# The Influence of a Network’s Spatial Symmetry, Topological Dimension, and Density on Its Percolation Threshold

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Percolation Properties of Network Structures and their Invariance

#### 2.1. Theoretical Methods within Percolation Theory

_{c}(L) can be defined by a given value of the probability of the network transitioning to a conducting state. In Figure 1, this probability is set at 0.5. However, it can be, for example, 0.95 or 0.99 (whereupon the percolation threshold corresponds to the specified criterion of data transmission reliability).

- (1)
- Find the distribution by size for clusters of blocked nodes within the network structure at a given probability of blocking the nodes.
- (2)
- Assess statistical characteristics of clusters, for example, the average size of clusters of blocked nodes within the network structure.
- (3)
- Explore how the percolation threshold value of the network structure depends on the density of the network (average number of links per node), as well as a number of other issues.

^{−τ}($n\left(s\right)~{s}^{-\tau}$), where τ is a parameter which considers the topology and spatial symmetry of the network (for example, for a square network τ = 187/91 ≈ 2.05). The power law $n\left(s\right)~{s}^{-\tau}$ shows that the ratio of the number of clusters of one size to the number of clusters of another size does not depend on S size, but only on the ratio of their sizes. Thus, percolation clusters are self-similar, or independent of scale, in the interval from the network step to the entire network size (invariant at scales change, i.e., preservation of proportionality, which, in fact, is a characteristic of the symmetry of properties).

_{i}is the polynomial coefficients, ξ is the proportion of blocked nodes, and ξ

_{c}(L) is the proportion of blocked nodes corresponding to the percolation threshold value depending on the network size L.

#### 2.2. Numerical Methods for Defining Percolation Thresholds

- Randomly select two nodes, A and B, taking into account a restriction that there is at least one intermediate node between them.
- Set a probability of blocking a single node (for the node problem) or a link (for the link problem), and randomly block the proportion of network nodes (or links) that is equal to this probability.
- Check if there is at least one “free” path in the network (a path formed by non-excluded nodes or links) from node A to node B. If at least one of the nodes A or B is excluded, then there is no free path (number of "free" paths is 0). Otherwise, write 1.
- Increase the value of probability of blocking a single node (for the node problem) or link (for the link problem) by some value and randomly block the proportion of network nodes (or links) equal to the specified probability value. Then, define which specific network nodes were blocked.
- Go back to step 3 until all nodes of the network are moved.
- Return to item № 2 and perform items № 3–№ 5—Q times (e.g., several hundred times) from the first to the last step (in cases where the entire network is blocked), for all experiments. Find the number of times in which at least one “free” path was found (let this number be ξ). For example, say at step h = 18, in 8, 12, 19, 56, 58, 76, 80 and 89 experiments of Q, at least one “free” path was found, then number ξ(5) = 18 (8 is the total number of “free” paths). Find value ρ (h) = ξ(h)/Q for each step, where h is step number. Calculate the average size of blocked node clusters, the number of such clusters, etc. (for all N experiments at each step). The average cluster size can be defined as the ratio of the sum of all average values obtained at this clustering step (for all Q experiments) to the total number of experiments Q. For clarification, let us consider an example. Given that, for h = 6 steps, 4 clusters comprising 15 nodes were found in the 1st experiment, in the 2nd 3 clusters were found, in the 3rd 2 clusters were found, etc., and in the 100th experiment, 20 clusters were found. Then the average number of clusters with a size of 10 blocked nodes will be equal to: (4 + 3 + 2 + … + 5)/100.
- Then, go back to item № 1 and repeat once more steps № 2–№ 6 W times. For each of the W experiment values, $\overline{{p}_{w}}\left(h\right)=\xi \left(h\right)/Q$ will be found. The W index defines which of the W tests we consider.
- After the simulation, for each h step, find a value $\langle \overline{\rho}\left(h\right)\rangle ={{\displaystyle \sum}}_{w=1}^{W=100}\overline{{p}_{w}}\left(h\right)/W$—the average value of probability of information transmission through the network as a whole using non-blocked nodes at each step (considering different possible configurations of all paths). Build graphical dependences of the average value of the probability that data or information will pass through the network $\langle \overline{\rho}\left(h\right)\rangle $ as a whole on the proportion of blocked network nodes.

