Figure 1

The generalized modularity does not distinguish the direction of links. Nodes $A$, $B$, $A^{\prime }$, and $B^{\prime }$ are four nodes in a directed binary network. The out-strength and in-strength of those nodes are $w_A^{\text{out}}=w_{A^{\prime }}^{\text{out}}=w_B^{\text{in}}=w_{B^{\prime }}^{\text{in}}=3$, and $w_A^{\text{in}}=w_{A^{\prime }}^{\text{in}}=w_B^{\text{out}}=w_{B^{\prime }}^{\text{out}}=1$. The contribution of the link between node $A$ and node $B$ is equal to the contribution of link between node $A^{\prime }$ and node $B^{\prime }$: $q_{AB}=q_{A^{\prime }{B}^{\prime }}=1-5/M$.Reuse & Permissions

Figure 2

(Color online) Maximizing our modified modularity finds the same community assignment as the method proposed by Rosvall and Bergstrom does. The weight of the bold links is twice the weight of normal links, and the color of a node indicates the community that the node belongs to. (a) Community assignment given by optimizing the modified modularity of our method or by optimizing the map equation of Rosvall and Bergstrom (Ref. [17]). The modified modularity for this community assignment is $Q^{lr}=0.42$, while the modularity used by Leicht and Newman (Ref. [18]) is $Q^d=0.25$. (b) Community assignment given by optimizing the modularity that is used by Leicht and Newman. Our modified modularity for this assignment is $Q^{lr}=0.33$, while the modularity used by Leicht and Newman is $Q^d=0.50$.Reuse & Permissions

Figure 3

(Color online) A model network that is designed to verify our method. This network is a directed network composed of $n$ subnetworks, and the subnetworks are embedded on a ring structure. Also, each subnetwork is a small ring composed of $m$ nodes. Each small ring has an entrance node, which is the node receiving a directed link from upper stream ring, and an exit node, which is giving a directed link to the down stream ring, and the entrance node and the exit node are placed at the opposite side of each other. The direction is chosen counterclockwise both in the small ring and in the big ring. The weight of link between subnetworks is a tunable parameter $w$, while the weight of link in every subnetwork is fixed as 1. According to our definition to community in directed networks, each small ring should be considered as a community as long as $w$ is not significantly large because the random walker would be more likely to be trapped in each small ring rather than freely moving between the small rings.Reuse & Permissions

Figure 4

(Color online) Variation of information and modularity as the function of $w$ for a model network of $m=n=8$. (a) Square (circle) symbols represent the VOI between the embedded community structure and the community assignment given by optimizing $Q^{ud}\left(Q^{lr}\right)$. (b) Square symbols and circle symbols represent the highest $Q^{ud}$ and the highest $Q^{lr}$ found by the simulated annealing algorithm during the community identifying process of both methods. The black dotted line, the black solid line, the red dotted line, and the red solid line are corresponding to the functions in Eqs. (19, 20, 21) separately.Reuse & Permissions

Figure 5

(Color online) Community assignments given by direction-ignoring method and our method in the model network of $m=n=8$. (a) The community assignment given by direction-ignoring method, when $w$ is smaller than 1.9. And the community assignment given by our method through the illustrated values of $w$. (b) The stationary community assignment given by direction-ignoring method, when $w$ is larger than 4.3.Reuse & Permissions