Francesco Sorrentino Department of Mechanical Engineering

Master stability function approaches to analyze stability of the synchronous evolution for hypernetworks and of synchronized clusters for networks with symmetries Francesco Sorrentino Department of Mechanical Engineering 203 IEEE International Workshop on Complex Systems and Networks, Vancouver, December the 2 th 203

Synchronization of Networks x ( t) F( x ( t)) x ( t) F( x2( 2 t )) x i ( t) F( x ( t)) i x N ( t) F( x ( t)) N The systems are characterized by the same What is the effect of the network structure on the dynamics (possibly chaotic), when uncoupled. dynamics? How do faults affect the network dynamics?

Synchronization of Complex Networks The dynamics of each node in the network can be described by the following equation:

Synchronization of Complex Networks The dynamics of each node in the network can be described by the following equation: x i ( t) F( x ( t)) i Individual dynamics (chaotic)

Synchronization of Complex Networks The dynamics of each node in the network can be described by the following equation: x i ( t) F( x ( t)) i N j A ij t Hx t H x j i Coupling term The matrix A{ A ij } is the network (weighted) adjacency matrix. The parameter measures the strengths of the network connections. The function H is an output function at each node. In principle H can be any function.

Synchronization of Complex Networks The dynamics of each node in the network can be described by the following equation: is the Laplacian matrix N j j ij i i t x H L t x F t x )) ( ( ) ( { ij } L L N N N N N d A A A d A A A d L 2 2 2 2 2 N j d i A ij

Synchronization of Complex Networks The dynamics of each node in the network can be described by the following equation: 2 3 4 is the Laplacian matrix N j j ij i i t x H L t x F t x )) ( ( ) ( { ij } L L 2 0 2 0 3 0 0 L L0 N N 2 0 The network equations admit a synchronous solution: t x t x t x t x s N 2. t F x t x s s where

Master Stability Function Reduced Linearized equation: [L. M. Pecora, T. L. Carroll, 98] s i s The Master Stability Function (MSF) approach studies transversal stability of the synchronous trajectory The sync solution is linearly stable iff the MSF is negative. Depending on the individual nodes dynamics, we can have three different kinds of MSF. Maximum Transverse Lyapunov exponent Λ(ν) DF( x ( t)) DH( x ( t)) x, x σλ 2,...,σλ N i

Parallel and transversal perturbations By construction there is always one eigenvalue 0 This eigenvalue is associated with the eigenvector [,,,], that is parallel to the synchronization manifold In order to evaluate stability, we are only concerned with transversal perturbations

Other Master Stability Functions Similar approaches have been used to analyze other types of synchronization. EXAMPLES )Pinning control of networks. F. Sorrentino, M. di Bernardo, F. Garofalo, and G. Chen, Phys. Rev.E, 75, 4, 04603 (2007).

Other Master Stability Functions Similar approaches have been used to analyze other types of synchronization. EXAMPLES 2) Network Synchronization of Groups F. Sorrentino and Edward Ott, Network Synchronization of Groups, Phys. Rev. E, 76, 0564 (2007).

Other Master Stability Functions Similar approaches have been used to analyze other types of synchronization. EXAMPLES 3) Adaptive synchronization of a time-evolving dynamical networks/ sensor network application F. Sorrentino, G. Barlev, A. B. Cohen, and E. Ott, Chaos, 20, 0303 (200) A. Cohen, B. Ravoori, F. Sorrentino, T. Murphy, E. Ott, and R. Roy, Chaos, 20, 04342 (200).

Other Master Stability Functions Similar approaches have been used to analyze other types of synchronization. EXAMPLES 4) Networks with couplings of different types F. Sorrentino, Synchronization of hypernetworks of coupled dynamical systems, New Journal of Physics, 4, 033035 (202) D. Irving and F. Sorrentino, Phys. Rev. E, Phys. Rev. E, 86, 05602 (202).

Motivation: chemical synapses and electrical gap junctions How does synchronization arises in this networks? Is a low dimensional approach still possible?

Mathematical formulation Two different coupling mechanisms: Stability?

Stability analysis Problem: In general, it is not possible to simultaneously diagonalize two matrices Question: Are there conditions under which these equations can be reduced in this form? The answer is yes in three cases.

Case I The matrices L A and L B commute Question: What are the graph properties for the Laplacians to commute?

Case II One of the two networks is fully connected and unweighted:

Case III One of the two networks is such that all the links originating from the same node have equal weights:

SBD approach What if none of these conditions (I-III) is satisfied? Danny Irving and I proposed an alternative approach based on simultaneous block-diagonalization (SBD) of matrices PROBLEM: Given the set of N-square real matrices = {L (),L (2),...,L (M) }, find the finest simultaneous block diagonalization (SBD) of. APPROACH: Find an invertible matrix P, such that P - L (i) P= j=,..n B j i and the dimensions of the blocks are minimal.

Method by Maehara & Murota (i) Let O (i) be the N 2 -matrix O (i) = I N L(i) L(i) I N. (ii) Construct the matrix S = Σ i O (i)t O (i) (iii) Let y be any N 2 -vector in the null subspace of the matrix S. Let y = [u T,u 2T,...,u NT ] T. (iv) Construct the matrix U=[u,u 2,,u N ]. (v) Output the matrix P whose columns are the eigenvectors of U. This procedure provides the finest simultaneous block diagonalization it provides the maximum reduction of the sync stability problem

Example I

Example II

Example III: undirected unweighted 3-motifs

Cluster synchronization in networks with symmetries L. Pecora, F. Sorrentino, A. Hagerstrom, T. Murphy, R. Roy Consider the following general equations: Where now the matrix A is completely arbitrary, e.g., non constant-row-sum QUESTION: Can synchronization be achieved? ANSWER: These equations are compatible with cluster synchronization: M synchronized motions {s,, s M }, where the clusters are determined by the network structure

Symmetries and Clusters Three -node random networks with the same number of edges These three networks are characterized by different numbers of symmetries (automorphisms) Equivalence relation partition into maximal disjoint sets of nodes

Sync solutions Graph symmetries Dynamical solutions: nodes that belong to the same cluster can synchronize, nodes that belong to different clusters cannot. The symmetries of the network form a group G. Each symmetry of the group can be described by a permutation matrix R g that re-orders the nodes in a way that leaves the dynamical equations unchanged (i.e., each R g commutes with A). Computation of the graph automorphisms enables us to find the possible cluster sync solutions corresponding to a given network structure - Are these solutions stable?

Sync solutions - stability Group theory provides a powerful way to transform the variational equations to a new coordinate system (the irreducible representation) in which the transformed coupling matrix B = TAT - has a block diagonal form that matches the cluster structure.

Experimental setup The dynamical oscillators that form the network are realized as square patches of pixels selected from a 32 x 32 tiling of the SLM array.

Stability results and Isolated Desynchronization

Application to power networks Geographical diagram of the Nepal Power Grid Network

Conclusions We are pushing the master stability function approach [Pecora and Carroll] to its limits In the case of networks with multiple type of connections, the stability problem can be reduced in a low-dimensional form by using SBD This guarantees that the problem is reduced in the lowest possible dimensional form In the case of arbitrary adjacency matrices, clusters may arise that correspond to a partition of the nodes based on the network symmetries Stability of each cluster can be studied independently from other clusters unless clusters are intertwined As a consequence we can experience isolated desynchronization of the cluster solutions Possible applications to networks of neurons and power grids