# An Efficient Graph-based Method for Long-term Land-use Change Statistics

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

#### 2.1. STDB for Land Management

**Figure 2.**STDB based on a typical spatio-temporal model (refer to Teng, 2005 [18]).

#### 2.2. Statistical Process in STDB

## 3. Analysis of Land Use Change in Graph Theory Approach

#### 3.1. Characterization of Spatio-Temporal Flow Network

**Figure 3.**(

**a**) Land-use snapshots at three different timestamps; (

**b**) corresponding spatio-temporal graph (refer to Yin, 2003 [23]).

#### 3.2. Modelling Long-Term Transition as Multi-Commodity Flow

#### 3.3. Constant Multi-Commodity Flow Condition

**Preliminary.**A transportation network with multiple sources and sinks is a 5-tuple (V, E, S, T, c) where m sources ${s}_{1},{s}_{2,\dots}{s}_{m}\in S$ and n sinks ${t}_{1},{t}_{2},\dots {t}_{n}\in T$ belong to the vertex set V, and $c:E\to {\mathbb{R}}_{+}$ is the capacity function for edges in the directed edge set E.

**Definition 1.**A vertex v is called mixed vertex if more than one source can reach v and v can reach more than one sink, i.e.:

**Definition 2.**A one-way cut is a cut $\left[X,\overline{X}\right]$ separating S from T where no directed edges have a tail in $\overline{X}$ and a head in $X$.

**Definition 3.**An S-T exclusive cut $\delta \left(X\right)$ is a one-way cut that only contains edges which can be reached from exactly one source and can reach exactly one sink, i.e.:

**Lemma 1.**

- $v\in T$, then $\left|R\left(v\right)\cap \text{}T\right|=1$
- $v\in V-T$, then v must be reachable from other sources than ${s}_{i}$ (otherwise it will be included in ${X}_{i}$), so it must connect only one sink (or it is a mixed vertex). $\left|R\left(v\right)\cap \text{}T\right|=1$

**Definition 4.**A multiflow f on a network is saturated if the total flow on each edge is maximized. i.e.,:

**Definition 5.**A network is capacity-balanced if for all internal vertices, the capacity sum of inflow edges equals the capacity sum of outflow edges. Let ${c}^{-}\left(v\right)\u2254{\sum}_{u\in V}c\left(u,v\right)$, ${c}^{+}\left(v\right)\u2254{\sum}_{u\in V}c\left(v,u\right)$. Hereby, the capacity-balanced constraint is represented as:

**Fact 1.**In a DACB network, the residual network for any given feasible flow is still a DACB network.

**Fact 2.**In a DACB network, if P is the only S-T path that includes edge (u, v), then (u, v) has the smallest capacity among edges along P.

**Proposition 1.**

**Lemma 2.**

**Lemma 3.**

**Theorem 1.**

- The saturated multi-commodity flow F is constant
- No mixed vertices exist
- An S-T exclusive cut exists

#### 3.4. Reducible or Unreducible

## 4. Description of the Graph-Based TSP Method

PartitionSTFN(N) /* Label T-Reachability */ Q ← ∅; |

## 5. Data Experiments

#### 5.1. Sample Data

**Figure 7.**Experimental land-use data in Hunan. The first picture displays base state at 2009, and the following five pictures display incremental data indicating the location and time of change events (2009–2014).

#### 5.2. Method for Evaluation

#### 5.3. Discussion of the Results

Query Condition | Basic TSP | Query-Optimized TSP | Graph-Based TSP | |||
---|---|---|---|---|---|---|

Amount | Ratio | Amount | Ratio | Amount | Ratio | |

2009–2010 | 37,162 | 1.0000 | 2105 | 0.0566 | 0 | 0.0000 |

2009–2011 | 37,254 | 1.0000 | 2530 | 0.0679 | 13 | 0.0003 |

2009–2012 | 37,311 | 1.0000 | 2613 | 0.0700 | 13 | 0.0003 |

2009–2013 | 37,320 | 1.0000 | 3723 | 0.0998 | 20 | 0.0005 |

2009–2014 | 38,938 | 1.0000 | 3914 | 0.1005 | 25 | 0.0006 |

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Zhang, Y.; Gao, Y.; Gao, B.; Pan, Y.; Yan, M. An Efficient Graph-based Method for Long-term Land-use Change Statistics. *Sustainability* **2016**, *8*, 9.
https://doi.org/10.3390/su8010009

**AMA Style**

Zhang Y, Gao Y, Gao B, Pan Y, Yan M. An Efficient Graph-based Method for Long-term Land-use Change Statistics. *Sustainability*. 2016; 8(1):9.
https://doi.org/10.3390/su8010009

**Chicago/Turabian Style**

Zhang, Yipeng, Yunbing Gao, Bingbo Gao, Yuchun Pan, and Mingyang Yan. 2016. "An Efficient Graph-based Method for Long-term Land-use Change Statistics" *Sustainability* 8, no. 1: 9.
https://doi.org/10.3390/su8010009