^{1,2}, Liang Zhao

^{2}and Zonghua Liu

^{1}

### Abstract

It has been revealed that un-weighted scale-free (SF) networks have an effect of amplifying weak signals [Acebrón *et al.*, Phys. Rev. Lett. **99**, 128701 (2007)]. Such a property has potential applications in neural networks and artificial signaling devices. However, many real and artificial networks, including the neural networks, are weighted ones with adaptive and plastic couplings. For this reason, here we study how the weak signal can be amplified in weighted SF networks by introducing a parameter to self-tune the coupling weights. We find that the adaptive weights can significantly extend the range of coupling strength for signal amplification, in contrast to the relatively narrow range in un-weighted SF networks. As a consequence, the effect of finite network size occurred in un-weighted SF networks can be overcome. Finally, a theory is provided to confirm the numerical results.

It is a challenging task to design an artificial structure or device for reproducing the signal amplification ability of biological systems.Stochastic resonance is one of the first attempts to this end, where the response of a nonlinear system to a weak signal can be enhanced by adding a suitable intensity of noise. Later on, complex networks are found as another way to push that goal forward. Particularly, it is uncovered that scale-free (SF) networks have the ability of signal amplification due to the heterogeneity of the degree structure. However, all those works consider the un-weighted networks, i.e., the couplings between all pairs of nodes are identical. On the other hand, many real and artificial networks, including the biological neural networks, are weighted ones with adaptive and plastic couplings. For this reason, here we study the signal amplification in weighted SF networks by introducing a parameter to self-tune the coupling weights. Our results show that the weighted SF networks can significantly extend the range of coupling strength for signal amplification and thus overcome the effect of finite network size occurred in un-weighted SF networks.

X. Liang was supported by the FAPESP of Brazil under Grant No. 2010/50614-7. L. Zhao was partially supported by CNPq of Brazil under Grant Nos. 306227/2011-8 and 560031/2010-6, and FAPESP of Brazil under Grant No. 2011/50151-0. Z. Liu was supported in part by the NNSF of China under Grant Nos. 10975053 and 11135001, and the FAPESP of Brazil under Grant No. 11/03631-6.

I. INTRODUCTION

II. MODEL DESCRIPTION AND ANALYTICAL SOLUTION

III. NUMERICAL SIMULATIONS

IV. DISCUSSIONS AND CONCLUSIONS

### Key Topics

- Oscillators
- 10.0
- Acoustic noise
- 3.0
- Brain
- 3.0
- Network topology
- 3.0
- Stochastic processes
- 3.0

## Figures

Parameters in Eq. (6) depend on the coupling strength *λ*, where (a) represents the amplitude *A*′ vs *λ*; (b) the parameter *a* vs *λ*; and (c) the oscillation center . Parameters *N* = 500 and *m* = 3 are used.

Parameters in Eq. (6) depend on the coupling strength *λ*, where (a) represents the amplitude *A*′ vs *λ*; (b) the parameter *a* vs *λ*; and (c) the oscillation center . Parameters *N* = 500 and *m* = 3 are used.

(a) *g* vs *λ* where the “square,” “circle,” and “triangle” traces are the simulation results of *α* = −0.5, 0, and 0.5, respectively. (b) *g* vs *α*, where the “square,” “circle,” and “triangle” traces are the simulations results of *λ* = 0.06, 0.1, and 1, respectively. (c) The optimal *g* as a function of *α* and *λ* from Eq. (2) . Parameters *N* = 500 and *m* = 3 are used.

(a) *g* vs *λ* where the “square,” “circle,” and “triangle” traces are the simulation results of *α* = −0.5, 0, and 0.5, respectively. (b) *g* vs *α*, where the “square,” “circle,” and “triangle” traces are the simulations results of *λ* = 0.06, 0.1, and 1, respectively. (c) The optimal *g* as a function of *α* and *λ* from Eq. (2) . Parameters *N* = 500 and *m* = 3 are used.

(a) The maximum responses (“circles”) and the analytical result *g* (“squares”) vs *λ*. (b) The optimal values (“circles”) and the analytical result *α* (“squares”) vs *λ*. (c) The degree *k _{L} * of the node with maximum signal response (“circles”) and the analytical result

*k*(“squares”) vs

_{L}*λ*. (d) The effective coupling strength vs

*λ*, where the circles denote , the squares denote , and the triangle denote . The dashed lines in (c) and (d) denote . Parameters

*N*= 500 and

*m*= 3 are used.

(a) The maximum responses (“circles”) and the analytical result *g* (“squares”) vs *λ*. (b) The optimal values (“circles”) and the analytical result *α* (“squares”) vs *λ*. (c) The degree *k _{L} * of the node with maximum signal response (“circles”) and the analytical result

*k*(“squares”) vs

_{L}*λ*. (d) The effective coupling strength vs

*λ*, where the circles denote , the squares denote , and the triangle denote . The dashed lines in (c) and (d) denote . Parameters

*N*= 500 and

*m*= 3 are used.

(a) The maximum response vs *λ*. (b) The optimal value *α* vs *λ*. (c) The degree *k _{L} * of the node with maximum signal response vs

*λ*. In (a)-(c), “circles” represent the results with

*N*= 1000 and

*m*= 3 and “squares” represent the results with

*N*= 500 and

*m*= 5. (d) The maximum response vs

*ω*, where “squares” represent the case of

*α*= −0.5 and

*λ*= 0.04, “circles” represent the case of

*α*= 0 and

*λ*= 0.08, and “triangles” represent the case of

*α*= 2 and

*λ*= 5. In (d), parameters

*N*= 500 and

*m*= 3 are used.

(a) The maximum response vs *λ*. (b) The optimal value *α* vs *λ*. (c) The degree *k _{L} * of the node with maximum signal response vs

*λ*. In (a)-(c), “circles” represent the results with

*N*= 1000 and

*m*= 3 and “squares” represent the results with

*N*= 500 and

*m*= 5. (d) The maximum response vs

*ω*, where “squares” represent the case of

*α*= −0.5 and

*λ*= 0.04, “circles” represent the case of

*α*= 0 and

*λ*= 0.08, and “triangles” represent the case of

*α*= 2 and

*λ*= 5. In (d), parameters

*N*= 500 and

*m*= 3 are used.

The optimal *g _{i} * vs

*k*for fixed

*α*where (a) to (d) represent the cases of

*α*= −0.5 and

*λ*= 0.04,

*α*= 0 and

*λ*= 0.08,

*α*= 0.5 and

*λ*= 0.16, and

*α*= 2 and

*λ*= 5, respectively. Parameters

*N*= 500 and

*m*= 3 are used.

The optimal *g _{i} * vs

*k*for fixed

*α*where (a) to (d) represent the cases of

*α*= −0.5 and

*λ*= 0.04,

*α*= 0 and

*λ*= 0.08,

*α*= 0.5 and

*λ*= 0.16, and

*α*= 2 and

*λ*= 5, respectively. Parameters

*N*= 500 and

*m*= 3 are used.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content