Small-world networkA small-world network is a mathematical graph in which most nodes are not neighbors of one another, but the neighbors of any given node are likely to be neighbors of each other. Due to this, most neighboring nodes can be reached from every other node by a small number of hops or steps. Specifically, a small-world network is defined to be a network where the typical distance L between two randomly chosen nodes (the number of steps required) grows proportionally to the logarithm of the number of nodes N in the network, that is: while the global clustering coefficient is not small.
Clustering coefficientIn graph theory, a clustering coefficient is a measure of the degree to which nodes in a graph tend to cluster together. Evidence suggests that in most real-world networks, and in particular social networks, nodes tend to create tightly knit groups characterised by a relatively high density of ties; this likelihood tends to be greater than the average probability of a tie randomly established between two nodes (Holland and Leinhardt, 1971; Watts and Strogatz, 1998). Two versions of this measure exist: the global and the local.
CentralityIn graph theory and network analysis, indicators of centrality assign numbers or rankings to nodes within a graph corresponding to their network position. Applications include identifying the most influential person(s) in a social network, key infrastructure nodes in the Internet or urban networks, super-spreaders of disease, and brain networks. Centrality concepts were first developed in social network analysis, and many of the terms used to measure centrality reflect their sociological origin.
Random permutationA random permutation is a random ordering of a set of objects, that is, a permutation-valued random variable. The use of random permutations is often fundamental to fields that use randomized algorithms such as coding theory, cryptography, and simulation. A good example of a random permutation is the shuffling of a deck of cards: this is ideally a random permutation of the 52 cards. One method of generating a random permutation of a set of size n uniformly at random (i.e.
Hamiltonian path problemIn the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path (a path in an undirected or directed graph that visits each vertex exactly once) or a Hamiltonian cycle exists in a given graph (whether directed or undirected). Both problems are NP-complete.
Null graphIn the mathematical field of graph theory, the term "null graph" may refer either to the order-zero graph, or alternatively, to any edgeless graph (the latter is sometimes called an "empty graph"). The order-zero graph, K_0, is the unique graph having no vertices (hence its order is zero). It follows that K_0 also has no edges. Thus the null graph is a regular graph of degree zero. Some authors exclude K_0 from consideration as a graph (either by definition, or more simply as a matter of convenience).
Ramsey theoryRamsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of the mathematical field of combinatorics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in Ramsey theory typically ask a question of the form: "how big must some structure be to guarantee that a particular property holds?" A typical result in Ramsey theory starts with some mathematical structure that is then cut into pieces.
Tree decompositionIn graph theory, a tree decomposition is a mapping of a graph into a tree that can be used to define the treewidth of the graph and speed up solving certain computational problems on the graph. Tree decompositions are also called junction trees, clique trees, or join trees. They play an important role in problems like probabilistic inference, constraint satisfaction, query optimization, and matrix decomposition. The concept of tree decomposition was originally introduced by .
Social network analysis softwareSocial network analysis (SNA) software is software which facilitates quantitative or qualitative analysis of social networks, by describing features of a network either through numerical or visual representation. Networks can consist of anything from families, project teams, classrooms, sports teams, legislatures, nation-states, disease vectors, membership on networking websites like Twitter or Facebook, or even the Internet. Networks can consist of direct linkages between nodes or indirect linkages based upon shared attributes, shared attendance at events, or common affiliations.
Planarity testingIn graph theory, the planarity testing problem is the algorithmic problem of testing whether a given graph is a planar graph (that is, whether it can be drawn in the plane without edge intersections). This is a well-studied problem in computer science for which many practical algorithms have emerged, many taking advantage of novel data structures. Most of these methods operate in O(n) time (linear time), where n is the number of edges (or vertices) in the graph, which is asymptotically optimal.
