Order parameter allows classification of planar graphs based on balanced fixed points in the Kuramoto model

Phase balanced states are a highly underexplored class of solutions of the Kuramoto model and other coupled oscillator models on networks. So far, coupled oscillator research focused on phase synchronized solutions. Yet, global constraints on oscillators may forbid synchronized state, rendering phase balanced states as the relevant stable state. If, for example, oscillators are driving the contractions of a fluid filled volume, conservation of fluid volume constrains oscillators to balanced states as characterized by a vanishing Kuramoto order parameter. It has previously been shown that stable, balanced patterns in the Kuramoto model exist on circulant graphs. However, which noncirculant graphs first of all allow for balanced states and what characterizes the balanced states is unknown. Here, we derive rules of how to build noncirculant, planar graphs allowing for balanced states from the simple cycle graph by adding loops or edges to it. We thereby identify different classes of small planar networks allowing for balanced states. Investigating the balanced states' characteristics, we find that the variance in basin stability scales linearly with the size of the graph for these networks. We introduce the balancing ratio as an order parameter based on the basin stability approach to classify balanced states on networks and evaluate it analytically for a subset of the network classes. Our results offer an analytical description of noncirculant graphs supporting stable, balanced states and may thereby help to understand the topological requirements on oscillator networks under global constraints.

Figure 1

Balanced graphs may be constructed by making use of the two different building blocks. (a) Balanced graph representing the symmetries of the fourth roots of unity with respect to the corresponding winding number $q_1=+1$ in the spectral drawing. (b) Graph consisting of two cycles connected to each other but with additional loops added in such a way that they are symmetric with respect to each of the big faces with winding numbers $q_{2,1}=q_{2,2}=+1$. (c) Two cycle graphs connected at more than one edge no longer possess any stable balanced fixed points. Fixed point shown with $q_{3,1}=q_{3,2}=+1$ is balanced if cycles are only connected at a single edge. (d) A graph that is not symmetric with respect to the winding number $q_{4,1}=+1$ of the big face. The stable state shown is not balanced. Faces with no winding number indicated carry winding number zero. All graphs shown here have automorphism groups with basis elements being rotation and reflection along horizontal axis.

Figure 2

(a) The cycle graph $C_{18}$ to which $k=6$ loops of size $m=1$ have been added with angular variables showing a stable, phase balanced fixed point (color bar). The fixed point has winding number $q=1$ in the central cycle. (b) Angular variables corresponding to the fixed point shown in (a) plotted on the unit circle. The angles in each of the loops are visible and form roots of unity themselves, resulting in a balanced fixed point.

Figure 3

Additional edges (red, dotted lines) added to cycle graphs may not change the overall stability properties of stable, balanced fixed points. (a) A stable, balanced fixed point in a cycle graph (top) with winding number $q_1=+2$ (center of graph) does not lose stability upon addition of an edge connecting two vertices having the same angular value (bottom). The overall winding number is distributed equally over the two resulting graphs (bottom) $q_{1,1}=q_{1,2}=+1$ (center of the graphs). (b) The same procedure may be applied for higher order fixed points (top) by connecting again vertices of equal phases (bottom). The overall winding number $q_2=+4$ is again distributed on the different resulting graphs with $q_{2,1}=+1,q_{2,2}=+2$, and $q_{2,3}=+1$ from left to right. (c) The same mechanism can be applied starting from winding number $q_3=+5$ (center) and connecting vertices of same phase. The last graph's automorphism group has only one element which is the reflection along the horizontal axis, whereas all other graphs have an additional rotational symmetry.

