Random walks and diffusion on networks

Masuda, Naoki; Porter, Mason A.; Lambiotte, Renaud

doi:10.1016/j.physrep.2017.07.007

Cited by 624 publications

(585 citation statements)

References 408 publications

(706 reference statements)

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“…The rate of convergence toward the steady state given by p s is exponential (asymptotically) with rate Re(Λ 2 ), where Λ 2 is the eigenvalue with the smallest nonzero real part (Lodato et al, 2007;Masuda et al, 2017;Tejedor et al, 2018). Equivalently, the timescale of convergence to steady state, τ, is inversely proportional to the rate of convergence τ ∝ 1= Re Λ 2 ð Þ ð Þ .…”

Section: /2018gl078355mentioning

confidence: 99%

Multiplex Networks: A Framework for Studying Multiprocess Multiscale Connectivity Via Coupled‐Network Theory With an Application to River Deltas

Tejedor

Longjas

Passalacqua

et al. 2018

Geophysical Research Letters

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Transport of water, nutrients, or energy fluxes in many natural or coupled human natural systems occurs along different pathways that often have a wide range of transport timescales and might exchange fluxes with each other dynamically. Although network approaches have been proposed for studying connectivity and transport properties on single-layer networks, theories considering interacting networks are lacking. We present a general framework for transport on multiscale coupled-connectivity systems, via multilayer networks which conceptualize the system as a set of interacting networks, each arranged in a separate layer, and with interactions across layers acknowledged by interlayer links. We illustrate this framework by examining transport in river deltas as a dynamic interaction of flow within river channels and overland flow on the islands, when controlled by the flooding level. We show the potential of the framework to answer quantitative questions related to the characteristic timescale of response in the system. Plain Language SummaryThe physical processes that shape landscapes leave behind patterns of connectivity along which fluxes occur via a multitude of processes, for example, flow through channels, subsurface or overland flow. The connectivity imposed by those processes (e.g., channel networks) exerts a significant control on the evolution and form of the underlying systems. We introduce a framework based on coupled networks, Multiplex, that allows to quantify the connectivity properties emerging from the simultaneous action of different processes, enabling thus to assess the overall system properties and dynamics. We illustrate this framework by examining the case of river deltas, where intermittent flooding and exchange of water, sediment, and nutrients between the channels and the islands maintains the delta top by trapping sediment, stabilizing banks, and enriching rivers with carbon and nutrients. By describing the delta system as a Multiplex-integrating the connectivity imposed by confined (in the channel network) and overland (on the islands) flows as well as the interactions (flux exchange) between them-we show the emergence of system transport properties and dynamics not foreseen by analyzing each process separately, and therefore revealing key information essential to predict the system response under changing forcing.

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Section: /2018gl078355mentioning

confidence: 99%

Multiplex Networks: A Framework for Studying Multiprocess Multiscale Connectivity Via Coupled‐Network Theory With an Application to River Deltas

Tejedor

Longjas

Passalacqua

et al. 2018

Geophysical Research Letters

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“…The dynamics describing how a piece of information diffuses through networked systems has been well studied for classical [24] and multilayer networks [25,26] (see Ref. [27] for a thorough review). The probability to find the random walker in any node after a certain amount of time τ is given by the solution of the master equationṗ…”

Section: Synchronization Dynamics Let Us Indicate With a Ijmentioning

confidence: 99%

Diffusion Geometry Unravels the Emergence of Functional Clusters in Collective Phenomena

Domenico

2017

Phys. Rev. Lett.

