Jean-Paul Rodrigue (2017), New York:
Routledge, 440 pages.
Graph Theory: Definition and Properties
Authors: Dr. Jean-Paul Rodrigue and Dr. Cesar Ducruet
1. Basic Graph Definition
A graph is a symbolic representation of a network and of its
connectivity. It implies an abstraction
of the reality so it can be simplified as a set of linked nodes.
Graph theory is a branch of mathematics concerned about how
networks can be encoded and their properties measured. It has been
enriched in the last decades by growing influences from studies
of social and complex networks.
The origins of graph theory can be traced to Leonhard Euler who
devised in 1735 a problem that came to be known as the "Seven Bridges of
Konigsberg". In this problem, someone had to cross once all the bridges
only once and in a continuous sequence, a problem the Euler proved to
have no solution by representing it as a set of nodes and links. This
led the the foundation of graph theory and its subsequent improvements.
In transport geography most networks have an obvious spatial foundation,
namely road, transit and rail networks, which tend to be defined more
by their links than by their nodes. This it is not necessarily the case
for all transportation networks. For instance, maritime and air networks
tend to be more defined more by their nodes than by their links since
links are often not clearly defined. A telecommunication system can
also be represented as a network, while its spatial expression can have
limited importance and would actually be difficult to represent. Mobile
telephone networks or the Internet, possibly to most complex graphs
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 phone
calls. Servers, the core of the Internet, can also be represented as
nodes within a graph while the physical infrastructure between them,
namely fiber optic cables, can act as links. Consequently, all transport
networks can be represented by graph theory in one way or the other.
The following elements are fundamental at understanding graph theory:
Graph. A graph G is
a set of vertex (nodes) v connected by edges (links) e.
Thus G=(v , e).
Vertex (Node). A node v is a terminal point or an
intersection point of a graph. It is the abstraction of a location
such as a city, an administrative division, a road intersection
or a transport terminal (stations, terminuses, harbors and airports).
Edge (Link). An edge e is a link between two nodes.
The link (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 commonly represented as an arrow. When an
arrow is not used, it is assumed the link is bi-directional.
Sub-Graph. A sub-graph is a subset of a graph G
where p is the number of sub-graphs. For instance G’
= (v’, e’) can be a distinct sub-graph of
G. Unless the global transport system is considered in its
whole, every transport network is in theory a sub-graph of another.
For instance, the road transportation network of a city is a sub-graph
of a regional transportation network, which is itself a sub-graph
of a national transportation network.
Buckle (Loop or self edge). A link that makes a node correspond to itself is
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 two-dimensional.
This is typically the case for power grids, road and railway networks,
although great care must be inferred to the definition of nodes
(terminals, warehouses, cities).
Non-planar Graph. A graph
where there are no vertices at the intersection of at least two
edges. This implies a third dimension in the topology of the graph
since there is the possibility of having a movement "passing
over" another movement such as for air and maritime transport.
A non-planar graph has potentially much more links than a planar
Simple graph. A graph that includes
only one type of link between its nodes. A road or rail
network are simple graphs.
Multigraph. A graph that includes
several types of links between its nodes. Some nodes can be
connected to one link type while others can be connected to
more than one that are running in parallel. A graph depicting a road and a rail network
with different links between nodes serviced by either or
both modes is a multigraph.
2. Links and their Structures
A transportation network enables flows of people, freight or information,
which are occurring along its links. Graph theory must thus offer the
possibility of representing movements as linkages, which can be considered
over several aspects:
Connection. A set of
two nodes as every node is linked to the other. Considers if a movement
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 same direction. For a path to exist between
two nodes, it must be possible to travel an uninterrupted sequence
of links. Finding all the possible paths in a graph is a fundamental
attribute in measuring accessibility and traffic flows.
Chain. A sequence of links having a connection in common
with the other. Direction does not matter.
Length of a Link, Connection or Path.
Refers to the label associated with a link, a connection or a path.
This label can be distance, the amount of traffic, the capacity
or any attribute of that link. The length of a path is the number
of links (or connections) in this path.
Cycle. Refers to a chain
where the initial and terminal node is the same and that does not
use the same link more than once is a cycle.
Circuit. A path where the
initial and terminal node corresponds. It is a cycle where all the
links are traveled in the same direction. Circuits are very important
in transportation because several distribution systems are using
circuits to cover as much territory as possible in one direction
Clique. A clique is a maximal complete subgraph
where all vertices are connected.
Cluster. Also called community, it refers to a
group of nodes having denser relations with each other than with
the rest of the network. A wide range of methods are used to reveal
clusters in a network, notably they are based on modularity measures
(intra- versus inter-cluster variance).
Ego network. For a given node, the ego
network corresponds to a sub-graph where only its adjacent
neighbors and their mutual 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 node) and a number of
subordinate nodes (which largest flow 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 only the heaviest links.
Dual graph. A method in space syntax
that considers edges as nodes and nodes as edges. In urban
street networks, large avenues made of several segments become
single nodes while intersections with other avenues or streets
become links (edges). This method is particularly useful to
reveal hierarchical structures in a planar network.
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 direction is also linked in the other. By
convention, a line without an arrow represents a link where it is
possible to move in both directions. However, both directions have
to be defined in the graph. Most transport systems are symmetrical
but asymmetry can often occur as it is the case for maritime (pendulum)
and air services. Asymmetry is rare on road transportation networks,
unless one-way streets are considered.
Assortativity and disassortativity. Assortative
networks are those characterized by relations among similar nodes,
while disassortative networks are found when structurally different
nodes are often connected. Transport (or technological) networks
are often disassortative when they are non-planar, due to the higher
probability for the network to be centralized into a few large hubs.
Completeness. A graph is complete if two nodes are linked
in at least one direction. 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 distinct pairs of
nodes there is a linking chain. Direction does not have importance
for a graph to be connected, but may be a factor 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 various levels of connectivity,
depending on the degree at which each pair of nodes is connected.
Complementarity. Two sub
graphs are complementary if their union results in a complete graph.
Multimodal transportation networks are complementary as each sub-graph
benefits from the connectivity of other 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 root is generally the starting
point of a distribution system, such as a factory or a warehouse.
Trees. A connected graph without
a cycle is a tree. A tree has the same number of links than nodes
plus one. (e = v-1). If a link is removed, the graph ceases
to be connected. If a new link between two nodes is provided, a
cycle is created. A branch of root r is a tree where no links
are connecting any node more than once. River basins are typical
examples of tree-like networks based on multiple sources
connecting towards a single estuary. This structure strongly
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 port or an airport, or an important hub of a
transportation network, which serves as a bottleneck.
It is also called a bridge node.
Isthmus. In a connected graph,
an isthmus is a link that is creating, when removed, two sub-graphs
having at least one connection.
Most central links in a complex network are often isthmuses,
which removal by reiteration helps revealing dense communities