Directed random geometric graphs: structural and spectral properties

We define the elements of the non-Hermitian adjacency matrix $\mathbf{A_{nH}}$ of the dRGGs as

$$\begin{equation} [A_{nH}]_{uv} = \left\lbrace \begin{array}{ll} \epsilon_{uu}& u=v, \\ \epsilon_{uv}& u\rightarrow v, \\ 0& \mbox{otherwise}, \end{array} \right. \end{equation} \tag{ 3 }$$

where ε_uv are independent random variables drawn from a normal distribution with zero mean and unit variance; $\epsilon_{uv}\neq \epsilon_{vu}$ since $G(n,\alpha,\ell_0)$ is directed. According to this definition, diagonal random matrices (known in RMT as the Poisson ensemble (PE) [27]) are obtained for $\ell_0 \rightarrow 0$, when the graphs consist mostly of disconnected vertices; whereas the real Ginibre ensemble (RGE) (i.e. full real and non-symmetric random matrices [28]) is recovered for $\ell_0 = \sqrt{2}$, when the graphs are complete. Therefore, we expect to observe a transition from the PE to the RGE for increasing $\ell_0$. Thus, the ensemble defined by $\mathbf{A_{nH}}$ corresponds to a diluted RGE.

Additionally, given the standard binary adjacency matrix B extracted from the directed graph $G(n,\alpha,\ell_0)$, we consider the following Hermitian adjacency matrix $\mathbf{A_H}$:

$$\begin{equation} \left[A_H\right]_{uv} = \left\{ \begin{array}{ll} \epsilon_{uu} & u = v , \\ \epsilon_{uv}B_{uv}& u\leftrightarrow v , \\ i\epsilon_{uv}B_{vu}& u \rightarrow v , \\ -i\epsilon_{uv}B_{vu}& v \rightarrow u , \\ 0 & \mbox{otherwise} . \end{array} \right. \end{equation} \tag{ 4 }$$

Again, ε_uv are independent random variables taken from a normal distribution with zero mean and variance one. It is clear that $[A_H]_{vu} = [A_H]_{uv}^*$ by construction. In fact, $\mathbf{A_H}$, with the name of magnetic adjacency matrix, was already used in [29] to represent randomly-weighted Erdös-Rényi directed graphs; while the unweighted version of $\mathbf{A_H}$ was introduced in [30]. The main advantage of $\mathbf{A_H}$ over $\mathbf{A_{nH}}$ is that the former has a real spectrum, thus some spectral measures from RMT can be computed easier.

Therefore, the magnetic random matrix ensemble (MRME) defined by $\mathbf{A_H}$ transits for increasing $\ell_0$ from the PE, when $\ell_0 \approx 0$, to the GOE (i.e. full real and symmetric random matrices [27]), when $\ell_0 = \sqrt{2}$. However, note that the MRME can not be considered as a diluted GOE since for any $\ell_0 \lt \sqrt{2}$, $\mathbf{A_H}$ is a complex Hermitian matrix.

In addition to the standard quantities used to characterize the structure of random graphs and networks, such as the statistical properties of the vertex degree k and the average number of nonisolated vertices V_x , here we also use topological indices.

Given a simple graph $G = (V(G), E(G))$ with V(G) and E(G) the vertex set and edge set, respectively, the Randić connectivity index is defined as [31]

$$\begin{equation} R(G) = \sum_{uv\in E(G)}\frac{1}{\sqrt{k_u k_v}}, \end{equation} \tag{ 5 }$$

while the Harmonic index is given by [32]

$$\begin{equation} H(G) = \sum_{uv\in E(G)}\frac{2}{k_u + k_v}. \end{equation} \tag{ 6 }$$

Here uv denotes the edge connecting vertices u and v and k_u the degree of vertex u:

$$\begin{equation} k_u = k_u^\mathrm{in} + k_u^\mathrm{out}, \end{equation} \tag{ 7 }$$

where $k_u^\mathrm{in}$ and $k_v^\mathrm{out}$ are the in-degree and out-degree of vertex u.

