# Modeling and Analyzing Urban Sensor Network Connectivity Based on Open Data

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## Abstract

**:**

## 1. Introduction

## 2. Network Modeling Algorithms

#### 2.1. Network Device Data to Slots of Space Nodes

Algorithm 1: Network device data to slots of space nodes. | |

Input: | |

$area\leftarrow (\left[{X}_{min},{X}_{max}\right],\left[{Y}_{min},{Y}_{max}\right])$ | // $\phantom{(}$network area of interest$\phantom{(}$ |

$classes\leftarrow {\left(clas{s}_{i}\right)}_{i\leftarrow 1}^{j}:$ | // $\phantom{(}$list of j distinct device classes$\phantom{(}$ |

$clas{s}_{i}\leftarrow {\left\{devic{e}_{k}\right\}}_{k\leftarrow 1}^{l}:$ | // $\phantom{(}$set of l devices of class i$\phantom{(}$ |

$devic{e}_{k},\leftarrow ($ | // $\phantom{(}$device k$\phantom{(}$ |

$instance{s}_{k},\leftarrow {\left(instanc{e}_{m}\right)}_{m\leftarrow 1}^{n}:$ | // $\phantom{(}$list of n time instances of device k$\phantom{(}$ |

$instanc{e}_{m}\leftarrow ($ | // $\phantom{(}$instance m$\phantom{(}$ |

$tim{e}_{m},$ | // $\phantom{(}$instance occurrence time$\phantom{(}$ |

$locatio{n}_{m}$ | // $\phantom{(}$instance location$\phantom{(}$ |

$),$ | |

$i{d}_{k},$ | // $\phantom{(}$device id$\phantom{(}$ |

$rol{e}_{k}$ | // $\phantom{(}$device role$\phantom{(}$ |

) | |

$slotLength\in {\mathbb{R}}_{>0}$ | // $\phantom{(}$length of time slots$\phantom{(}$ |

$topologyLength\in {\mathbb{N}}^{+}$ | // $\phantom{(}$number of subsequent time slots of the time topology$\phantom{(}$ |

$topologyStart$ | // $\phantom{(}$start time of the topology$\phantom{(}$ |

$windows\leftarrow {\left(windo{w}_{i}\right)}_{i\leftarrow 1}^{j}$ | // $\phantom{(}$list of j time window lengths corresponding to the respective j$\phantom{(}$ |

device classes | |

Output: | |

$slots\leftarrow {\left(slo{t}_{p}\right)}_{p\leftarrow 1}^{topologyLength}:$ | // $\phantom{(}$list of subsequent time slots$\phantom{(}$ |

$slo{t}_{p}\leftarrow {\left\{nod{e}_{r}\right\}}_{r\leftarrow 1}^{s}:$ | // $\phantom{(}$set of all s nodes of slot p$\phantom{(}$ |

$nod{e}_{r}\leftarrow ($ | // $\phantom{(}$node r of slot p$\phantom{(}$ |

$clas{s}_{r},$ | // $\phantom{(}$node class$\phantom{(}$ |

$i{d}_{r},$ | // $\phantom{(}$node id$\phantom{(}$ |

$locatio{n}_{r},$ | // $\phantom{(}$node geographic location$\phantom{(}$ |

$rol{e}_{r},$ | // $\phantom{(}$node role$\phantom{(}$ |

$slotNumbe{r}_{r}$ | // $\phantom{(}$node slot number$\phantom{(}$ |

) | |

1 $slots\leftarrow assignDevicesInstancesToSlots\left(\right)$ | // $\phantom{(}$assign devices instances to slots of nodes$\phantom{(}$ |

2 output slots |

Procedure assignDevicesInstancesToSlots |

#### 2.2. Slots of Space Nodes to Space Connectivity List

Algorithm 2: Slots of space nodes to space connectivity list |

#### 2.3. Space Connectivity List to Space–Time Connectivity Graph

Algorithm 3: Space connectivity list to space–time connectivity graph |

- $intraSlotTimeEdgeSpaceUnitCost$—space unit cost of a time edge within a slot:
- −
- default value: 0;
- −
- meaning: nonzero value stands for unit space cost related to the node (device) operating within a time slot. It can be used, for example, to model the cost of receiving location beacons;

