(c) Use all these dominant interactions to unidirectionally link the network nodes, and draw the dominant phase-advanced path pattern, which represents the dynamic geodesic of the complex network, showing the most significant (or say the shortest) driving paths in the gi ven oscillatory states. This pattern reduce the original network G ( V , E ) to a new graph G ( V , E $^{′}$) with E ′ being a subset of E ( E ′ ∈ E ). Note, we have M ′ = N since each node must have one and only one dominant phase-advanced driving, and thus this graph is nothing but the 1 D TFL pattern of Fig. 1, where the loop is the core topology for the oscillation serving as the source loop and the unidirectional links indicate the wave propagation pathways. All the above steps rely on only the necessary regulating topology under the general condition: any nonoscillatory node can oscillate only if it is driven by phase-advanced interactions from other nodes. They are widely applicable in diverse fields for self-sustained o scillations of complex networks of individually nonoscillatory nodes, independing of node dynamics (excitable or nonexcitable), coupling forms (active interactions or repressive interactions, directional or symmetric couplings), and network structures (small- world networks, purely random networks, or scale free networks). The particular meanings of $^{′′}$advanced phase ′′ and $^{′′}$dominant phase-advanced driving′′ should be properly defined, according to realistic physical , chemical and biological interaction mechanisms in each individual system. In the following sections we will apply this method to particular systems of oscillatory excitable small-world networks. 4 III. SELF-SUSTAINED TARGET WAVES IN EXCITABLE SMALL-WORLD NETWORKS We take a two-dimensional (2 D ) B¨ a r model [22] as our example where f ( u$_{i}$$_{,}$$_{j}$ ) = 0 for u$_{i}$$_{,}$$_{j}$ < 1 $_{3}$; f ( u$_{i}$$_{,}$$_{j}$ ) = 1 − 6 . 75 u$_{i}$$_{,}$$_{j}$ ( u$_{i}$$_{,}$$_{j}$ − 1) 2 for 1 3 ≤ u$_{i}$$_{,}$$_{j}$ ≤ 1; and f ( u$_{i}$$_{,}$$_{j}$ ) = 1 for u$_{i}$$_{,}$$_{j}$ > 1. The system parameters are kept throughout this paper as a = 0 . 84 , b = 0 . 07 , ε = 0 . 04 and D$_{u}$ = 1 . 0 just for the sake that the local cell follows excitable dynamics. In the present paper we consider 100 × 100 cells in the 2 D regular lattice with constant and homogeneous nearest couplings. With this coupling topology, spiral waves can be easily observed for random initial conditions. It is well known (and also it is verified in our simulations) that target waves can never be observed in the asymptotic states of Eq. (1) unless some persistent external pacings provide the wave sources. Many systems of practical importance, such as neural ˙ u$_{i}$$_{,}$$_{j}$ = − 1 $_{ε}$u$_{i}$$_{,}$$_{j}$ ( u$_{i}$$_{,}$$_{j}$ − 1)( u$_{i}$$_{,}$$_{j}$ − v$_{i}$$_{,}$$_{j}$ + b a ) + D$_{i}$$_{,}$$_{j}$ , (1a) ˙ v$_{i}$$_{,}$$_{j}$ = f ( u$_{i}$$_{,}$$_{j}$ ) − v$_{i}$$_{,}$$_{j}$ , (1b) D$_{i}$$_{,}$$_{j}$ = D$_{u}$ ( u$_{i}$$_{−}$$_{1}$$_{,}$$_{j}$ + u$_{i}$$_{+}$$_{1}$$_{,}$$_{j}$ + u$_{i}$$_{,}$$_{j}$$_{−}$$_{1}$ + u$_{i}$$_{,}$$_{j}$$_{+}$$_{1}$ − 4 u$_{i}$$_{,}$$_{j}$ ) .