## 3. Dependence of Percolation Properties of Network Structures on Density, Symmetry Groups, and Topological Dimension

- (1)
- Neumann’s Principle: the symmetry elements of any physical property of a crystal (including network structures) should contain symmetry elements of their point group [46].
- (2)
- Pierre Curie’s symmetry (dissymmetry) principle [46]: at the superposition of several phenomena of different nature, each of which having own symmetry, only matching elements of symmetry of these phenomena will be preserved in the same system. This principle can be written as: ${G}_{1}\subseteq {G}_{2}$, where symmetry of the cause is described by group G
_{1}, and symmetry of the consequence by group G_{2}.

#### 3.1. D and 3D Networks with Regular Structures

^{2}network (see Table 1); centered square and hexagonal 2D networks; a centered 2D network with 3.12

^{2}; a 3D hexagonal honeycomb structure; 3D parallel 3.12

^{2}structures with vertical links; 3D hexagonal and a number of others, percolation thresholds were defined using numerical simulation methods. For other structures listed in Table 1, percolation threshold values were cited from other studies.

_{i}. coefficients. This research proposes not to use methods of algebraic Hodge geometry [42] and Kadanoff–Wilson similarity theory [43,44] with renormalization groups (e.g., in [19]), which do not consider the spatial symmetry of the network structure. The essence of the present approach is that it is possible to express dependence of the i-polynomial degree $S\left(\xi ,L\right)$ on the conditional probability Y(ξ, L) of conductivity in the network and to define the effect of topological factors on this dependence. Using formula (1) we obtain: $\mathrm{lnY}\left(\mathsf{\xi},\mathrm{L}\right)=-\mathrm{ln}\left\{1+{e}^{-S\left(\xi ,L\right)}\right\}$, where $S\left(\xi ,L\right)={\displaystyle \sum}_{i}{a}_{i}\left\{{\xi}^{i}-{\xi}_{c}^{i}\left(L\right)\right\}$ is a i-polynomial degree, a

_{i}is its coefficients, ξ is the current value of the proportion of blocked nodes, and ξ

_{c}(L) is the proportion of blocked nodes corresponding to the percolation threshold value (this depends on the network size L). Taking into account the fact that near the percolation threshold, ξ ≈ ξ

_{c}(L), the value of the $S\left(\xi ,L\right)$ polynomial is small and ${e}^{-S\left(\xi ,L\right)}$ can be expanded into a series, limiting these to two terms. After the conversion, we obtain:

#### 3.2. D and 3D Random Structure Networks

- (1)
- Set the number of nodes N and number of links E.
- (2)
- Generate the S list consisting of N nodes with random coordinates (x, y).
- (3)
- Select the n
_{0}node with the smallest x–coordinates; if there are several such nodes, then select the node with the largest y–coordinate. - (4)
- Sort the S list of nodes by increasing distance L from node n
_{0}:$$L=\sqrt{{\left({n}_{0x}-{n}_{ix}\right)}^{2}+{\left({n}_{0y}-{n}_{iy}\right)}^{2}}$$ - (5)
- Combine the first three nodes ${n}_{0},{n}_{1},{n}_{2}$ from S list into the first triangle by adding edges. Passing the nodes clockwise, beginning from the edge between the first and second nodes of the list, add the edges of the triangle to the cyclical list H.
- (6)
- Sequentially process all nodes from the S list.
- Find the first raw node n
_{i}. - In the H list, take the last edge V whose nodes n
_{a}and n_{b}together with n_{i}form a left turn (the following condition is satisfied:$$\left({n}_{ix}\u2013{n}_{ax}\right)\ast \left({n}_{by}-{n}_{ay}\right)\u2013\left({n}_{iy}\u2013{n}_{ay}\right)\ast \left({n}_{bx}-{n}_{ax}\right)0$$ - Among the H edges, find the first edge V
_{L}which does not satisfy the left turn condition (located before the edge V to the left of it). - Among the H edges, find the first edge V
_{R}which does not satisfy the left turn condition (located after the edge V to the right of it). - Sequentially process the edges in the H list between V
_{L}and V_{R}. Each of these edges together with n_{i}node forms a new triangle by adding new edges between them. - Remove from the H list all edges between V
_{L}and V_{R}. - From the first added triangle, find the edge between n
_{i}and point of the edge that is not included in the next processed triangle and add it to the H list. - From the last added triangle, find the edge between n
_{i}and point of the edge that is not included in the previous processed triangle and add it to the H list.

- (7)
- Remove edges from the current graph until their number is equal to E. Choose edges randomly, but remove them only if there is a path between the nodes of this edge even without this edge.

