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Networks: Structure and Dynamics

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Encyclopedia of Complexity and Systems Science

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Definition and Relevance

Scientific research has had a long history of bottom-up approaches, which break the system into small or elementary constituents and map outinteractions between these components. The Standard Model describing elementary particles and the four types of interactions governing our world isperhaps the most successful example. Biology has developed a very detailed description of cellular components such as the DNA molecule or the variousproteins and metabolites. Furthermore, many of the interactions that govern a cells life have been investigated in great detail, but mainly in isolation:transcription of DNA, protein assembly, enzyme function, etc. Perhaps not surprisingly, the first attempts to understand complexity in physics were focused on small,simple system with complex dynamics: chaos theory. Nonetheless, large natural or social systems, like a cell, an ecosystem or the Internet are muchmore intuitive examples of complex systems. A meaningful description of...

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Abbreviations

Simple graph or network :

A group of N nodes (vertices) among which there exist L undirected connections (links, edges), identical in strength.

Directed graph :

A group of nodes among which connections are directed.

Weighted network :

A group of nodes among which connections are not identical in strength, but carry a weight.

Bipartite network :

A network with more than one type of node, in which connections only exist between different node types (the definition can be relaxed to a network were most, but not all links run between vertices of different types).

Adjacency matrix A :

An \( { N\times N } \) matrix representing the network, whose elements \( { a_{ij} } \) are equal to 1 when there is a link from node i to j, zero otherwise.

Degree distribution \( { P(k) } \) :

The probability that a node of a network, chosen uniformly at random, has degree k.

Scale-free network :

A network in which the tail of the degree distribution follows a power law (strictly speaking, the term scale-free implies \( { P(k) \sim k^{-\gamma} } \), however, it is often used for networks where the tail of the distribution follows a power-law).

Degree exponent γ:

The power law exponent of the (tail of the) degree distribution

Scale-free model :

A growing network model proposed by Barabási and Albert [15]. The model builds a simple graph starting from a small connected group of nodes, to which new nodes are added one by one. These new nodes connect to m old nodes with probabilities that increase linearly with the degree of the old nodes.

Shortest path (geodesic path):

The smallest collection of links that form a path through the network from one vertex to another.

Diameter D :

The length of the largest geodesic path in a network.

Small-world network :

A network in which the average shortest path length grows logarithmically (or slower) with N.

Node betweenness (betweenness centrality or load):

The number of shortest paths between nodes of the network that run through a given node [62].

Edge betweennes :

The number of shortest paths between nodes of the network that run through a given edge.

Clustering coefficient C :

The fraction of connections that are realized between the neighbors of a node:

$$ C_i = \frac{2 \, n_i}{k_i\,(k_i - 1)}\;, $$

where \( { n_i } \) denotes the number of links connecting the \( { k_i } \) neighbors of node i. (The average clustering coefficient is given by \( { \langle C \rangle\, = \frac{1}{N}\,\sum_i C_i } \). An alternative global measure of clustering, also called transitivity, is the fraction of node triples that are linked into triangles.)

Assortativity coefficient :

A measure of the tendency of links to run among nodes that are similar in some respect. If the similarity is described by a scalar quantity (most often the node' s degree), then the assortativity coefficient is given by

$$ r=\frac{\sum_{x,y}\, xy\,(e_{x,y}-a_x b_y)}{\sigma_a \, \sigma_b}\;, $$

where x (y) is the scalar at the origin (end) of a link, \( { e_{x,y} } \) denotes the fraction of all edges in the network that go from nodes with value x to ones with value y, \( { a_x } \) (\( { b_y } \)) is the fraction of edges that start (end) at a link with values x (y), and \( { \sigma_a } \) (\( { \sigma_b } \)) is the standard deviations of the distributions of \( { a_x } \) (\( { b_y } \)) values [107].

Modularity Q :

The number of links between nodes within the same community minus the number expected by chance:

$$ Q= \frac{1}{2L}\sum_{i=1}^N\sum_{j=1}^N (A_{ij} - P_{ij})\, \delta_{g_i,g_j}\;, $$

where node i (j) belongs to the community \( { g_i } \) (\( { g_j } \)). \( { P_{ij} } \) gives the expected number of links between two nodes if the network is random with respect to communities [110]. In the simplest case, in which the null model is a random network, \( { P_{ij} =2L/N^2 } \). A more suitable assumption is \( { P_{ij} =k_i\, k_j/2L } \), which preserves the degree distribution of the network in question (the expected degree of node i is \( { \sum_j P_{ij} = k_i } \)) [109].

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  1. Department of Medicine, Beth Israel Deaconess Medical Center, Boston, USA

    Erzsébet Ravasz Regan

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    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Regan, E.R. (2009). Networks: Structure and Dynamics. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_356

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