In recent years, there has been great progress in understanding and modeling the structure of complex networks that emerge naturally in certain systems. A statistical analysis of their connectivity has found universal properties in the architecture of social networks, technological networks (Internet, power-grid, etc.), and biological networks (neural networks, gene regulatory networks, etc.). However, only a few studies have considered the dynamics of processes taking place on networks and even fewer have produced analytic expressions to describe them.

Schematic representation of the connectivity used to compute network dynamics. The incoming links (arrows) determine the nodes' (boxes) evolution in time.

In the context of my NSF project "Collective Behavior of Complex Systems with Long-Range effective Interactions: A Network Approach" (DMS-0507745), we have analyzed certain complex systems by separating their dynamics into an internal one (that describes the evolution of each element's internal state) and an external one (that determines which elements interact). The idea is to then replace the external dynamics by a fixed network of effective interactions capturing the long-range effects that emerge from local contributions. In order to develop a better understanding of the typical dynamics on these networks, we analyzed the stationary solutions, collective behaviors, and phase transitions in systems with various internal rules interacting through network connections.

By carrying out mean-field-like calculations, we computed analytically the stationary solutions of network systems with different Boolean interaction rules and with rules equivalent to the ones used in swarming models. In every case considered, we found that the qualitative characteristics and order of its phase transition were the same in the network case and in the corresponding proximity-based interaction case. Furthermore, the stationary behavior of the full dynamical system is exactly the same as the network one if the agents' positions are randomized at every time-step.

Our analysis of dynamics on networks has also allowed us to study the difference in the effect produced on swarming systems by internal and external noise sources (affecting the agents' displacements or measurements, respectively) by finding the analytical solution of a network system that combines both noise mechanisms simultaneously. Our results show that this system can only produce a discontinuous phase transition if the internal noise is very low. Finally, we have shown that a one time-step mapping of the order parameter in the full swarming simulation has the same features as the equivalent rules interacting through its effective network connections and that the swarming solutions converge to the network solutions in the limit of high agent speed.

Here are a few papers in which we have published some of these results:

M. Aldana, C. Huepe, H. Larralde, and J. A. Pimentel.
Noise mechanism effects on the phase transitions of swarming systems and related network models.
Submitted to: Phys. Rev. E (2008). [pdf]

M. Aldana, V. Dossetti, C. Huepe, V. M. Kenkre, and H. Larralde.
Phase Transitions in Systems of Self-Propelled Agents and Related Network Models.
Phys. Rev. Lett. 98 095702 (2007). [link] [pdf]

M. Aldana and C. Huepe.
Phase transitions in self-driven many-particle systems and related non-equilibrium models: A network approach.
J Stat Phys 112 (1-2), 135-153 (2003). [link] [pdf]

C. Huepe and M. Aldana.
Dynamical phase transition in a neural network model with noise: An exact solution.
J Stat Phys 108 (3-4), 527-540 (2002). [link] [pdf]

Swarming models are used to study the collective motion of various groups of biological agents such as bird flocks, fish schools, herds of quadrupeds, and bacteria colonies. They capture in an idealized setting the dynamics of groups of self-propelled agents. Their goal is to help understand how simple local interactions can produce large-scale coherent group motion and what the advantage of such collective behavior is. This knowledge could improve our ability to probe and control swarming systems that can strongly affect our lives such as locust plagues, epidemics, or fish schools.

Snapshots of two swam simulations. (A) Ordered phase: groups head in the same direction. (B) Disordered phase: groups head in different directions.

In the context of my NSF project "Collective Behavior of Complex Systems with Long-Range effective Interactions: A Network Approach" (DMS-0507745), we have considered simple swarming models as test cases for our network approach, where the dynamics of a complex system is separated into an internal one (that describes the evolution of each element's internal state) and an external one (that determines which elements interact). In this framework, we have investigated various aspects of the dynamics of swarming models.