From definitions (5) and (6), when $\ell_0 \approx 0$ (i.e. when the model of dRGGs produces mostly isolated vertices), $R(G) \approx 0$ and $H(G) \approx 0$; while for $\ell_0 = \sqrt{2}$ (i.e. when the graph model produces complete graphs), $R(G) = N/2$ and $H(G) = N/2$. That is, when applying R(G) and H(G) to ensembles of random graphs characterized by the parameter set $(n,\alpha,\ell_0)$, we expect to observe a transition from $\langle R(G)\rangle\approx 0$ and $\langle H(G)\rangle\approx 0$ to $\langle R(G)\rangle = N/2$ and $\langle H(G)\rangle = N/2$ when increasing $\ell_0$ from $\ell_0\approx 0$ to $\sqrt{2}$.

We stress that the random weights we impose to the adjacency matrices $\mathbf{A_{nH}}$ and $\mathbf{A_H}$, as defined respectively in equations (3) and (4), do not play any role in the computation of the degree-based topological indices.

In fact, within a statistical approach to random graphs, it has been recently shown that the average values of R(G) and H(G), normalized to the order of the graph n, scale with the average degree $\left\langle k \right\rangle$; see e.g. [33–35]. That is, $\left\langle R(G) \right\rangle/n$ and $\left\langle H(G) \right\rangle/n$ are functions of $\left\langle k \right\rangle$ only. Moreover, it was also found that $\left\langle R(G) \right\rangle$ and $\left\langle H(G) \right\rangle$ are highly correlated with the Shannon entropy of the eigenvectors of the adjacency matrix of RGGs [36]. This is a notable result because it puts forward the application of degree-based topological indices beyond mathematical chemistry (it is relevant to add that random graphs have also been studied by means of eigenvalue-based topological indices such as the Estrada index, the Laplacian Estrada index, and Rodriguez-Velazquez indices, see e.g. [37, 38]). Therefore, the study of R(G) and H(G) on dRGGs seems to be pertinent.

We use standard RMT measures to characterize spectral and eigenvector properties of the non-Hermitian $\mathbf{A_{nH}}$ and Hermitian $\mathbf{A_H}$ adjacency matrices defined, respectively, in equations (3) and (4).

Regarding spectral properties, for $\mathbf{A_{nH}}$ which has complex spectra, we use the ratio between nearest- and next-to-nearest-neighbor spacings $r_{\mathbb{C}}$, with the ith ratio defined as [39]

$$\begin{equation} r_{\mathbb{C}}^i = \frac{\vert \lambda_i^{\mathrm{NN}}-\lambda_i\vert}{\vert \lambda_i^{\mathrm{NNN}}-\lambda_i\vert} ; \end{equation} \tag{ 8 }$$

where $\lambda_i^{\mathrm{NN}}$ and $\lambda_i^{\mathrm{NNN}}$ are, respectively, the nearest and the next-to-nearest neighbors of λ_i in $\mathbb{C}$. While, given the real and ordered spectrum of $\mathbf{A_H}$, $\lambda_1\gt\lambda_2\gt\ldots \gt\lambda_n$, we compute the ratio between consecutive level spacings $r_{\mathbb{R}}$, where the ith ratio is given by [40]

$$\begin{equation} r_{\mathbb{R}}^i = \frac{\mathrm{min}(\lambda_{i+1}-\lambda_i,\lambda_{i}-\lambda_{i-1})}{\mathrm{max}(\lambda_{i+1}-\lambda_i,\lambda_i-\lambda_{i-1})} . \end{equation} \tag{ 9 }$$

Note that $r_{\mathbb{C}}$ can also be computed for real spectra, so it works for both $\mathbf{A_{nH}}$ and $\mathbf{A_H}$.

Regarding eigenvector properties, given the normalized eigenvectors $\Psi^i$ (i.e. $\sum_{m = 1}^n\vert\Psi_m^i\vert^2 = 1$) of $\mathbf{A_{nH}}$ or $\mathbf{A_H}$, we compute the inverse participation ratios [41]

$$\begin{equation} \mathrm{IPR}_i = \left[\sum_{m=1}^n\vert\Psi_m^i\vert^4\right]^{-1} \end{equation} \tag{ 10 }$$

and the Shannon entropies [42]

$$\begin{equation} S_i = \sum_{m=1}^n\vert\Psi_m^i\vert^2 \ln\vert\Psi_m^i\vert^2. \end{equation} \tag{ 11 }$$

Both $\mathrm{IPR}$ and S measure the extension of eigenvectors on a given basis, so in the case of random graphs they can be used to prove percolation transitions; see e.g. [24, 36].

