# Network Rewiring in the r-K Plane

^{*}

## Abstract

**:**

## 1. Introduction

- Is it possible to build any desired assortative or disassortative network, defined at the probabilistic level through a suitable “theoretical” ${e}_{jk}$ matrix, using a Newman rewiring, at least at the ensemble level? What is in this respect the role of the asymptotic constraints on r? Can the asymptotic constraints tell us in advance that a certain theoretical ${e}_{jk}$ is impossible to be implemented in a real network?
- Among the networks obtained through a Xulvi–Brunet–Sokolov rewiring or our $\Delta r$ rewiring, will one find the desired assortative or disassortative network? If yes, with what accuracy is this possible, compared to the Newman rewiring?
- How do the results (and their level of fluctuations and uncertainty) change if we modify the degree distribution, and especially the probability of the nodes with the lowest degree? Will a giant component always be present? If the network is much fragmented, what are the consequences for diffusion processes?

## 2. The Function “Average Degree of the First Neighbors” ${\mathit{K}}_{\mathit{N}}$

#### 2.1. Uncorrelated Networks

#### 2.2. Correlated Networks

#### 2.3. Local Variation of K

#### 2.4. Relation between the Local Variations of K and r

## 3. The Average Number of Second Neighbors ${\overline{\mathit{z}}}_{\mathbf{2},\mathit{B}}$

`Friends`” list, of the first neighbors of each node. The list is updated and used in many parts of the program, for instance after the first wiring of the stubs, in order to check that the nodes’ degrees match the prescribed degree distribution. It is also used at the end of the rewiring cycles, in order to find the giant component of the final network, and possibly for the numerical solution of diffusion equations in first or second closure approximation. It is straightforward to use the

`Friends`list to also obtain the number of second neighbors of each node, because the degrees of the nodes do not change in the rewiring and are stored in a vector “

`Degrees[i]`”, with $i=1,\dots ,N$, fixed from the degree distribution before the wiring. The contribution to ${\overline{z}}_{2,B}$ from each node is obtained as the sum of (degree of each friend − 1). The total network average ${\overline{z}}_{2,B}$ is the sum of the contributions of all nodes, divided by N. One can check that the exact value obtained in this way is well approximated by the probabilistic value in Equation (11).

**Property**

**1**

**Property**

**2**

## 4. The Rewiring Algorithm

## 5. Results

#### 5.1. Trajectories of Assortative and Disassortative Rewiring in the r-K Plane at Low T

#### 5.2. Equilibrium Rewiring at Variable T

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Scheme of a rewiring which involves the nodes $a,b,c,d$ and leaves their degrees $A,B,C,D$ unchanged. Before the rewiring the links are $(a,b)$, $(c,d)$. After the rewiring the links are $(a,c)$, $(b,d)$. Each node has in general other neighbors, not depicted; the sum of the degrees of the neighbors of a not involved in the rewiring is denoted in the text as ${\sigma}_{a}$, and similarly for $b,c,d$.

**Figure 2.**Trajectories of assortative rewiring in the r-K plane. Each trajectory begins from the left (initial network obtained by uncorrelated rewiring) and converges to a maximally assortative network on the right. Between two dots on the same trajectory there are 100 rewiring steps with return probability $exp(-\Delta r/T)$, $T={10}^{-6}$. The number of nodes is $N=1500$.

**Figure 3.**Trajectories of disassortative rewiring in the r-K plane. Each trajectory begins from the right (initial network obtained by uncorrelated rewiring) and converges to a maximally disassortative network on the left. Between two dots on the same trajectory there are 100 rewiring steps with return probability $exp(\Delta r/T)$, $T={10}^{-6}$. The number of nodes is $N=1500$.

**Figure 4.**Basic assortative building blocks classified through k (node degree) and ${z}_{2}$ (number of second neighbors). Looking at the table which gives for each node of a strongly assortative network the number ${z}_{2}$ of its second neighbors, we find a large number of nodes with $k=1$, ${z}_{2}=0$ (corresponding to Graph (1), isolated couples), of nodes with $k=2$, ${z}_{2}=2$ (corresponding to Graph (2), chains), and of nodes with $k=3$, ${z}_{2}=6$ (corresponding to Graph (3), linked 3-stars).

