* National Natural Science Foundation of China.11305078 National Natural Science Foundation of China.61877030
Abstract It has been found that a hub node is better able to amplify weak external signals than leaf nodes in star networks. But hub-enhanced amplification is only attained by a single hub node and is limited to weak coupling strength. We show here that random initial phases in external weak signals do not affect the hub-enhanced amplification at weak coupling strength. Instead, they can improve the responses of all the leaf nodes to external signals at large coupling strength, resulting in a double resonance-like signal amplification. We use a reduced model to analyze the influence of the star structure and random initial phases on the emergence of double resonance. Keywords：signal amplification;random initial phases;star networks;bistable units
PDF (487KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Yang Ge, Xiao-Ming Liang. Double resonance in star networks of bistable units with disordered signals*. Communications in Theoretical Physics, 2020, 72(5): 055601- doi:10.1088/1572-9494/ab7ec7
1. Introduction
Over the last three decades much effort has been devoted to finding conditions for the improvement of signal amplification [1–7]. A remarkable example is stochastic resonance (SR), wherein the response of a nonlinear system to a weak external signal is optimized by the presence of noise [8, 9]. The SR response of an isolated system can be further enhanced by coupling it into an array, which is known as array-enhanced SR [10, 11]. Moreover, the effect of array-enhanced SR can be maximized at a certain array size, i.e., system size resonance [12, 13]. In addition to regular arrays, many real systems can be well modeled by complex networks [14–17]. Recent studies have shown that the topology of complex networks can significantly affect the performance of SR. For instance, small-world networks with small numbers of long-range links can greatly improve SR and synchronization [18–20]; scale-free networks with star structures can generate double SR on hub nodes and single SR on leaf nodes [21–23].
Previous works on signal amplification have focused on identical signals; that is, the initial phases of the external signals are assumed to be the same. However, if the source of the signal is at different distances, the initial phases of received signals may be different, and such phase differences are found to be important for hunting prey in biological systems [24–26]. For instance, to determine the prey’s angle, surface-feeding fish can discriminate different arrival times of a target signal between distributed lateral organs [27]. To pinpoint prey in the dark, the barn owl turns its head toward the source of the sound to obtain the optimal interaural time difference between its two ears [28]. Accordingly, it is interesting to investigate the effect of random initial phases on signal amplification.
To this end, we study the signal amplification in star networks of bistable units subjected to disordered signals. The reason for considering star networks is that the star-type structure is an important naturally occurring motif that performs cognitive and sensory functions, ranging from a single neuron to neural networks [29–32]. We find that a double resonance emerges when star networks are forced by disordered signals with random initial phases, in contrast to a single resonance when forced by identical signals with the same initial phase. Moreover, we find that the first resonance appears at weak coupling strength and is attained by the hub node, while the secondary resonance appears at large coupling strength and is contributed by the leaf nodes. In addition, we find that the first resonance is more sensitive to network size than the secondary, and both resonances are sensitive to the frequency of the external signals. The analytical results show that the interplay of star structure and random initial phases results in the emergence of double resonance.
2. Model and method
We consider a star network of N+1 bistable units written as$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{\dot{x}}_{0} & = & {x}_{0}-{x}_{0}^{3}+\lambda \displaystyle \sum _{i=1}^{N}({x}_{i}-{x}_{0})+A\sin (\omega t+{\phi }_{0}),\\ {\dot{x}}_{i} & = & {x}_{i}-{x}_{i}^{3}+\lambda ({x}_{0}-{x}_{i})+A\sin (\omega t+{\phi }_{i}),\,i=1,\cdots ,N,\end{array}\end{array}\,\end{eqnarray}$where x0 and xi denote the dynamics of the hub and leaf nodes, respectively. λ is the coupling strength, and $A\sin (\omega t+{\phi }_{i})$ is the external signal with amplitude A, frequency ω and initial phase φi.
Without coupling, i.e., λ=0, the bistable unit can generate two distinct responses to the external signal, depending on the signal amplitude A. When A is below a threshold Ac (subthreshold signal), the bistable unit jiggles about one of its two stable fixed points xs=±1; otherwise, it oscillates about the unstable fixed point xu=0 if $A\geqslant {A}_{c}$ (suprathreshold signal). For a low frequency ω, the threshold amplitude is ${A}_{c}\approx 0.41$ (see figure 3(d)). To model weak disordered signals, we set the signal amplitude to be A=0.35 and frequency ω=π/50 unless mentioned otherwise, and assign the initial phase φi a random value drawn from a uniform distribution over (0, 2π).