- (1)
- Find the center of the $\overline{R}=\frac{{{\displaystyle \sum}}_{i}{\overline{r}}_{i}}{i}$ polygon.
- (2)
- Shift all vertices so that the center is at the root of coordinates.
- (3)
- Find a reference point (for example, radius vector OA = (0, 1) see Figure 4).
- (4)
- Assess angles between vectors from the center to each vertex and OA (the angles should be within the range [0‒360)).
- (5)
- Sort the angles from smallest to largest.

#### 3.3. Analysis of Percolation Threshold Values for Regular 2D and 3D Structures and Selection of Components Responsible for Symmetry Elements

^{о}, see shape III in Figure 7) consists of equilateral triangular elements of the coating, each of which has a perpendicular third-order symmetry axis—

**C**and three symmetry planes in which the axis lies (3mm point group). For a square network, the coating element has a fourth-order symmetry axis—

_{3}**C**and four symmetry planes in which the axis lies (4mm point group). Similarly, the honeycomb structure element of the coating has a sixth-order symmetry axis—

_{4}**C**and six symmetry planes in which the axis lies (6 mm point group). It is easy to see that, upon transition from 3mm to 4mm, and from 4mm to 6mm, the percolation threshold value for symmetrical 2D structures changes by 0.03 units. Given that the

_{6}**C**axis does not exist, and, for

_{5}**C**, there is no planar element to create the network, it can be assumed that transition between symmetry axes orders, in the presence of corresponding symmetry planes, also has an invariant that contributes to the percolation threshold value.

_{2}^{2}structure of (shape g in Figure 2) can be considered as a 3.12

^{2}network (shape f in Figure 2), to which an inversion center for three pairs of homogeneous groups of nodes (each including an equilateral triangle on the outer side of the general structure ring) is added. The presence of such an inversion center causes the percolation threshold to change by 0.07 units (see Table 3).

## 4. Discussion

- At the same density of links, 3D networks have smaller percolation threshold values than 2D ones. Regardless of the type of network and symmetry, at the transition from 2D to 3D structures, the percolation threshold changes to the value equal to exp{–(d − 1)}, where d = topological dimension (this value for any network is invariant at the transition between structures). Thus, an increase in topological dimension by one unit reduces the percolation threshold value of the networks and increases the potential for information transmission.
- For random planar (2D) structures, the percolation threshold value, at the same network density, is less than for 2D structures with symmetry. Thus, for 2D structures, the presence of symmetry leads to an increase in the required proportion of nodes at which information conductivity will occur.
- If 2D or 3D networks have structures which can be transformed (transformed into each other) by deformation (elongation or contraction) without breaking and forming new links, they have the same percolation threshold. There is symmetry of similarity (A.V. Shubnikov), in which not only truly equal figures are symmetrical, but also all those which are similar to them. Thus, the presence of some elements of symmetry contributes to the percolation threshold value, certain others do not.
- The presence of axes of symmetry and corresponding number of planes of symmetry in which they lie affects the percolation threshold value. A third-order axis with three planes of symmetry has an increased invariant (which is preserved for structures of any kind) of the percolation threshold by a value of 0.02 units. At the transition to a fourth-order axis with four planes of symmetry, this invariant increases by 0.03 units, and at the transition from a fourth-order axis to a sixth-order axis, it increases by 0.03 units more. Thus, the transition between the orders of the axes of symmetry, in the presence of corresponding planes of symmetry, has an invariant which contributes to the percolation threshold value.
- Percolation analyses of regular 2D structures show that the presence of inversion centers in symmetry elements reduces the percolation threshold value. Moreover, the greater the number of pairs of the structure elements which have inversion, the more the presence of such centers of symmetry contribute to the fraction of the percolation threshold. However, if the center of symmetry lies in the plane of mirror symmetry separating 3D structure layers, the mutual presence of this group of symmetry elements will not affect the percolation threshold value. This corresponds to Curie’s dissymmetry principle (upon superposition of several phenomena of different natures, each of which having its own symmetry, only matching elements of symmetry of these phenomena will be preserved in the same system), since, in this case, the center of inversion performs the same role as the plane of mirror symmetry.
- The numerical values of percolation threshold fractions (see Table 3), resulting from the influence of symmetry elements, show that, in general, the presence of spatial symmetry and density of networks has a stronger influence on the percolation threshold value in 2D structures than in 3D ones.

## 5. Further Activities

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Dependency of the possibility of percolation occurrence on the value of the proportion of conductive nodes (or connections).

