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Complex Network Zeta Function
Different definitions have been given for the dimension of a complex network or graph. For example, metric dimension is defined in terms of the resolving set for a graph. Dimension has also been defined based on the box covering method applied to graphs. Here we describe the definition based on the complex network zeta function. This generalises the definition based on the scaling property of the volume with distance. The best definition depends on the application. Definition One usually thinks of dimension for a set which is dense, like the points on a line, for example. Dimension makes sense in a discrete setting, like for graphs, only in the large system limit, as the size tends to infinity. For example, in Statistical Mechanics, one considers discrete points which are located on regular lattices of different dimensions. Such studies have been extended to arbitrary networks, and it is interesting to consider how the definition of dimension can be extended to cover these cases ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Network
In the context of network theory, a complex network is a graph (network) with non-trivial topological features—features that do not occur in simple networks such as lattices or random graphs but often occur in networks representing real systems. The study of complex networks is a young and active area of scientific research (since 2000) inspired largely by empirical findings of real-world networks such as computer networks, biological networks, technological networks, brain networks, climate networks and social networks. Definition Most social, biological, and technological networks display substantial non-trivial topological features, with patterns of connection between their elements that are neither purely regular nor purely random. Such features include a heavy tail in the degree distribution, a high clustering coefficient, assortativity or disassortativity among vertices, community structure, and hierarchical structure. In the case of directed networks these f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Dimension (graph Theory)
In graph theory, the metric dimension of a graph ''G'' is the minimum cardinality of a subset ''S'' of vertices such that all other vertices are uniquely determined by their distances to the vertices in ''S''. Finding the metric dimension of a graph is an NP-hard problem; the decision version, determining whether the metric dimension is less than a given value, is NP-complete. Detailed definition For an ordered subset W = \ of vertices and a vertex ''v'' in a connected graph ''G'', the representation of ''v'' with respect to ''W'' is the ordered ''k''-tuple r(v, W) = (d(v,w_1), d(v,w_2),\dots,d(v,w_k)), where ''d''(''x'',''y'') represents the distance between the vertices ''x'' and ''y''. The set ''W'' is a resolving set (or locating set) for ''G'' if every two vertices of ''G'' have distinct representations. The metric dimension of ''G'' is the minimum cardinality of a resolving set for ''G''. A resolving set containing a minimum number of vertices is called a basis (or reference s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractal Dimension On Networks
Fractal analysis is useful in the study of complex networks, present in both natural and artificial systems such as computer systems, brain and social networks, allowing further development of the field in network science. Self-similarity of complex networks Many real networks have two fundamental properties, scale-free property and small-world property. If the degree distribution of the network follows a power-law, the network is scale-free; if any two arbitrary nodes in a network can be connected in a very small number of steps, the network is said to be small-world. The small-world properties can be mathematically expressed by the slow increase of the average diameter of the network, with the total number of nodes N, \left\langle l\right\rangle\sim\ln where l is the shortest distance between two nodes. Equivalently, we obtain: N\sim e^ where l_0 is a characteristic length. For a self-similar structure, a power-law relation is expected rather than the exponential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski–Bouligand Dimension
450px, Estimating the box-counting dimension of the coast of Great Britain In fractal geometry, the Minkowski–Bouligand dimension, also known as Minkowski dimension or box-counting dimension, is a way of determining the fractal dimension of a set ''S'' in a Euclidean space R''n'', or more generally in a metric space (''X'', ''d''). It is named after the Polish mathematician Hermann Minkowski and the French mathematician Georges Bouligand. To calculate this dimension for a fractal ''S'', imagine this fractal lying on an evenly spaced grid and count how many boxes are required to cover the set. The box-counting dimension is calculated by seeing how this number changes as we make the grid finer by applying a box-counting algorithm. Suppose that ''N''(''ε'') is the number of boxes of side length ''ε'' required to cover the set. Then the box-counting dimension is defined as : \dim_\text(S) := \lim_ \frac . Roughly speaking, this means that the dimension is the exponent ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shortcut Model
An important question in statistical mechanics is the dependence of model behaviour on the dimension of the system. The shortcut model was introduced in the course of studying this dependence. The model interpolates between discrete regular lattices of integer dimension. Introduction The behaviour of different processes on discrete regular lattices have been studied quite extensively. They show a rich diversity of behaviour, including a non-trivial dependence on the dimension of the regular lattice. In recent years the study has been extended from regular lattices to complex networks. The shortcut model has been used in studying several processes and their dependence on dimension. Dimension of complex network Usually, dimension is defined based on the scaling exponent of some property in the appropriate limit. One property one could use is the scaling of volume with distance. For regular lattices \textstyle \mathbf Z^d the number of nodes \textstyle j within a dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