**Figure 5.**An example of a maximally assortative network with $\gamma =3$, ${k}_{min}=2$. The minimum degree has been taken greater than 1 in this example in order to avoid the formation of isolated couples, which make up the large majority of maximally assortative networks if ${k}_{min}=1$. The length of the chains and their connection or disconnection to the the giant component have only a little effect on the total r coefficient, and can therefore vary in each realization, even at a very low temperature (here $T={10}^{-6}$).

**Figure 6.**Maximally disassortative network with $\gamma =2.75$, ${k}_{min}=1$. The network is completely fragmented into “stars” (compare [29]). For all hubs the number of second neighbors is exactly zero.

**Figure 7.**Assortative rewiring in equilibrium at different temperatures T. (Network of 1000 nodes, with $\gamma =2.5$, ${k}_{min}=1$.) The entropy per node $S/N$ is shown, along with the average degree of the first neighbors K, the Newman assortativity coefficient r multiplied by 10 (note the slight structural disassortativity at high temperature), and the fractional size of the giant component multiplied by 10.

**Table 1.**An example of values of K for Markovian networks with 1000 nodes in dependence on the scale-free exponent $\gamma $ and the minimum degree ${k}_{min}$, in the uncorrelated case, assortative case (Vazquez–Weigt recipe, Equation (4), with $r=0.5$) and disassortive case (Porto–Weber recipe as employed in [23]).

$\phantom{\rule{4pt}{0ex}}\mathit{\gamma}\phantom{\rule{4pt}{0ex}}$ | ${\mathit{k}}_{\mathit{min}}$ | N | n | ${\mathit{c}}_{\mathit{n},\mathit{\gamma}}$ | $\langle \mathit{k}\rangle $ | K (unc.) | K (ass.) | K (dis.) | ${\mathit{r}}_{\mathit{dis}}$ |
---|---|---|---|---|---|---|---|---|---|

2.5 | 1 | 1000 | 93 | 1.34 | 1.79 | 7.43 | 4.61 | 8.52 | −0.088 |

2.5 | 4 | 1000 | 15 | 0.0896 | 6.23 | 7.39 | 6.81 | 20.1 | −0.140 |

2.75 | 1 | 1000 | 43 | 1.26 | 1.50 | 3.63 | 2.56 | 5.15 | −0.062 |

2.75 | 4 | 1000 | 6 | 0.0413 | 4.64 | 4.77 | 4.70 | 16.0 | −0.120 |

**Table 2.**An example of the numerical factors involved in the relation of Equation (9) between $\Delta K$ and $\Delta r$, for scale-free networks with a fixed number of nodes. These factors explain why the rewiring trajectories in the r-K plane have different slopes. (See Figures 2 and 3 and the description in Section 5.1.) N is the number of nodes, L the number of links. ${\sigma}_{q}^{2}$ is the variance of the excess-degree distribution (Equation (8)). ${\langle (AD+BC)/\left(ABCD\right)\rangle}_{rew}$ denotes the average along an assortative rewiring trajectory with low temperature $T={10}^{-6}$. The hubs of the networks are defined by the cumulative probability method.

$\phantom{\rule{4pt}{0ex}}\mathit{\gamma}\phantom{\rule{4pt}{0ex}}$ | N | L | ${\mathit{\sigma}}_{\mathit{q}}^{2}$ | $\frac{\mathit{L}\mathit{\sigma}}{2\mathit{N}}$ | ${\langle \frac{\mathit{AD}+\mathit{BC}}{\mathit{ABCD}}\rangle}_{\mathit{rew}}$ |
---|---|---|---|---|---|

2.25 | 715 | 798 | 193 | 108 | 0.24 |

2.5 | 715 | 630 | 81.6 | 36 | 0.31 |

2.75 | 715 | 536 | 30.2 | 11.3 | 0.35 |

3 | 715 | 481 | 12.4 | 4.2 | 0.43 |

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Bertotti, M.L.; Modanese, G.
Network Rewiring in the *r*-*K* Plane. *Entropy* **2020**, *22*, 653.
https://doi.org/10.3390/e22060653

**AMA Style**

Bertotti ML, Modanese G.
Network Rewiring in the *r*-*K* Plane. *Entropy*. 2020; 22(6):653.
https://doi.org/10.3390/e22060653

**Chicago/Turabian Style**

Bertotti, Maria Letizia, and Giovanni Modanese.
2020. "Network Rewiring in the *r*-*K* Plane" *Entropy* 22, no. 6: 653.
https://doi.org/10.3390/e22060653