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Influence of Technological Innovations on Industrial Production: A Motif Analysis on the Multilayer Network^{ †}

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Bipartite Configuration Model (BiCM) and Assist-Matrix Null Model

- Through a constrained maximum-entropy approach, we define ensemble $\Omega $ of bipartite networks that are maximally random, apart from the ensemble average of the node degrees on both layers of the bipartite network that are constrained to generic fixed values. Such an ensemble is thus an instance of an Exponential Random Binary Graph (ERBG).
- In order to determine the ERBG that best represents the empirical bipartite network, we use a maximal-likelihood argument showing that the mean values of the node degrees have to be taken equal to the observed ones in the empirical network [30]: ${\langle {\tilde{k}}_{a}\rangle}_{\Omega}={k}_{a}$, $\forall a\in A$ and ${\langle {\tilde{k}}_{b}\rangle}_{\Omega}={k}_{b}$, $\forall b\in B$, where we have indicated with k the observed degrees in the real network, and with $\tilde{k}$ the degrees in a generic configuration of the null model. We remind that ${k}_{a}={\sum}_{b}{M}_{ab}$ and ${k}_{b}={\sum}_{a}{M}_{ab}$, and analogously for "tilded" quantities.

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**Figure 1.**Mean signal $\varphi $ computed over 5500 combinations of t, ${t}^{\prime}$, and p chosen at random, for different values of time lag $\mathsf{\Delta}y$. Error bars represent the standard deviation over year pairs giving the same time lag, whereas the dotted line is significance level $\alpha $.

**Figure 2.**Mean signal $\varphi $ computed over 5500 combinations of t, ${t}^{\prime}$, and p, chosen for specific regions of the assist matrix. Red circles: technological codes in the G09F-G11B region—related to sector “physics: instruments of communications, acoustics, optics”, and products in the region 8401–8518 “machinery and metals”. Green squares: technological codes in the F22G-F23N region—related to sector “engineering: various types of machines including steam and combustion”, and products in the 8401–8518 region “machinery and metals”. Blue diamonds: technological codes in the C09H-C10L region—related to sector “chemistry: macromolecular and inorganic compounds, gas and petroleum products”, and products in the 2706–3104 region “inorganic and organic chemicals, pharmaceuticals”. In all cases, error bars represent the standard deviation over the year pairs giving the same time lag, whereas the dotted line is significance level $\alpha $.

**Figure 3.**Mean signal $\varphi $ computed over 5500 combinations of t, ${t}^{\prime}$, and p chosen for specific regions of the assist matrix. Cyan upper triangles: technological codes in the C07–C30 region related to sector “chemistry” (i.e., a much larger region than that represented in the bottom panel of Figure 2) and products in the 2706–3104 region “inorganic and organic chemicals, pharmaceuticals”. Orange lower triangles: technological codes in the H01P-H02M region—related to sector “physics: electricity”, and products in the 4411–5516 region “textiles”. In all cases, error bars represent the standard deviation over the year pairs giving the same time lag, whereas the dotted line is significance level $\alpha $.

**Table 1.**Examples of highly significant $\mathsf{\Lambda}$ motifs. p-value is averaged over all year pairs ${y}_{1}={y}_{2}$ giving $\mathsf{\Delta}y=0$. To make this selection, we picked the most significant pairs (individual links) $(t,p)$, and then chose ${t}^{\prime}$ within the region where the average p-value was highest.

p-Value | p, t, ${\mathbf{t}}^{\prime}$ |
---|---|

4701: Wood pulp. | |

$2\xb7{10}^{-4}$ | C05B: Lime; magnesia; slag; cements. |

C09K: Materials for applications not otherwise provided for. | |

2605: Mineral products. | |

$5\xb7{10}^{-4}$ | C21D: Modifying the physical structure of ferrous metals. |

F04F: Pumping of fluid by direct contact of another fluid or by using inertia of fluid to be pumped. | |

2605: Mineral products. | |

$5\xb7{10}^{-4}$ | C21D: Modifying the physical structure of ferrous metals. |

F04F: Working metallic powder. | |

8443: Printing machine. | |

$5\xb7{10}^{-4}$ | D02H: Mechanical methods or apparatus in the manufacture of artificial filaments. |

G01T Measurement of nuclear or x-radiation. | |

4703: Chemical wood pulp. | |

$7\xb7{10}^{-4}$ | D21F: Decorating textiles |

B27C: Planing, drilling, milling, turning, or universal machines. | |

2605: Mineral products. | |

$8\xb7{10}^{-4}$ | C21D: Modifying the physical structure of ferrous metals. |

FF15B: Systems acting by means of fluids in general. | |

4703: Chemical wood pulp. | |

$2\xb7{10}^{-3}$ | D21F: Paper-making machines. |

F03D: Wind motors. | |

4703: Chemical wood pulp. | |

$2\xb7{10}^{-3}$ | D21F: Paper-making machines. |

D06Q: Decorating textiles. | |

8519: Sound recording or reproducing apparatus. | |

$3\xb7{10}^{-3}$ | G10K: Sound-producing devices. |

G01T: Capacitors, rectifiers, detectors, switching devices. | |

8519: Sound recording or reproducing apparatus. | |

$3\xb7{10}^{-3}$ | G10K: Sound-producing devices. |

G04f: Time-interval measuring. |

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**MDPI and ACS Style**

Formichini, M.; Cimini, G.; Pugliese, E.; Gabrielli, A.
Influence of Technological Innovations on Industrial Production: A Motif Analysis on the Multilayer Network. *Entropy* **2019**, *21*, 126.
https://doi.org/10.3390/e21020126

**AMA Style**

Formichini M, Cimini G, Pugliese E, Gabrielli A.
Influence of Technological Innovations on Industrial Production: A Motif Analysis on the Multilayer Network. *Entropy*. 2019; 21(2):126.
https://doi.org/10.3390/e21020126

**Chicago/Turabian Style**

Formichini, Martina, Giulio Cimini, Emanuele Pugliese, and Andrea Gabrielli.
2019. "Influence of Technological Innovations on Industrial Production: A Motif Analysis on the Multilayer Network" *Entropy* 21, no. 2: 126.
https://doi.org/10.3390/e21020126