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Recent empirical and theoretical works on collective behaviors based on a topological interaction are beginning to offer some explanations as for the physical reasons behind the selection of a particular number of nearest neighbors locally affecting each individual's dynamics. Recently, flocking starlings have been shown to topologically interact with a very specific number of neighbors, between six to eight, while metric-free interactions were found to govern human crowd dynamics. Here, we use network- and graph-theoretic approaches combined with a dynamical model of locally interacting self-propelled particles to study how the consensus reaching process and its dynamics are influenced by the number k of topological neighbors. Specifically, we prove exactly that, in the absence of noise, consensus is always attained with a speed to consensus strictly increasing with k. The analysis of both speed and time to …