To characterize the response of each node to the external signal, we use the signal amplification factor defined by [33]$\begin{eqnarray}{g}_{i}\equiv \displaystyle \frac{{\max }_{t}{x}_{i}(t)-{\min }_{t}{x}_{i}(t)}{2},\end{eqnarray}$where gi is the amplitude of the ith node. Then, the maximum signal amplification of the star network is defined as$\begin{eqnarray}{G}_{k}\equiv {\max }_{i=0}^{N}\ {g}_{i},\end{eqnarray}$where k represents the index of the node with the maximum response. If k=0, the maximum signal amplification is generated by the hub node, while it is produced by one of the leaf nodes if $k\ne 0$. In numerical simulations, the initial conditions of the nodes are chosen at random from xs=±1, and the maximum signal amplification Gk is averaged over 100 different realizations.
3. Numerical results
Let us firstly recall what happens if all the nodes receive identical signals with the same initial phase, i.e., ${\phi }_{0}\,=\,\cdots \,=\,{\phi }_{N}$. Figure 1(a) shows the maximum signal amplification Gk as a function of coupling strength λ with a fixed N=500. As can be seen, a maximal Gk is obtained as λ increases to λ1≈0.0003. Figures 1(b) and (c) show that this maximal amplification is obtained by the hub node since g0>gi, i=1, ⋯, N. Such hub-enhanced amplification at weak coupling strength is similar to the observation shown in [33]. One difference, however, is that a small secondary peak of Gk appears when λ approaches a critical coupling strength λ2≈0.063. But after that point, Gk drops to a small constant value. In contrast to the primary peak generated by the hub node, the small secondary peak at λ2 is attributed to the transition to network synchronization. That is, the hub and leaf nodes reach complete synchronization when λ>λ2. Then, all nodes have the same response to the external signals, i.e., g0=⋯=gN; see figure 1(d). From figure 1, the star networks can largely amplify the weak identical signals, but the enhanced amplification is solely attained by the hub node and is limited for weak coupling strength.
Figure 1.
New window|Download| PPT slide Figure 1.Signal amplification of equation (1) for identical signals. (a) The maximum signal amplification Gk versus λ, where the dashed line denotes the theoretical approximation of equation (17); (b) ${g}_{0}\gt {g}_{i},i=1,\cdots ,N$ at λ=0.0001; (c) g0>gi, i=1, ⋯, N at λ=0.001; (d) g0=⋯=gN at λ=0.1. Parameter N=500 is considered.
We next turn our attention to the case of disordered signals with random initial phases, i.e., ${\phi }_{i}\ne {\phi }_{j}$. Figure 2(a) displays the maximum signal amplification Gk as a function of coupling strength λ, in the presence of random initial phases. It can be found that the star network now generates a double resonance-like amplification of the disordered signals. The first resonance appears at weak coupling strength and is produced by the hub node since g0>gi, i=1, ⋯, N (see figure 2(b)), similar to the case for identical signals. Interestingly, the secondary resonance emerges at large coupling strength and is contributed by the leaf nodes since g0<gi, i=1, ⋯, N (see figures 2(c) and (d)). In addition, the secondary resonance ceases to exist when λ>0.4, since the star network attains high levels of synchronization above that coupling strength (all the nodes oscillate about the same fixed point xs=1 or xs=−1). By comparison, the random initial phases do not destroy the hub-enhanced amplification at the weak coupling strength; instead, they can further improve the responses of the leaf nodes to external signals at large coupling strength, which improves the applicability of star networks in signal amplification.
Figure 2.
New window|Download| PPT slide Figure 2.Signal amplification of equation (1) for disordered signals. (a) The maximum signal amplification Gk versus λ, where the dashed line denotes the theoretical approximation of equation (33); (b) g0>gi, i=1, ⋯, N at λ=0.001; (c) g0<gi, i=1, ⋯, N at λ=0.2; (d) g0<gi, i=1, ⋯, N at λ=0.7. Parameter N=500 is considered.