- $interSlotTimeEdgeSpaceUnitCost$—space unit cost of a time edge between slots:
- −
- default value: 0;
- −
- meaning: nonzero value stands for unit space cost related to the node (device) transitioning between time slots. It can be used, for example, to model the cost of transmitting location beacons;

- $intraSlotSpaceEdgeTimeUnitCost$—time unit cost of a space edge within a slot:
- −
- default value: 0;
- −
- meaning: nonzero value indicates unit time cost related to transmitting a message between two devices, e.g., due to technology-dependent buffering or delays. It can be used, for instance, together with $intraSlotSpaceEdgeTimeUnitCost$ to favor intra-slot time edges over intra-slot space edges by path-finding algorithms. It will lead to maximizing buffering time in a single relay, minimizing the number of inter-node transmissions, and hence, the nodes involved. Although, it can happen at the expense of an overall increase in the space cost of the constructed $STCG$;

- $intraSlotTimeEdgeTimeUnitCost$—time unit cost of a time edge within a slot:
- −
- default value: 0;
- −
- meaning: should be considered in relation to $intraSlotSpaceEdgeTimeUnitCost$ for given modeling scenario. It can also be used with $interSlotTimeEdgeTimeUnitCost$ to shape time-path cost properties of $STCG$, e.g., as a tie-breaker;

- $interSlotTimeEdgeTimeUnitCost$—time unit cost of a time edge between slots:
- −
- default value: 1;
- −
- meaning: indicates unit cost related to transitioning (buffering) a message over time by a node. It is of key significance for path searching scenarios that aim to optimize the message delivery time, e.g., to minimize the total time cost of a path. If set to 0 it may lead to unexpected or erroneous results in optimization algorithms which are based on ordering the weights of the edges. It can be of use though when consciously used with properly selected values of $intraSlotSpaceEdgeTimeUnitCost$ and $intraSlotTimeEdgeTimeUnitCost$.

Procedure addTimeNodeInstancesAndEdges(i, SCG, STCE, STCV) |

Procedure makeTimeNodeInstance(i, spaceNodeInstance, timeInstanceType) |

Procedure addTimeEdges(STCE, STCV, timeNodeInstances) |

Procedure addSpaceEdges(i, SCG, STCE) |

#### 2.4. Space–Time Connectivity Graph to First-Contact Graph

Algorithm 4: Space–time connectivity graph to first-contact graph | |

Input: | |

$STCG\leftarrow ($ | // $\phantom{(}$directed space–time connectivity graph |

$STCE,$ | // $\phantom{(}$set of space–time connectivity edges$\phantom{(}$ |

$STCV$ | // $\phantom{(}$set of space–time connectivity vertices$\phantom{(}$ |

) | |

Output: | |

$FCG\leftarrow ($ | // $\phantom{(}$directed first-contact graph$\phantom{(}$ |

$FCE,$ | // $\phantom{(}$set of first-contact edges$\phantom{(}$ |

$FCV$ | // $\phantom{(}$set of first-contact vertices$\phantom{(}$ |

) | |

1 $spaceNodes\leftarrow findFirstInstancesAndTimeNeighbors\left(STCG\right)$ | |

2 $FCG\leftarrow buildFirstContactGraph(STCG.STCV,spaceNodes)$ | |

3 output FCG |

Procedure findFirstInstancesAndTimeNeighbors(STCG) |

Procedure buildFirstContactGraph(STCV, spaceNodes) |

## 3. Simulation and Analysis Methodology

#### 3.1. Comparative Study Methodology

#### 3.2. Statistical Analysis and Visualization

#### 3.3. Simulation Data Sources and Node Classes

- mobile advanced class ⇒ mobile relays;
- stationary advanced class ⇒ stationary relays;
- stationary simple class ⇒ stationary destinations.