From definitions (8)–(11), when $\ell_0 \approx 0$ (i.e. when $\mathbf{A_{nH}}$ and $\mathbf{A_H}$ are mostly diagonal), $r_{\mathbb{C}}(G) \approx (r_{\mathbb{C}})_{\textrm{PE}}$, $r_{\mathbb{R}}(G) \approx (r_{\mathbb{R}})_{\textrm{PE}}$, $\mathrm{IPR}(G) \approx 1$, and $S(G) \approx 0$; while for $\ell_0 = \sqrt{2}$ (i.e. when $\mathbf{A_{nH}}$ and $\mathbf{A_H}$ are full random matrices), all spectral and eigenvector measures are expected to acquire their corresponding full random matrix values.

We start by exploring some statistical properties of the vertex degree k of the model of dRGGs. In figure 2 we present examples of degree distributions P(k) for different graph sizes n (different columns correspond to different values of n) and two values of α: α = 1.5, panels (a–d), and α = 7, panels (i–l). We note that, for fixed α, the support of P(k) grows with n while keeping the shape; indeed the shape of P(k) evolves with α. Specifically, while for small α, P(k) develops a two-peak structure (one broad peak for k → 1 and one sharp peak at $k = n-1$, see e.g. panels (a–d) for α = 1.5), P(k) is unimodal for large α (see e.g. panels (i–l) for α = 7). Moreover, we observe a smooth transition in the shape of P(k) from the two-peak structure to the unimodal form for increasing α; see for example figures 3(a)–(e), where we show P(k) for several values of α. Notice, though, that in figures 3(a)–(e) we are plotting distributions of the normalized degree $\overline{k} = (k - \langle k \rangle)$ $/\sigma(k)$, where $\sigma^2(k)$ is the variance of k, to show that the shape of $P(\overline{k})$ depends on α only.

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Also, in figure 2 we plot the average degree $\langle k\rangle$ as a function of $\ell_0$ for dRGGs characterized by α = 1.5, panels (e)–(h), and α = 7, panels (m)–(p). In particular, we noticed that the curves $\langle k\rangle$ vs. $\ell_0$ for α = 7 can be well described by the expression for $\langle k(n,\ell) \rangle$, derived in [9], for standard RGGs when replacing $\ell$ by $\ell_0$:

$$\begin{equation} \langle k\rangle = (n-1) \ell_0^2 \left( \pi - \frac{8}{3}\ell_0 + \frac{1}{2}\ell_0^2 \right) ; \end{equation} \tag{ 12 }$$

see the magenta dashed lines in figures 2(m)–(p). This fact makes sense to us because, for large α, $f(\ell)$ is strongly peaked at $\ell_0$ so most radii get the value of $\ell_0$. Moreover, taking into account that we are dealing with a distribution of radii $\ell$, we found that

$$\begin{equation} \langle k\rangle = (n-1) \langle \ell \rangle^2 \left( C_1 - C_2\langle \ell \rangle + C_3\langle \ell \rangle^2 \right) \end{equation} \tag{ 13 }$$

describes the average degree of dRGGs with α > 3 even better than equation (12). In equation (13), C_i are fitting constants and $\langle \ell \rangle$ is the average connection radius that we computed directly from equation (1) as

$$\begin{equation} \langle \ell \rangle = \left\lbrace \begin{array}{ll} \eta\left[ {2}^{(2-\alpha)/2} - {\ell_{0}}^{2-\alpha} \right][2-\alpha]^{-1} & \alpha \neq 1, 2 , \\ \eta\left[ {2}^{(2-\alpha)/2} - {\ell_{0}}^{2-\alpha} \right] & \alpha = 1, \\ \eta\ln(\sqrt{2}/\ell_{0}) & \alpha = 2, \end{array} \right. \end{equation} \tag{ 14 }$$

where η is given in equation (2). An important consequence of equation (13) is that the ratio $\langle k\rangle/(n-1)$ does not depend on n, as it is verified in figures 3(f)–(j) for all values of α (not only for α > 3): there, in each panel, we present four curves for different graph sizes n that fall one on top of the other, as anticipated. Thus, in figures 3(f)–(j) we plot equation (13) (as magenta dashed lines) and see a very good correspondence with the numerical data for α > 3, while for smaller α we see good correspondence for large $\ell_0$ only. Notice form table 1 that the value of the fitting constant C₁ is approximately equal to π for all α; therefore, the prediction for $\langle k(n,\ell) \rangle$ for standard RGGs from [9] with $\ell\to \langle \ell \rangle$ approximately coincides with equation (13) to the leading order.

