Introduction
Nowadays we are surrounded by assorted large information networks. For example, the phone network has all users as vertices which are interconnected by phone calls from one user to another. The Web can be viewed as a network with webpages as vertices which are then linked to other webpages. There are various biological networks arising from numerous databases, such as the gene network which represents the regulatory effect among genes. Of interest are many social networks expressing various types of social interactions. Some noted examples include the Collaboration graph (denoting coauthorship among mathematicians) and the Hollywood graph (consisting of actors/actresses and their joint appearances in feature films), among others.
How are these networks formed? What are basic structures of such large networks? How do they evolve? What are the underlying principles that dictate their behaviors?
To answer these questions, graph theory comes into play. Random graphs have...
Abbreviations
- Random graph:
-
a graph which is chosen from a certain probability distribution over the set of all graphs satisfying some given set of constraints. Statements about random graphs are thus statements about “typical” graphs for the given constraints.
- Graph diameter:
-
the maximum distance between any pair of vertices in a graph. Here the distance between two vertices is the length of any shortest path joining the vertices.
- Node degree distribution n k :
-
the number of vertices having degree k.
- Power law distribution:
-
a node degree distribution which (at least approximately) obeys \( { n_k \propto k^{-\beta} } \) with fixed positive exponent β.
- Erdős–Rényi random graph \( { {\mathcal G}(n,p) } \) :
-
a graph with n vertices, in which each possible edge between the \( { \binom n 2 } \) pairs of vertices is present with probability p. The fact that the edges are chosen independently makes it relatively easy to compute properties of \( { {\mathcal G}(n,p) } \).
- Random graph model \( { {\mathcal G}({\mathbf w}) } \) :
-
for expected degree sequence \( { \mathbf w } \):] a model which generates random graphs which, on average, will have a prescribed degree sequence \( { {\mathbf w} = (w_1, w_2, \dots, w_n) } \).
- On‐line random graph:
-
a graph which grows in size over time, according to given probabilistic rules, starting from a start graph G 0. One can make statements about these graphs in the limit of long time (and hence large vertex number n, as n approaches infinity).
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Chung, F. (2009). Random Graphs, A Whirlwind Tour of. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_442
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