Figure 4

(a) Scaling of the variance ${\sigma }^2$ characterizing the Gaussian distribution describing the basin stability with the number of vertices $N$ in different graphs. Linear fits (dotted lines) and data points from Monte Carlo sampling (markers) are shown for cycle graphs $C_N$ (dark purple dots), cycle graphs with loops $G_l$ (light purple squares), a cycle graph with a central edge added $G_c$ (blue diamonds), and prism graphs $Y_N$ (green stars). Fit parameters for linear relationship ${\sigma }^2\left(N\right)=mN+b$ are $m_{C_N}=3.2\times {10}^{-2},b_{C_N}=-1.7\times {10}^{-3}$, $m_{G_l}=3.1\times {10}^{-2},b_{G_l}=-4.9\times {10}^{-2}$, $m_{G_c}=1.6N\times {10}^{-2},b_{G_c}=-1.1\times {10}^{-3}$, and $m_{Y_N}=1.3\times {10}^{-2},b_{Y_N}=-1.0\times {10}^{-2}$ from top to bottom. Note that for $G_l$, fit was performed in terms of the number of vertices in the central cycle. (b) Scaling of share of basin stability occupied by the stable fixed points with all winding numbers in outer loops equal to zero $A_G\left(k\right)$ in dependence of the number of loops $k$ added to a cycle graph. Data points from Monte Carlo sampling represented by dots and analytical approximation as dotted line are shown. Errors in both figures take a maximal value of $\sigma ({\mathcal{S}}_{\mathcal{B}}\left(\mathit{\vartheta }\right))=2\times {10}^{-3}$ for each dot and are thus too small to be visible in the figure.

Figure 5

Balancing ratio $b\left(G\right)$ plotted for different balanced graphs against total number of vertices in graph $N$. Color code and symbols represent different prototypes for the graphs analyzed here as shown in legend, right. For the cycle graph with loops, purple numbers 2, 3, and 5 correspond to the number of loops added. For graphs on the left side of the legend, only the central cycle grows with increasing $N$ whereas the added loops, if they exist, stay constant in size. For graphs on the right side of the legend, loops grow accordingly with increasing $N$ (top). Graph at the bottom right of the legend marked by yellow color is the only asymmetric graph analyzed here. Results marked by symbols were obtained using Monte Carlo sampling performed for respective graph except for analytical results indicated by colored lines. See text for detailed information on how the respective graph was constructed exactly. Errors take a maximal value of $\sigma ({\mathcal{S}}_{\mathcal{B}}\left(\mathit{\vartheta }\right))=2\times {10}^{-3}$ for each dot and are thus too small to be visible in the figure.

Figure 6

Winding numbers multiple of the number of loops induce asymmetries in the phase variables at stable fixed points. (a) Phase variables at the stable fixed point depicted in (b). The asymmetry toward $\vartheta =0$ is clearly visible and indicated by the Kuramoto order parameter in Eq. (3) (black square), which assumes a nonzero value for this configuration. (b) Cycle graph $C_{28}$ to which $k=2$ loops have been added. The stable fixed point shown here with central winding number $q=2$ is not balanced.

Figure 7

Automorphisms for a cycle-like graph with $k=5$ loops symmetrically added to the graph and a total of $N=25$ vertices. Numbers on vertices (light blue circles) label vertices uniquely which are connected through edges (dotted, black lines). The graph displayed (a) is modified by applying two different automorphisms, namely, the two basis elements of its automorphism group. (b) The first basis element is a reflection along a symmetry axis of the graph (straight, dotted line, reflection indicated by arrow). (c) The second one is a rotation (gray, dotted arrow) by a length corresponding to the distance between individual loops. Together, these symmetries represent the dihedral symmetries.

Figure 8

A balanced graph with empty automorphism group may be created by combining the two building blocks of adding loops to cycle graphs and connecting copies of the resulting graphs. (a) A balanced graph with $N=12$ vertices and three symmetrically placed loops. (b) A balanced graph with $N=36$ vertices created by joining together three copies of the graph in (a) resulting in a graph with empty automorphism group.

Figure 9

Sketch of a cycle graph with $k=6$ loops. The winding numbers in the loops for balanced fixed points are not limited to be all zero (top left) or all unity (bottom left, unity value indicated by color), but also patterns where the loop winding numbers are prime divisors of $k$ may result in balanced fixed points (center and right plots).