To complete the statistical inspection of k, in figures 3(k)–(o), we show that the variance of k grows with $\ell_0$, approaches a maximum, and then decreases to zero for $\ell\to \sqrt{2}$; where the maximum is higher the larger the value of α is. Also, in figures 3(p)–(t), we plot the ratio $\sigma^2(k)/\langle k\rangle^2$ as a function of $\ell_0$. It is interesting to note that for small α, $\sigma^2(k)/\langle k\rangle^2$ remains constant for increasing graph size (that is, the curves in panels (p, q) fall one on top of the other for different n), meaning that k is not a self-averaging quantity. On the contrary, for large α and small $\ell_0$, the ratio $\sigma^2(k)/\langle k\rangle^2$ decreases with n (see the left part of the curves in panels (s, t)); so, under these conditions, k is a self-averaging quantity.

Now, we recall that increasing $\ell_0$ from $\ell_0\approx 0$ to $\ell_0 = \sqrt{2}$ drives the model of dRGGs from mostly isolated vertices to complete graphs. This transition, as a function of $\ell_0$, could be well characterized by computing the average number of nonisolated vertices $\langle V_x(G)\rangle$ as well as the average values of the Randić and the Harmonic indices, see e.g. [36]. Certainly, we expect $\langle V_x(G)\rangle\approx 0$ when $\ell_0\approx 0$ and $\langle V_x(G)\rangle = n$ for $\ell_0 = \sqrt{2}$.

Then, in figures 4(a), (d) and (g) we present, respectively, the average number of nonisolated vertices $\langle V_x(G)\rangle$, the average Randić index $\langle R(G)\rangle$, and the average Harmonic index $\langle H(G)\rangle$ as a function of $\ell_0$ of dRGGs characterized by α = 3. We show curves for four different graph sizes n in each panel. As can be clearly seen, the three quantities $\langle X(G)\rangle$ (where X represents V_x , R or H) undergo a smooth transition (in log scale) from $\langle X(G)\rangle\approx 0$ for small $\ell_0$ to the corresponding maxima $\mbox{max}[X(G)]$ at $\ell_0 = \sqrt{2}$. Moreover, the normalized curves $\langle X(G)\rangle/\mbox{max}[X(G)]$ vs. $\ell_0$ have a very similar functional form but they are displaced to the left on the $\ell_0$–axis for increasing n; see figures 4(b), (e) and (h). In fact it was shown, for standard RGGs [36] and also for non-uniform RGGs [14], that these three normalized quantities scale with $\langle k\rangle$; that is, the curves $\langle X(G)\rangle/\mbox{max}[X(G)]$ vs. $\langle k\rangle$ fall one on top of the other for different graph sizes. Therefore, we validate this scaling for dRGGs in figures 4(c), (f) and (i) where we plot $\langle V_x(G)\rangle/n$, $\langle R(G)\rangle/(n/2)$ and $\langle H(G)\rangle/(n/2)$ vs. $\langle k\rangle$ in log–log scale (main panels) as well as in semilog scale (insets). While we observe a very good scaling for the average number of nonisolated vertices, the scaling of both the Randić and the Harmonic indices is good for $\langle k\rangle\lt1$ only.

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Furthermore, in figure 5 we explore the scaling of $\langle X(G)\rangle/\mbox{max}[X(G)]$ with $\langle k\rangle$ for dRGGs characterized by values of α ranging from 1.5 to 7. We do confirm that $\langle V_x(G)\rangle/n$ shows a perfect scaling with $\langle k\rangle$ for all the values of α examined in this work; see figures 5(a)–(e). In contrast, the scaling of $\langle R(G)\rangle/n$ and $\langle H(G)\rangle/n$ with $\langle k\rangle$ is reasonably good for α > 3 only; indeed for α < 3 these two indices do not scale in any range of $\langle k\rangle$, see figures 5(f), (g), (k) and (l). We surmise that the lack of scaling of $\langle R(G)\rangle/n$ and $\langle H(G)\rangle/n$ with $\langle k\rangle$, for small α, may be related to the non self-averaging property of k for those values of α.

