Random Walks on Simplicial Complexes and the Normalized Hodge 1-Laplacian

Schaub, Michael T.; Benson, Austin R.; Horn, Paul; Lippner, Gábor; Jadbabaie, Ali

doi:10.1137/18m1201019

Cited by 185 publications

(179 citation statements)

References 84 publications

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“…Different generalizations of the Laplacian operator have been proposed so far in the literature to include higher orders of interactions: from the simplest versions for uniform hypergraphs [45,46], to those more complicated associated with simplicial complexes [47][48][49] and Hodge Laplacians [25,50], to mention a few. Let us notice that these Laplacians describe the hierarchy among building blocks of the topology, and different orders are associated with Laplacian matrices of different sizes, where the order zero is the traditional node point of view; the first order represents the edge perspective where the Laplacian size is equal to the number of pairwise connections, and each entry is associated with edge adjacency; the second-order Laplacian has a different size again, being based on the existing triangles, and so on.…”

Section: Multiorder Laplacianmentioning

confidence: 99%

“…A stream of research has recently focused on correctly characterizing the structure of systems with higher-order interactions [14][15][16][17][18][19][20]. Interestingly, considering this additional level of complexity sometimes leads to changes in the emerging dynamics of complex systems, including social contagions [21,22], activity-driven models [23], diffusion [24,25], random walks [26,27], and evolutionary games [28].…”

Section: Introductionmentioning

confidence: 99%

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Multiorder Laplacian for synchronization in higher-order networks

Lucas

Cencetti

Battiston³

2020

Phys. Rev. Research

126

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The emergence of synchronization in systems of coupled agents is a pivotal phenomenon in physics, biology, computer science, and neuroscience. Traditionally, interaction systems have been described as networks, where links encode information only on the pairwise influences among the nodes. Yet, in many systems, interactions among the units take place in larger groups. Recent work has shown that the presence of higher-order interactions between oscillators can significantly affect the emerging dynamics. However, these early studies have mostly considered interactions up to four oscillators at time, and analytical treatments are limited to the all-to-all setting. Here, we propose a general framework that allows us to effectively study populations of oscillators where higher-order interactions of all possible orders are considered, for any complex topology described by arbitrary hypergraphs, and for general coupling functions. To this end, we introduce a multiorder Laplacian whose spectrum determines the stability of the synchronized solution. Our framework is validated on three structures of interactions of increasing complexity. First, we study a population with all-to-all interactions at all orders, for which we can derive in a full analytical manner the Lyapunov exponents of the system, and for which we investigate the effect of including attractive and repulsive interactions. Second, we apply the multiorder Laplacian framework to synchronization on a synthetic model with heterogeneous higher-order interactions. Finally, we compare the dynamics of coupled oscillators with higher-order and pairwise couplings only, for a real dataset describing the macaque brain connectome, highlighting the importance of faithfully representing the complexity of interactions in real-world systems. Taken together, our multiorder Laplacian allows us to obtain a complete analytical characterization of the stability of synchrony in arbitrary higher-order networks, paving the way toward a general treatment of dynamical processes beyond pairwise interactions.

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Section: Multiorder Laplacianmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Multiorder Laplacian for synchronization in higher-order networks

Lucas

Cencetti

Battiston³

2020

Phys. Rev. Research

126

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“…For instance scientific authors naturally team up in larger groups to complement the expertise of different members 22 , neurons send and receive stimuli from multiple adjacent partners at the same time 21 , 23 , and the stability of large ecosystems relies on mutual and cooperative partnerships often involving three or more species 24 , 25 . Besides, higher-order interactions were shown to significantly modify the collective behavior of many dynamical processes, from diffusion 26 , 27 and synchronization 28 – 30 to spreading 31 , 32 , social dynamics 33 , 34 and games 35 . For a thorough introduction on the structure and dynamics of these higher-order systems, we refer the interested reader to the comprehensive overview provided in Ref 19 .…”

Section: Introductionmentioning

confidence: 99%

Temporal properties of higher-order interactions in social networks

Cencetti

Battiston

Lepri

et al. 2021

Sci Rep

101

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Human social interactions in local settings can be experimentally detected by recording the physical proximity and orientation of people. Such interactions, approximating face-to-face communications, can be effectively represented as time varying social networks with links being unceasingly created and destroyed over time. Traditional analyses of temporal networks have addressed mostly pairwise interactions, where links describe dyadic connections among individuals. However, many network dynamics are hardly ascribable to pairwise settings but often comprise larger groups, which are better described by higher-order interactions. Here we investigate the higher-order organizations of temporal social networks by analyzing five publicly available datasets collected in different social settings. We find that higher-order interactions are ubiquitous and, similarly to their pairwise counterparts, characterized by heterogeneous dynamics, with bursty trains of rapidly recurring higher-order events separated by long periods of inactivity. We investigate the evolution and formation of groups by looking at the transition rates between different higher-order structures. We find that in more spontaneous social settings, group are characterized by slower formation and disaggregation, while in work settings these phenomena are more abrupt, possibly reflecting pre-organized social dynamics. Finally, we observe temporal reinforcement suggesting that the longer a group stays together the higher the probability that the same interaction pattern persist in the future. Our findings suggest the importance of considering the higher-order structure of social interactions when investigating human temporal dynamics.

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“…This has profound consequences for network models of relational data— a cornerstone in the interdisciplinary study of complex systems. For example, higher-order dependencies have been shown to either speed up or slow down dynamical processes [8, 9, 10], change node rankings [11, 12, 13], and alter community structures [12, 14, 15, 16, 17, 18].…”

mentioning

confidence: 99%