Mandala Network

A Mandala network refers a family of networks that allow for fast communication and are cost-effective yet are robust against failures and attacks. They are built up in layers (or shells) and their name derives from their visual similarity to Mandala images.

They are defined by construction in [ref Sampaio et al] as mathematical graph with certain rules for the distribution and connectedness (or degree) of the nodes and edges in each shell. They are characterised by being

• Ultra-small-world
• Highly sparse

In the basic method for construction, Mandala networks are cahracterised by three paramaters (n_1, b, lambda). Here n_1 is the number of nodes in the first generation, b is the number of new nodes added to each external shell, and lambda is the scale factor. Once these parameters have been chosen, a Mandala network is uniquely determined by the total number of shells g. The total number of nodes in each shell is labelled n_i and total number of nodes on the network is given by

N = Sum i=1 to g n_i .

In the first shell the nodes form a connected graph. A subsequent shell is created where each node is connected to its ancestor and to other nodes in the same shell. This process is repeated creating multiple shells. The degree of each node at the ith shell in a network with a total of g shells is given by

Kig = b lambda^{g-i} + (i-1)

The mean shortest path length is given by

<l> = alpha + O(N)/N^2

where alpha is a constant which may be determined for each network. Note that alpha > 1 as only in the case of a connected graph do we have alpha = 1.

References

• 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