To show the influence of network size on double resonance, figure 3(a) depicts the two resonance peaks ${G}_{k}^{i},i=1,2$ as a function of N. The first resonance peak Gk1 does not change with N. In contrast, the second resonance peak Gk2 increases with N and reaches a saturated value at large N. Meanwhile, figure 3(b) shows the optimal coupling strength λi corresponding to Gki as a function of N. The first optimal coupling strength λ1 decreases with N, satisfying ${\lambda }_{1}\sim {N}^{-1}$. This relationship implies that the sensitivity of star networks to external signals is improved by a larger number of leaf nodes. Unlike λ1, the secondary optimal coupling strength is more robust to the network size, since λ2 remains constant at ${\lambda }_{2}\approx 0.25$. In addition, figure 3(c) displays the dependences of the resonance peaks Gki on the signal frequency ω. The first resonance peak Gk1 decreases monotonically with ω, while the secondary resonance peak Gk2 decreases to a constant when ω>0.1. Finally, figure 3(d) compares the maximum signal amplification Gk as a function of signal amplitude A between identical and disordered signals, keeping coupling strength λ2=0.25 and network size N=500. Clearly, when A is close to but below the threshold Ac, the network with disordered signals exhibits a larger response than that with identical signals. However, when $A\geqslant {A}_{c}$, the network with identical signals generates a larger response since the external signals become suprathreshold and the network reaches complete synchronization. From these observations, the double resonance induced by disordered signals is sensitive to the signal frequency ω as well as the amplitude A.
Figure 3.
New window|Download| PPT slide Figure 3.Effect of network size, signal frequency and signal amplitude on weak signal amplification with disordered signals. (a) Resonance peaks ${G}_{k}^{i},i=1,2$ as a function of N; (b) the optimal coupling strength λi corresponding to Gki as a function of N, where the dashed line denotes ${\lambda }_{1}\sim {N}^{-1}$; (c) the secondary resonance peak Gk2 as a function of ω, with a fixed N=500; (d) the maximum signal amplification Gk as a function of A for incidental and disordered signals, with fixed λ=0.25 and N=500.
4. Analysis
4.1. Identical signals
We first explain the single resonance for identical signals. Without loss of generality, we assume the initial phase φi=0 for 1≤i≤N. As mentioned above, the star network becomes completely synchronized when the coupling strength is above λ2. Since λ2 is a small value, before this critical coupling strength the force λx0 is tiny compared to that of the external signal. For this reason, the force λx0 can be neglected in the dynamics of the leaf nodes, yielding$\begin{eqnarray}{\dot{x}}_{i}=(1-\lambda ){x}_{i}-{x}_{i}^{3}+A\sin (\omega t),\,i=1,\cdots ,N.\end{eqnarray}$The potential function of equation (4) is $V(x,t)\,=-(1-\lambda ){x}^{2}/2-{x}^{4}/4-{Ax}\sin (\omega t)$, which is modulated by the coupling strength λ since the signal amplitude A is fixed. When λ increases above a critical coupling strength ${\lambda }_{{\ell }}$, the potential barrier of V(x, t) disappears. The critical coupling strength λℓ is given by the formula$\begin{eqnarray}{\left(\displaystyle \frac{A}{2}\right)}^{2}={\left(\displaystyle \frac{1-{\lambda }_{{\ell }}}{3}\right)}^{3}.\end{eqnarray}$Equation (5) shows that ${\lambda }_{{\ell }}$ is determined only by the signal amplitude A. For A=0.35, we obtain$\begin{eqnarray}{\lambda }_{\ell }=1-3{\left(\displaystyle \frac{A}{2}\right)}^{2/3}\approx 0.061,\end{eqnarray}$which is close to the critical coupling strength ${\lambda }_{2}\approx 0.063$ for network synchronization. The reason is that, when $\lambda \geqslant {\lambda }_{{\ell }}$, the external signals become suprathreshold in equation (4), which forces the leaf nodes as well as the hub node to synchronize. But when the synchronization is achieved, the external signals return to subthreshold again, since the coupling terms $\lambda ({x}_{0}-{x}_{i})=0$ vanish in equation (1). As a result, equation (1) behaves as a single bistable unit driven by the subthreshold signal$\begin{eqnarray}{\dot{x}}_{i}={x}_{i}-{x}_{i}^{3}+A\sin (\omega t),\,i=0,\cdots ,N.\end{eqnarray}$Its approximate solution is given by$\begin{eqnarray}{x}_{i}(t)\approx {x}_{s}+\displaystyle \frac{A}{\sqrt{4+{\omega }^{2}}}\sin (\omega t+{\psi }_{1}),\end{eqnarray}$where $\Psi$1 denote the phase shift.
