How far can a rumor travel without shortcuts?

We consider a rumor model in which the network is divided into three classes of agents: ignorant, spreader, and stifler. A spreader transmits the rumor to each of its ignorant neighbors at rate one, and at the same rate, it becomes a stifler after interacting with other spreaders or stiflers. The overall process is described by a continuous-time Markov chain that represents the state of each node at any given time. The underlying network is a ring lattice with n nodes, where each node is connected to its 2k nearest neighbors. This structure has often been used as the foundation for small-world network models, which are typically generated by rewiring or adding edges to introduce shortcuts. It is well known that when a rumor process takes place on such modified networks, the system undergoes a transition between localization and propagation at a finite mean degree. This transition illustrates the strong influence of shortcuts on the spreading of information. In this work, we adopt a complementary perspective by focusing on the rumor process within the pure ring lattice, without adding any shortcuts. Our aim is to show that even in this simplified setting, the model can exhibit behavior regarding the proportion of nodes reached by the rumor that is comparable to what is observed in homogeneously mixed populations. To this end, we identify the value of k as a function of n for which this behavior emerges and demonstrate that it scales as logn. Our conclusions are drawn from the analysis of contrasting examples and from a broader examination of the general case through numerical simulations.