Eigenvalues of Random Signed Graphs with Cycles: A Graph-Centered View of the Method of Moments with Practical Applications

Abstract

We illustrate a simple connection between the cycles in a graph and eigenvalues its the adjacency matrix. Then we use this connection to derive properties of the eigenvalues of random graphs with short cyclic motifs and circulant graphs with random signs. We find that the eigenvalue distributions that emerge from those structures are surprisingly beautiful. Finally, we illustrate their practical relevance in the field Reservoir Computing.

A Appendix

1.1 A.1 Reservoir Computing

In the most common implementation of RC [JH04], the reservoir is described by

$$\begin{aligned} \begin{aligned} \mathbf {x}(t) = \tanh \left[ \mathbf {W}\mathbf {x}(t-1) + \mathbf {w_{\text {in}}} u(t) \right] \end{aligned} \end{aligned}$$

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where $\mathbf {x}(t)$ is the state of the reservoir at time t given by the output of all neurons at that time, $\mathbf {W}$ is the adjacency matrix of the reservoir network, $\mathbf {w_{\text {in}}}$ is a vector determining how the input is fed into the reservoir network and u(t) is the input time series to be processed.

The output y(t) is then given by a linear combination of the neurons’ outputs with coefficients given by the readout vector $\mathbf {r}$,

$$\begin{aligned} y(t) = \mathbf {r}\mathbf {x}(t) \end{aligned}$$

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which is usually obtained by a linear regression or similar methods.

Beyond the mathematical description, it is useful to realize that a reservoir can be seen as a set of nonlinear filters that extract features of the input signal while keeping traces of previous inputs in its dynamics and a linear readout can then be used to extract relevant information by training it with linear methods. Thus, having the right filters for a certain task – by tuning their memory or resonances, for instance – can help improve the reservoirs’ performance.

1.2 A.2 Poles and Eigenvalues

Here we look at the relationship between eigenvalues and poles in a linear infinite impulse response (IIR) filter. As illustrated in Fig. 7 an IIR filter can be implemented as a cyclic network with the appropriate cycle length: the eigenvalues of the network will then correspond to the poles of the filter and the phases of the pole tell us which frequencies are enhanced while their distance to the origin tells us how strong the enhancement is.

Fig. 7.

Simple example of the relationship between poles and eigenvalues: An IIR filter with delay 2 and feedback weight $w_1 w_2 w_3$ (left) can be implemented as a cycle of length 3 with weights $w_1, w_2, w_3$, and the poles of the transfer function are given by the eigenvalues of the adjacency matrix. The plot in the complex plane represents thus the poles (or eigenvalues) of this system.