# Analyzing Spatial Behavior of Backcountry Skiers in Mountain Protected Areas Combining GPS Tracking and Graph Theory

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Data

#### 2.2.1. Inventory of Skiing Zones System

#### 2.2.2. Mobility Data of Backcountry Skiers

#### 2.3. Data Analysis

- GPS data pre-processing
- Creation of the structural network (undirected graph)
- Creation of the functional network (directed graph)
- Quantification of network connectivity indices
- Calculation of centrality measures of network nodes.

#### 2.3.1. Pre-Processing of GPS Data

#### 2.3.2. Creating the Structural Network

#### 2.3.3. Creating the Functional Network

#### 2.3.4. Calculating Network Connectivity Indices

- Kansky index $\beta =\text{}\frac{e}{v}$ where $e$ is the number of edges, $v$ is the number of vertices (the higher the value of $\beta $, the more coherent the network);
- Kansky index $\gamma =e/\left(3\ast \left(v-2\right)\right)$, defining the ratio of the existing number of edges (e) to the maximum possible number of edges resulting from the number of vertices (v). The $\gamma $ value ranges from 0 to 1, where the value of 1 indicates a completely connected network.
- Kansky index $\alpha =\frac{\mu}{2v-5}$, where $\mu $ is a cyclomatic measure calculated as $\mu =e-v+p$, where p is the number of isolated subgraphs. An $\alpha $ value of 1 indicates a completely meshed network, and 0 indicates a very simple network.

#### 2.3.5. Calculating Node Centrality Measures

## 3. Results

#### 3.1. General Characteristics of the Structural and Functional Networks

#### 3.2. Network Coherence (Connectivity Indices)

#### 3.3. Relative Importance of Network Nodes (Centrality Measures)

## 4. Discussion and Conclusions

#### 4.1. General Meaning of the Findings

#### 4.2. Comparison of the Structural and Functional Networks in the Study Area

#### 4.3. Limitations of the Proposed Methodology

#### 4.4. Meaning of the Findings for Visitor Management in Protected Areas

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Characteristics of the study area: designated backcountry skiing zones within the border of Tatra National Park (TNP), Poland.

**Figure 2.**Translation of the designated backcountry skiing areas system into a structural network (undirected graph). The geographic coordinates (longitude and latitude) were defined in WGS84 spatial reference system and are expressed in decimal degrees.

**Figure 3.**Three-dimensional visualization of the structural skiing network (undirected graph). The geographic coordinates (longitude and latitude) were defined in WGS84 spatial reference system and are expressed in decimal degrees. Altitude is expressed in meters above sea level (m.a.s.l.). The elevation data was used only for visualization purposes.

**Figure 4.**Example of a skiing route. (

**a**) Recorded GPS track of a TNP visitor; (

**b**) representation of the corresponding directed graph G = (V,E) where the vertices are V = {44,…,66} and the edges are E = {(50, 49), (49, 51), (51, 59), (59,62), (62, 61), (61, 66), (66, 61), (61,54), (54, 53), (53, 58), (58, 60), (60, 54), (54, 48), (48, 44), (44, 45), (45, 49), (49, 50)}. In the final graph the loop, (66, 66) was deleted, since a property of directed graphs is vi ≠ vj.

**Figure 5.**Structural network (undirected graph) depicting the designated backcountry skiing system in TNP. The network is composed of 93 nodes and 133 links.

**Figure 6.**Functional network (directed graph) of the designated backcountry skiing system; graph based on the recorded visitors’ trip itineraries. Arrows indicate movement direction; color scale (grey scale) illustrates the intensity of skiing traffic. A darker color means higher use intensity.