**Figure 3.**Dependence of natural logarithms of percolation thresholds lnP(x) versus inverse network density (1/x) for regular structures (line 1 is for 2D topological dimension, line 2 is for 3D topological dimension).

**Figure 5.**Dependence of the natural logarithm of percolation threshold lnP(x) of 2D random networks versus inverse network density (1/x) in comparison with regular networks.

**Figure 6.**Dependence of the natural logarithm of the percolation threshold lnP(x) of 3D random networks versus inverse network density (1/x) in comparison with regular networks.

**Table 1.**Characteristics and values of percolation thresholds (for the node blocking problem) of some regular network structures.

№ | Topological Dimension | Network Structure | Hermann–Mogen Spatial Symmetry Group and/or Point Group (in brackets) | Number of Links per Node (Density). Values of Inverse Density Are Presented in Brackets. | Value of Percolation Threshold (General Proportion of Conducting Links Necessary for Conductivity to Occur). Values of Natural Logarithms of Percolation Threshold Values are Presented in Brackets. |
---|---|---|---|---|---|

1. | 2D | Network 3.12^{2} (shape f in Figure 2). | Group of symmetry elements contains: sixth-order symmetry axis and six symmetry planes. | 2.7 (0.37) | 0.74 (−0.30) |

2. | Hexagonal honeycomb structure (shape d in Figure 2). | Point group $\frac{6}{\mathrm{m}}\mathrm{mm}$ | 3 (0.33) | 0.70 (−0.36) [18,37] | |

3. | Centered network with 3.12^{2} (shape g in Figure 2). | Group of symmetry elements contains: sixth-order symmetry axis, six symmetry planes, and inversion center. | 3.40 | 0.64 | |

4. | Square network (shape a in Figure 2). | Point group 4mm | 4 (0.25) | 0.59 (−0.53) [18,36] | |

5. | Structure presented with a shape e in Figure 2—“starfish”. | Group of symmetry elements contains: sixth-order symmetry axis, six symmetry planes, and inversion center. | 4.5 | 0.56 | |

6. | Centered square network (shape b in Figure 2). | Point group $\frac{4}{\mathrm{m}}\mathrm{mm}$ | 6 | 0.50 | |

7. | Hexagonal network (shape c in Figure 2). | Point group 6mm | 6 (0.17) | 0.50 (−0.69) [18,37] | |

8. | 3D | Structure in the shape of diamond (shape m in Figure 2). | $\mathrm{Fd}\overline{3}\mathrm{m}$ (point group cF8) | 4 (0.25) | 0.43 (−0.84) [18,36] |

9. | Parallel 3.12^{2} structures with vertical links | Not defined | 4.7 (0.21) | 0.38 (−0.98) | |

10. | Hexagonal honeycomb structure (shape k in Figure 2). | $\mathrm{P}{6}_{3}/\mathrm{mmc}$ (point group hP4) | 5 (0.2) | 0.36 (−1.03) | |

11. | Cubic network (shape h in Figure 2). | $\mathrm{Pm}\overline{3}\mathrm{m}$ (point group cP1) | 6 (0.17) | 0.31 (−1.17) [18,37] | |

12. | Cubic volume-centered network (shape i in Figure 2). | $\mathrm{Im}\overline{3}\mathrm{m}$ (point group cl1) | 8 (0.125) | 0.25 (−1.39) [18,37] | |

13. | Hexagonal network (shape l in Figure 2). | P6/mmm (point group hP1) | 8 (0.125) | 0.27 (−1.31) | |

14. | Cubic face-centered network (shape j in Figure 2). | $\mathrm{Fm}\overline{3}\mathrm{m}$ (point group cF4) | 12 (0.08) | 0.20 (−1.61) [18,37] |

№ | Topological Dimension | Number of Links Per node (Density). Values of Inverse Density are Presented in Brackets. | Value of the Percolation Threshold (Proportion of Conducting Links Necessary for Conductivity to Occur in General). Values of Natural Logarithms of Percolation Thresholds are Presented in Brackets. |
---|---|---|---|