Before network synchronization, i.e., $\lambda \lt {\lambda }_{{\ell }}$, the external signal is subthreshold in equation (4). The approximate solution of equation (4) is then given by$\begin{eqnarray}\begin{array}{rcl}{x}_{i}(t) & \approx & \pm \sqrt{1-\lambda }+\displaystyle \frac{A}{4{\left(1-\lambda \right)}^{2}+{\omega }^{2}}\\ & & \times [2(1-\lambda )\sin (\omega t)-\omega \cos (\omega t)],\end{array}\end{eqnarray}$where $\pm \sqrt{1-\lambda }$ denote the oscillation centers (stable fixed points of equation (4) at A=0) depending on the initial conditions. For simplicity, we assume that half of the leaf nodes have the same positive initial conditions and the other half have the same negative initial conditions, i.e., the initial conditions of the leaf nodes are equally split. As a result, we obtain$\begin{eqnarray}\sum _{i=1}^{N}{x}_{i}=\displaystyle \frac{{AN}}{4{\left(1-\lambda \right)}^{2}+{\omega }^{2}}[2(1-\lambda )\sin (\omega t)-\omega \cos (\omega t)].\end{eqnarray}$Substituting equation (10) into equation (1), the dynamics of the hub node can be rewritten as$\begin{eqnarray}{\dot{x}}_{0}=(1-N\lambda ){x}_{0}-{x}_{0}^{3}+\sqrt{{a}^{2}+{b}^{2}}\sin (\omega t+{\psi }_{2}),\end{eqnarray}$where $a\equiv A[2N\lambda {(1-\lambda )+4(1-\lambda )}^{2}+{\omega }^{2}]$/$[4(1-{\lambda )}^{2}\,+{\omega }^{2}]$, $b\equiv -{AN}\lambda \omega /[4{(1-\lambda )}^{2}+{\omega }^{2}]$ and $\Psi$2 denotes the phase shift. Similarly, there exists a critical coupling strength λh above which the periodic signal becomes suprathreshold in equation (11), where λh is determined by$\begin{eqnarray}{\left(\displaystyle \frac{\sqrt{{a}^{2}+{b}^{2}}}{2}\right)}^{2}={\left(\displaystyle \frac{1-N{\lambda }_{h}}{3}\right)}^{3}.\end{eqnarray}$For N=500, ω=π/50 and A=0.35, equation (12) gives$\begin{eqnarray}{\lambda }_{h}\approx 0.0001.\end{eqnarray}$This critical coupling strength corresponds well to the numerical result λ≈0.0002 at which the first resonance occurs.
The signal is subthreshold for the hub node when λ<λh. Then the approximate solution of equation (11) is obtained as$\begin{eqnarray}{x}_{0}(t)\approx \pm \sqrt{1-N\lambda }+\displaystyle \frac{\sqrt{{a}^{2}+{b}^{2}}}{\sqrt{4{\left(1-N\lambda \right)}^{2}+{\omega }^{2}}}\sin (\omega t+{\psi }_{3}),\end{eqnarray}$where $\pm \sqrt{1-N\lambda }$ are the stable fixed points of equation (11) at A=0 for λ≤N−1, and $\Psi$3 is the phase shift.
