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The algebraic connectivity (also known as Fiedler value or Fiedler eigenvalue after
connected_graph_
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.Miroslav Fiedler
Miroslav Fiedler (7 April 1926 – 20 November 2015) was a Czech mathematician known for his contributions to linear algebra, graph theory and algebraic graph theory.
His article, "Algebraic Connectivity of Graphs", published in the ''Czechosl ...
) of a graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
''G'' is the second-smallest eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
(counting multiple eigenvalues separately) of the Laplacian matrix
In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph La ...
of ''G''. This eigenvalue is greater than 0 if and only if ''G'' is a connected graph
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgrap ...
. This is a corollary to the fact that the number of times 0 appears as an eigenvalue in the Laplacian is the number of connected components in the graph. The magnitude of this value reflects how well connected the overall graph is. It has been used in analyzing the robustness and synchronizability of networks.
Properties
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connected graph
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgrap ...
. Furthermore, the value of the algebraic connectivity is bounded above by the traditional (vertex) connectivity (graph theory)">connectivity of the graph, . If the number of vertices of an undirected connected graph with nonnegative edge weights is ''n'' and the Distance (graph theory)">diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...is ''D'', the algebraic connectivity is also known to be bounded below by , and in fact (in a result due to Brendan McKay (mathematician)">Brendan McKay) by .Bojan Mohar
The Laplacian Spectrum of Graphs
in ''Graph Theory, Combinatorics, and Applications'', Vol. 2, Ed. Y. Alavi, G. Chartrand, O. R. Oellermann, A. J. Schwenk, Wiley, 1991, pp. 871–898. For the graph with 6 nodes show above (n=6,D=3) these bound means, 4/18 = 0.222 ≤ algebraic connectivity 0.722 ≤ connectivity 1. Unlike the traditional connectivity, the algebraic connectivity is dependent on the number of vertices, as well as the way in which vertices are connected. In
random graph
In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which generates them. The theory of random graphs ...
s, the algebraic connectivity decreases with the number of vertices, and increases with the average degree.
The exact definition of the algebraic connectivity depends on the type of Laplacian used. Fan Chung has developed an extensive theory using a rescaled version of the Laplacian, eliminating the dependence on the number of vertices, so that the bounds are somewhat different.
In models of synchronization
Synchronization is the coordination of events to operate a system in unison. For example, the conductor of an orchestra keeps the orchestra synchronized or ''in time''. Systems that operate with all parts in synchrony are said to be synchronou ...
on networks, such as the Kuramoto model
The Kuramoto model (or Kuramoto–Daido model), first proposed by , is a mathematical model used to describing synchronization. More specifically, it is a model for the behavior of a large set of coupled oscillators. Its formulation was motivated b ...
, the Laplacian matrix arises naturally, so the algebraic connectivity gives an indication of how easily the network will synchronize. Other measures, such as the average distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
(characteristic path length) can also be used, and in fact the algebraic connectivity is closely related to the (reciprocal of the) average distance.
The algebraic connectivity also relates to other connectivity attributes, such as the isoperimetric number
In mathematics, the Cheeger constant (also Cheeger number or isoperimetric number) of a graph is a numerical measure of whether or not a graph has a "bottleneck". The Cheeger constant as a measure of "bottleneckedness" is of great interest in ma ...
, which is bounded below by half the algebraic connectivity.
Fiedler vector
The original theory related to algebraic connectivity was produced byMiroslav Fiedler
Miroslav Fiedler (7 April 1926 – 20 November 2015) was a Czech mathematician known for his contributions to linear algebra, graph theory and algebraic graph theory.
His article, "Algebraic Connectivity of Graphs", published in the ''Czechosl ...
.M. Fiedler. "Laplacian of graphs and algebraic connectivity", Combinatorics and Graph Theory (Warsaw, 1987), ''Banach Center Publications'' 25(1) (1989), 57–70. In his honor the eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
associated with the algebraic connectivity has been named the Fiedler vector. The Fiedler vector can be used to partition a graph.
Partitioning a graph using the Fiedler vector
For the example graph in the introductory section, the Fiedler vector is . The negative values are associated with the poorly connected vertex 6, and the neighbouring articulation point Articulation may refer to:
Linguistics
* Articulatory phonetics, the study of how humans produce speech sounds via the interaction of physiological structures
** Manner of articulation, how speech organs involved in making a sound make contact
* ...
, vertex 4; while the positive values are associated with the other vertices. The ''signs of the values in the Fiedler vector'' can therefore be used to partition this graph into two components: . Alternatively, the value of 0.069 (which is close to zero) can be placed in a class of its own, partitioning the graph into three components: or moved to the other partition , as pictured. The squared values of the components of the Fiedler vector, summing up to one since the vector is normalized, can be interpreted as probabilities of the corresponding data points to be assigned to the sign-based partition.
See also
*Connectivity (graph theory)
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgrap ...
* Graph property
In graph theory, a graph property or graph invariant is a property of graphs that depends only on the abstract structure, not on graph representations such as particular labellings or drawings of the graph..
Definitions
While graph drawing an ...
References
{{reflist Algebraic graph theory Graph connectivity Graph invariants