Graph theory boils down to places to go, and ways to get there.
A graph refers to a collection of nodes and a collection of edges that connect pairs of nodes.
Nodes: Places to be
Edges: Ways to get there
In the Königsberg example, the land masses and islands are nodes, and the bridges are edges.
An introduction to networks
A network is simply a collection of connected objects. We refer to the objects as nodes or vertices, and usually draw them as points. We refer to the connections between the nodes as edges, and usually draw them as lines between points.
In mathematics, networks are often referred to as graphs, and the area of mathematics concerning the study of graphs is calledgraph theory. Unfortunately, the term graph can also refer to a graph of a function, but we won’t use that use of the term when talking about networks. Here, we’ll use the terms network and graph interchangeably.
Networks can represent all sorts of systems in the real world. For example, one could describe the Internet as a network where the nodes are computers or other devices and the edges are physical (or wireless, even) connections between the devices. The World Wide Web is a huge network where the pages are nodes and links are the edges. Other examples include social networks of acquaintances or other types of interactions, networks of publications linked by citations, transportation networks, metabolic networks, and communication networks. You can click on the following images for more information about their respective networks.
Network of connections between devices within the Internet. Colors indicate operator of network. Structure determined by sending a storm of IP packets out randomly across the network. Each packet is programmed to self-destruct after a delay, and when this happens, the packet failure notice reports back the path the packet took before it died.
Types of networks
When one tries to model systems such as those mentioned above, one quickly realizes that the simple network model with identical nodes and edges cannot describe important features of real networks. One problem is the edges in this simplest network model are undirected.
In some networks, not all nodes and edges are created equal. For example, in metabolic networks, nodes may indicate different enzymes which have a wide variety of behaviors, and edges may indicate vastly different types of interactions. To model such difference, one can introduce different types of nodes and edges in the network, as illustrated by the different colors and edge styles, above. In networks where the differences among nodes and edges can be captured by a single number that, for example, indicates the strength of the interaction, a good model may be a weighted graph. One can represent a weighted graph by different sizes of nodes and edges.
In some contexts, one may work with graphs that have multiple edges between the same pair of nodes. One might also allow a node to have a self-connection, meaning an edge from itself to itself. An example of such a network is shown, below.
For simplicity, we will focus primarily on unweighted graphs with a single type of node and a single type of edge. We will consider both directed and undirected graphs, but won’t allow multiple connections or self-connections.
Creating a Network Graph with Gephi
Exploratory Data Analysis: intuition-oriented analysis by networks manipulations in real time.
Link Analysis: revealing the underlying structures of associations between objects.
Social Network Analysis: easy creation of social data connectors to map community organizations and small-world networks.
Biological Network analysis: representing patterns of biological data.
Poster creation: scientific work promotion with hi-quality printable maps.