On the other hand, when ${\lambda }_{h}\leqslant \lambda \lt {\lambda }_{{\ell }}$, the signal is suprathreshold for the hub node. In this situation, the fixed points $\pm \sqrt{1-N\lambda }$ lose stability, which allows the hub node to generate large oscillations about xs=0. Then the approximate solution of equation (11) becomes$\begin{eqnarray}{x}_{0}(t)\approx 2\sqrt{\displaystyle \frac{1-N\lambda }{3}}\cosh \left[\displaystyle \frac{1}{3}{\rm{arccosh}}\left(\sqrt{\displaystyle \frac{27({a}^{2}+{b}^{2})}{4{\left(1-N\lambda \right)}^{3}}}\right)\right]\sin (\omega t)\end{eqnarray}$for ${\lambda }_{h}\leqslant \lambda \leqslant {N}^{-1}$, and becomes$\begin{eqnarray}{x}_{0}(t)\approx 2\sqrt{\displaystyle \frac{N\lambda -1}{3}}\sinh \left[\displaystyle \frac{1}{3}{\rm{arcsinh}}\left(\sqrt{\displaystyle \frac{27({a}^{2}+{b}^{2})}{4{\left(N\lambda -1\right)}^{3}}}\right)\right]\sin (\omega t)\end{eqnarray}$for ${N}^{-1}\lt \lambda \lt {\lambda }_{l}$.
Using equation (8) together with equations (14)–(16), the maximum signal amplification Gk of the star network with identical signals is given by$\begin{eqnarray}\begin{array}{c}\begin{array}{l}{G}_{k}\,\approx \\ \left\{\begin{array}{c}{\textstyle \tfrac{\sqrt{{a}^{2}+{b}^{2}}}{\sqrt{4{\left(1-N\lambda \right)}^{2}+{\omega }^{2}}}}\,{\rm{i}}{\rm{f}}\,\lambda \lt {\lambda }_{h},\\ 2\sqrt{{\textstyle \tfrac{1-N\lambda }{3}}}\cosh \left[{\textstyle \tfrac{1}{3}}{\rm{a}}{\rm{r}}{\rm{c}}{\rm{c}}{\rm{o}}{\rm{s}}{\rm{h}}\left(\sqrt{{\textstyle \tfrac{27({a}^{2}+{b}^{2})}{4{\left(1-N\lambda \right)}^{3}}}}\right)\right]\,{\rm{i}}{\rm{f}}\,{\lambda }_{h}\leqslant \lambda \leqslant {N}^{-1},\\ 2\sqrt{{\textstyle \tfrac{N\lambda -1}{3}}}\sinh \left[{\textstyle \tfrac{1}{3}}{\rm{a}}{\rm{r}}{\rm{c}}{\rm{s}}{\rm{i}}{\rm{n}}{\rm{h}}\left(\sqrt{{\textstyle \tfrac{27({a}^{2}+{b}^{2})}{4{\left(N\lambda -1\right)}^{3}}}}\right)\right]\,{\rm{i}}{\rm{f}}\,{N}^{-1}\lt \lambda \lt {\lambda }_{\ell },\\ {\textstyle \tfrac{A}{\sqrt{4+{\omega }^{2}}}}\,{\rm{i}}{\rm{f}}\,\lambda \geqslant {\lambda }_{\ell }.\end{array}\right.\end{array}\end{array}\,\end{eqnarray}$The prediction of equation (17) is in good agreement with the simulation result shown in figure 1. Therefore, the star structure and the equally split initial conditions of the leaf nodes cause the instability of the stable fixed points of the hub node in the coupling region ${\lambda }_{h}\leqslant \lambda \lt {\lambda }_{{\ell }}$, resulting in the hub-enhanced amplification above there.