1. | 2D | 5.99 (0.167) | 0.500 (−0.693) |

2. | 5.40 (0.185) | 0.533 (−0.629) | |

3. | 4.80 (0.208) | 0.570 (−0.562) | |

4. | 4.50 (0.222) | 0.593 (−0.523) | |

5. | 4.20 (0.238) | 0.618 (−0.481) | |

6. | 3.90 (0.256) | 0.650 (−0.431) | |

7. | 3.60 (0.278) | 0.683 (−0.381) | |

8. | 3.42 (0.292) | 0.708 (−0.345) | |

9. | 3.18 (0.314) | 0.750 (−0.288) | |

10. | 2.94 (0.340) | 0.793 (−0.232) | |

11. | 2.70 (0.370) | 0.852 (−0.160) | |

12. | 2.46 (0.407) | 0.925 (−0.078) | |

13. | 3D | 9.31 (0.107) | 0.217 (−1.530) |

14. | 8.27 (0.121) | 0.230 (−1.470) | |

15. | 7.09 (0.141) | 0.250 (−1.386) | |

16. | 6.47 (0.155) | 0.280 (−1.273) | |

17. | 5.27 (0.190) | 0.340 (−1.079) | |

18. | 4.89 (0.204) | 0.375 (−0.981) | |

19. | 4.54 (0.220) | 0.399 (−0.920) | |

20. | 4.31 (0.232) | 0.405 (−0.904) | |

21. | 3.94 (0.254) | 0.470 (−0.755) | |

22. | 3.34 (0.299) | 0.543 (−0.610) |

**Table 3.**Fractions of percolation thresholds of 2D and 3D regular networks, resulting from the influence of spatial symmetry groups (for the node blocking problem).

№ | Topological Dimension | Network Structure | Hermann–Mogen Spatial Symmetry Group and/or Point Group | Fraction of Percolation Threshold, Resulting from the Influence of Symmetry Elements |
---|---|---|---|---|

1. | 2D | Network 3.12^{2} (shape f in Figure 2). | Group of symmetry elements contains: sixth-order symmetry axis and six symmetry planes. | 0.12 |

2. | Hexagonal honeycomb structure (shape d in Figure 2). | Point group $\frac{6}{\mathrm{m}}\mathrm{mm}$ | 0.08 | |

3. | Centered network with 3.12^{2} (shape g in Figure 2). | Group of symmetry elements contains: sixth-order symmetry axis, six symmetry planes, and inversion center. | 0.05 | |

4. | Square network (shape a in Figure 2). | Point group 4mm | 0.05 | |

5. | Structure presented with a shape e in Figure 2—“starfish”. | 0.03 | ||

6. | Centered square network (shape b in Figure 2). | Point group $\frac{4}{\mathrm{m}}\mathrm{mm}$ | 0.02 | |

7. | Hexagonal network (shape c in Figure 2). | Point group 6 mm | 0.02 | |

8. | 3D | Structure in the shape of diamond (shape m in Figure 2). | $\mathrm{Fd}\overline{3}\mathrm{m}$ (point group cF8) | 0.02 |

9. | Parallel 3.12^{2} structures with vertical links | Undefined | −0.01 | |

10. | Hexagonal honeycomb structure (shape k in Figure 2). | $\mathrm{P}{6}_{3}/\mathrm{mmc}$ (point group hP4) | −0.01 | |

11. | Cubic network (shape h in Figure 2). | $\mathrm{Pm}\overline{3}\mathrm{m}$ (point group cP1) | −0.01 | |

12. | Cubic volume-centered network (shape i in Figure 2). | $\mathrm{Im}\overline{3}\mathrm{m}$ (point group cl1) | −0.01 | |

13. | Hexagonal network (shape l in Figure 2). | P6/mmm (point group hP1) | −0.03 | |

14. | Cubic face-centered network (shape j in Figure 2). | $\mathrm{Fm}\overline{3}\mathrm{m}$ (point group cF4) | −0.01 |

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**MDPI and ACS Style**

Zhukov, D.O.; Andrianova, E.G.; Lesko, S.A.
The Influence of a Network’s Spatial Symmetry, Topological Dimension, and Density on Its Percolation Threshold. *Symmetry* **2019**, *11*, 920.
https://doi.org/10.3390/sym11070920

**AMA Style**

Zhukov DO, Andrianova EG, Lesko SA.
The Influence of a Network’s Spatial Symmetry, Topological Dimension, and Density on Its Percolation Threshold. *Symmetry*. 2019; 11(7):920.
https://doi.org/10.3390/sym11070920

**Chicago/Turabian Style**

Zhukov, Dmitry O., Elena G. Andrianova, and Sergey A. Lesko.
2019. "The Influence of a Network’s Spatial Symmetry, Topological Dimension, and Density on Its Percolation Threshold" *Symmetry* 11, no. 7: 920.
https://doi.org/10.3390/sym11070920