# Automatic Detection of Multilevel Communities: Scalable, Selective and Resolution-Limit-Free +
## 2 Method
### On the Hierarchical Ravas-Barabasi (RB) Networks  
We first of all introduce the structure of the Ravasz-Barabasi (RB) networks [20]. The smallest RB network is RB5, which is a complete graph consisting of 5 nodes and 10 edges, see Figure 2a. To facilitate our discussion, we call node 0 the _central node_, while all other nodes are _peripheral nodes_ of the RB5 network. RB5 is a basic unit to constitute larger RB networks. Five RB5 units, one in the center and four on the periphery, constitute an RB25 network, as shown in Figure 2b,c. According to [20], these RB5 units should be connected in such a way: every peripheral node of the peripheral RB5 units is connected to the central node of the central RB5 unit, but the peripheral RB5 units themselves are not connected to one another. Note that in Figure 2b,c, for easy drawing we did not shift the central node of each RB5 unit slightly off its center as in Figure 2a: each RB5 unit in Figure 2b,c is still a complete graph containing 10 edges, only the diagonal edges are invisible by overlapping with other edges. Following the same way, five RB25 units constitute an RB125 network, as shown in Figure 3. Obviously, the RB networks are fractal-like and hierarchical, which can grow infinitely. We choose these networks to test our method because they can provide hierarchical structures of any depth--deep enough to test the limit of any multiresolution method for community detection. As synthetic/artificial networks, the implanted community structures within the RB networks are apparent: an RB network with 5\({}^{n}\) nodes (hereafter, we call it an \"RB5\({}^{n}\) network\") is expected to be naturally divided into 5\({}^{n-1}\) RB5 units on the lowest community level, or 5\({}^{n-2}\) RB25 units on a higher level, and so on, constructing a hierarchical structure of \(n\) levels. However, such \"natural\" divisions should not be taken for granted. One problem is, within each RB5\({}^{m}\) unit (\(2\leq m\leq n\)), the central node (e.g., see the hollow red circles in Figures 2c and 3b-d is connected to every peripheral node of the same unit. By the natural division, this central node always has a larger out-degree than in-degree (for example, the central node of an RB25 network, as in Figure 2b, has an out-degree of 16 but an in-degree of 10), which violates the customary definition of community in a strong sense [50]. Moreover, within larger networks, this problem gets even more serious: when \(n\geq 3\) the natural division further violates the definition of community in a weak sense [50]. Figure 3a shows an example: the central RB5 community (solid red circles) within an RB125 network has a total out-degree of 80 (all contributed by its central node), but a total in-degree of only 20. Therefore, it seems reasonable to break up the central RB5 community and segregate its central node as an individual community, as we did in Figure 2c, which has also been suggested previously by [4]. Through literature investigation (such as on [4]), as well as numerical simulation, for the RB networks, we propose a plausible revision to their \"natural divisions\", i.e., on each community level, preserve the peripheral communities, but further divide the central communities. More specifically, on the lowest community level, instead of 5 communities (as in Figure 2b), we divide each RB25 unit into 6 communities (as in Figure 2c). Thus, an RB5\({}^{n}\) network (\(n\geq 2\)) will be divided into 6 \(\times\) 5\({}^{n-2}\) communities (since it contains 5\({}^{n-2}\) RB25 units); Figure 3b shows an example that an RB125 network is so divided into 30 communities. Similarly, on a higher level, we find each RB125 unit tends to split into 10 communities as in Figure 3c: 4 peripheral RB25 units make 4 communities, while the central RB25 unit splits to 6 communities as in Figure 2c. On this level, an RB5\({}^{n}\) network (\(n\geq 3\)) will be divided into 10 \(\times\) 5\({}^{n-3}\) communities.