical formulas for the random and targeted node removal thresholds for network collapse. We find, when $N\gg 1$, that (i) the robustness against random and targeted node removal for this multimodal network is controlled by a single combination of variables, $N^{1∕(m-1)}$, (ii) the robustness of the multimodal network against targeted node removal decreases rapidly when the number of modes becomes larger than a critical value that is of the order of $\ln N$, and (iii) the values of exponent ${\lambda }^{\mathrm{opt}}$ that characterizes the scale-free degree distribution of the multimodal network that maximize the robustness against both random and targeted node removal fall between 2.5 and 3.

Figure 1

Schematic representation of the degree distribution for the scale-free multimodal network and the part of the degree distribution removed by a targeted attack on the higher degree modes between $k_l$ and $k_m$.

Figure 2

(a) The dependence of the values of ${\ln }_N{a}^{\mathrm{opt}}$ for the values of $a$ that maximize the total thresholds on the total number of modes $m$. The thick curve stands for $1∕(m-1)$. (b) The dependence of the values of ${\ln }_N{b}^{\mathrm{opt}}$ for the values of $b$ that maximize the total thresholds on the total number of modes $m$. The thick curve stands for $-1∕2(m-1)$.

Figure 3

(a) Optimal values for the sum of the thresholds, $f_T\equiv f_r+f_t$, versus mode number for $⟨k⟩=2.8$ for several values of $N$. (b) The optimal values for the measure, $f_T^{\mathrm{opt}}$, for $⟨k⟩=2.8$ are replotted in terms of the scaled mode number $m^{*}\equiv (m-1)∕{\log }_{10}N$. The collapse of the data on a single curve is seen. The curve changes its profile at a certain value of $m^{*}$, which is about $0.7$ for $⟨k⟩=2.8$, and the value is denoted by $m_c^{*}$.

Figure 4

The values for the threshold against random node removal $f_r$ and the threshold against targeted node removal $f_t$ for the optimal configuration for $⟨k⟩=2.8$ are plotted in terms of the scaled variable $m^{*}\equiv (m-1)∕{\log }_{10}N$ for $0<m^{*}<m_c^{*}\approx 0.7$.

Figure 5

The behavior of the exponent $\alpha$, which is defined as ${\ln }_N{q}^{\mathrm{opt}}$, where $q^{\mathrm{opt}}$ is the number of the highest degree nodes, and the behavior of the highest degree $k_m^{\mathrm{opt}}$ for the optimal configuration for $⟨k⟩=2.8$ in terms of the scaled mode number $m^{*}$ smaller than the critical value $m_c^{*}\approx 0.7$. The values of $\alpha$ smoothly decrease from the values for the optimal bimodal network, which are approximately equal to $0.25$, and reach zero, which is the smallest possible value corresponding to $q=1$, at $m^{*}=m_c^{*}$. For the values of the highest degree for the optimal configuration the relation $k_m^{\mathrm{opt}}\approx k_{\max }\equiv \sqrt{⟨k⟩N}$ holds for $0<m^{*}<m_c^{*}$.

Figure 6

The dependence of the critical values of the scaled mode number $m_c^{*}$ on the average degree $⟨k⟩$. The value of $m_c^{*}$ for each $⟨k⟩$ is the average over all $m_c^{*}$ for different values of the total node number $N$. The critical value $m_c^{*}$ is the value of the scaled mode number at which the scale-free multimodal network completely loses its robustness against targeted node removal. The broken line is the fit $m_c^{*}=0.62⟨k⟩-1.0$.

Figure 7

The exponent in the multimodal degree distribution for the optimal configuration ${\lambda }^{\mathrm{opt}}$ for the scale-free multimodal degree distribution, versus the scaled variable $m^{*}\equiv (m-1)∕{\log }_{10}N$ of Eq. (37). Note that, as argued in Sec. 6C, the values of ${\lambda }^{\mathrm{opt}}$ reside mainly in the interval between 2.5 and 3.