### Network models

Values of many network measures are greatly influenced by basic network characteristics, such as the number of nodes and links, and the degree distribution. Consequently, the significance of network statistics should often be established by comparison with statistics calculated on null network models. Null network models have simple random or ordered topologies but preserve basic characteristics of the original network. The most commonly used null network model has a random topology but shares the size, density and binary degree distribution of the original network.

### Null network models

• Network randomization: These functions randomize the network, while preserving the degree distribution (the indegree and outdegree distributions in directed networks).

randomizer_bin_und.m (BU networks): This function directly searches for rewirable edge pairs (rather than trying to rewire edge pairs at random), and hence avoids long loops and works especially well in dense matrices.

randmio_und.m; randmio_und_connected.m; randmio_und_signed.m (BU, WU networks): These functions do not preserve the out-strength distribution in weighted networks. The connected version of the function ensures that the randomized network does not disconnect. The signed version of the function separately preserves the degree distributions of positive and negative weights.

randmio_dir.m; randmio_dir_connected.m; randmio_dir_signed (BD, WD networks): These functions preserve the out-strength but not the in-strength distributions in weighted networks. The connected version of the function ensures that the randomized network does not disconnect.

null_model_und_sign.m (WU networks), null_model_dir_sign.m (WD networks): These functions simultaneously preserve the degree-, weight- and strength- distributions in networks with positive and negative weights.

randomize_graph_partial_und.m (BU, WU networks): This function produces a constrained randomization, by avoiding randomization of and into specified links.

Contributors: MR, JP, DB, OS, RB.

• Network latticization: These functions latticize the network, while preserving the degree distribution (the indegree and outdegree distributions in directed networks).

latmio_und.m; latmio_und_connected.m (BU, WU networks): These functions do not preserve the out-strength distribution in weighted networks. The connected version of the function ensures that the latticized network does not disconnect.

latmio_dir.m; latmio_dir_connected.m (BD, WD networks): These functions preserve the out-strength but not the in-strength distributions in weighted networks. The connected version of the function ensures that the latticized network does not disconnect.

Contributors: MR, OS.

• Unbiased sampling of networks with soft constraints: This function returns an ensemble of unbiasedly sampled networks with weighted node-strength and module-weight constraints. These constraints are soft in that they are satisfied on average for the full network ensemble but not, in general, for each individual network.

mleme_constraint_model.m (BU, BD, WU, WD networks). NB: The weights matrices must contain only integer weights (but real-valued weights may be linearly mapped to integer weights with arbitrary precision).

Contributor: MR.
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### Generative network models

• Generative growth models: These functions generate a wide range of synthetic networks using a combination of spatial and topological growth rules, and evaluate the accuracy of correspondence between the input networks and the generated synthetic.

generative_model.m (BU networks);
evaluate_generative_model.m (BU networks).

Contributor: RB.

• Structural-functional generative models: These functions predict and generate functional connectivity networks from structural connectivity network measures.

predict_fc.m (BU, WU networks);
generate_fc.m (BU, WU networks).

Contributors: AA, JG, OS.

• Random networks: These functions generate random networks with specified numbers of nodes and links.

makerandCIJ_und.m (BU networks); makerandCIJ_dir.m (BD networks).
Contributor: OS.

• Random networks with specified degree sequence: This function generates a directed random network with a specified in-degree and out-degree sequence.

makerandCIJdegreesfixed.m (BD networks).
Contributor: AR.

• Non-ring lattice networks: This function generates a directed lattice network without toroidal boundary counditions (i.e. without ring-like "wrapping around").

makelatticeCIJ.m (BD networks).
Contributor: OS.

• Ring lattice networks: This function generates a directed lattice network with toroidal boundary counditions (i.e. with ring-like "wrapping around").

makeringlatticeCIJ.m (BD networks).
Contributor: OS.

• Modular small-world networks: This function generates a random, directed network with a specified number of fully connected modules linked together by randomly distributed between-module connections.

makeevenCIJ.m (BD networks).
Contributor: OS.

• Hierarchical modular small-world networks: This function generates a directed network with a hierarchical modular small-world organization.

makefractalCIJ.m (BD networks).
Contributor: OS.

• Gaussian drop-off of connectivity: This function generates a directed network with a Gaussian drop-off in edge density with increasing distance from the main diagonal. There are no toroidal boundary counditions (i.e. no ring-like "wrapping around").

maketoeplitzCIJ.m (BD networks).
Contributor: OS.
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