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Collective phenomena emerge from the interaction of natural or artificial units with a complex organization. The interplay between structural patterns and dynamics might induce functional clusters that, in general, are different from topological ones. In biological systems, like the human brain, the overall functionality is often favored by the interplay between connectivity and synchronization dynamics, with functional clusters that do not coincide with anatomical modules in most cases. In social, socio-technical and engineering systems, the quest for consensus favors the emergence of clusters.Despite the unquestionable evidence for mesoscale organization of many complex systems and the heterogeneity of their inter-connectivity, a way to predict and identify the emergence of functional modules in collective phenomena continues to elude us. Here, we propose an approach based on random walk dynamics to define the diffusion distance between any pair of units in a networked system. Such a metric allows to exploit the underlying diffusion geometry to provide a unifying framework for the intimate relationship between metastable synchronization, consensus and random search dynamics in complex networks, pinpointing the functional mesoscale organization of synthetic and biological systems.The absence of a central authority coordinating the interactions among units of a complex system might lead to interesting collective phenomena, such as synchronization [1] in biological systems or consensus [2] in social and technological networks. This type of self-organization is affected by the underlying structure, which for a wide variety of real systems is highly heterogenous [3] and modular [4,5]. Understanding the interplay between structure and dynamics of such systems has been, and still is, a major challenge in the study of complex systems. Empirical observations, confirmed by numerical simulations and theoretical predictions, suggest that complex systems with hierarchical and/or modular mesoscale organization of their units [6] are characterized by topological scales [7] and the emergence of functional clusters that might be, in general, different from topological ones.In this letter, we show that such functional clusters might be predicted and identified for a wide variety of complex networks. More specifically, for biological systems which can be modeled as networks of oscillators, and for systems of individuals or sensors attempting to reach consensus. The unifying picture is provided by diffusion geometry [8], developed one decade ago for nonlinear dimensionality reduction of complex data. This approach uses Markov processes to integrate local similarities at different scales, allowing to approximate the manifold which better describes the data while preserving their topological features. From a physical perspective, this approach relies on topological information gathered by random searches across time, a principle that has been used successfully in network science to unravel the topological mesoscale organization of ...

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“…Other real-world networks, such as transportation hubs or social interactions, are more suited to study by the basic topological network methods. A generic starting point for studying the stochastic aspects of signal spread is a random walk on the structural network [29]. This captures multi-step and secondary pathways and, importantly, also captures the stochastic nature of the underlying processes.…”

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confidence: 99%

Network analysis of mesoscale mouse brain structural connectome yields modular structure that aligns with anatomical regions and sensory pathways

Pailthorpe

2019

Preprint

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The Allen mesoscale mouse brain structural connectome is analysed using standard network methods combined with 3D visualizations. The full region-to-region connectivity data is used, with a focus on the strongest (top 40%) structural links. The spatial embedding of links and time evolution of signalling is incorporated, with twostep links included. Modular decomposition using the Infomap method produces 8 network modules that correspond approximately to major brain anatomical regions and system functions. These modules align with the anterior and posterior primary sensory systems and association areas. 3D visualization of network links is facilitated by using a set of simplified schematic coordinates that reduces visual complexity. Selection of key nodes and links, with a focus on primary sensory pathways and cortical association areas, together reveal structural features of the mouse structural connectome consistent with biological functions in the sensory-motor systems, and selective roles of the anterior and posterior cortical association areas of the mouse brain. The fabric of weaker links generally are longer range with some having brainwide reach. Cortical gradients are evident along sensory pathways within the structural network. Author's SummaryNetwork models incorporating spatial embedding and signalling delays are used to investigate the mouse structural connectome. Here computational method work like experimental probes to uncover biologically relevant features. I use the Infomap method, which follows random walks on the network, to decompose the directed, weighted network into 8 modules that align with classical brain anatomical regions and system functions. Primary sensory pathways and cortical association areas are separated into individual modules. Strong, short range links form the sensory-motor paths while weaker links spread brain-wide, possibly coordinating many regions. modern experimental probes (e.g. see [8]). The challenge is to find appropriate network models that can realistically reproduced/predict brain structural and functional connectivity data.Here the Allen Institute for Brain Science (AIBS) dataset of a mesoscale structural connectome for the mouse brain [9] is studied using a combination of data analytical, network and visualization methods. While the data is brain wide it is at the meoscale (resolution of 0.1 mm voxels and 0.35 mm coronal plane imaging), in which links involving many neurons are probed. Connectivity data at finer scales, at the level of individual neurons and synapses, are available for smaller samples, such as with mouse retina [10,11]. Higher resolution data also is emerging at the level of cortical layers [14]. These studies shed light on the correlation between structural and functional connections between nodes in the mouse brain. The nodes are located at known 3D coordinates from the Allen Atlas [13]. Typically the sources and targets of the probed connections are classical anatomical regions that, given sampling protocols, vary significantly in size and fu...

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Random walks and diffusion on networks

Cited by 624 publications

References 408 publications

Multiplex Networks: A Framework for Studying Multiprocess Multiscale Connectivity Via Coupled‐Network Theory With an Application to River Deltas

Multiplex Networks: A Framework for Studying Multiprocess Multiscale Connectivity Via Coupled‐Network Theory With an Application to River Deltas

Diffusion Geometry Unravels the Emergence of Functional Clusters in Collective Phenomena

Network analysis of mesoscale mouse brain structural connectome yields modular structure that aligns with anatomical regions and sensory pathways

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