Transportation Geography and Network Science/Shared micromobility networks

Background
Shared Micromobility is a range of small and lightweight vehicles made available for shared use to individuals on a short-term basis for a price or free . The station-based bicycle-sharing system allows people to borrow a bike from a station and return it at another station belonging to the same system. There are docks in each station as special racks to lock bikes. The dock controlled by the computer will release the bike after the user enters payment information. To return the bike, the user needs to lock it by moving it into an empty dock. Other systems are the short-term checkout, the long-term checkout, the coin deposit stations and the dockless.

Devices
Shared Micromobility devices can be human-powered or electric. The e-scooter, bike and e-bike are the mainstream in 2020. Human-powered devices are considered to be low speed (top speed of 8 miles per hour (mph)), while electric devices can have sped up 20 mph . Any vehicle with an internal combustion engine cannot be defined as micromobility, nor can devices with top speeds above 20mph.

Usage in the United States
This bar graph presents the usage of shared micromobility in the United States from 2010 to 2019. Usage measures the number of the trip generated by riding those devices. In the first three years, the usage of shared micromobility gradually increased year by year, while the data is doubled in the fourth year, then it backs to steady development. With the introduction of new technologies scooter and dockless bike share in 2018, more users entered the market, and the trip increased dramatically from this year. In 2019, Station based bike share maintained a continuous growth trend and reached 40 million trips, and dockless e-bikes had the same growth trend (10M trips), while the usage of scooters surged to 86 million trips .

Shared Micromobility as A Network
Wu and Kim (2020) demonstrate how shared micromobility networks can be represented as a directed and weighted graph $$G = (V, E, w)$$. Their work focuses on the station-based bike sharing system, but these concepts can also be applied to dockless system and other sharing systems.

Vertices
Vertices represent bike stations (nodes).

Edges
Edges represents links connecting any two bike stations.

The weight of edges
Weight is defined as the cycling distance between two stations.

The Availability of the Trip between Two Stations
A = {aij} is an adjacent matrix representing the graph in a new way. The aij ranges from 0 to 1, indicating if there is at least one trip from the station i to station j.

Network structure
The degree distribution is the proportion of k-degree nodes over the whole network, which is written as:
 * $$P(k) = \frac{n(k)}{\sum\limits_{j=1}^\infty n(j)}$$

where $$n(k)$$ is the number of nodes that have a degree $$k$$ .

The average path length is the average number of steps along the shortest paths for all node pairs in the entire network, which is measured by:
 * $$L = \frac{1}{n*(n-1)}\sum\limits_{i \neq j} d_{ij}$$

where $$d_{ij}$$is the number of edges through the shortest path between node $$i$$ and node $$j$$.

The local clustering coefficient is used to measure the probability that the neighborhood of $$i$$ can be connected, calculated by:
 * $$C_{i} = \frac{E_{i}}{r_{i}(r_{i}-1)}, k_{i} \geq 2$$

where $$Ei$$ is the actual number of edges between the connected neighbors of the node $$i$$, and $$r_{i}$$ is the number of neighbors of the node $$i$$.

The neighbors Ni of the node $$i$$ are expressed as:
 * $$ N_{i} = \{v_{j}: a_{ij} = 1 \lor a_{ji} = 1\}$$

The global clustering coefficient is the ratio of the number of closed triplets to the total number of triplets, calculated by:
 * $$ C = \frac{number\; of\; closed\; triplets}{number\; of\; all\; triplets}, 0 \leqslant C \leqslant 1 $$

Node centrality
In order to measure connectivity, accessibility, and intermediateness of stations in bike-sharing networks respectively, Wu and Kim measured following centrality indices.

Degree centrality:
 * $$C_{D}(i) = k_{i}^{in} + k_{i}^{out}$$
 * $$k_{i}^{in} = \sum\limits_{j=1}^n a_{ji}$$
 * $$k_{i}^{in} = \sum\limits_{j=1}^n a_{ij}$$

(if there is a trip from node $$i$$ to node $$j$$, $$a_{ij} = 1$$, otherwise $$a_{ij} = 0$$)

Closeness centrality:
 * $$C_{C}(i) = \sum\limits_{j=1}^n \frac{1}{d_{ji}}$$

where $$d_{ji}$$ is the shortest path length from node $$j$$ to the given node $$i$$.

Betweenness centrality :
 * $$C_{B}(i) = \sum\limits_{j \neq k} \frac{n_{kj}(i)}{n_{kj}}$$

where $$n_{kj}$$ is the total number of shortest paths from node $$k$$ to node $$j$$, and $$n_{kj}(i)$$ is the number of shortest paths that pass through the node $$i$$.

Spatial autocorrelation analysis
Wu and Kim conducted this analysis to measure the degree of clustering among observations in geographical space.

The global spatial autocorrelation indicator, called Global Moran's I, can be measured by:
 * $$I = \frac{n}{S_{0}} \frac{\sum\limits_{i=1}^n \sum\limits_{j=1}^n w_{i,j}(x_{i} - \bar x)}{\sum\limits_{i=1}^n(x_{i} - \bar x)^2}$$

where $$n$$ is the number of observations;

$$x$$ is the variable of interest;

$$i,j$$ is the spatial units;

$$\bar x$$ is the mean of $$x$$;

$$w_{i,j}$$ is the matrix of spatial weights which is expressed by the inverse distance between $$x_{i}$$ and $$x_{j}$$;

$$S_{0}$$ is the sum of all spatial weights.

The local indicator of spatial association (LISA) can be measured by:
 * $$I_{i} = \frac{(x_{i}- \bar x)}{m} \sum\limits_{j=1}^n w_{i,j}(x_{j}- \bar x)$$

where $$m$$ is the variance of $$x$$.

Small World
Wu and Kim summarized the network structure measures of the five bike-sharing networks compared to those of random networks with the same number of nodes and edges. The findings show these bike-sharing networks have a shorter average path length and a larger global clustering coefficient, which indicate that bike-sharing networks have small-world properties.

Not a Scale-Free Network
Wu and Kim also looked for scale-free network features among five bike-sharing networks, and found that the cumulative degree distributions of the network in Washington, D.C. follows a power-law with exponential cutoff distribution, while other networks follow an exponential distribution.