# Mining Topological Dependencies of Recurrent Congestion in Road Networks

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. Congestion Analysis

#### 2.2. Spatio-Temporal Data Mining

#### 2.3. Data Sources

## 3. Problem Statement and Formalisation

- the subgraphs are located in spatial proximity,
- the subgraphs are typically simultaneously affected by Recurrent Congestion or
- the road network topology causes the correlation of the congestion on these subgraphs.

**Transportation graph**. We represent the road network as a directed multi-graph $TG:=(V,U)$, referred to as a transportation graph. U is a set of edges (i.e., road segments); V is a set of nodes (i.e., junctions). We refer to an edge of the transportation graph as a unit $u\in U$.

**Unit load.**We denote the traffic flow observed on a unit at a particular time point as unit load. Formally, $ul(u,t)$ is the traffic load on the unit u at the time point $t\in \mathcal{T}$, where $\mathcal{T}$ denotes the set of time points.

**unit load**$ul(u,t)\in [0,1]$ as the relative speed reduction at unit u at time point t with respect to the speed limit $lim\left(u\right)$ of the corresponding edge of the transportation graph:

**Affected unit.**We denote a unit that exhibits an abnormally high traffic load at a certain time point t as an affected unit at t. Formally, affected(u,t) $U\times \mathcal{T}\mapsto \{True,False\}$ indicates whether unit u is affected at time point t.

**Subgraph.**We refer to a subgraph of the transportation graph as subgraph $sg:=({V}^{\prime},{U}^{\prime})$ with ${V}^{\prime}\subset V$, ${U}^{\prime}\subset U$.

**Affected subgraph**. An affected subgraph represents a subgraph that exhibits an abnormally high traffic load at a certain time point t (e.g., a congested highway section). Formally, $\mathit{affected}(sg,t):SG\times \mathcal{T}\mapsto \{True,False\}$ indicates whether a subgraph $sg$ is affected at time point t, where $SG=\left\{\mathcal{P}\right(V)\times \mathcal{P}(U\left)\right\}$ denotes the set of possible subgraphs and $\mathcal{P}(\xb7)$ denotes the power set.

## 4. Proposed Solution

#### 4.1. Identification of Affected Units

#### 4.2. Identification of Affected Subgraphs

#### 4.3. Spatial Merging of Affected Subgraphs

Algorithm 1 Merge Subgraphs |

Input: C: Set of subgraphs |

Output: SG: Set of merged subgraphs |

1: $SG\leftarrow C$ |

2: $commonUnits\leftarrow \left[\right]$ |

3: for all $sg\in SG$ do |

4: for all $u\in sg.U$ do |

5: $commonUnits\left[u\right]\leftarrow commonUnits\left[u\right]\cup sg$ |

6: changed← True |

7: while changed do |

8: changed ← False
{Generate candidates} |

9: candidates $\leftarrow \varnothing $ |

10: for all $u\in commonUnits$ do |

11: candidates ← candidates $\cup {\left[\mathcal{P}\left(commonUnits\left[u\right]\right)\right]}^{2}$ {Compute similarities} |

12: $s\left[\right]\leftarrow \varnothing $ |

13: for all $(s{g}_{1},s{g}_{2})\in $ candidates do |

14: $s\left[(s{g}_{1},s{g}_{2})\right]\leftarrow $ similarity $(s{g}_{1},s{g}_{2})$ {Merge subgraphs} |

15: visited $\leftarrow \varnothing $ |

16: for all $(s{g}_{1},s{g}_{2})\in $ ordered $\left(s\right)$ do |

17: if $s{g}_{1}\in $ visited $\vee s{g}_{2}\in $ visited then |

18: continue |

19: if $s\left[(s{g}_{1},s{g}_{2})\right]\ge t{h}_{sim}$ then |

20: $s{g}_{1}\leftarrow s{g}_{1}\cup s{g}_{2}$ |

21: $SG\leftarrow SG\setminus s{g}_{2}$ |

22: visited ← visited $\cup \{s{g}_{1},s{g}_{2}\}$ |

23: update(commonUnits) |

24: changed ← True |

25: return $SG$ |

#### 4.4. Identification of Spatio-Temporal Dependencies

`ordered`()).

Algorithm 2 Determine Spatio-Temporal Subgraph Dependencies |

Input: $SG$: Set of subgraphs $\mathcal{T}$: Set of time points |

Output: ${P}_{dependent}$ Set of pairs of subgraphs, ordered by dependency score |

1: $occ\left[\right]\left[\right]\leftarrow \varnothing $ |

2: for all $t\in \mathcal{T}$ do |

3: for all $sg\in \mathcal{S}\mathcal{G}$ do |

4: if $\exists u\in SG:iqr(u,t)$ then |

5: $occ\left[t\right]\left[sg\right]\leftarrow 1$ |

6: else |

7: $occ\left[t\right]\left[sg\right]\leftarrow 0$ {Determine candidate pairs} |

8: candidates $\leftarrow \varnothing $ |

9: for all $(s{g}_{1},s{g}_{2})\in {\left[\mathcal{P}\left(SG\right)\right]}^{2}$ do |

10: if $(s{g}_{1},s{g}_{2})\in $ candidates then |

11: continue |

12: for all $t\in \mathcal{T}$ |

13: if $occ\left[t\right]\left[s{g}_{1}\right]=1\wedge occ\left[t\right]\left[s{g}_{2}\right]=1$ then |

