^{1,a)}, Shiqun Zhu

^{1,b)}and Xiaoqin Luo

^{1}

### Abstract

The diversity-induced resonance in weighted scale-free networks is investigated numerically and analytically. The couplings are weighted according to node degree. It is found that self-organized phase shifts occur between large-degree and small-degree nodes, although there is no time delay in coupling terms. Such spontaneous phase shifts are relevant to the improvement of signal resonance amplification. This finding may help to better understand how the collective behavior of individual units promotes the response of the whole system to external signals.

Stochastic resonance in complex networks has been extensively studied due to its potential applications in various fields. Many real networks display a strong heterogeneity in the capacity and the intensity of the connections. These systems can be better described in terms of weighted networks. However, most studies focused on unweighted networks. The resonance phenomenon in realistic weighted networks is of substantial interest to study. In this paper the diversity-induced resonance in weighted networks is investigated. It is shown that signal amplification can be enhanced due to weighted couplings, and the optimal weighting scheme depends on the overall coupling strength. The time traces of individual nodes are studied in detail. It is found that a forehead-laggard relation spontaneously occurs between nodes of different degrees. This phase-shift relation may influence the optimal weighting scheme for resonance. The analytical results show a good agreement with those of numerical simulations. In many natural and artificial systems, a number of signaling elements are connected to form networks. Their information-processing potential usually depends on the response to weak external signals. In biological systems, for example, cells in living organisms make response to their environment in virtue of an interconnected network of receptors, messengers, protein kinases, and other signaling molecules. Their proper functioning always implies sensitive response to external signals. Our results might have potential importance in understanding the relation between collective behaviors of units and the sensitivity of a whole system to external signals.

I. INTRODUCTION

II. THEORETICALMODEL

III. NUMERICAL RESULTS

IV. ANALYTIC EXPLANATION

V. CONCLUSIONS

### Key Topics

- Networks
- 21.0
- Mean field theory
- 4.0
- Oscillators
- 4.0
- Stochastic processes
- 4.0
- Topology
- 4.0

## Figures

Diversity-induced resonance in weighted Barabási–Albert networks (Ref. 40), , . (a) and (b) show the spectral amplification factor as a function of diversity for different values of weighting parameter . (a) A weak overall coupling of . (b) A strong overall coupling of . The inset of (b) shows the differences in positions and values of the maxima more clearly. (c) The contour plot of maximum signal amplification factor on the parameter plane . Each cross marks the optimal weighting parameter for a given overall coupling strength . Each circle marks the optimal coupling strength for a given weighting parameter . (d) The optimal diversity as a function of for different values of and the optimal coupling strength .

Diversity-induced resonance in weighted Barabási–Albert networks (Ref. 40), , . (a) and (b) show the spectral amplification factor as a function of diversity for different values of weighting parameter . (a) A weak overall coupling of . (b) A strong overall coupling of . The inset of (b) shows the differences in positions and values of the maxima more clearly. (c) The contour plot of maximum signal amplification factor on the parameter plane . Each cross marks the optimal weighting parameter for a given overall coupling strength . Each circle marks the optimal coupling strength for a given weighting parameter . (d) The optimal diversity as a function of for different values of and the optimal coupling strength .

Diversity-induced resonance on a wheel-like network consisting of a hub node and 99 fringe nodes. (a) Time traces of the hub node and the average of the fringe nodes , respectively. A phase shift take places between them. The insets show the phase shift more clearly during the two successive jumping events, respectively. (b) Phase plot of and . The upper part is the time period from 728 to 732, corresponding to the lower-left inset of (a). The lower part is the time period from 843 to 847, corresponding to the upper-right inset of (a). The phase plot crosses x-axis first at , crosses y-axis later at .

Diversity-induced resonance on a wheel-like network consisting of a hub node and 99 fringe nodes. (a) Time traces of the hub node and the average of the fringe nodes , respectively. A phase shift take places between them. The insets show the phase shift more clearly during the two successive jumping events, respectively. (b) Phase plot of and . The upper part is the time period from 728 to 732, corresponding to the lower-left inset of (a). The lower part is the time period from 843 to 847, corresponding to the upper-right inset of (a). The phase plot crosses x-axis first at , crosses y-axis later at .

The contour plot of maximum signal amplification factor on the parameter plane calculated according to Eq. (1). The network and the parameters are the same as that in Fig. 2. Each cross marks the optimal weighting parameter for a given coupling strength . The solid line is the analytical result of Eq. (8).

The contour plot of maximum signal amplification factor on the parameter plane calculated according to Eq. (1). The network and the parameters are the same as that in Fig. 2. Each cross marks the optimal weighting parameter for a given coupling strength . The solid line is the analytical result of Eq. (8).

Forehead-laggard relation between the hub node and its neighboring set on a BA network. (a) Time traces of the hub node and its neighboring set . The arrows indicate leave for the other fixed point ahead of . (b) Time traces of the velocities and . (c) Three separated parts of . . represents the bistable oscillation function . is the part induced by the couplings. represents the part caused by an external signal . Since is subthreshold, the jump events will not occur without the phase-shift-induced driving force. The parameters are chosen as . The degree of the hub is 175.

Forehead-laggard relation between the hub node and its neighboring set on a BA network. (a) Time traces of the hub node and its neighboring set . The arrows indicate leave for the other fixed point ahead of . (b) Time traces of the velocities and . (c) Three separated parts of . . represents the bistable oscillation function . is the part induced by the couplings. represents the part caused by an external signal . Since is subthreshold, the jump events will not occur without the phase-shift-induced driving force. The parameters are chosen as . The degree of the hub is 175.

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