Authors: Dr. Jean-Paul Rodrigue and Dr. Cesar Ducruet
Graph theory is a branch of mathematics concerned about how networks can be encoded, and their properties measured .
1. basic Graph Definition
A graph is a symbolic representation of a network and its connectivity. It implies an abstraction of reality so that it can be simplified as a fructify of linked nodes. The origins of graph theory can be traced to Leonhard Euler, who devised in 1735 a trouble that came to be known as the “ Seven Bridges of Konigsberg ”. In this problem, person had to cross all the bridges only once, and in a continuous sequence, a problem the Euler proved to have no solution by representing it as a hardened of nodes and links. This led to the foundation of graph theory and its subsequent improvements. It has been enriched in the last decades by growing influences from studies of social and complex networks .
In transport geography, most networks have an obvious spatial foundation, namely road, transportation system, and rail networks, which tend to be defined more by their links than by their nodes. This is not necessarily the casing for all transportation system networks. For exemplify, nautical and air networks tend to be more defined by their nodes than by their links since links are often not distinctly defined. A telecommunication system can besides be represented as a network, while its spatial expression can have limited importance and would be unmanageable to represent. Mobile call networks or the Internet, possibly to most building complex graph to be considered, are relevant cases of networks having a structure that can be difficult to symbolize. however, cellular phones and antennas can be represented as nodes, while the links could be individual earphone calls. Servers, the congress of racial equality of the Internet, can besides be represented as nodes within a graph while the physical infrastructure between them, namely roughage ocular cables, can act as links. consequently, all transport networks can be represented by graph theory in one way or the other .
The adopt elements are fundamental to understanding graph hypothesis :
Graph. A graph G is a determine of vertices ( nodes ) v connected by edges ( links ) e. Thus G=(v, e) .
Vertex (Node). A node v is a terminal bespeak or an overlap point of a graph. It is the abstraction of a localization such as a city, an administrative class, a road intersection or a conveyance terminal ( stations, terminuses, harbors and airports ) .
Edge (Link). An edge e is a link between two nodes. The yoke ( i, j ) is of initial extremity i and of terminal extremity j. A link is the abstraction of a transport infrastructure supporting movements between nodes. It has a direction that is normally represented as an arrow. When an arrow is not used, it is assumed the connect is bi-directional .
Sub-Graph. A sub-graph is a subset of a graph G where p is the number of sub-graphs. For exemplify, G ’ = ( five ’, e ’ ) can be a distinct sub-graph of G. Unless the ball-shaped transport organization is considered in its whole, every enchant net is in theory a sub-graph of another. For example, the road exile network of a city is a sub-graph of a regional exile network, which is itself a sub-graph of a national transportation network .
Buckle (Loop or self edge). A yoke that makes a node represent to itself is a buckle .
Planar Graph. A graph where all the intersections of two edges are a vertex. Since this graph is located within a plane, its topology is planar. This is typically the case for power grids, road and railway networks, although bang-up care must be inferred to the definition of nodes ( terminals, warehouses, cities ) .
Non-planar Graph. A graph where there are no vertices at the overlap of at least two edges. Networks that can be considered in a planar manner, such as roads, can be represented as non-planar networks. This implies a third dimension in the topology of the graph since there is the possibility of having a movement “ ephemeral over ” another motion such as for atmosphere and nautical transport, or an overpass for a road. A non-planar graph has potentially much more links than a planar graph .
Simple graph. A graph that includes only one type of link between its nodes. A road or vilify network are dim-witted graph .
Multigraph. A graph that includes several types of links between its nodes. Some nodes can be connected to one yoke type while others can be connected to more than one that are running in twin. A graph depicting a road and a rail network with different links between nodes serviced by either or both modes is a multigraph .
Graph Representation of a Real Network
Basic Graph Representation of a Transport Network
Planar and Non-Planar Graphs
Simple and Multigraph
A transportation net enables flows of people, freight or data, which are occurring along its links. Graph theory must therefore offer the possibility of representing movements as linkages, which can be considered over several aspects :
Connection. A laid of two nodes as every node is linked to the other. Considers if a motion between two nodes is possible, whatever its direction. Knowing connections makes it possible to find if it is possible to reach a node from another node within a graph .
Path. A sequence of links that are traveled in the like management. For a path to exist between two nodes, it must be possible to travel an uninterrupted sequence of links. Finding all the potential paths in a graph is a fundamental assign in measuring approachability and traffic flows .
Chain. A sequence of links having a connection in common with the early. management does not matter .
Length of a Link, Connection or Path. Refers to the pronounce associated with a link, a joining or a way. This tag can be distance, the amount of traffic, the capability or any relevant attribute of that connection. The duration of a path is the number of links ( or connections ) in this way .
