Models signal dispersion by representing the radio propagation environment by a graph
Propagation graphs are a mathematical modelling method for radio propagation channels. A propagation graph is a signal flow graph in which vertices represent transmitters, receivers or scatterers. Edges in the graph model propagation conditions between vertices. Propagation graph models were initially developed by Troels Pedersen, et al. for multipath propagation in scenarios with multiple scattering, such as indoor radio propagation. It has later been applied in many other scenarios.
Mathematical definition.
A propagation graph is a simple directed graph formula_1 with vertex set formula_2 and edge set formula_3.
The vertices models objects in the propagation scenario. The vertex set formula_2 is split into three disjoint sets as 
formula_5 where formula_6 is the set of transmitters, 
formula_7 is the set of receivers and 
formula_8 is the set of objects named "scatterers".
The edge set formula_9 models the propagation models propagation conditions between vertices. Since formula_10 is assumed simple, formula_11 and an edge may be identified by a pair of vertices as formula_12
An edge formula_12 is included in formula_3 if a signal emitted by vertex formula_15 can propagate to formula_16. In a propagation graph, transmitters cannot have incoming edges and receivers cannot have outgoing edges.
Two propagation rules are assumed
The definition of the vertex gain scaling and the edge transfer functions can be adapted to accommodate particular scenarios and should be defined in order to use the model in simulations. A variety of such definitions have been considered for different propagation graph models in the published literature. 
The edge transfer functions (in the Fourier domain) can be grouped into transfer matrices as
where formula_24 is the frequency variable.
Denoting the Fourier transform of the transmitted signal by formula_25, the received signal reads in the frequency domain
formula_26
Transfer function.
The transfer function formula_27 of a propagation graph forms an infinite series
formula_28
The transfer function is a Neumann series of operators. Alternatively, it can be viewed pointwise in frequency as a geometric series of matrices. This observation yields a closed form expression for the transfer function as
formula_29
where formula_30 denotes the identity matrix and formula_31 is the spectral radius of the matrix given as argument. The transfer function account for propagation paths irrespective of the number of 'bounces'.
The series is similar to the Born series from multiple scattering theory.
The impulse responses formula_32 are obtained by inverse Fourier transform of formula_27
Partial transfer function.
Closed form expressions are available for partial sums, i.e. by considering only some of the terms in the transfer function. The partial transfer function for signal components propagation via at least formula_34 and at most formula_35 interactions is defined as
formula_36 where
formula_37
Here formula_38 denotes the number of interactions or the "bouncing order".
The partial transfer function is then 
formula_39
Special cases:
One application of partial transfer functions is in hybrid models, where propagation graphs are employed to model part of the response (usually the higher-order interactions).
The partial impulse responses formula_47 are obtained from formula_48 by the inverse Fourier transform.
Propagation graph models.
The propagation graph methodology have been applied in various settings to create radio channel models. Such a model is referred to as a "propagation graph model". Such models have been derived for scenarios including 
Calibration of propagation graph models.
To calibrate a propagation graph model, its parameters should be set to reasonable values. Different approaches can be taken.
Certain parameters can be derived from simplified geometry of the room. In particular, reverberation time can be computed via room electromagnetics. Alternatively, the parameters can ben set according to measurement data using inference techniques such as method of moments (statistics), approximate Bayesian computation., or deep neural networks
Related radio channel model types.
The method of propagation graph modeling is related to other methods. Noticeably, 
References.
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