Balanced Hodge Laplacians optimize consensus dynamics over simplicial complexes

Ziegler, Cameron; Skardal, Per Sebastian; Dutta, Haimonti; Taylor, Dane

doi:10.1063/5.0080370

Cited by 25 publications

(26 citation statements)

References 37 publications

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“…Besides the computational demonstrations, the scenarios are treated mathematically. Consensus dynamics over higher-order networked systems can be investigated through the concept of generalized Hodge Laplacians for the instances in which the weights for lower- and higher-order interactions between simplices are different [ 94 ]. Using the Hodge decomposition, convergence can be analysed and thereafter with the techniques of algebraic topology the role of simplicial complex homology can be studied.…”

Section: Social Dynamicsmentioning

confidence: 99%

Dynamics on higher-order networks: a review

Majhi

Perc

Ghosh

2022

J. R. Soc. Interface.

279

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Network science has evolved into an indispensable platform for studying complex systems. But recent research has identified limits of classical networks, where links connect pairs of nodes, to comprehensively describe group interactions. Higher-order networks, where a link can connect more than two nodes, have therefore emerged as a new frontier in network science. Since group interactions are common in social, biological and technological systems, higher-order networks have recently led to important new discoveries across many fields of research. Here, we review these works, focusing in particular on the novel aspects of the dynamics that emerges on higher-order networks. We cover a variety of dynamical processes that have thus far been studied, including different synchronization phenomena, contagion processes, the evolution of cooperation and consensus formation. We also outline open challenges and promising directions for future research.

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Section: Social Dynamicsmentioning

confidence: 99%

Dynamics on higher-order networks: a review

Majhi

Perc

Ghosh

2022

J. R. Soc. Interface.

279

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“…The research on multiplex and higher-order networks has rapidly expanded in recent years and it has emerged that going beyond simple pairwise interactions significantly enrich our ability to describe the interplay between network structure and dynamics. Indeed dynamics on multiplex network [1] and on higher-order networks [5,6,8] display significantly different behaviours than the corresponding dynamics defined on single pairwise networks and has effect on percolation [10,11], contagion models [12][13][14][15][16][17] game theory models [18], synchronisation [19][20][21][22][23][24][25][26][27] and diffusion models [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43].…”

Section: Introductionmentioning

confidence: 99%

“…Diffusion on higher-order networks is also attracting large attention with recent works related to consensus models [35,36], random walks on higher-order networks [37], diffusion and higher-order diffusion on simplicial complexes [38][39][40][41][42] and diffusion on hypergraphs and oriented hypergraph [14,[32][33][34]. Of special interest for this work is the recent use of applied topology and spectral graph theory to treat diffusion on oriented hypergraphs that has been recently proposed and investigated in Refs.…”

Section: Introductionmentioning

confidence: 99%

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Hyper-diffusion on multiplex networks

Ghorbanchian¹,

Latora²,

Bianconi³

2022

Preprint

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Multiplex networks describe systems whose interactions can be of different nature, and are fundamental to understand complexity of networks beyond the framework of simple graphs. Recently it has been pointed out that restricting the attention to pairwise interactions is also a limitation, as the vast majority of complex systems include higher-order interactions and these strongly affect their dynamics. Here, we propose hyper-diffusion on multiplex network, a dynamical process in which the diffusion on each single layer is coupled with the diffusion in other layers thanks to the presence of higher-order interactions occurring when there exists link overlap in different layers. We show that hyper-diffusion on a duplex (a network with two layers) is driven by Hyper-Laplacians in which the relevance of higher-order interactions can be tuned by a continuous parameter δ 11 . By combining tools of spectral graph theory, applied topology and network science we provide a general understanding of hyperdiffusion on duplex networks, including theoretical bounds on the Fiedler and the largest eigenvalue of Hyper-Laplacians and the asymptotic expansion of their spectrum for δ 11 1 and δ 11 1. Although hyper-diffusion on multiplex networks does not imply a direct "transfer of mass" among the layers (i.e. the average state of replica nodes in each layer is a conserved quantity of the dynamics), we find that the dynamics of the two layers is coupled as the relaxation to the steady state becomes synchronous when higher-order interactions are taken into account and the Fiedler eigenvalue of the Hyper-Laplacian is not localized on a single layer of the duplex network.

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“…Moreover edge signals might also represent a number of climate data such as currents in the ocean and velocity of wind that can be projected on a suitable triangulation of the Earth surface [35,36]. Topological signals can undergo higher-order simplicial synchronization [37][38][39][40][41][42][43], and higher-order diffusion [39,44,45]. Moreover datasets of topological signals can be treated with topological signal processing [33,36,46] and with topological machine learning tools [47][48][49][50].…”

Section: Introductionmentioning

confidence: 99%

Diffusion-driven instability of topological signals coupled by the Dirac operator

Giambagli¹,

Calmon²,

Muolo³

et al. 2022

Preprint

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The study of reaction-diffusion systems on networks is of paramount relevance for the understanding of nonlinear processes in systems where the topology is intrinsically discrete, such as the brain. Until now reactiondiffusion systems have been studied only when species are defined on the nodes of a network. However, in a number of real systems including the brain and the climate, dynamical variables are not only defined on nodes but also on links, triangles and higher-dimensional simplices, leading to topological signals. In this work we study reaction-diffusion processes of topological signals coupled through the Dirac operator. The Dirac operator allows topological signals of different dimension to interact or cross-diffuse as it projects the topological signals defined on simplices of a given dimension to simplices of one dimension up or one dimension down. By focusing on the framework involving nodes and links we establish the conditions for the emergence of Turing patterns and we show that the latter are never localized only on nodes or only on links of the network. Moreover when the topological signals display Turing pattern their projection does as well. We validate the theory hereby developed on a benchmark network model and on square lattices with periodic boundary conditions.

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Balanced Hodge Laplacians optimize consensus dynamics over simplicial complexes

Cited by 25 publications

References 37 publications

Dynamics on higher-order networks: a review

Dynamics on higher-order networks: a review

Hyper-diffusion on multiplex networks

Diffusion-driven instability of topological signals coupled by the Dirac operator

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