**Figure 7.**Node centrality measures in the structural (

**a**) and functional networks (

**b**) of designated backcountry skiing zones in TNP.

Centrality Measure | Description | Mathematical Equation |
---|---|---|

Input degree | Number of edges entering to vertex i. | ${d}_{i}^{+}=\text{}{\sum}_{j\in G}{x}_{ij}$ ${x}_{ij}$ signal the position between node i and node j. |

Output degree | Number of edges leaving vertex i. | ${d}_{i}^{-}=\text{}{\displaystyle \sum _{j\in G}}{x}_{ij}$ |

Degree (all) | Total number of edges connected to the vertex i. | ${D}_{i}={d}^{+}+{d}^{-}$ |

Closeness | Inverse sum of distances from a given vertex to all other vertices in the graph. | ${C}_{i}={\sum}_{j\text{}\ne i\in G}{(d\left(i,j\right))}^{-1}$ where $d\left(i,j\right)$ is a topological distance between vertices $i\text{}\mathrm{and}\text{}j.$ |

Betweenness | A number of times a vertex is crossed by shortest paths in the graph. | $B\left(i\right)={\sum}_{j\text{}\text{}k}{g}_{jk}\left(i\right)/{g}_{jk}$ where ${g}_{jk}$ is the number of geodesics connecting jk, and ${g}_{jk}\left(i\right)$ is the number that geodesics i is on. |

Proximity prestige | Expresses the influence domain of a vertex by the average distance from all vertices in the influence domain. Pp value of 0 indicates that node i is unreachable; whereas Pp = 1 if all nodes are directly connected to node i. | $P{p}_{i}=\frac{Ii}{g-1}{\displaystyle \sum _{j\text{}\ne i\in G}}{(d\left(i,j\right)/Ii)}^{-1}$ |

**Table 2.**Structural network node centrality measures of the 10 nodes with the highest “all degree” values.

Node Number | Degree | Closeness | Betweenness |
---|---|---|---|

79 | 6 | 0.14 | 0.14 |

24 | 5 | 0.15 | 0.08 |

48 | 5 | 0.18 | 0.43 |

61 | 5 | 0.17 | 0.30 |

69 | 5 | 0.14 | 0.02 |

16 | 4 | 0.11 | 0.12 |

18 | 4 | 0.12 | 0.17 |

27 | 4 | 0.15 | 0.09 |

44 | 4 | 0.17 | 0.09 |

45 | 4 | 0.15 | 0.14 |

**Table 3.**Functional network node centrality and prestige measures of the 10 nodes with the highest “weighted all degree” values.

Node Number | Weighted Input Degree | Weighted Output Degree | Weighted All Degree | Closeness | Betweenness | Proximity Prestige |
---|---|---|---|---|---|---|

45 | 350 | 532 | 882 | 0.19 | 0.08 | 0.19 |

49 | 351 | 149 | 500 | 0.20 | 0.13 | 0.20 |

48 | 197 | 194 | 391 | 0.19 | 0.06 | 0.19 |

44 | 152 | 149 | 301 | 0.18 | 0.01 | 0.17 |

61 | 109 | 90 | 199 | 0.17 | 0.10 | 0.17 |

62 | 93 | 100 | 193 | 0.15 | 0.05 | 0.15 |

41 | 86 | 84 | 170 | 0.17 | 0.04 | 0.17 |

54 | 82 | 88 | 170 | 0.17 | 0.06 | 0.17 |

7 | 79 | 78 | 157 | 0.12 | 0.07 | 0.12 |

51 | 56 | 83 | 139 | 0.17 | 0.02 | 0.17 |

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## Share and Cite

**MDPI and ACS Style**

Taczanowska, K.; Bielański, M.; González, L.-M.; Garcia-Massó, X.; Toca-Herrera, J.L.
Analyzing Spatial Behavior of Backcountry Skiers in Mountain Protected Areas Combining GPS Tracking and Graph Theory. *Symmetry* **2017**, *9*, 317.
https://doi.org/10.3390/sym9120317

**AMA Style**

Taczanowska K, Bielański M, González L-M, Garcia-Massó X, Toca-Herrera JL.
Analyzing Spatial Behavior of Backcountry Skiers in Mountain Protected Areas Combining GPS Tracking and Graph Theory. *Symmetry*. 2017; 9(12):317.
https://doi.org/10.3390/sym9120317

**Chicago/Turabian Style**

Taczanowska, Karolina, Mikołaj Bielański, Luis-Millán González, Xavier Garcia-Massó, and José L. Toca-Herrera.
2017. "Analyzing Spatial Behavior of Backcountry Skiers in Mountain Protected Areas Combining GPS Tracking and Graph Theory" *Symmetry* 9, no. 12: 317.
https://doi.org/10.3390/sym9120317