Figure 1

(a–c) Instantaneous frequencies ${\dot{\varphi }}_i(t)$ of each one of the 100 oscillators in $A$ (light lines) [in $B$ (dark lines)] vs time, obtained from simulation of Eq. (1) for (a) $d=3.25$, (b) $d=5.75$, and (c) $d=9.75$. (d) Log-log plot of the period $T_O$ of the switching process in the interface vs ${\omega }_{\Delta }$, for $d=3.5$. The solid line represents a linear fit with $T_O\sim 120/{\omega }_{\Delta }$. Each point is the average of 5 independent realizations. Other parameters reported in the text.Reuse & Permissions

Figure 2

(a) Instantaneous frequencies ${\dot{\varphi }}_i(t)$ vs time from simulation of Eq. (1) with $d=0.1$ (other parameters and stipulations are reported in the text). Squares, diamonds, and full circles represent, respectively, nodes belonging to $A$, $B$, and $O$. (b) Log-log plot of the switching period $T_O$ of the oscillations in the frequency of the nodes in $O$ vs the frequency difference ${\omega }_{\Delta }$. The solid line represents a linear fit with slope $-1.002\pm 0.0069$. (c) Solution of Eq. (2) for ${\omega }_1=0.5$, ${\omega }_2=0.25$ [representing the frequencies of the two main clusters in (a)], $K=0.1$, $\tilde{A}=0.019$, $\tilde{B}=0.25$.Reuse & Permissions

Figure 3

Effects of coupling strength (a–b) and asymmetry (c). (a) $d=0.5$; (b) $d=0.95$. (c) $d=0.1$, but 2 (5) links of the nodes in the interface go to nodes of cluster $A$ ($B$). As in Fig. 2, squares, diamonds, and full circles refer, respectively, to nodes belonging to $A$, $B$, and $O$.Reuse & Permissions

Figure 4

(a) Graphical illustration of the constructed modular network, where the overlapping region of two circles (communities) represents the overlapping community. (b) $C$ (see text for definition) vs node index in $G$. Overlapping nodes 1, 2, 3 (101, 102, 103), labeled, respectively, with triangle, diamond, and star, have [1,5], [2,4], [3,3] connections with nodes of [$B$, $A$] ([$A$, $B$]), while 4–100 (104–200) labeled with squares belong functionally to $A$ ($B$) and $d=0.15$. (c) Overlapping clusters each consisting of complete graphs of nodes 1–4 (101–104), labeled with stars, have symmetrical connections with $A$ and $B$, while nodes 5–100 (105–200), labeled with squares, belong functionally to $A$ ($B$); here $d=0.1$.Reuse & Permissions

Figure 5

(a) $C$ (see text for definition) vs node index in the Zachary karate club network [8]. The values of $C_i$ are visibly split into two main groups that identify exactly the split of the club that was originally observed in 8. Node 3 (further labeled with a square) is ambiguously classified by different classical algorithms [2] and displays a value of $C_i$ consistently different from that of any other node. (b) Instantaneous frequencies ${\dot{\varphi }}_i(t)$ vs time from simulation of Eq. (1) with $d=0.1$. The dynamics of node 3 (open circles) is the only one that reflects the typical oscillating behavior of the interfaces.Reuse & Permissions