Cycle. Refers to a chain where the initial and terminal node is the like and that does not use the lapp connect more than once is a hertz.
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Circuit. A way where the initial and concluding node corresponds. It is a cycle where all the links are traveled in the like steering. Circuits are very important in transportation because several distribution systems are using circuits to cover a a lot territory as possible in one focus ( delivery route ) .
Clique. A clique is a maximal complete subgraph where all vertices are connected .
Cluster. besides called community, it refers to a group of nodes having dense relations with each other than with the pillow of the network. A wide-eyed range of methods are used to reveal clusters in a network, notably they are based on modularity measures ( intra- versus inter-cluster variability ) .
Ego network. For a given node, the self network corresponds to a sub-graph where alone its adjacent neighbors and their common links are included .
Nodal region. A nodal region refers to a subgroup ( tree ) of nodes polarized by an independent node ( which largest flow link connects a smaller lymph node ) and several subordinate nodes ( which largest menstruation link connects a larger node ). Single or multiple linkage analysis methods are used to reveal such regions by removing secondary links between nodes while keeping lone the heaviest links .
Dual graph. A method in space syntax that considers edges as nodes and nodes as edges. In urban street networks, big avenues made of respective segments become single nodes while intersections with other avenues or streets become links ( edges ). This method acting is peculiarly utilitarian to reveal hierarchical structures in a planar net .
Common neighbor. For two or more nodes, the number of nodes that they are normally connected two .
Connections and Paths
Length of a Link, Connection or Path
Cycles and Circuits
3. basic structural Properties
The organization of nodes and links in a graph conveys a structure that can be described and labeled. The basic structural properties of a graph are :
Symmetry and Asymmetry. A graph is symmetrical if each pair of nodes linked in one steering is besides linked in the other. By convention, a line without an arrow represents a connect where it is possible to move in both directions. however, both directions have to be defined in the graph. Most transmit systems are harmonious, but asymmetry can often occur as it is the encase for nautical ( pendulum ) and air services. Asymmetry is rare on road exile networks, unless one-way streets are considered .
Assortativity and disassortativity. Assortative networks are those characterized by relations among exchangeable nodes, while disassortative networks are found when structurally different nodes are frequently connected. Transport ( or technological ) networks are much disassortative when they are non-planar, due to the higher probability for the net to be centralized into a few large hubs .
Completeness. A graph is complete if two nodes are linked in at least one management. A complete graph has no sub-graph and all its nodes are interconnected .
Connectivity. A complete graph is described as connected if for all its clear-cut pairs of nodes there is a linking chain. commission does not have importance for a graph to be connected but may be a gene for the level of connectivity. If p > 1 the graph is not connected because it has more than one sub-graph ( or component ). There are assorted levels of connectivity, depending on the academic degree at which each pair of nodes is connected .
Complementarity. Two substitute graphs are complemental if their coupling results in a complete graph. Multimodal transportation system networks are complementary color as each sub-graph ( modal network ) benefits from the connectivity of early sub-graphs .
Root. A node r where every other node is the extremity of a path coming from r is a root. Direction has an importance. A beginning is generally the starting point of a distribution organization, such as a factory or a warehouse.
Trees. A connect graph without a bicycle is a tree. A tree has the lapp number of links than nodes plus one. ( e = v-1 ). If a radio link is removed, the graph ceases to be connected. If a new connection between two nodes is provided, a bicycle is created. A branch of etymon r is a tree where no links are connecting any node more than once. river basins are distinctive examples of tree-like networks based on multiple sources connecting towards a one estuary. This social organization powerfully influences river transportation systems .
Articulation Node. In a connected graph, a node is an articulation node if the sub-graph obtained by removing this node is no longer connected. It therefore contains more than one sub-graph ( p > 1 ). An articulation node is generally a larboard or an airport, or an crucial hub of a transportation network, which serves as a constriction. It is besides called a bridge node .
Isthmus. In a connect graph, an isthmus is a associate that is creating, when removed, two sub-graphs having at least one association. Most central links in a complex network are frequently isthmuses, which removal by reduplication helps revealing dense communities ( clusters ) .
Connectivity in a Graph
- Arlinghaus, S.L., W.C. Arlinghaus, and F. Harary (2001) Graph Theory and Geography: An Interactive View. New York: John Wiley & Sons.
- Garrison, W. and D. Marble (1974) “Graph theoretic concepts” in Transportation Geography: Comments and Readings, New York: McGraw Hill, pp. 58-80.
- Nystuen J.D. and M.F. Dacey (1961) “A graph theory interpretation of nodal regions”, Regional Science Association, Papers and Proceedings 7, p. 29-42.