From networks to optimal higher-order models of complex systems

Lambiotte, Renaud; Rosvall, Martin; Scholtes, Ingo

doi:10.1038/s41567-019-0459-y

Cited by 386 publications

(285 citation statements)

References 61 publications

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“…e study of higher-order structures in networks is an increasingly rich area of research [29,[39][40][41][42], often focusing on constructing networks that better capture the data they represent. Here, we introduce a formal and generalized way to recast networks at a higher scale while preserving random walk dynamics.…”

Section: Discussionmentioning

confidence: 99%

The Emergence of Informative Higher Scales in Complex Networks

Klein

Hoel

2020

Complexity

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The connectivity of a network contains information about the relationships between nodes, which can denote interactions, associations, or dependencies. We show that this information can be analyzed by measuring the uncertainty (and certainty) contained in paths along nodes and links in a network. Specifically, we derive from first principles a measure known as effective information and describe its behavior in common network models. Networks with higher effective information contain more information in the relationships between nodes. We show how subgraphs of nodes can be grouped into macronodes, reducing the size of a network while increasing its effective information (a phenomenon known as causal emergence). We find that informative higher scales are common in simulated and real networks across biological, social, informational, and technological domains. These results show that the emergence of higher scales in networks can be directly assessed and that these higher scales offer a way to create certainty out of uncertainty.

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Section: Discussionmentioning

confidence: 99%

The Emergence of Informative Higher Scales in Complex Networks

Klein

Hoel

2020

Complexity

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“…Recently, the network science community has turned its attention to network geometry [6-9] to better represent the kinds of interactions that one can find beyond typical pairwise interactions.These higher-order interactions are encoded in geometrical structures that describe the different kinds of simplex structure present in the network: a filled clique of m + 1 nodes is known as an m-simplex, and together a set of 1-simplexes (links), 2-simplexes (filled triangles), etc., comprise the simplicial complex. While simplicial complexes have been proven to be very useful for the analysis and computation in high dimensional data sets, e.g., using persistent homologies [10][11][12][13][14], little is understood about their role in shaping dynamical processes, save for a handful of examples [15][16][17][18].A more accurate description of dynamical processes on complex systems necessarily requires a new paradigm where the network structure representation helps to include higher-order interactions [19]. Simplicial geometry of complex networks is a natural way to extend manybody interactions in complex systems.…”

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confidence: 99%

“…A more accurate description of dynamical processes on complex systems necessarily requires a new paradigm where the network structure representation helps to include higher-order interactions [19]. Simplicial geometry of complex networks is a natural way to extend manybody interactions in complex systems.…”

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confidence: 99%

Abrupt phase transition of epidemic spreading in simplicial complexes

Matamalas

Gómez

Arenas

2020

Phys. Rev. Research

140

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Recent studies on network geometry, a way of describing network structures as geometrical objects, are revolutionizing our way to understand dynamical processes on networked systems. Here, we cope with the problem of epidemic spreading, using the Susceptible-Infected-Susceptible (SIS) model, in simplicial complexes. In particular, we analyze the dynamics of SIS in complex networks characterized by pairwise interactions (links), and three-body interactions (filled triangles, also known as 2-simplices). This higher-order description of the epidemic spreading is analytically formulated using a microscopic Markov chain approximation. The analysis of the fixed point solutions of the model, reveal an interesting phase transition that becomes abrupt with the infectivity parameter of the 2-simplices. Our results are of outmost importance for network theorists to advance in our physical understanding of epidemic spreading in real scenarios where diseases are transmitted among groups as well as among pairs, and to better understand the behavior of dynamical processes in simplicial complexes.The collective behavior of dynamical systems on networks, has been a major subject of research in the physics community during the last decades [1][2][3][4][5]. In particular, our understanding of both natural and man-made systems has significantly improved by studying how network structures and dynamical processes combined shape the overall systems' behavior. Recently, the network science community has turned its attention to network geometry [6-9] to better represent the kinds of interactions that one can find beyond typical pairwise interactions.These higher-order interactions are encoded in geometrical structures that describe the different kinds of simplex structure present in the network: a filled clique of m + 1 nodes is known as an m-simplex, and together a set of 1-simplexes (links), 2-simplexes (filled triangles), etc., comprise the simplicial complex. While simplicial complexes have been proven to be very useful for the analysis and computation in high dimensional data sets, e.g., using persistent homologies [10][11][12][13][14], little is understood about their role in shaping dynamical processes, save for a handful of examples [15][16][17][18].A more accurate description of dynamical processes on complex systems necessarily requires a new paradigm where the network structure representation helps to include higher-order interactions [19]. Simplicial geometry of complex networks is a natural way to extend manybody interactions in complex systems. The standard approach so far consists in understanding the coexistence of two-body (link) interactions and three-body interactions (filled triangles). Note that this approach is different from considering pairwise interactions among three elements of a triangle, it refers to the interaction of the three elements, in the filled triangle, at unison.Here we present a probabilistic formalization of the higher-order interactions of an epidemic process, represented by the well-known Suscep...

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“…Although the context is quite different, the closest work to the construction of this section is Ser-Giacomi et al 2015, which shows that the most probable paths in a Markovian model of a very complicated temporal network (viz., ocean water transport in the Mediterranean) suffice to describe the network's key features. Other works have looked at higher-order models in discrete time as a way to finesse the challenges of continuous time modeling as discussed here (Lambiotte et al 2019;Rosvall et al 2014). Despite the many differences of detail, our own model likewise shows that the most probable paths/flows suffice for capturing the essential dynamics of directed contact networks.…”

Section: Markov Chain Models For Dcnsmentioning

confidence: 82%

Analytics for directed contact networks

Cybenko

Huntsman

2019

Appl Netw Sci

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Directed contact networks (DCNs) are temporal networks that are useful for analyzing and modeling phenomena in transportation, communications, epidemiology and social networking. Specific sequences of contacts can underlie higher-level behaviors such as flows that aggregate contacts based on some notion of semantic and temporal proximity. We describe a simple inhomogeneous Markov model to infer flows and taint bounds associated with such higher-level behaviors, and also discuss how to aggregate contacts within DCNs and/or dynamically cluster their vertices. We provide examples of these constructions in the contexts of information transfers within computer and air transportation networks, thereby indicating how they can be used for data reduction and anomaly detection.

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From networks to optimal higher-order models of complex systems

Abstract: Standfirst Rich data is revealing that complex dependencies between the nodes of a network may escape models based on pairwise interactions. Higher-order network models go beyond these limitations, offering new perspectives for understanding complex systems.

Cited by 386 publications

References 61 publications

The Emergence of Informative Higher Scales in Complex Networks

The Emergence of Informative Higher Scales in Complex Networks

Abrupt phase transition of epidemic spreading in simplicial complexes

Analytics for directed contact networks

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