Article Outline
Glossary
Definition of the Subject
Introduction
Graphs, Networks, and Complex Networks
Structure of Networks
Dynamical Network Structures
Dynamical Processes on Networks
Graph Visualization
Future Directions
Bibliography
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Abbreviations
- Directed/Undirected graph :
-
A set of vertices connected by directed or undirected edges. A directed edge is one-way \( { (A\to B) } \), while an undirected edge is two‐way or symmetric: \( { A-B } \).
- Network :
-
For our purposes, a network is defined identically to a graph: it is an abstract object composed of vertices (nodes) joined by (directed or undirected) edges (links). Hence we will use the terms ‘graph’ and ‘network’ interchangeably.
- Graph topology:
-
The list of nodes i and edges \( { (i,j) } \) or \( { (i \to j) } \) defines the topology of the graph.
- Graph structure:
-
There is no single agreed definition for what constitutes the “structure” of a graph. To the contrary: this question has been the object of a great deal of research—research which is still ongoing.
- Node degree distribution :
-
One crude measure of a graph's structure.If n k is the number of nodes having degree k in a graph with N nodes, then the set of n k is the node degree distribution—which is also often expressed in terms of the frequencies \( { p_{k}=n_{k}/N } \).
- Small‐worlds graph :
-
A “small‐worlds graph” has two properties: it has short path lengths (as is typical of random graphs)—so that the “world” of the network is truly “small”, in that every node is within a few (or not too many) hops of every other; and secondly, it has (like real social networks, and unlike random graphs) a significant degree of clustering—meaning that two neighbors of a node have a higher‐than‐random probability of also being linked to one another.
- Graph visualization :
-
The problem of displaying a graph's topology (or part of it) in a 2D image, so as to give the viewer insight into the structure of the graph. We see that this is a hard problem, as it involves both the unsolved problem of what we mean by the structure of the graph, and also the combined technological/psychological problem of conveying useful information about a (possibly large) graph via a 2D (or quasi‐3D) layout.Clearly, the notion of a good graph visualization is dependent on the use to which the visualization is to be put—in other words, on the information which is to be conveyed.
- Section:
-
Here, a ‘bookkeeping’ definition. This article introduces the reader to all of the other articles in the Section of the Encyclopedia which is titled “Complex Networks and Graph Theory”. Therefore, whenever the word ‘Section’ (with a large ‘S’) is used in this ‘roadmap’ article, the word refers to that Section of the Encyclopedia. To avoid confusion, the various subdivisions of this roadmap article will be called ‘parts’.
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Canright, G. (2012). Complex Networks and Graph Theory. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_37
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