On the extremal spectral properties of random graphs

An Erdős-Rényi (ER) graph, denoted by $G(N,p)$, is an undirected random graph with $N$ independent vertices connected with probability $p$. Given two vertices $u$ and $v$, $p$ is the probability of an edge from vertex $u$ to vertex $v$, so $p\in[0,1]$. When $p=0$, the graph consists of $N$ isolated vertices; when $p=1$, it becomes a complete graph. We can generate graphs between these two extremes by varying the value of $p$ between $0$ and $1$. It is important to note that a given pair of parameters $(N,p)$ represents an infinite set of random graphs. Therefore, calculating a given property for a single graph may not be representative of the entire model. Instead, we can obtain more relevant information by calculating the average of that property over an ensemble of random graphs characterized by the same pair of parameters $(N,p)$. Although this statistical approach is a common practice in RMT, it is not as common in graph theory; however, it has been recently applied to several random graph models MMRS20 ; AHMS20 ; AMRS20 ; MMRS21 ; MAMRP15 ; PRRCM20 ; PM23 .

As already said above, in this work we study spectral properties of ER graphs. Moreover, to enable the comparison with standard results from RMT, we assign Gaussian random weights to the entries of the adjacency matrix $\mathbf{A}$ of $G(N,p)$ as

Here, we choose $\epsilon_{ij}$ as statistically independent random variables drawn from a normal distribution with zero mean and variance one. Also, $\epsilon_{ij}=\epsilon_{ji}$, since $G$ is undirected. According to this definition, diagonal random matrices are obtained for $p=0$ (Poisson ensemble (PE) in RMT terms), whereas the GOE (i.e. full real and symmetric random matrices) is recovered when $p=1$ M04 . Therefore, a transition from the PE to the GOE can be observed by increasing $p$ from zero to one for any given graph size $N$. Indeed, to characterize this PE–to–GOE transition, several RMT measures based on the eigenvalues and eigenvectors of the adjacency matrix of Eq. (1) have already been used; see e.g. Refs. MAM15 ; TML18 ; TFM19 ; AMR20 .

In what follows, we call the ER random graphs represented by the randomly-weighted adjacency matrix of Eq. (1) as randomly-weighted ER graphs.

We first examine the spectrum of randomly-weighted ER graphs by calculating the spectral density, which is the distribution of eigenvalues of the adjacency matrix. Indeed, for a finite graph, the spectral density can be expressed as a sum of Dirac delta functions:

where $\lambda_{j}$ is the $j-$th eigenvalue of the graph’s adjacency matrix.

A well-known result in RMT is Wigner’s semicircle law W58 . This law states that for a real, symmetric, $N\times N$ random matrix $A$ with uncorrelated elements, where $\langle A_{ij}\rangle=0$ and $\langle A_{ij}^{2}\rangle=\sigma^{2}$ for $i\neq j$, and where each moment of $\lvert A_{ij}\rvert$ remains finite as $N$ increases, the spectral density of $A/\sqrt{N}$ converges to the semicircular distribution

in the limit $N\to\infty$.

Therefore, we expect the spectrum of randomly-weighted ER graphs to approach Wigner’s semicircle law when $p\to 1$. In contrast, notice that when $p\to 0$, the adjacency matrices are diagonal with random entries drawn from a Gaussian distribution, resulting in a spectral density following that Gaussian distribution. Thus, in the transition from isolated to complete graphs, we expect to observe the transition from a Gaussian spectral density to the Wigner’s semicircle law. For intermediate values of $p$, the average variance of the elements of the adjacency matrix is strongly affected by the number of elements equal to zero, which is closely related to the average degree of the graph.

