Simple spinning graphs and cluster-constant equilibria

Let a weighted network mean a finite simple graph on an ordered nonempty vertex set $V$ with positive integer edge weights $w_{uv}$. Fix a positive integer $k\ge \max w_{uv}$, with maximum zero if there are no edges. Write ${\mathbb{Z}}_k=\mathbb{Z}\mathrm{/}k$ for integers modulo $k$, with addition of residue classes; $i+r$ means $i+[r{]}_k$. Its spinning graph $S_k$ has vertex set $V\times {\mathbb{Z}}_k$. Each cluster $\{u\}\times {\mathbb{Z}}_k$ is a clique, meaning that every unordered pair of distinct vertices within it is an edge. For every edge with $u<v$, add the undirected edges $\{(u,i),(v,i+r)\}$ for $i\in {\mathbb{Z}}_k$ and integer offsets $0\le r<w_{uv}$. Fixing this orientation once is part of the construction; the reverse adjacency is the transpose relation. All edges have unit weight and there are no loops. For phases ${\theta }_u$ modulo $2\pi$, the cluster-constant lift is ${\mathrm{\Theta }}_{u,i}={\theta }_u$.

For either graph, set $w_{uv}=w_{vu}$ equal to its edge weight on edges and zero on nonedges and on the diagonal; in the spinning graph the edge weights are one. Indices in the following formulas run over the vertex set of the graph in question, and unordered-pair sums run over its edges. Its Kuramoto equation is ${\dot{\theta }}_u={\sum }_v{w}_{uv}\sin ({\theta }_v-{\theta }_u)$ and its potential is $U(\theta )={\sum }_{\{u,v\}}{w}_{uv}(1-\cos ({\theta }_u-{\theta }_v))$. An equilibrium has zero right-hand side at every vertex. At a phase configuration its Hessian has off-diagonal entries $-w_{uv}\cos ({\theta }_u-{\theta }_v)$ and diagonal entries the sums of the corresponding edge coefficients. The quadratic form is ${\sum }_{\{u,v\}}{w}_{uv}\cos ({\theta }_u-{\theta }_v)(x_u-x_v{)}^2$.

Given: The finite weighted graph, its ordering and the positive integer $k$ in the Definition.

Goal: Verify simplicity, degree, and equilibrium lifting.

1.1 By [F3], each cluster is nonempty and has $k$ vertices. For fixed $i\in {\mathbb{Z}}_k$, equality $i+[r{]}_k=i+[s{]}_k$ implies $k\mid (r-s)$; for $0\le r,s<w_{uv}\le k$ this forces $r=s$. Thus distinct offsets give distinct neighbours on the forward side of each cross block. On the transpose side the neighbours are indexed by $j-[r{]}_k$ for fixed $j$, and the same divisibility argument gives exactly $w_{uv}$ distinct neighbours there too. The orientation is fixed once per base edge, preventing accidental union with a second, reverse offset convention. Within-cluster edges join all distinct pairs and are disjoint from cross edges. Thus the construction is finite and simple. Each vertex of cluster $u$ has $k-1+{\sum }_v{w}_{uv}$ neighbours. [F1, F3, given, construct, algebra]

1.2 By [F4], $v(t)=1-\cos t$ is even and $2\pi$-periodic. The derivative rules in [F5] and identities in [F4] give $v^{\mathrm{\prime }}(t)=\sin t$ and $v^{\mathrm{\prime }\mathrm{\prime }}(t)=\cos t$. Since sine and cosine are differentiable on $\mathbb{R}$ by [F4], they are continuous by [F5]. Thus $v$ is twice continuously differentiable and satisfies every function hypothesis of [F2]. With the symmetric zero-extended weights from the Definition, each unordered edge appears twice in the ordered-pair sum, so its half-sum potential is exactly $U$. The gradient formula in [F2] and sine oddness in [F4] give the negative-gradient equations in the Definition. Differentiating the gradient component using [F5] and [F4] gives the stated off-diagonal and diagonal Hessian entries; [F2]'s quadratic-form identity, with $v^{\mathrm{\prime }\mathrm{\prime }}=\cos$ and the same double counting of edges, gives the stated quadratic form. [F1, F2, F4, F5, given, algebra]

2.1 At the lifted phase, within-cluster sine terms vanish because $\sin 0=0$ by [F4]. Every neighbour cluster $v$ supplies exactly $w_{uv}\sin ({\theta }_v-{\theta }_u)$ at each vertex in cluster $u$, by step 1.1. Thus the lifted right-hand side at each vertex of cluster $u$ equals the base right-hand side at $u$. Since each cluster is nonempty, the lifted configuration is an equilibrium exactly when the original one is. [F4, step 1.1, step 1.2, algebra] ∎

This gives an explicitly oriented version of Canale §II.A, p. 4, and the direct calculation of Proposition 1, p. 7. The local justification uses the declared network, residue-arithmetic, trigonometric and differentiation suppliers. It does not assert a blanket stability transfer for arbitrary $k$.

This gives an explicitly oriented version of Canale §II.A, p. 4, and the direct calculation of Proposition 1, p. 7. The local justification uses the declared network, residue-arithmetic, trigonometric and differentiation suppliers. It does not assert a blanket stability transfer for arbitrary $k$.