Mandala Network

They are defined by construction in [1] as a mathematical graph with certain rules for the distribution of nodes and edges in each shell, and how they connect to nodes in the shell below. They are characterised by being

• Ultra-small-world
• Highly sparse

In the basic method for construction, Mandala networks are characterised by three paramaters {\displaystyle (n_{1},b,\lambda )}, where {\displaystyle n_{1}} is the number of nodes in the first generation, {\displaystyle b} is the number of new nodes added to each node in subsequent shells, and {\displaystyle \lambda } is the number of connections between nodes in the same shell (other than the first shell). The choice of these parameters determines a type of Mandala network, where a unique Mandala network is determined by type and total number of shells {\displaystyle g}.

In the first shell there are {\displaystyle n_{1}} nodes that form a connected graph. A second shell is created by connecting each node in the first shell with {\displaystyle b} nodes in the second shell, and connecting each node in the second shell to {\displaystyle \lambda } nodes in the second shell. This method is used to create a third shell where, in addition, each node is also connected to its ancestor node in the first shell. This process can be repeated itaratively to create {\displaystyle g} shells. Because each node is connected directly to a node in the first shell, and each node in the first shell is directly to another node in the first shell, the maximum shortest path length between nodes is 3.

If the number of nodes in each shell is labelled by {\displaystyle n_{i}} then the total number of nodes on the network is given by

{\displaystyle N=\sum _{i=1}^{g}n_{i}}.

Due to the symmetry of the construction, the mean shortest path length is given by

{\displaystyle \langle l\rangle =\sum _{i=1}^{g}n_{i}\phi _{i}}

where {\displaystyle \phi _{i}} is the sum of the shortest path lengths connecting a node in the {\displaystyle i}-th shell with all other nodes in the network. It can be shown that

{\displaystyle \langle l\rangle =\alpha +{\frac {O(N)}{N^{2}}}}

where {\displaystyle \alpha } is a constant which may be determined for each network. It can be shown that {\displaystyle 1\leq \alpha <{\frac {8}{3}}}, where {\displaystyle \alpha \to {\frac {8}{3}}} as {\displaystyle N\to \infty }.

References

(1) Mandala Networks: ultra-small-world and highly sparse graphs, Sampaio Filho, C., Moreira, A., Andrade, R. et al. Sci Rep 5, 9082 (2015) doi:10.1038/srep09082