Fig. 1: Schematic illustration of the Local Search (LS) algorithm.

a An example network where digits on nodes and size of nodes indicate the degree. b The identification of local leaders based on local dominance by creating a forest of directed acyclic graphs (DAGs) as indicated by short dashed directed edges. For each node u, it points to any adjacent neighbor v with k_v ≥ k_u and \({k}_{v}=\max \{{k}_{z}| z\in {{{{{{{\bf{V}}}}}}}}(u)\}\), where V(u) is the set of neighboring nodes. In this example, nodes are traversed by their lexicographical order, when node b is traversed, it points to m as \({k}_{m}=\max \{{k}_{z}| z\in {{{{{{{\bf{V}}}}}}}}(b)\}\ge {k}_{b}\); later, when m is traversed, it has no out-going link, and so m is identified as a local leader: it does not point to any of its followers and its remaining neighbors all have smaller degrees. When there are more than one neighbor with the same largest degree, more than one directed edge is temporarily added, e.g., node c points to both b and m as \({k}_{b}={k}_{m}=\max \{{k}_{z}| z\in {{{{{{{\bf{V}}}}}}}}(c)\}\ge {k}_{c}\); nodes d and l also have more than one outgoing link. The local leaders, which are potential community centers, are f, m, and p (indicated by dark gray color). c Each node randomly retains just one out-going edge shown as a short dashed directed edge (e.g., c can point to b or m with an equal probability, similarly for l and d). Then, for each local leader u, a local-BFS is performed to find its nearest local leader with k_v≥k_u, and the shortest path length on network d_uv, ∀ v is designated by l_u. Here, p → f with l_p = 2, and f → m with l_f = 4. In (c), short-dash arrows and long-dash arrows correspond to pure followers (whose l_u = 1) and local leaders (whose l_u≥2), respectively. Each node has at most one out-going link (u → v), which can go beyond direct connections. The local leader(s) with the maximal degree has no out-going link (here node m). d The corresponding tree structure formed by local dominance. The scale on the left is a visual aid for calculating l_i between connected nodes in the DAG. e The scatter plot of k_i and l_i for all nodes. Community centers are of both a larger degree k_i and a longer l_i. f The decision graph for quantitatively determining community centers (indicated by triangles) based on the product of rescaled degree \({\tilde{k}}_{i}\) and rescaled distance \({\tilde{l}}_{i}\) (see more details in Supplementary Note 1.2). Community centers can be detected by a visual inspection for obvious gaps or sophisticated automatic detection methods. Here, two centers, nodes m and f, are identified. The color of nodes in (c) and (d) represents the community partition, and community centers are highlighted by a darker hue of the same color.