Fig. 1

Illustrative examples of send-receive communication asymmetry. The toy network is spatially embedded, unweighted and undirected. Communication efficiency from node i to j under measure x ∈ {sp, nav, si, dif} is denoted E_x(i, j). Shortest path and navigation efficiencies are computed as the inverse of the number of connections comprising shortest and navigation paths, respectively. Diffusion efficiency relates to how quickly, on average, a random walker can travel between two nodes, while search information relates to the probability that a random walker will travel between two nodes via the shortest path linking them. The path identified under each communication model is designated with green (i → j) and mauve (j → i) arrows. Send-receive communication asymmetry refers to E_x(i, j) ≠ E_x(j, i). a Shortest path efficiency is always symmetric in undirected networks, and thus E_sp(i, j) = E_sp(j, i). b Navigation routes information by progressing to the next directly connected node that is closest in distance to the target node. This results in the i-c-b-j and j-b-i navigation paths, with respective efficiency E_nav(i, j) = 0.33 and E_nav(j, i) = 0.5. Hence, navigation is more efficient from node j to node i. c Arrows denote the symmetric shortest paths between i and j. Arrows are annotated with the probabilities that a random walker will traverse their respective connections based on node degree (e.g., each of the 3 connections of node i has approximately 0.33 probability to be traversed by a random walker leaving i). We have E_si(i, j) ∝ 0.33 × 0.25 = 0.0825 and E_si(j,i) ∝ 1 × 0.25 = 0.25. Hence, a random walker has higher probability of traveling via the shortest path in the j → i direction, characterizing search information asymmetry between i and j. Similarly, on average, a random walker is expected to visit fewer nodes traveling from i to j than from j to i. Hence, E_dif(j, i) > E_dif(i, j), characterizing diffusion efficiency asymmetry