4.2. Disordered signals
Based on the above analysis, we now explain the mechanism of the double resonance for disordered signals. We assume that the oscillations of the leaf nodes can be expressed as ${x}_{i}(t)={\xi }_{i}+B\sin (\omega t+{\phi }_{i})$, where ξi denotes the oscillation center of the ith leaf node. Under this assumption, we have$\begin{eqnarray}\sum _{i=1}^{N}{x}_{i}=\sum _{i=1}^{N}{\xi }_{i}+B\sin (\omega t)\sum _{i=1}^{N}\cos ({\phi }_{i})+B\cos (\omega t)\sum _{i=1}^{N}\sin ({\phi }_{i}).\end{eqnarray}$Moreover, we assume that half the leaf nodes have the same positive initial conditions and the other half have the same negative initial conditions. In this way, we get ${\sum }_{i=1}^{N}{\xi }_{i}=0$ for low levels of network synchronization (λ≤0.4) and ${\sum }_{i=1}^{N}{\xi }_{i}=\pm N$ for high levels (λ>0.4). In addition, we assume that the initial phases are uniformly distributed in the interval (0, 2π); then we obtain ${\sum }_{i=1}^{N}\cos ({\phi }_{i})=0$ and ${\sum }_{i=1}^{N}\sin ({\phi }_{i})=0$. As a result, equation (18) becomes$\begin{eqnarray}\left\{\begin{array}{ll}{\sum }_{i=1}^{N}{x}_{i}=0 & \quad \mathrm{if}\ \lambda \leqslant 0.4,\\ {\sum }_{i=1}^{N}{x}_{i}=\pm N & \quad \mathrm{if}\ \lambda \gt 0.4.\end{array}\right.\end{eqnarray}$
When λ≤0.4, equation (1) becomes$\begin{eqnarray}{\dot{x}}_{0}=(1-N\lambda ){x}_{0}-{x}_{0}^{3}+A\sin (\omega t+{\phi }_{0}).\end{eqnarray}$Analogously, the external signal becomes suprathreshold if the coupling strength is above a critical coupling strength ${\lambda }_{h^{\prime} }$, which is given by$\begin{eqnarray}{\lambda }_{{h}^{{\rm{{\prime} }}}}=\displaystyle \frac{1}{N}\left[1-3{\left(\displaystyle \frac{A}{2}\right)}^{2/3}\right].\end{eqnarray}$Clearly, ${\lambda }_{h^{\prime} }$ is inversely proportional to N, which well explains the increased sensitivity to external signals as N increases (see figure 3(b)). For A=0.35 and N=500, equation (21) gives$\begin{eqnarray}{\lambda }_{h^{\prime} }\approx 0.0001.\end{eqnarray}$Figure 2 shows that this critical coupling strength fits well to the numerical result λ≈0.0002 at which the first resonance occurs.
When $\lambda \lt {\lambda }_{h^{\prime} }$, the external signal is subthreshold for the hub node, and thus the approximate solution of equation (20) is given by$\begin{eqnarray}{x}_{0}(t)\approx \pm \sqrt{1-N\lambda }+\displaystyle \frac{A}{\sqrt{4{\left(1-N\lambda \right)}^{2}+{\omega }^{2}}}\sin (\omega t+{\psi }_{4}),\end{eqnarray}$where $\pm \sqrt{1-N\lambda }$ denote the stable fixed points of equation (20) at A=0, and $\Psi$4 represents the phase shift.
When $\lambda \geqslant {\lambda }_{h^{\prime} }$, the external signal is suprathreshold in equation (20), and the approximate solution of x0(t) is$\begin{eqnarray}\begin{array}{c}\begin{array}{l}{x}_{0}(t)\approx 2\sqrt{\displaystyle \frac{1-N\lambda }{3}}\cosh \left[\displaystyle \frac{1}{3}{\rm{a}}{\rm{r}}{\rm{c}}{\rm{c}}{\rm{o}}{\rm{s}}{\rm{h}}\left(\sqrt{\displaystyle \frac{27{A}^{2}}{4{\left(1-N\lambda \right)}^{3}}}\right)\right]\\ \,\times \,\sin (\omega t+{\phi }_{0})\end{array}\end{array}\end{eqnarray}$for ${\lambda }_{h^{\prime} }\leqslant \lambda \leqslant {N}^{-1}$, and$\begin{eqnarray}\begin{array}{c}\begin{array}{l}{x}_{0}(t)\approx 2\sqrt{\displaystyle \frac{N\lambda -1}{3}}\sinh \left[\displaystyle \frac{1}{3}{\rm{a}}{\rm{r}}{\rm{c}}{\rm{s}}{\rm{i}}{\rm{n}}{\rm{h}}\left(\sqrt{\displaystyle \frac{27{A}^{2}}{4{\left(N\lambda -1\right)}^{3}}}\right)\right]\\ \,\times \,\sin (\omega t+{\phi }_{0}).\end{array}\end{array}\end{eqnarray}$for N−1<λ≤0.4.