The degree of a node is the number of edges connected to it. According to several studies, see e.g. MAM15 ; AHMS20 ; AMRS20 ; MMRS20 ; MMRS21 ; PRRCM20 ; PM23 ; MAMRP15 , the average degree is a quantity that characterizes several spectral and topological properties of graphs and networks. In the case of ER-type graphs, the average degree is given by

However, in our specific case, the graphs include self-loops, which means that even when the connection probability $p$ is zero, each node is still connected to itself. This results in a modified expression for the average degree:

The random weights of the adjacency matrix $\mathbf{A}$ have unit variance. However, for decreasing connection probability, the number of zero entries in the adjacency matrix increases, effectively reducing the overall variance of the matrix. This variance reduction can be directly connected to the average degree $\langle k\rangle$. In particular, since the model includes self-connections, the percolation threshold is reached at $\langle k\rangle=2$; at this point, connected components emerge more frequently. Specifically, we find that $\sigma=\langle k\rangle$ for $\langle k\rangle<2$, while $\sigma=\langle k\rangle/2$ for $\langle k\rangle>2$.

So, following Wigner’s law and considering the relationship between variance and average degree, we propose the following expression for the spectral density of randomly-weighted ER graphs:

if $\langle k\rangle<2$, while

if $\langle k\rangle>2$. In Fig. ERGs1, we present the spectral density of randomly-weighted ER graphs for several values of $\langle k\rangle$, as specified in the caption. Indeed, Eqs. (6) (blue lines) and (7) (red lines) fit the distributions well for small and large values of $\langle k\rangle$, respectively.

From Fig. ERGs1 we observe that, unlike the binary case, for randomly-weighted ER graphs there is no clear separation between the largest and second-largest eigenvalues. Furthermore, for large values of $\langle k\rangle$, the largest eigenvalue appears to be directly related to the radius of the spectral density. Next, we analyze the behavior of the largest eigenvalue as a function of the model parameters.

In Figs. ERGs2(a) and ERGs2(b) we plot the average largest eigenvalue $\left\langle\lambda_{1}\right\rangle$ of randomly-weighted ER graphs as a function of $p$ and $N$, respectively. In the inset of Fig. ERGs2(a), we present $\left\langle\lambda_{1}\right\rangle$ as a function of the average degree and observe that it depends solely on it. Moreover, according to Eq. (7), the largest eigenvalue can be expressed as

The inset of Fig. ERGs2(a) shows that Eq. (8) (dashed black line) provides a good description of the numerical data for $k>2$, which works even better for $k>10$ (brown dashed line). Fig. ERGs2(b) shows the average largest eigenvalue $\langle\lambda_{1}\rangle$ as a function of the network size $N$ for different values of the connection probability $p$, as indicated in the legend. Green symbols correspond to higher values of $p$, while red symbols indicate lower values. We can see that $\langle\lambda_{1}\rangle$ follows a power-law dependence on $N$. Based on Eq. (8), by substituting the expression for the average degree, we obtain the approximation $\lambda_{1}\approx\sqrt{2Np}$. This approximation is indicated by dashed lines in the figure, particularly for higher values of $p$. We observe that this approximation agrees better with the numerical results when $p$ is bigger.

Moreover, we also explore the distribution of $\lambda_{1}$ for different parameter combinations. To ease the comparison of results, we normalize the largest eigenvalue as:

where $\langle\lambda_{1}\rangle$ is the mean value of the largest eigenvalue and $\sigma_{\lambda_{1}}$ its standard deviation.

Figure ERGs3 shows the average variance of the largest eigenvalue $\left\langle\sigma_{\lambda_{1}}\right\rangle$ of randomly-weighted ER graphs as a function of (a) $p$ and (b) $N$. From Fig. ERGs3(a) we can see that $\left\langle\sigma_{\lambda_{1}}\right\rangle$ is not a simple function of $p$: For small $p$ it remains almost constant, then decreases as $p$ grows, reaching a minimum, and finally increases with $p$. As a function of the graph size $N$, see Fig. ERGs3(b)), $\left\langle\sigma_{\lambda_{1}}\right\rangle$ follows a decreasing power law, where the power law depends on the value of $p$.

