Schematic representation of the $S^D$ model for D=1 (left) and for D=2 (right)

1. Multidimensional network model
2. Usage
3. Examples
4. Publications

The $\mathbb{S}^D$ model is the geometric soft configuration model which is able to explain many fundamental features of real networks such as small-world property, heterogeneous degree distributions, high level of clustering, and self-similarity.

In the $\mathbb{S}^D$ model, a node $i$ is assigned two hidden variables:

• a hidden degree $\kappa_i$, quantifying its popularity, influence, or importance,
• a position in $D$-dimensional similarity space chosen uniformly at random, and represented as a point on a $D$-dimensional sphere. For instance for $D=1$ the similarity space is represented as a circle, for $D=2$ as a sphere (see Figure above).

The connection probability between any pair of nodes increases with the product of their hidden degrees (i.e., their combined popularities), and decreases with the angular distance between the two nodes. Specifically, nodes $i$ and $j$ are connected with the probability

$$p_{ij} = \left(1 + \frac{(R\Delta\theta_{ij})^\beta}{\left(\mu \kappa_i \kappa_j\right)^{\frac{\max(D, \beta)}{D}}}\right)^{-1}$$

where $\Delta\theta_{ij}$ is the angular distance between nodes $i$ and $j$. Parameters $\mu$ and $\beta$ (also called inverse temperature) control the average degree and the clustering coefficient, respectively.

The $\mathbb{S}^D$ model can be expressed as a purely geometric model $\mathbb{H}^{D+1}$ in the hyperbolic space by mapping the hidden degree of each node into a radial coordinate as

$$r_i = \hat{R} - \frac{2\max(D, \beta)}{D\beta \zeta} \ln\left(\frac{\kappa_i}{\kappa_0}\right) \text{ with } \hat{R} = \frac{2}{\zeta} \ln \left(\frac{2R}{(\mu \kappa_0^2)^{\frac{\max(D, \beta)}{D\beta}}}\right)$$

where $\hat{R}$ is the radius of the hyperbolic disk and $\zeta = \sqrt{-K}$ and $K$ is the constant negative curvature of the hyperbolic disk. For $\beta < D$ we choose $\zeta=\frac{1}{\beta}$ and $\beta > D$ we set $\zeta=1$. The connection probability then reads

$$p_{ij} = \frac{1}{1 + e^{{\frac{\beta\zeta}{2} (x_{ij} - \hat{R})}}}$$

where

$$x_{ij} = r_i + r_j + 2\ln \frac{\Delta\theta_{ij}}{2}$$

is a good approximation of the hyperbolic distance between two nodes separated by an angular distance $\Delta\theta_{ij}$ with radial coordinates $r_i$ and $r_j$.

There are two types of possible usage:

1. Generate synthetic networks with specified parameters
2. Infer a set of hidden degrees and parameter $\beta$ from the input network. Based on them generate synthetic networks from the model.

Ad. 1.

Compile code (c++17 or higher is required)

g++ -O3 -std=c++17 -lboost_system -lboost_math_c99 src/generatingSD_unix.cpp -o genSD

Run the code

./genSD -d <dimension_value> -b <beta_value> -n <network_size> -g <gamma> -k <mean_degree>

The program accepts the following parameters:

• -d - Dimension of the model
• -b - Value of the inverse temperature $\beta$ (positive value)
• -n - Size of the network
• -g - Exponent of the power-law distribution for hidden degrees.
• -k - Mean degree of the network
• -v - (optional) Save generated coordinates to .gen_coord file

The set of kappas will be generated from the power-law degree distribution with specified $\gamma$ value, i.e., $P(\kappa) \sim \kappa^{-\gamma}$.

If you want to use your own set of kappas, you can run code as

./generateSD -d <dimension_value> -b <beta_value> -l <path_to_kappas>

In this case parameters -n, -g, and -k are unnecessary.

Ad. 2.

Compile code (c++17 or higher is required)

gfortran -O3 include/trapecis_Sd.f -c -o include/trapecis_Sd.o
g++ -O3 -std=c++17 -lgfortran -I include/ include/trapecis_Sd.o  src/infer_kappas_beta_unix.cpp -o infer_kappas_beta

Infer the hidden degrees and parameter $\beta$

./infer_kappas_beta -d <dimension_value> <edgelist_filename>

The code generates two files:

• .inf_log with network information and value of inferred $\beta$
• .kappas with a set of inferred hidden degrees

Later you can use .kappas file as the input to the ./generateSD program. See Ad. 1 for more information.

See also tutorial

Generate network with specified parameters:

./generateSD -d 1 -b 1.5 -n 100 -g 2.7 -k 10

It will create a $\mathbb{S}^1$ model with 100 nodes. The $\beta$ parameter is set to 1.5 and the exponent of the powerlaw distribution of the hidden degrees is 2.7 with a mean value of 10.

Generate network with predefined hidden degrees:

for i in `seq 1 1000`; do; echo 10; done > kappas.txt
./generateSD -d 2 -b 3 -l kappas.txt

It will create a $\mathbb{S}^2$ model with 1000 nodes. The $\beta$ parameter is set to 3 and all nodes with hidden degree $\kappa=10$.

Infer hidden degrees and parameter $\beta$ from a real network and generate network replicas

./infer_kappas_beta -d 2 <path to edgelist>
./generateSD -d 2 -b <inferred beta> <path to .kappas file>

It will create a $\mathbb{S}^2$ model with size of input network <path to edgelist> with inferred set of $\kappa$-s and parameter $\beta$.

• Serrano, M. Á., Krioukov, D., & Boguná, M. (2008). Self-similarity of complex networks and hidden metric spaces. Physical review letters, 100(7), 078701.

• Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A., & Boguná, M. (2010). Hyperbolic geometry of complex networks. Physical Review E, 82(3), 036106.

• García-Pérez, G., Allard, A., Serrano, M. Á., & Boguñá, M. (2019). Mercator: uncovering faithful hyperbolic embeddings of complex networks. New Journal of Physics, 21(12), 123033.

• Jankowski, R., Allard, A., Boguñá, M., & Serrano, M. Á. (2023). The D-Mercator method for the multidimensional hyperbolic embedding of real networks. Nature Communications, 14(1), 7585.