Prime stars multiplexes

Abstract: This work investigates the class of prime star multiplexes, in which each of its layers $i$, $i=1,2, \ldots M$, consists of a regular cycle graph where any node has $2J_i$ neighbors. In a process that does not affect the cyclic topology, it is assumed that, before the multiplex is assembled, the nodes are labeled differently in each individual layer. As the setup requires that all representations of the same node in the different $M$ layers must be linked by inter-layers connections, the resulting multiplex pattern can be highly complex. This can be better visualized if one assumes that in one layer the nodes are labeled in the sequentially ascending order and that the nodes with the same label are drawn on the top of the other, so that all inter-layer connections are represented by vertical lines. In such cases, the other $M-1$ layers are characterized by long distance shortcuts. As a consequence, in spite of sharing the same internal topological structure, the multiplex ends up with very dissimilar layers. For prime number of nodes, a regular star geometry arises by requiring that the neighbor labels of the $M-1$ layers differ by a constant value $p_i>1$. For $M=2$, we use analytical and numerical approaches to provide a thorough characterization of the multiplex topological properties, of the inter layer dissimilarity, and of the diffusive dynamical processes taking place on them. For the sake of definitiveness, it is considered that each node in the sequentially labeled layer is characterized by $J_1\geq1$. In the other layer, we fix $J_2\equiv1$, while $p>1$ becomes a proxy of layer dissimilarity.