#### 3.4. Simulation Areas and Example Modeled Networks

#### 3.4.1. Gdańsk

- Population:
- Public transport day routes: around 80 [37];
- Simulation area:
- −
- Latitude: 54.34398–54.36191;
- −
- Longitude: 18.62036–18.66666;

- Example space connectivity graph in Figure 11:
- −
- Slot length: 6 s;
- −
- Radio range: 100 m;
- −
- Nodes: 121—mobile relays: 9, stationary relays: 20, stationary destinations: 92;
- −
- Average node degree: 2.45, edges: 148, space cost: 6177 m, connected components: 66.

#### 3.4.2. Poznań

- Population:
- Public transport day routes: around 70 [39];
- Simulation area:
- −
- Latitude: 52.39853–52.41645;
- −
- Longitude: 16.88965–16.93389;

- Example space minimum spanning forest in Figure 12:
- −
- Time interval: 6 s;
- −
- Radio range: 100 m;
- −
- Nodes: 171—mobile relays: 38, stationary relays: 17, stationary destinations: 116;
- −
- Average node degree: 1.20, edges: 103, space cost: 4904.00 m, connected components: 68.

#### 3.4.3. Warsaw

- Population:
- Public transport day routes: around 190 [41];
- Simulation area:
- −
- Latitude: 52.22082–52.23879;
- −
- Longitude: 20.97058–21.01454;

- Example space maximum spanning forest in Figure 13:
- −
- Slot length: 6 s;
- −
- Radio range: 100 m;
- −
- Nodes: 213—mobile relays: 49, stationary relays: 4, stationary destinations: 160;
- −
- Average node degree: 1.10, edges: 117, space cost: 8199.00 m, connected components: 96.

#### 3.4.4. Wrocław

- Population:
- Public transport day routes: around 85 [43];
- Simulation area:
- −
- Latitude: 51.10015–51.11813;
- −
- Longitude: 17.01273–17.05570;

- Example first-contact graph in Figure 14:
- −
- Slot length: 6 s;
- −
- Radio range: 100 m;
- −
- Nodes: 217—mobile relays: 145, stationary relays: 2, stationary destinations: 70;
- −
- Average node degree: 1.66, edges: 180, space cost: 3089.00 m, connected components: 120.

#### 3.5. Simulation Architecture and Parameters

- Algorithm 1, network device data to slots of space nodes (NDD-SSN):
- −
- period: 27 November 2019 from 3:00 p.m. to 5:00 p.m.;
- −
- areas: 4 $\Rightarrow J=4$;
- *
- $are{a}_{1}$: $\left(\right[54.34398,54.36191],[18.62036,18.66666\left]\right)$—Gdańsk;
- *
- $are{a}_{2}$: $\left(\right[52.39853,52.41645],[16.88965,16.93389\left]\right)$—Poznań;
- *
- $are{a}_{2}$: $\left(\right[52.22082,52.23879],[20.97058,21.01454\left]\right)$—Warsaw;
- *
- $are{a}_{2}$: $\left(\right[51.10015,51.11813],[17.01273,17.05570\left]\right)$—Wrocław;

- −
- topology lengths: (75, 150, 300, 600, 1200);
- *
- durations: (7.5 min, 15 min, 30 min, 60 min, 120 min) $\Rightarrow L=5$;
- *
- topologies: (16, 8, 4, 2, 1) $\Rightarrow N=31$;

- −
- slot length: 6 s $\Rightarrow S=1200$;
- −
- classes: (mobile advanced, stationary simple, stationary advanced);
- −
- windows: (10 s, 24 h, 24 h);
- −
- relays: (mobile advanced, stationary advanced).

- Algorithm 2, slots of space nodes to space connectivity list (SSN-SCL):
- −
- radio coverage: omnidirectional;
- *
- radio ranges: (25 m, 50 m, 100 m) $\Rightarrow Q=3$;
- *
- space distance: great-circle distance between two nodes.