14: candidates ← candidates $\cup \left\{(s{g}_{1},s{g}_{2})\right\}$ |

15: break {Compute dependency} |

16: ${P}_{dependent}\leftarrow \left[\right]$ |

17: for all $(s{g}_{1},s{g}_{2})\in candidates$ do |

18: score ←dependency $(s{g}_{1},s{g}_{2},\mathcal{T})$ |

19: ${P}_{dependent}\left[(s{g}_{1},s{g}_{2})\right]\leftarrow score$ |

20: return ordered $\left({P}_{dependent}\right)$ |

## 5. Datasets

#### 5.1. OpenStreetMap

#### 5.2. Traffic Dataset

## 6. Experiments and Discussion

#### 6.1. Structural Dependencies

#### 6.2. Analysis of Affected Units and Subgraphs

#### 6.2.1. Distribution of Affected Units

#### 6.2.2. Distribution of Affected Subgraphs

#### 6.2.3. Temporal Persistence of Affected Subgraphs

#### 6.3. Evaluation of Subgraph Merging

## 7. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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## Short Biography of Authors

**Nicolas Tempelmeier**is a Ph.D. student at the L3S Research Center in Hannover, Germany. He received his M.Sc. in Computer Science from the Leibniz University Hannover. His research interests lie in the intersection of artificial intelligence and geographic information systems. His research focuses on data-driven analysis of spatial, temporal and Web data.

**Udo Feuerhake**is a post-doc at the Institute of Cartography and Geoinformatics of the Leibniz Universität Hannover. After studying Computer Aided Engineering, he received his PhD degree for his work on object tracking and movement pattern recognition. His current research interests are in the analysis of spatio-temporal data in a variety of domains, where soccer and transportation stand out.

**Oskar Wage**is currently a research assistant at the Institute of Cartography and Geoinformatics at the Leibniz University Hannover. Previously, he completed his Bachelor and Master of Science in Geodesy and Geoinformatics there. Already during his studies, he worked on different applications of route guidance and navigation. At the institute, he is doing research in the field of urban transport by analyzing trajectories and other spatial data, especially about alternative concepts for urban logistics and bicycle infrastructure and mobility.

**Prof. Dr. Elena Demidova**is Professor of Computer Science and head of the Data Science and Intelligent Systems Group (DSIS) at the University of Bonn since October 2020. She received her PhD from the Leibniz University of Hannover in 2013, and her MSc degree in Information Engineering from the University of Osnabrück in a joint program with the University of Twente in 2006. Previously, she worked as Research Group Leader at the L3S Research Center at the Leibniz University of Hannover and as Senior Research Fellow at the University of Southampton. Her main research interests are in Data Science, Machine Learning, the Web and Semantic Web.

**Figure 1.**Example of structural dependencies in an urban road network observed near Gehrden, Germany.

**Figure 3.**Precision@k with respect to k and $t{h}_{sim}$ of the identified structural subgraph dependencies.

**Figure 4.**Examples of the identified dependencies between subgraphs in Hanover. Dependent subgraphs are marked in blue and purple. Map images: ©OpenStreetMap contributors, ODbL.

**Figure 5.**Examples of identified dependencies between subgraphs in Brunswick. Dependent subgraphs are marked in blue and purple. Map images: ©OpenStreetMap contributors, ODbL.

**Figure 7.**The number and the average size of the identified affected subgraphs show opposite trends dependent on the chosen threshold ${d}_{u,max}$. Subgraph size is measured as the number of edges.

**Figure 8.**The existence time and the size of the affected subgraphs dependent on the value of the ${d}_{u,max}$ threshold. The axes are in logarithmic scale.

**Figure 9.**Influence of $t{h}_{sim}$ in Algorithm 1 on the number of subgraphs and an average subgraph size.

**Figure 10.**Example of affected subgraphs calculated for a major junction in Hanover with varying values of $t{h}_{sim}$. Colours of the road segments indicate the subgraph assignment of the segments. Map images: ©OpenStreetMap contributors, ODbL.

Hanover | Brunsiwck | |
---|---|---|

No. Units | 23,125 | 7678 |

No. Records | $195\times {10}^{6}$ | $43\times {10}^{6}$ |

Avg. No. Records/Unit | 8422.79 | 5674.91 |

Time Span | October 2017–January 2018 | December 2018–January 2019 |

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## Share and Cite

**MDPI and ACS Style**

Tempelmeier, N.; Feuerhake, U.; Wage, O.; Demidova, E.
Mining Topological Dependencies of Recurrent Congestion in Road Networks. *ISPRS Int. J. Geo-Inf.* **2021**, *10*, 248.
https://doi.org/10.3390/ijgi10040248

**AMA Style**

Tempelmeier N, Feuerhake U, Wage O, Demidova E.
Mining Topological Dependencies of Recurrent Congestion in Road Networks. *ISPRS International Journal of Geo-Information*. 2021; 10(4):248.
https://doi.org/10.3390/ijgi10040248

**Chicago/Turabian Style**

Tempelmeier, Nicolas, Udo Feuerhake, Oskar Wage, and Elena Demidova.
2021. "Mining Topological Dependencies of Recurrent Congestion in Road Networks" *ISPRS International Journal of Geo-Information* 10, no. 4: 248.
https://doi.org/10.3390/ijgi10040248