For the GOE, when $N\to\infty$, the largest normalized eigenvalue $\tilde{\lambda}_{1}$ follows a distribution that converges to the Tracy-Widom distribution of type 1 TW02 . In fact, for the GOE, the asymptotic expected value of the largest eigenvalue is given by $\langle\lambda_{1}\rangle=\sqrt{2N}$ and the scale of fluctuations around this expected value is given by $\sigma_{\lambda_{1}}=N^{-1/6}$ for large matrices; this value has been plotted in dotted lines in Fig. ERGs3(a) for $n=100$, $400$ and $1600$. More specifically, if $\lambda_{1}$ is the largest eigenvalue of a GOE $N\times N$ matrix, then the distribution of

converges to the Tracy-Widom distribution of type 1.

Then, in Fig. ERGs4 (top panels), we present distributions of the largest eigenvalue $P(\tilde{\lambda}_{1})$ of randomly-weighted ER graphs. $\lambda_{1}$ is normalized according with Eq. (9). Given that the average degree serves as a scaling parameter of several structural and spectral properties of graphs AMRS20 ; PM23 ; MAM15 ; MMS24 , we decided to examine $P(\tilde{\lambda}_{1})$ for fixed values of $\langle k\rangle$. To this end, we choose values of $\langle k\rangle$ from weakly to highly connected graphs: $\langle k\rangle=1.25$, 5, 50, and 100; as indicated on top of the panels. In addition, in Fig. ERGs4 (lower panels), we present distributions of the second-largest eigenvalue. We note that the distributions of both (normalized) eigenvalues tend to the Tracy-Widom distribution of type 1 for increasing $\langle k\rangle$, see the green dashed lines in Fig. ERGs4(d,h).

To better characterize the overall behavior of the eigenvalue distributions of Fig. ERGs4, we compute their excess kurtosis and skewness. Then, in Figs. ERGs5(a) and ERGs5(b), respectively, we plot the excess kurtosis and the skewness of the distribution of the largest eigenvalue $\lambda_{1}$ as a function of the connection probability $p$ of randomly-weighted ER graphs. Both quantities are normalized to the corresponding values of the Tracy-Widom distribution of type 1, which is approached when $p\to 1$; see Fig. ERGs4. In Figs. ERGs5(c) and ERGs5(d) we also plot the excess kurtosis and the skewness of $P(\lambda_{2})$, respectively.

From Figs. ERGs5(a) and ERGs5(b), when $p$ is small, we observe that the excess kurtosis and skewness of $P(\lambda_{1})$ deviate considerably from the Tracy-Widom value. However, as $p$ increases, as expected, both quantities progressively approach the reference Tracy-Widom values; although fluctuations persist at large values of $p$. In the case of $P(\lambda_{2})$, the situation is different: While both the excess kurtosis and the skewness decrease with $p$, they do not approach the reference Tracy-Widom values; see Figs. ERGs5(c) and ERGs5(d). This may be expected, even with the apparent good correspondence between $P(\lambda_{2})$ and the Tracy-Widom distribution shown in Fig. ERGs4(h), since Tracy-Widom–type statistics is expected for $\lambda_{1}$ only.

Now, we also compute the correlation coefficient between the first and second eigenvalues, $c_{\lambda_{1},\lambda_{2}}$, defined by

$c_{\lambda_{1},\lambda_{2}}$ as a function of $p$ of randomly-weighted ER graphs is shown in Fig. ERGs6. From this figure, we observe that, for small $p$, the correlation coefficient remains approximately constant and close to $c_{\lambda_{1},\lambda_{2}}=0.65$. This may be regarded as the PE value. Then, for further increasing $p$, $c_{\lambda_{1},\lambda_{2}}$ grows, reaches a maximum, and finally decreases until reaching the predicted value for the GOE, $c_{\lambda_{1},\lambda_{2}}\approx 0.53$ CBP23 .

