Synchronization versus neighborhood similarity in complex networks of nonidentical oscillators

Does the assignment order of a fixed collection of slightly distinct subsystems into given communication channels influence the overall ensemble behavior? We discuss this question in the context of complex networks of nonidentical interacting oscillators. Three types of connection configurations are considered: Similar, Dissimilar, and Neutral patterns. These different groups correspond, respectively, to oscillators alike, distinct, and indifferent relative to their neighbors. To construct such scenarios we define a vertex-weighted graph measure, the total dissonance, which comprises the sum of the dissonances between all neighbor oscillators in the network. Our numerical simulations show that the more homogeneous a network, the higher tend to be both the coupling strength required for phase locking and the associated final phase configuration spread over the circle. On the other hand, the initial spread of partial synchronization occurs faster for Similar patterns in comparison to Dissimilar ones, while neutral patterns are an intermediate situation between both extremes.

Figure 1

Examples of Similar, Neutral, and Dissimilar patterns related to $N$-node graphs following network topology: 4-Regular ($N$ RE); Barabási-Albert ($N$ BA), with $E=100$ edges and a community graph ($N$ CO) with $E=441$. Vertex color is presented according to its natural frequency ${\mathcal{W}}_i$, ranging from $-\pi$ (black) to $\pi$(white); vertex size is proportional to node degree. The associate total dissonance ${\nu }_{\text{Total}}$ is displayed.

Figure 2

Total Dissonance ${\nu }_{\text{Total}}$ distribution chart of different topologies with Similar (light gray), Neutral (gray), and Dissimilar (black) patterns. Data correspond to the single members from categories 50 RE, 500 RE, and 105 CO and 100 different elements for the others. Colored bars indicate the overall range of ${\nu }_{\text{Total}}$ obtained, while mean values of each distribution are joined by a black line within each category.

Figure 3

Mean order parameter after transient $\langle R\rangle$ and partial synchronization index $S$, solid and dashed lines respectively, as a function of coupling strength $\varepsilon$ for different graph topologies. Average values of all graphs simulated within each category are shown. Similar, Neutral, and Dissimilar cases are respectively plotted in light gray, gray, and black. Lines are drawn to $\varepsilon$ equal to the respective average critical phase locking ${\varepsilon }_{\text{PL}}$.