Finally, when λ>0.4, equation (1) becomes$\begin{eqnarray}{\dot{x}}_{0}=(1-N\lambda ){x}_{0}-{x}_{0}^{3}\pm N\lambda +A\sin (\omega t+{\phi }_{0}).\end{eqnarray}$Its approximate solution is given by$\begin{eqnarray}{x}_{0}(t)\approx \pm {x}_{s}+\displaystyle \frac{A}{\sqrt{{\left(2+N\lambda \right)}^{2}+{\omega }^{2}}}\sin (\omega t+{\psi }_{2}).\end{eqnarray}$Equation (27) indicates that the oscillation of the hub node is tiny, since the oscillation amplitude $A/\sqrt{{\left(2+N\lambda \right)}^{2}+{\omega }^{2}}\approx 0$ when λ>0.4.
To analyze the signal amplification of the leaf nodes, we note that the maximum signal amplification Gk is dominated by the leaf nodes when λ>0.005 (see figure 2(a)). In addition, equations (25) and(27) show that the oscillation amplitude of the hub node is small when λ>N−1. Based on these observations, we can obtain the approximations λx0≈0 for 0.005<λ≤0.4 and λx0≈±λ for λ>0.4. In the former case, the dynamics of the leaf nodes simplify to$\begin{eqnarray}{\dot{x}}_{i}=(1-\lambda ){x}_{i}-{x}_{i}^{3}+A\sin (\omega t+{\phi }_{i}),\,i=1,\cdots ,\,N,\end{eqnarray}$which is the same as equation (4). So the approximate solution of equation (28) for λ<λℓ is given by$\begin{eqnarray}{x}_{i}(t)\approx \pm \sqrt{1-\lambda }+\displaystyle \frac{A}{\sqrt{4{\left(1-\lambda \right)}^{2}+{\omega }^{2}}}\sin (\omega t+{\phi }_{i}+{\psi }_{5}),\end{eqnarray}$where $\pm \sqrt{1-\lambda }$ are the stable fixed points of equation (28) at A=0 and $\Psi$5 is the phase shift. For ${\lambda }_{{\ell }}\leqslant \lambda \lt 0.4$, the fixed points $\pm \sqrt{1-\lambda }$ lose stability and the approximate solution of equation (28) becomes$\begin{eqnarray}\begin{array}{c}\begin{array}{l}{x}_{i}(t)\approx 2\sqrt{\displaystyle \frac{1-\lambda }{3}}\cosh \left[\displaystyle \frac{1}{3}{\rm{a}}{\rm{r}}{\rm{c}}{\rm{c}}{\rm{o}}{\rm{s}}{\rm{h}}\left(\sqrt{\displaystyle \frac{27{A}^{2}}{4{\left(1-\lambda \right)}^{3}}}\right)\right]\\ \,\times \,\sin (\omega t+{\phi }_{i}).\end{array}\end{array}\end{eqnarray}$From equations (25) and (30), all the leaf nodes can generate larger oscillations to disordered signals than the hub node in the coupling region $({\lambda }_{{\ell }},0.4)$.
In the latter case, i.e., $\lambda {x}_{0}\approx \pm \lambda $ for $\lambda \gt 0.4$, the dynamics of the leaf nodes become$\begin{eqnarray}{\dot{x}}_{i}=(1-\lambda ){x}_{i}-{x}_{i}^{3}\pm \lambda +A\sin (\omega t+{\phi }_{i}),\,i=1,\cdots ,N.\end{eqnarray}$The approximate solution of equation (31) is obtained as$\begin{eqnarray}{x}_{i}(t)\approx {x}_{s}+\displaystyle \frac{A}{\sqrt{{\left(2+\lambda \right)}^{2}+{\omega }^{2}}}\sin (\omega t+{\phi }_{i}+{\psi }_{6}),\end{eqnarray}$where $\Psi$6 is some phase shift. In contrast to equation (27), the oscillation amplitudes of the leaf nodes are larger than that of the hub node when λ>0.4.
