Nonlinearity + Networks: A 2020 Vision

Porter, Mason A.

doi:10.48550/arxiv.1911.03805

Cited by 4 publications

(5 citation statements)

References 187 publications

(307 reference statements)

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“…b) Electronic mail: juanga@colorado.edu gion mechanisms (i.e., contagion processes that can not be described solely by pairwise interactions) has received much attention 16 . It has been shown that higher-order interactions in networks (i.e., interactions involving multiple nodes) can have profound effects on dynamical network processes 17 such as opinion formation 18 , synchronization [19][20][21] and population dynamics 22 . Efforts to map higher-order interactions in realworld networks have uncovered rich structure 23 which is only now starting to be appreciated.…”

Section: Introductionmentioning

confidence: 99%

The effect of heterogeneity on hypergraph contagion models

Landry

Restrepo

2020

Chaos: An Interdisciplinary Journal of Nonlinear Science

118

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The dynamics of network social contagion processes such as opinion formation and epidemic spreading are often mediated by interactions between multiple nodes. Previous results have shown that these higher-order interactions can profoundly modify the dynamics of contagion processes, resulting in bistability, hysteresis, and explosive transitions. In this paper, we present and analyze a hyperdegree-based mean-field description of the dynamics of the susceptible–infected–susceptible model on hypergraphs, i.e., networks with higher-order interactions, and illustrate its applicability with the example of a hypergraph where contagion is mediated by both links (pairwise interactions) and triangles (three-way interactions). We consider various models for the organization of link and triangle structures and different mechanisms of higher-order contagion and healing. We find that explosive transitions can be suppressed by heterogeneity in the link degree distribution when links and triangles are chosen independently or when link and triangle connections are positively correlated when compared to the uncorrelated case. We verify these results with microscopic simulations of the contagion process and with analytic predictions derived from the mean-field model. Our results show that the structure of higher-order interactions can have important effects on contagion processes on hypergraphs.

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Section: Introductionmentioning

confidence: 99%

The effect of heterogeneity on hypergraph contagion models

Landry

Restrepo

2020

Chaos: An Interdisciplinary Journal of Nonlinear Science

118

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“…While graphs provide a useful way to model pairwise relationships, many complex systems and datasets are characterized by higher-order relationships that are better modeled by hypergraphs [13,33,65,80,84]. For example, in scientific computing, nodes may represent rows in a sparse matrix and hyperedges encode the nonzero pat-terns of each column [9].…”

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confidence: 99%

Hypergraph Cuts with General Splitting Functions

Veldt¹,

Benson²,

Kleinberg³

2020

Preprint

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The minimum s-t cut problem in graphs is one of the most fundamental problems in combinatorial optimization, and graph cuts underlie algorithms throughout discrete mathematics, theoretical computer science, operations research, and data science. While graphs are a standard model for pairwise relationships, hypergraphs provide the flexibility to model multi-way relationships, and are now a standard model for complex data and systems. However, when generalizing from graphs to hypergraphs, the notion of a "cut hyperedge" is less clear, as a hyperedge's nodes can be split in several ways. Here, we develop a framework for hypergraph cuts by considering the problem of separating two terminal nodes in a hypergraph in a way that minimizes a sum of penalties at split hyperedges. In our setup, different ways of splitting the same hyperedge have different penalties, and the penalty is encoded by what we call a splitting function.Our framework opens a rich space on the foundations of hypergraph cuts. We first identify a natural class of cardinality-based hyperedge splitting functions that depend only on the number of nodes on each side of the split. In this case, we show that the general hypergraph s-t cut problem can be reduced to a tractable graph s-t cut problem if and only if the splitting functions are submodular. We also identify a wide regime of non-submodular splitting functions for which the problem is NP-hard.We also analyze extensions to multiway cuts with at least three terminal nodes and identify a natural class of splitting functions for which the problem can be reduced in an approximation-preserving way to the node-weighted multiway cut problem in graphs, again subject to a submodularity property. Finally, we outline several open questions on general hypergraph cut problems.

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“…Current approaches include network-aware dynamical systems theory, linear stability analysis, and matrix spectral theory [35]. Still, mathematical ideas concerning empirically relevant networked dynamics are needed, particularly when the dynamics on the network and of the network have similar time scales, or when networks have multiple layers or higher-order interactions [36]. We believe advances in algebraic geometry, computational topology, and graph limit theory will prove useful in overcoming this challenge.…”

Section: Limit Objects For Sparse Graphsmentioning

confidence: 99%

Bridging the gap between graphs and networks

Iñiguez

Battiston²,

Karsai³

2020

Commun Phys

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Network science has become a powerful tool to describe the structure and dynamics of real-world complex physical, biological, social, and technological systems. Largely built on empirical observations to tackle heterogeneous, temporal, and adaptive patterns of interactions, its intuitive and flexible nature has contributed to the popularity of the field. With pioneering work on the evolution of random graphs, graph theory is often cited as the mathematical foundation of network science. Despite this narrative, the two research communities are still largely disconnected. In this commentary, we discuss the need for further crosspollination between fieldsbridging the gap between graphs and networksand how network science can benefit from such influence. A more mathematical network science may clarify the role of randomness in modeling, hint at underlying laws of behavior, and predict yet unobserved complex networked phenomena in nature.The mathematics of a networked reality Behind the history of mankind's scientific progress there is a story of mathematical theory building. One where scientists not only uncover the behavior of empirical phenomena through experiments and data analysis, but develop theories that abstract such behavior mathematically 1 . This process amounts to scientific understanding when mathematics is "sufficiently isomorphic" to reality 2 ; when theory includes the mechanisms deemed most important and, despite ignoring some features, still allows for logical inferences that accurately describe the phenomena of interest 3 . A sufficient similarity in structure between mathematics and reality allows scientists to forecast events, control natural systems, and even predict behavior that has not yet been confirmed empirically 4 .Up until recent times, scientific efforts towards a mathematical representation of reality have mostly focused on elements of the "simplest" natural systems at both the smallest and largest scales: the physical, chemical, and biological entities comprising the microscopic world, as well as thermodynamic systems and the large-scale structure of the Universe. The success of this

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Nonlinearity + Networks: A 2020 Vision

Cited by 4 publications

References 187 publications

The effect of heterogeneity on hypergraph contagion models

The effect of heterogeneity on hypergraph contagion models

Hypergraph Cuts with General Splitting Functions

Bridging the gap between graphs and networks

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