Miroslav Kramar

Topology of force networks in compressed granular media

Using numerical simulations, we investigate the evolution of the structure of force networks in slowly compressed model granular materials in two spatial dimensions. We quantify the global properties of the force networks using the zeroth Betti number B0, which is a topological invariant. We find that B0 can distinguish among force networks in systems with frictionlessvs. frictional disks and varying size distributions. In particular, we show that 1) the force networks in systems composed of frictionless, monodisperse disks di!er significantly from those in systems with frictional, polydisperse disks and we isolate the e!ect (friction, polydispersity) leading to the di!erences; 2) the structural properties of force networks change as the system passes through the jamming transition; and 3) the force network continues to evolve as the system is compressed above jamming,e.g., the size of connected clusters with forces larger than a given threshold decreases significantly with increasing packing fraction. Copyright c !EPLA, 2012

Topological data analysis of contagion maps for examining spreading processes on networks

Social and biological contagions are influenced by the spatial embeddedness of networks. Historically, many epidemics spread as a wave across part of the Earth’s surface; however, in modern contagions long-range edges—for example, due to airline transportation or communication media—allow clusters of a contagion to appear in distant locations. Here we study the spread of contagions on networks through a methodology grounded in topological data analysis and nonlinear dimension reduction. We construct “contagion maps” that use multiple contagions on a network to map the nodes as a point cloud. By analyzing the topology, geometry, and dimensionality of manifold structure in such point clouds, we reveal insights to aid in the modeling, forecast, and control of spreading processes. Our approach highlights contagion maps also as a viable tool for inferring low-dimensional structure in networks.

Quantifying force networks in particulate systems

Abstract We present mathematical models based on persistent homology for analyzing force distributions in particulate systems. We define three distinct chain complexes of these distributions: digital , position , and interaction , motivated by different types of data that may be available from experiments and simulations, e.g. digital images, location of the particles, and the forces between the particles, respectively. We describe how algebraic topology, in particular, homology allows one to obtain algebraic representations of the geometry captured by these complexes. For each complex we define an associated force network from which persistent homology is computed. Using numerical data obtained from discrete element simulations of a system of particles undergoing slow compression, we demonstrate how persistent homology can be used to compare the force distributions in different systems, and discuss the differences between the properties of digital, position, and interaction force networks. To conclude, we formulate well-defined measures quantifying differences between force networks corresponding to the different states of a system, and therefore allow to analyze in precise terms dynamical properties of force networks.

Analysis of Kolmogorov Flow and Rayleigh-B enard Convection using Persistent Homology

We use persistent homology to build a quantitative understanding of large complex systems that are driven far-fromequilibrium; in particular, we analyze image time series of ow eld patterns from numerical simulations of two important problems in uid dynamics: Kolmogorov ow and Rayleigh-B enard convection. For each image we compute a persistence diagram to yield a reduced description of the ow eld; by applying dierent metrics to the space of persistence diagrams, we relate characteristic features in persistence diagrams to the geometry of the corresponding ow patterns. We also examine the dynamics of the ow patterns by a second application of persistent homology to the time series of persistence diagrams. We demonstrate that persistent homology provides an eective method both for quotienting out symmetries in families of solutions and for identifying multiscale recurrent dynamics. Our approach is quite general and it is anticipated to be applicable to a broad range of open problems exhibiting complex spatio-temporal behavior.

Evolution of force networks in dense particulate media.

We discuss sets of measures with the goal of describing dynamical properties of force networks in dense particulate systems. The presented approach is based on persistent homology and allows for extracting precise, quantitative measures that describe the evolution of geometric features of the interparticle forces, without necessarily considering the details related to individual contacts between particles. The networks considered emerge from discrete element simulations of two-dimensional particulate systems consisting of compressible frictional circular disks. We quantify the evolution of the networks for slowly compressed systems undergoing jamming transition. The main findings include uncovering significant but localized changes of force networks for unjammed systems, global (systemwide) changes as the systems evolve through jamming, to be followed by significantly less dramatic evolution for the jammed states. We consider both connected components, related in a loose sense to force chains, and loops and find that both measures provide a significant insight into the evolution of force networks. In addition to normal, we consider also tangential forces between the particles and find that they evolve in the consistent manner. Consideration of both frictional and frictionless systems leads us to the conclusion that friction plays a significant role in determining the dynamical properties of the considered networks. We find that the proposed approach describes the considered networks in a precise yet tractable manner, making it possible to identify features which could be difficult or impossible to describe using other approaches.