From the above analysis, the maximum signal amplification Gk of the star network with disordered signals is obtained as$\begin{eqnarray}\begin{array}{c}\begin{array}{l}{G}_{k}\,\approx \\ \left\{\begin{array}{c}{\textstyle \tfrac{A}{\sqrt{4{\left(1-N\lambda \right)}^{2}+{\omega }^{2}}}}\,{\rm{i}}{\rm{f}}\,\lambda \lt {\lambda }_{{h}^{{\rm{{\prime} }}}},\\ 2\sqrt{{\textstyle \tfrac{1-N\lambda }{3}}}\cosh \left[{\textstyle \tfrac{1}{3}}{\rm{a}}{\rm{r}}{\rm{c}}{\rm{c}}{\rm{o}}{\rm{s}}{\rm{h}}\left(\sqrt{{\textstyle \tfrac{27{A}^{2}}{4{\left(1-N\lambda \right)}^{3}}}}\right)\right]\,{\rm{i}}{\rm{f}}\,{\lambda }_{{h}^{{\rm{{\prime} }}}}\leqslant \lambda \leqslant {N}^{-1},\\ 2\sqrt{{\textstyle \tfrac{N\lambda -1}{3}}}\sinh \left[{\textstyle \tfrac{1}{3}}{\rm{a}}{\rm{r}}{\rm{c}}{\rm{s}}{\rm{i}}{\rm{n}}{\rm{h}}\left(\sqrt{{\textstyle \tfrac{27{A}^{2}}{4{\left(N\lambda -1\right)}^{3}}}}\right)\right]\,{\rm{i}}{\rm{f}}\,{N}^{-1}\lt \lambda \lt 0.005,\\ {\textstyle \tfrac{A}{\sqrt{4{\left(1-\lambda \right)}^{2}+{\omega }^{2}}}}\,{\rm{i}}{\rm{f}}\,0.005\lt \lambda \lt {\lambda }_{\ell },\\ 2\sqrt{{\textstyle \tfrac{1-\lambda }{3}}}\cosh \left[{\textstyle \tfrac{1}{3}}{\rm{a}}{\rm{r}}{\rm{c}}{\rm{c}}{\rm{o}}{\rm{s}}{\rm{h}}\left(\sqrt{{\textstyle \tfrac{27{A}^{2}}{4{\left(1-\lambda \right)}^{3}}}}\right)\right]\,{\rm{i}}{\rm{f}}\,{\lambda }_{\ell }\leqslant \lambda \leqslant 0.4,\\ {\textstyle \tfrac{A}{\sqrt{{\left(2+\lambda \right)}^{2}+{\omega }^{2}}}}\,{\rm{i}}{\rm{f}}\,\lambda \gt \mathrm{0.4.}\end{array}\right.\end{array}\end{array}\,\end{eqnarray}$Figure 2(a) shows that this prediction agrees with the numerical result. Here, the star structure and the equally split initial conditions lead to the emergence of the first resonance, which is the same as for identical signals. In addition, the random initial phases have two effects on the emergence of the secondary resonance. The first is to avoid complete synchronization between the hub and leaf nodes. The second is to reduce the force from the hub to the leaf nodes, i.e., λx0=0, which results in the instability of the fixed points of the leaf nodes. As a result, the leaf nodes can generate significant amplifications at large coupling strength. However, large coupling strength will suppress the disordered effect of random initial phases, leading to the elimination of the secondary resonance.
We also look at the influence of the distribution of the initial phases on the double resonance in star networks, such as a Gaussian distribution with mean π and deviation σ. We find that the observed phenomenon remains if the deviation σ is large. The reason is that the Gaussian distribution becomes narrow for small σ, which resembles the same initial phase; in contrast, the distribution becomes flat for large σ, which resembles a uniform distribution.
5. Summary
In summary, we have compared the signal amplification of star networks subjected to identical and disordered signals, respectively. For identical signals with the same initial phase, star networks display a single resonance at weak coupling strength, which is induced by the hub node. However, for disordered signals with random initial phases, star networks exhibit a double resonance at both weak and large coupling strengths, where the first resonance is also the hub-enhanced amplification but the secondary one is produced by the leaf nodes. Since almost all of the nodes in star networks are leaf nodes, the secondary resonance suggests that the response of the entire network is improved by the random initial phases. In addition, the secondary resonance appearing at large coupling strength means that the coupling region for signal amplification is also enlarged by the random initial phases. These findings imply that heterogeneity in network structure as well in external signals may benefit weak signal amplification.
,School of Physics and Electronic Engineering, 
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