Consensus dynamics on random rectangular graphs

• The consensus dynamics model is applied to a random rectangular graph (RRG).
• An RRG generalizes the random geometric graph by embedding the nodes in a unit rectangle.
• Bounds for the diameter and the algebraic connectivity of RRG are obtained.
• It is proved that when the rectangle is elongated the RRGs become ‘large worlds’ with poor connectivity.
• It is proved that when the rectangle is elongated the time for consensus in the RRG grows to infinity.

A random rectangular graph   (RRG) is a generalization of the random geometric graph (RGG) in which the nodes are embedded into a rectangle with side lengths aa and b=1/ab=1/a, instead of on a unit square [0,1]2[0,1]2. Two nodes are then connected if and only if they are separated at a Euclidean distance smaller than or equal to a certain threshold radius rr. When a=1a=1 the RRG is identical to the RGG. Here we apply the consensus dynamics model to the RRG. Our main result is a lower bound for the time of consensus, i.e., the time at which the network reaches a global consensus state. To prove this result we need first to find an upper bound for the algebraic connectivity of the RRG, i.e., the second smallest eigenvalue of the combinatorial Laplacian of the graph. This bound is based on a tight lower bound found for the graph diameter. Our results prove that as the rectangle in which the nodes are embedded becomes more elongated, the RRG becomes a ’large-world’, i.e., the diameter grows to infinity, and a poorly-connected graph, i.e., the algebraic connectivity decays to zero. The main consequence of these findings is the proof that the time of consensus in RRGs grows to infinity as the rectangle becomes more elongated. In closing, consensus dynamics in RRGs strongly depend on the geometric characteristics of the embedding space, and reaching the consensus state becomes more difficult as the rectangle is more elongated.