Graph partitions and cluster synchronization in networks of oscillators

Schaub, Michael T.; Neave, Guy; Billeh, Yazan N.; Delvenne, Jean‐Charles; Lambiotte, Renaud; Barahona, Mauricio

doi:10.1063/1.4961065

Cited by 165 publications

(160 citation statements)

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“…In the literature, many versions of cluster synchronization scenarios have been reported in various settings, for instance, for unidirectional coupling, time delays and some special network structures [41,42]. Some numerical algorithms are required to identify synchronized clusters [43] or using graph partitions [44]. Recently, computational group theory has been proposed to characterize cluster synchronization, which hinges on the decomposition of the networked nodes into clusters with the help of network symmetries [25,26].…”

Section: Discussionmentioning

confidence: 99%

Fully solvable lower dimensional dynamics of Cartesian product of Kuramoto models

et al. 2019

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Implementing a positive correlation between the natural frequencies of nodes and their connectivity on a single star graph leads to a pronounced explosive transition to synchronization, additionally presenting hysteresis behavior. From the viewpoint of network connectivity, a star has been considered as a building motif to generate a big graph by graph operations. On the other hand, we propose to construct complex synchronization dynamics by applying the Cartesian product of two Kuramoto models on two star networks. On the product model, the lower dimensional equations describing the ensemble dynamics in terms of collective order parameters are fully solved by the Watanabe-Strogatz method. Different graph parameter choices lead to three different interacting scenarios of the hysteresis areas of two individual factor graphs, which further change the basins of attraction of multiple fixed points. Furthermore, we obtain coupling regimes where cluster synchronization states are often present on the product graph and the number of clusters is fully controlled. More specifically, oscillators on one star graph are synchronized while those on the other star are not synchronized, which induces clustered state on the product model. The numerical results agree perfectly with the theoretic predictions. critical coupling is denoted as l c b . In addition, a clear hysteresis area has been observed between these synchronization and desynchronization processes, which is further denoted asRecently, star graphs have been applied to understand the global explosive behavior of the order parameter r which shed insights for other cases of more general heterogeneous network settings [10,13,16]. The two fundamental results of explosive synchronization (discontinuity and hysteresis) have been delineated by the Watanabe-Strogatz (WS) approach [17,18]. More specifically, the exact nonlinear equation for the order parameter r of the high dimensional coupled system has been explicitly obtained and the different synchronized states correspond to different steady states of the equation. Furthermore, different stability conditions of coexisting fixed points in the parameter space lead to the hysteresis behavior and the discontinuous transitions in both the forward and backward continuation curves [16].From the viewpoint of network topologies, star graphs are considered as building motifs to generate a big graph by several graph operations, e.g. Cartesian product, direct tensor product and strong product [19,20]. For example, the Cartesian product of graphs is a commutative, associative binary operation on graphs [21]. It has many useful properties, most of which can be derived from the factors. Furthermore, several multilayer network properties have been obtained by graph product operations [20]. On the other hand, from the viewpoint of dynamics on top of networks, it remains largely undisclosed that the synchronization process is obtained by similar graph product operations except some discussions on eigenspectra [22,23]. In our work [24], we ...

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Section: Discussionmentioning

confidence: 99%

Fully solvable lower dimensional dynamics of Cartesian product of Kuramoto models

et al. 2019

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“…It is very similar to the notion of the equitable partition of Definition 5.1, except only edges joining different sets in the partition are considered. A 5 Almost equitable partitions are also known [2,43] as external equitable partitions. Note that the singleton partition is almost equitable and an equitable partition is almost equitable.…”

Section: 4mentioning

confidence: 99%

“…The following proposition is a straightforward consequence of the definitions and the fact that the Laplacian matrix for each arrow type satisfies j A i,j = 0. See [2,15,40,43]. The main application of exo-balanced partitions to coupled cell networks involves those cases where two cells do not interact when they are in the same state.…”

Section: 5mentioning

confidence: 99%

“…Another class of systems considered in [40,43] has Lagrangian coupling. They considered only one arrow type with a symmetric adjacency matrix, but the generalization to multiple arrow types is straightforward.…”

Section: Proposition 529 Consider a Coupled Cell Network With A Sinmentioning

confidence: 99%

“…Further background and references for the subject of network controllability, particularly as pertains to synchrony and almost equitable partitions, can be found in [4,18,30,43].…”

Section: Network Controllabilitymentioning

confidence: 99%

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Invariant Synchrony Subspaces of Sets of Matrices

Neuberger¹,

Sieben²,

Swift³

2020

SIAM J. Appl. Dyn. Syst.