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We also recall that an expression for $V_x(G)$ is known for standard RGGs [43]:

$$\begin{equation} V_x(G) \approx n[1-\exp(-n\pi \ell^2)] , \end{equation} \tag{ 15 }$$

which was recently generalized for both RGGs and non-uniform RGGs as [14]

$$\begin{equation} \langle V_x(G)\rangle \approx n[1-\exp(-\langle k\rangle)] . \end{equation} \tag{ 16 }$$

Then, given the perfect scaling of $\langle V_x(G)\rangle/n$ with $\langle k\rangle$ (meaning that $\langle V_x(G)\rangle/n$ is a function of $\langle k\rangle$ only), we test equation (16) on dRGGs by plotting it in figures 5(a)–(e); see the magenta dahsed lines. Indeed, we do confirm that equation (16) describes perfectly well our numerical data for all α. This is a remarkable result because it stresses the universality of equation (16) on RGGs models.

We now explore some spectral properties of dRGGs. We perform exact numerical diagonalization to obtain the eigenvalues λ_i and eigenvectors $\Psi^i$ $(i = 1, \ldots, n)$ of large ensembles of randomly-weighted adjacency matrices $\mathbf{A_{nH}}$ and $\mathbf{A_H}$ characterized by the parameter set $(n,\alpha,\ell_0)$.

In figures 6(a), (d) and (g) we present, respectively, the average ratio between consecutive eigenvalue spacings $\langle r_{\mathbb{C}}(G)\rangle$, the average inverse participation ratio $\langle \mathrm{IPR}(G)\rangle$, and the average Shannon entropy $\langle S(G)\rangle$ as a function of $\ell_0$ for dRGGs characterized by α = 3. Note that here the dRGGs are represented by the diluted RGE; i.e. by the non-Hermitian adjacency matrices $\mathbf{A_{nH}}$ of equation (3). In each panel we report results for graphs of four different sizes n. As for the structural measures (see figures 4(a), (d) and (g) as a reference), the three spectral measures $\langle X(G)\rangle$ (where X represents $r_{\mathbb{C}}$, $\mathrm{IPR}$ or S) undergo a smooth transition (in log scale) from $\langle X(G)\rangle\approx X_{\textrm{PE}}$ for $\ell_0\approx 0$ to the corresponding maxima $\mbox{max}[X(G)] = X_{\textrm{RGE}}$ at $\ell_0 = \sqrt{2}$. Here, $(r_{\mathbb{C}})_{\textrm{PE}}\approx 0.5$ [29], $(r_{\mathbb{C}})_{\textrm{RGE}}\approx 0.737$ [29], $\mathrm{IPR}_{\textrm{PE}} = 1$, $\mathrm{IPR}_{\textrm{RGE}} \approx n/2.04$, $S_{\textrm{PE}} = 0$, and $S_{\textrm{RGE}} \approx \ln(n/1.56)$ [29]. We note that since we did not find an expression for $\mathrm{IPR}_{\textrm{RGE}}$ in the literature we computed it numerically for the graph sizes used in this work.

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To ease the analysis of these average quantities we conveniently normalize them as

$$\begin{equation*} \langle \overline{r}_{\mathbb{C}}(G)\rangle \equiv \frac{\langle r_{\mathbb{C}}(G)\rangle - (r_{\mathbb{C}})_{\textrm{PE}}}{(r_{\mathbb{C}})_{\textrm{RGE}}-(r_{\mathbb{C}})_{\textrm{PE}}}, \end{equation*}$$

$$\begin{equation*} \langle \overline{\mathrm{IPR}}(G)\rangle \equiv \frac{\langle \mathrm{IPR}(G)\rangle - \mathrm{IPR}_{\textrm{PE}}}{\mathrm{IPR}_{\textrm{RGE}} - \mathrm{IPR}_{\textrm{PE}}} , \end{equation*}$$

and

$$\begin{equation*} \langle \overline{S}(G)\rangle \equiv \frac{\langle S(G)\rangle}{S_{\textrm{RGE}}} . \end{equation*}$$