Structure of force networks in tapped particulate systems of disks and pentagons. I. Clusters and loops.

The force network of a granular assembly, defined by the contact network and the corresponding contact forces, carries valuable information about the state of the packing. Simple analysis of these networks based on the distribution of force strengths is rather insensitive to the changes in preparation protocols or to the types of particles. In this and the companion paper [Kondic et al., Phys. Rev. E 93, 062903 (2016)10.1103/PhysRevE.93.062903], we consider two-dimensional simulations of tapped systems built from frictional disks and pentagons, and study the structure of the force networks of granular packings by considering networks topology as force thresholds are varied. We show that the number of clusters and loops observed in the force networks as a function of the force threshold are markedly different for disks and pentagons if the tangential contact forces are considered, whereas they are surprisingly similar for the network defined by the normal forces. In particular, the results indicate that, overall, the force network is more heterogeneous for disks than for pentagons. Such differences in network properties are expected to lead to different macroscale response of the considered systems, despite the fact that averaged measures (such as force probability density function) do not show any obvious differences. Additionally, we show that the states obtained by tapping with different intensities that display similar packing fraction are difficult to distinguish based on simple topological invariants.

Structure of force networks in tapped particulate systems of disks and pentagons. II. Persistence analysis.

In the companion paper [Pugnaloni et al., Phys. Rev. E 93, 062902 (2016)10.1103/PhysRevE.93.062902], we use classical measures based on force probability density functions (PDFs), as well as Betti numbers (quantifying the number of components, related to force chains, and loops), to describe the force networks in tapped systems of disks and pentagons. In the present work, we focus on the use of persistence analysis, which allows us to describe these networks in much more detail. This approach allows us not only to describe but also to quantify the differences between the force networks in different realizations of a system, in different parts of the considered domain, or in different systems. We show that persistence analysis clearly distinguishes the systems that are very difficult or impossible to differentiate using other means. One important finding is that the differences in force networks between disks and pentagons are most apparent when loops are considered: the quantities describing properties of the loops may differ significantly even if other measures (properties of components, Betti numbers, force PDFs, or the stress tensor) do not distinguish clearly or at all the investigated systems.

Erratum: Topological data analysis of contagion maps for examining spreading processes on networks (Nature Communications (2015) 6 (7723) DOI: 10.1038/ncomms8723)

Social and biological contagions are influenced by the spatial embeddedness of networks. Historically, many epidemics spread as a wave across part of the Earth’s surface; however, in modern contagions long-range edges—for example, due to airline transportation or communication media—allow clusters of a contagion to appear in distant locations. Here we study the spread of contagions on networks through a methodology grounded in topological data analysis and nonlinear dimension reduction. We construct ‘contagion maps’ that use multiple contagions on a network to map the nodes as a point cloud. By analysing the topology, geometry and dimensionality of manifold structure in such point clouds, we reveal insights to aid in the modelling, forecast and control of spreading processes. Our approach highlights contagion maps also as a viable tool for inferring low-dimensional structure in networks. DOI: 10.1038/ncomms8723

The order of bifurcation points in fourth order conservative systems via braids

In second order Lagrangian systems bifurcation branches of periodic solutions preserve certain topological invariants. These invariants are based on the observation that periodic orbits of a second order Lagrangian lie on 3-dimensional (noncompact) energy manifolds and the periodic orbits may have various linking and knotting properties. The main ingredients defining the topological invariants are the discretization of second order Lagrangian systems that satisfy the twist property and the theory of discrete braid invariants developed in [R. W. Ghrist, J. B. Van den Berg, and R. C. Vandervorst, Invent. Math., 152 (2003), pp. 369–432]. In the first part of this paper we recall the essential theory of braid invariants, and in the second part this theory is applied to second order Lagrangian systems and in particular to the Swift–Hohenberg equation. We show that the invariants yield forcing relations on bifurcation branches. We quantify this principle via an order relation on the topological type of a bifurc...

Evolution of force networks in dense granular matter close to jamming

When dense granular systems are exposed to external forcing, they evolve on the time scale that is typically related to the externally imposed one (shear or compression rate, for example). This evolution could be characterized by observing temporal evolution of contact networks. However, it is not immediately clear whether the force networks, defined on contact networks by considering force interactions between the particles, evolve on a similar time scale. To analyze the evolution of these networks, we carry out discrete element simulations of a system of soft frictional disks exposed to compression that leads to jamming. By using the tools of computational topology, we show that close to jamming transition, the force networks evolve on the time scale which is much faster than the externally imposed one. The presentation will discuss the factors that determine this fast time scale.