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A synchrony subspace of R n is defined by setting certain components of the vectors equal according to an equivalence relation. Synchrony subspaces invariant under a given set of square matrices form a lattice. Applications of these invariant synchrony subspaces include equitable and almost equitable partitions of the vertices of a graph used in many areas of graph theory, balanced and exo-balanced partitions of coupled cell networks, and coset partitions of Cayley graphs. We study the basic properties of invariant synchrony subspaces and provide many examples of the applications. We also present what we call the split and cir algorithm for finding the lattice of invariant synchrony subspaces. Our theory and algorithm is further generalized for non-square matrices. This leads to the notion of tactical decompositions studied for its application in design theory.2000 Mathematics Subject Classification. 15A72, 34C14, 34C15, 37C80, 06B23, 90C35, 1 0 0 5 6 0 1 0 7 8 0 0 1 7 8 0 0 0 0 0 of [P (A) | Q] has leading columns in the first three columns. So Col(Q) ⊆ Col(P (A)) = sys(A). Proposition 6.4. Let M = {M 1 , . . . , M m } ⊆ R m×n , A ∈ Π(m), and B ∈ Π(n). Then M l sys(B) ⊆ sys(A) for all l if and only if A ≤ ψ( M 1 P (B) · · · M m P (B) ). Proof. First assume that M l sys(B) ⊆ sys(A) for all l. Then Col(M l P (B)) = M l Col(P (B)) ⊆ Col(P (A))for all l. Proposition 6.1 (3) and (1) implyNow assume that A ≤ ψ( M 1 P (B) · · · M m P (B) ). Then A ≤ ψ(M l P (B)) by Proposition 6.1(2). Hence Col(M l P (B)) ⊆ sys(A) by Lemma 6.2.The following result is the special case of Proposition 6.4 when m = n and B = A. Coarsest invariant refinementIn this section we develop an algorithm that finds the coarsest invariant partition that is not larger than a given partition.Proposition 7.1. If M ⊆ R n×n and A ∈ Π(n), then (Π M (n)∩ ↓ A) is in Π M (n).Proof. Note that the down-set ↓ A is taken in Π(n). Proposition 3.5 implies thatThe following is a generalization of [49, Theorem 5]. It is a recursive process for finding cir M (A), so we refer to it as the cir algorithm. Proposition 7.3. Let M = {M 1 , . . . , M m } ⊆ R n×n . Given a partition A 0 := A ∈ Π(n), recursively define A k+1 := ψ( P (A k ) Q k ), where Q k := M 1 P (A k ) · · · M m P (A k ) .There is a k 0 such that A k = cir M (A) for all k ≥ k 0 .

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Structured Networks and Coarse‐Grained Descriptions

Delvenne

Lambiotte

Barahona

2019

Advances in Network Clustering and Blockmodeling

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This chapter discusses the interplay between structure and dynamics in complex networks. Given a particular network with an endowed dynamics, our goal is to find partitions aligned with the dynamical process acting on top of the network. We thus aim to gain a reduced description of the system that takes into account both its structure and dynamics.In the first part, we introduce the general mathematical setup for the types of dynamics we consider throughout the chapter. We provide two guiding examples, namely consensus dynamics and diffusion processes (random walks), motivating their connection to social network analysis, and provide a brief discussion on the general dynamical framework and its possible extensions.In the second part, we focus on the influence of graph structure on the dynamics taking place on the network, focussing on three concepts that allow us to gain insight into this notion. First, we describe how time scale separation can appear in the dynamics on a network as a consequence of graph structure. Second, we discuss how the presence of particular symmetries in the network give rise to invariant dynamical subspaces that can be precisely described by graph partitions. Third, we show how this dynamical viewpoint can be extended to study dynamics on networks with signed edges, which allow us to discuss connections to concepts in social network analysis, such as structural balance.In the third part, we discuss how to use dynamical processes unfolding on the network to detect meaningful network substructures. We then show how such dynamical measures can be related to seemingly different algorithm for community detection and coarse-graining proposed in the literature. We conclude with a brief summary and highlight interesting open future directions.

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Graph partitions and cluster synchronization in networks of oscillators

Cited by 165 publications

References 65 publications

Fully solvable lower dimensional dynamics of Cartesian product of Kuramoto models

Fully solvable lower dimensional dynamics of Cartesian product of Kuramoto models

Invariant Synchrony Subspaces of Sets of Matrices

Structured Networks and Coarse‐Grained Descriptions

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