Then, in figures 6(b), (e) and (h) we plot, respectively, the normalized average measures $\langle \overline{r}_{\mathbb{C}}(G)\rangle$, $\langle \overline{\mathrm{IPR}}(G)\rangle$, and $\langle \overline{S}(G)\rangle$ as a function of $\ell_0$. Now the curves $\langle \overline{X}(G)\rangle$ vs. $\ell_0$ have a very similar functional form but they are displaced to the left on the $\ell_0$–axis for increasing n. Following the previous subsection, in figures 6(c), (f) and (i) we test whether the curves $\langle \overline{X}(G)\rangle$ scale with $\langle k\rangle$. Indeed we observe a reasonable good scaling of $\langle \overline{r}_{\mathbb{C}}(G)\rangle$ and $\langle \overline{S}(G)\rangle$ with $\langle k\rangle$; that is, the curves of $\langle \overline{r}_{\mathbb{C}}(G)\rangle$ and $\langle \overline{S}(G)\rangle$ vs. $\langle k\rangle$ fall one on top of the other for different graph sizes n. In contrast, we note that the scaling of $\langle \overline{\mathrm{IPR}}(G)\rangle$ is not good, see figure 6(f). However, note that in figure 6 we are reporting results for dRGGs characterized by α = 3 only. Then, in figure 7 we explore the scaling of $\langle \overline{X}(G)\rangle$ with $\langle k\rangle$ for dRGGs characterized by several values of α. As for the topological indices reported in figure 5, here we observe better scaling of spectral and eigenvector properties the larger the value of α is, even for $\langle \overline{\mathrm{IPR}}(G)\rangle$; see figures 7(f)–(j).

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Recall that the results in figures 6 and 7 are for dRGGs represented by the non-Hermitian adjacency matrices $\mathbf{A_{nH}}$ of equation (3). The question now is whether the Hermitian adjacency matrices $\mathbf{A_H}$ of equation (4) provide equivalent results. Then in figure 8 we plot the average normalized ratios $\langle r_{\mathbb{C}}(G)\rangle$ and $\langle r_{\mathbb{R}}(G)\rangle$ (recall that $r_{\mathbb{C}}$ can be computed for real spectra too) for dRGGs represented by the MRME and characterized by several values of the power-law decay α. In figure 8 we have used the following normalizations:

$$\begin{equation*} \langle \overline{r}_{\mathbb{C}}(G)\rangle \equiv \frac{\langle r_{\mathbb{C}}(G)\rangle - (r_{\mathbb{C}})_{\textrm{PE}}}{(r_{\mathbb{C}})_{\textrm{MRME}(\ell\to\sqrt{2})}-(r_{\mathbb{C}})_{\textrm{PE}}} \end{equation*}$$

and

$$\begin{equation*} \langle \overline{r}_{\mathbb{R}}(G)\rangle \equiv \frac{\langle r_{\mathbb{R}}(G)\rangle - (r_{\mathbb{R}})_{\textrm{PE}}}{(r_{\mathbb{R}})_{\textrm{MRME}(\ell\to\sqrt{2})}-(r_{\mathbb{R}})_{\textrm{PE}}}. \end{equation*}$$

We just want to add that since the MRME reproduces the GOE only when the corresponding graphs are complete, the values for spectral and eigenvector measures have different values for $\ell\to\sqrt{2}$ and exactly at $\ell = \sqrt{2}$, as shown in figure 8 where we can see an abrupt decay of the curves at $\ell = \sqrt{2}$. Specifically, while $(r_{\mathbb{C}})_{\textrm{MRME}(\ell\to\sqrt{2})}\approx 0.6175$ and $(r_{\mathbb{R}})_{\textrm{MRME}(\ell\to\sqrt{2})}\approx 0.5995$ [36], $(r_{\mathbb{C}})_{\textrm{MRME}(\ell = \sqrt{2})} = (r_{\mathbb{C}})_{\textrm{GOE}}\approx 0.5688$ [36] and $(r_{\mathbb{R}})_{\textrm{MRME}(\ell = \sqrt{2})} = (r_{\mathbb{R}})_{\textrm{GOE}}\approx 0.5307$ [40].

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From figure 8 it is clear that both ratios, $\langle r_{\mathbb{C}}(G)\rangle$ and $\langle r_{\mathbb{R}}(G)\rangle$, scale better with $\langle k \rangle$ the larger the value of α is. However, even for small α (see figure 8(a), (f) corresponding to α = 1.5) the scaling is reasonably good. This is in contrast with $\langle r_{\mathbb{C}}(G)\rangle$ when computed on $\mathbf{A_{nH}}$, see figure 7(a), where the curves of $\langle r_{\mathbb{C}}(G)\rangle$ for different graph sizes n are clearly displaced on the $\langle k \rangle$–axis. We have also verified (see the appendix) that the eigenvector properties of the MRME scale with $\langle k \rangle$ better than those computed for the diluted RGE.