- Algorithm 3, space connectivity list to space–time connectivity graph (SCL-STCG):
- −
- unit cost:
- *
- intra-slot time edge space unit cost: 0;
- *
- inter-slot time edge space unit cost: 0;
- *
- intra-slot space edge time unit cost: 0;
- *
- intra-slot time edge time unit cost: 0;
- *
- inter-slot time edge time unit cost: 1.

#### 3.6. Simulation Study Metrics

- Space connectivity:
- (a)
- nodes:
- stationary destination nodes—the number of stationary destinations;
- stationary relay nodes—the number of stationary relays;
- mobile relay nodes—the number of mobile relays;
- mobile relay nodes to all nodes ratio—the percentage of mobile relay nodes as compared to the number of all nodes;
- all nodes—total number of nodes;
- connected components—the number of sets of nodes that are connected with each other by direct or indirect paths;
- nodes per component—average number of nodes in a component;

- (b)
- edges:
- average node degree—average number of edges adjacent to a node;
- edges—total number of edges;
- cost—the sum of space distances of all edges;

- Space–time connectivity:
- (a)
- nodes
- instances per node—the number of time nodes (instances) per unique device (space node)
- instances per mobile relay node—the number of time nodes (instances) per unique mobile relay device (space node).

## 4. Space Connectivity Analysis

#### 4.1. Space Connectivity Nodes

#### 4.2. Space Connectivity Edges

#### 4.3. Space Connectivity Relationships

#### 4.3.1. Space Connectivity in Gdańsk

#### 4.3.2. Space Connectivity in Poznań

#### 4.3.3. Space Connectivity in Warsaw

#### 4.3.4. Space Connectivity in Wrocław

## 5. Space–Time Connectivity Analysis

## 6. Summary

- Large-scale network modeling with space connectivity graphs (SCGs) and multidimensional analysis can be conducted based on open data;
- The unique topology and infrastructure features of each urban area influence the networks that are constructed;
- When conducting analyses of the data obtained for individual cities, it can be seen that the highest overall number of nodes does not necessarily correspond to the population density of the individual cities.
- Mobile relays are present in the areas of interest in varying numbers and with varying distributions, which depend on the infrastructure of the cities concerned, including the number of daily public transport lines, vehicle frequency, street topology, and density of stops;
- Changing the radio range of the nodes affects the modeled networks differently, depending on the distribution of the nodes. Immediate topologies are more fragmented for smaller radio ranges. In more fragmented areas, increased radio coverage is required to connect more nodes;
- The radio coverage affects the costs of space connectivity graphs and their associated minimum and maximum spanning forests. Studies have shown that as radio coverage increases, the number of edges and the cost of structures also increase. These increases are exponential, so the radio parameters of the designed networks must be carefully planned. In this way, excessive use of the wireless medium, as well as the computing and storage resources of the nodes, can be avoided;

- Space–time connectivity graphs (STCGs), which consist of as many as hundreds of thousands of nodes, can be modeled based on space connectivity lists (SCLs) and used as the intermediate structure in space–time network modeling;
- First-contact graphs (FCGs) are valuable compact-form indicators that capture how space–time networks develop over time and how the adjacencies occur for the first time;
- The structures become more complex when network duration and radio coverage grows. This means that topology and connectivity increase as well;
- The sets of mobile nodes present in the areas change over time—some enter and some leave at different moments. In a space–time connectivity graph, there are more unique space nodes (individual devices) than in a single space connectivity graph. As a result, the number of mobile relays in first-contact graphs is at least a few times higher than in the related space connectivity graphs;
- The average node degrees and the number of connected components are much higher in first-contact graphs than they are in space connectivity graphs. This proves that using mobile relays to construct space–time networks enables the momentarily disconnected parts of a network to be connected over time. As a result, larger and denser time-spanning topologies are constructed.

