Resonance in complex networks

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Overview

Resonance is the physical process by which the amplitude of oscillation of a dynamical system increases as the frequency of an applied periodic force approaches the natural frequency of the system. Complex systems can have different resonant frequencies. Consider a network of interacting entities. These entities may be of different types and form networks of different natures, such as biological, mechanical, technological, or social networks. Regardless of their nature, one of the simplest models for representing their interaction is to consider a graph whose nodes are coupled identical harmonic oscillators. Now suppose that a periodic force is applied to a specific node in the network. The network will respond differently depending on the frequency of that force, the node to which it is applied, and the node that is affected.

Let $\mathbf{𝐀}$ be the adjacency matrix describing the topological structure of the network and $\mathbf{𝐋}$ the corresponding Laplacian matrix. Then the resonant frequencies of the network are given by ${\omega }_i=\sqrt{1+{\mu }_i}$ where ${\mu }_i$ are the eigenvalues of the Laplacian $\mathbf{𝐋}$.

Details

To model the local interactions between $n$ identical harmonic oscillators, let us refer to an undirected graph $G=(V,E)$ without loops, where $V$ is the set of nodes and $E\subset V\times V$ is the set of edges. The Laplacian matrix is defined as $\mathbf{𝐋}=\mathbf{𝐊}-\mathbf{𝐀}$, where the adjacency matrix $\mathbf{𝐀}$ encodes the actual configuration of such interactions and $\mathbf{𝐊}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{k_i\}$ is the diagonal matrix of the degrees of the network nodes. Let us denote by ${\mu }_i$ and ${\varphi }_i,i=1,\dots ,n$, respectively the eigenvalues and eigenvectors of $\mathbf{𝐋}$, with ${\mu }_1>{\mu }_2>\dots >{\mu }_n=0$.

Let us set $\mathbf{𝐲}=\left(\begin{array}{c}\mathbf{𝐱}\\ \mathbf{𝐯}\\ \end{array}\right)\in {\mathbb{ℝ}}^{2n}$, where $\mathbf{𝐱}$ and $\mathbf{𝐯}$ are, respectively, the vector of the positions and the vector of the velocities of the $n$ oscillators. The general equation describing the network driven oscillations is given by $$\{\begin{array}{l}\dot{\mathbf{y}}=\mathbf{𝐆}\mathbf{𝐲}+\mathbf{𝐛}(t)\\ \mathbf{𝐲}(0)={\mathbf{𝐲}}_0\\ \end{array}$$ where $$\mathbf{𝐆}=\left[\begin{array}{cc}\mathbf{𝟎}& \mathbf{𝐈}\\ -(\mathbf{𝐈}+\mathbf{𝐋})& \mathbf{𝟎}\\ \end{array}\right]$$ and $$\mathbf{𝐛}(t)=f(t)\left[\begin{array}{c}\mathbf{𝟎}\\ {\mathbf{𝐞}}_h\\ \end{array}\right]$$ Moreover, $\mathbf{𝐈}\in {\mathbb{ℝ}}^{n\times n}$ represents the identity matrix, $f(t)$ the periodical driving force and ${\mathbf{𝐞}}_h=[0,0,\dots ,1,\dots ,0{]}^T$, with $1$ in position $h$, is in the standard basis of ${\mathbb{ℝ}}^n$.

The general solution for a sinusoidal driving force $f(t)=F_0\sin \omega t$ acting on node $h$, $h=1,\dots ,n$, and with $\mathbf{𝐲}(0)={\mathbf{𝐲}}_0=\mathbf{𝟎}$, is given by

$$\mathbf{𝐲}(t)=F_0\sum _{i=1}^n\frac{\omega }{{\omega }_i^2-{\omega }^2}{\varphi }_i(h)\left[\begin{array}{c}(\frac{\sin \omega t}{\omega }-\frac{\sin {\omega }_{i}t}{{\omega }_i}){\varphi }_i\\ (\cos \omega t-\cos {\omega }_{i}t){\varphi }_i\\ \end{array}\right]$$

where ${\omega }_i=\sqrt{1+{\mu }_i}$ are the resonance frequencies.

Specifically, the position variables are expressed as $$x_k(t)=F_0\omega \sum _{i=1}^{n-1}\frac{{\varphi }_i(h){\varphi }_i(k)}{{\omega }^2-{\omega }_i^2}(\frac{\sin {\omega }_{i}t}{{\omega }_i}-\frac{\sin \omega t}{\omega })+\frac{F_0}{n}\frac{\omega }{{\omega }^2-1}(\sin t-\frac{\sin \omega t}{\omega })$$

Note that, on the one hand, if ${\varphi }_{\tilde{ı}}(h)=0$, then an oscillation of frequency $\omega ={\omega }_{\tilde{ı}}$ in the source node $h$ cannot be spread throughout the network to any node and no resonance phenomenon can be induced from such a node with that frequency. The only frequency that is able to create a resonant state in the entire network is ${\omega }_{\tilde{ı}}={\omega }_n=1$. On the other hand, if ${\varphi }_{\tilde{ı}}(k)=0$, then, even if $\omega \to {\omega }_{\tilde{ı}}$, the node $k$ cannot resonate with the source node $h$ at such a frequency ${\omega }_{\tilde{ı}}$.

References

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