From the results presented so far, we have observed that the statistical properties of the largest eigenvalue (and also of the second largest eigenvalue) of the model of randomly-weighted ER graphs shows a clear PE–to–GOE transition as a function of the graph connectivity, or more precisely, as a function of the average degree. Although the PE–to–GOE transition as a function of $\langle k\rangle$ has been reported previously by the use of the full spectrum, see e.g. MAMRP15 , it is important to highlight that our results demonstrate that the PE–to–GOE transition could also be probed by the use of the extreme eigenvalues only. To further verify this statement, we now compute the distribution of the normalized distance between the largest and second-largest eigenvalues, defined as

and the distribution of the ratio between higher consecutive eigenvalues spacings

In Fig. ERGs7 we present $P(s)$ (top panels) and $P(r)$ (bottom panels) of randomly-weighted ER graphs for different values of $\langle k\rangle$, as indicated at the top of the panels. Note that we are using the same values of $\langle k\rangle$ reported in Fig. ERGs4. In Fig. ERGs7 we are also including the RMT predictions of both $P(s)$ and $P(r)$ for the PE and the GOE, which are given by

and

However, it is important to stress that Eqs. (14-17) are expected to work at the bulk of the spectrum. Then, from Fig. ERGs7 we observe that: (i) for fixed $\langle k\rangle$ the shape of both $P(s)$ and $P(r)$ remains invariant, i.e. they do not depend on the graph size; except for intermediate values of $\langle k\rangle$, see Figs. ERGs7(b,f); (ii) for small [large] values of $\langle k\rangle$, the shapes of $P(s)$ and $P(r)$ are well described by $P_{\mbox{\tiny PE}}(s)$ and $P_{\mbox{\tiny PE}}(r)$ [$P_{\mbox{\tiny GOE}}(s)$ and $P_{\mbox{\tiny GOE}}(r)$], respectively; (iii) both $P(s)$ and $P(r)$ can be used to probe the PE–to–GOE transition.

To further explore the properties of the edge of the spectrum, here we compute the inverse participation ratio (IPR) of the eigenvectors corresponding to the extreme eigenvalues. Given the normalized eigenvectors $\Psi^{i}$, its IPR is defined as

In fact, we compute distributions of IPRs of eigenvectors of randomly-weighted ER graphs, as shown in Fig. ERGs8. Notice that each column in Fig. ERGs8 corresponds to a fixed value of $\langle k\rangle$; we used the same values of $\langle k\rangle$ reported in Figs. ERGs4 and ERGs7. Specifically, in Fig. ERGs8 we show the distributions of the IPRs of the eigenvectors corresponding to the largest eigenvalue, see panels (a-d), and to the second largest eigenvalue, see panels (e-h). Moreover, for comparison purposes, we also show $P(\mbox{IPR})$ of the eigenvectors corresponding to the central eigenvalue, see panels (i-l), the smallest eigenvalue, see panels (m-p), and the average IPR over the full spectrum, see panels (q-t).

From Fig. ERGs8 we observe that all IPR distributions display a clear delocalization transition as the average degree of the graph increases. For small values of $\langle k\rangle$, see the left panels corresponding to $\langle k\rangle=5$, the IPR distributions show a pronounced peak at $\mbox{IPR}=3$, a signature of strongly localization and a characteristic of the PE. For large values of $\langle k\rangle$, see the right panels corresponding to $\langle k\rangle=100$, the IPR distributions follow a Gaussian-like bell shape centered at $\mbox{IPR}\propto N$, indicating delocalization which is the main characteristic of the GOE. For intermediate values of $\langle k\rangle$, we observe a transition between the PE and the GOE regimes. The eigenvectors corresponding to the extreme eigenvalues (largest, second largest, and smallest) exhibit similar localization characteristics. This, in contrast with the eigenvector corresponding to the central eigenvalue $\lambda_{N/2}$, which also displays the delocalization transition but in a different manner. The distributions of the average IPR, shown in panels (q-t), included for comparison purposes, clearly show that, on average, the localization properties of the bulk eigenvalues dominate.

We recall then that the localization properties of the eigenvectors are not uniform across the spectrum. In general, the eigenvectors at the center of the spectrum are more delocalized than those at the edges. Also, extreme states tend to be more closely aligned with the topological structure of the graph than those at the spectrum center, see e.g. FDBV01 . Indeed, the spectral heterogeneity is key to understanding dynamical processes in graphs, such as diffusion, synchronization, or stability, where different modes can have differentiated contributions depending on their degree of localization; see e.g. ADKMZ08 ; PC98 .