- Urban sensor network graph-based modeling algorithms:
- −
- space connectivity modeling;
- −
- space–time connectivity modeling;
- −
- first-contact modeling;

- Simulation study of introduced models and algorithms:
- −
- methodology for multidimensional modeling and analysis;
- −
- simulation architecture and custom-developed environment;
- −
- comparative investigation and observations for four Polish cities.

- Urban sensor network modeling:
- −
- Development of a reliable parametric network topology generator based on long-term observations of open data and timetables;
- −
- Construction of movement traces based on gathered node location open data;
- −
- Use of introduced modeling architecture as an element of stationary relay deployment planning;
- −
- Use of presented modeling algorithms in other fields:
- *
- social trends analysis;
- *
- multi-criteria route optimization for public service vehicles;
- *
- urban planning of bicycle paths, street infrastructure, etc.;

- Graph-based study of presented algorithms:
- −
- Investigation of extended scale and scope:
- *
- more data sources, including the closed ones, when available;
- *
- more areas of different locations and sizes, e.g., suburbs, small cities, countrysides, etc.;

- −
- Advanced radio connectivity modeling:
- *
- use of actual connectivity data, when available;
- *
- use of complex radio coverage models.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Examples of connected devices in cities in Poland. (

**a**) Electric rental car and parking lot in Wrocław in December 2019; (

**b**) bus, scooter, and bike rental station in Poznań in August 2019; (

**c**) electric kick scooters in Gdańsk in November 2019.

**Figure 9.**Open-data-based network modeling architecture [9].

City | Class | Scope | Format | Updates | Provider |
---|---|---|---|---|---|

Gdańsk | Mobile advanced | Buses and trams [23] | JSON | 20 s | Open Gdańsk |

Stationary simple | Public transport stops [24] | JSON | 24 h | Open Gdańsk | |

Stationary advanced | Ticket machines [25] | JSON | 24 h | Open Gdańsk | |

Poznań | Mobile advanced | Buses and trams [26] | protobuf | Continuous | ZTM Poznań |

Stationary simple | Public transport stops [27] | JSON | Infrequent | Poznan City Hall | |

Stationary advanced | Ticket machines [28] | JSON | Infrequent | Poznan City Hall | |

Warsaw | Mobile advanced | Buses and trams [29] | JSON | 10 s | City of Warsaw |

Stationary simple | Public transport stops [30] | JSON | Infrequent | City of Warsaw | |

Wrocław | Mobile advanced | Buses and trams [31] | JSON | Continuous | Open Data Wrocław |

Stationary simple | City bike rental stations [32] | JSON | 5 min | Open Data Wrocław | |

Stationary simple | Vozilla parking lots [33] | JSON | Continuous | Open Data Wrocław | |

All | Stationary advanced | Air quality meters [34] | JSON | Continuous | Airly |

Category | Type | Number of Graphs |
---|---|---|

Space | Connectivity | 14,400 |

Minimum spanning forest | 14,400 | |

Maximum spanning forest | 14,400 | |

Space–time | Connectivity | 372 |

First-contact | 372 | |

In total | 43,944 |

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**MDPI and ACS Style**

Musznicki, B.; Piechowiak, M.; Zwierzykowski, P.
Modeling and Analyzing Urban Sensor Network Connectivity Based on Open Data. *Sensors* **2023**, *23*, 9559.
https://doi.org/10.3390/s23239559

**AMA Style**

Musznicki B, Piechowiak M, Zwierzykowski P.
Modeling and Analyzing Urban Sensor Network Connectivity Based on Open Data. *Sensors*. 2023; 23(23):9559.
https://doi.org/10.3390/s23239559

**Chicago/Turabian Style**

Musznicki, Bartosz, Maciej Piechowiak, and Piotr Zwierzykowski.
2023. "Modeling and Analyzing Urban Sensor Network Connectivity Based on Open Data" *Sensors* 23, no. 23: 9559.
https://doi.org/10.3390/s23239559