1. Comparison of minimal swarming models using novel global quantities: In one of our recent studies we implemented a new set of global quantities (not previously used in this context) to analyze three well established minimal swarming algorithms. All three derive from the simple model first introduced by Vicsek et al. in 1995, describing particles that tend to head in the mean direction of motion of its close-by neighbors while advancing at a constant speed. By computing the number of interacting neighbors per agent, distance to closest neighbor, and distribution of cluster sizes in numerical simulations for more than 300 sets of parameters, we characterized the agents' spatial distribution, identified which cases best describe realistic swarms, and distinguished between two implementations of the Vicsek model that had previously been confused in the literature. We also found that all these minimal models overestimate the typical mean number of interacting agents observed in nature. In addition, we showed that agents organize into clusters, which have a size-distribution that is close to exponential at high noise levels, and approaches a power-law as the noise is reduced. Finally and unexpectedly, we found evidence of a universal critical behavior (where this distribution becomes a power-law) that could be of great interest for future studies. An important aspect of our analysis is that the global quantities introduced can be used for real swarms. This is of current relevance since new experiments that collect detailed data of every agent in a swarm are currently being implemented (such as those by the group of Prof. I. Couzin at the Dept. of Ecol. and Evol. Bio., Princeton U).
2. Intermittency and Clustering in swarming systems: In previous work, we were the first to report and analyze the intermittent behavior observed in the Vicsek model (described above). In the ordered phase, where agents mostly move in the same approximate direction, the system displays a series of intermittent bursts during which this order is temporarily lost. We characterized this intermittency in numerical simulations and also described analytically its statistical properties for a reduced system of only two particles. For larger systems, the particles aggregate into clusters that play an essential role in the intermittent dynamics. Our study of the cluster statistics for one specific set of parameters (which we now believe followed the observed behavior because of its proximity to the critical state) showed that both the cluster sizes and the transition probability between them have a power-law distribution. We also found that the exchange of particles between clusters satisfies detailed balance.
3. Proof of convergence for swarming systems with informed agents: We developed a different aspect of our analysis of swarming systems in a recent collaboration with Prof. F. Cucker (Dept. of Mathematics, City U. of Hong Kong). We extended the Cucker-Smale swarming model to include informed agents with a preferred direction of motion by adding a term that accelerates them in this direction as long as their neighbors have not reached velocity consensus. The Cucker-Smale model consists of a set of ordinary differential equations that tends to vanish the agents' velocity differences with time. While not particularly realistic, the main interest of this approach is to allow mathematical proofs of convergence. For the case with informed agents we were able to prove that, under certain conditions, agents converge exponentially fast towards an ordered swarming state. We are also studying numerically how informed agents influence the group?s direction of motion.

These are some of my papers that resulted from the work described above:

C. Huepe and M. Aldana.
New tools for characterizing swarming systems: A comparison of minimal models.
Physica A 387 (12) 2809-2822 (2008). [link] [pdf]

M. Aldana, V. Dossetti, C. Huepe, V. M. Kenkre, and H. Larralde.
Phase Transitions in Systems of Self-Propelled Agents and Related Network Models.
Phys. Rev. Lett. 98 095702 (2007). [link] [pdf]

C. Huepe and M. Aldana.
Intermittency and clustering in a system of self-driven particles.
Phys. Rev. Lett. 92 (16) Art. No. 168701 (2004). [link] [pdf]

M. Aldana and C. Huepe.
Phase transitions in self-driven many-particle systems and related non-equilibrium models: A network approach.
J Stat Phys 112 (1-2), 135-153 (2003). [link] [pdf]

An important application of swarming theory is the design of decentralized control algorithms for coordinating groups of (microscopic or macroscopic) robots that can perform collectively a task better than single agents. These can be used, for example, to monitor the environment by deploying autonomous sensor arrays or to organize self-assembling structures. The challenge here is to develop robust, scalable, and leaderless algorithms that achieve a given objective in various contexts, using different numbers of robots, and even if some of the agents fail.

Simulation of a control algorithm designed to achieve translating (top) and rotating (bottom) agent formations while satisfying constraints on their shape. (Images courtesy of Daniel Kiefer.)

In the context of my NSF project "Collective Behavior of Complex Systems with Long-Range effective Interactions: A Network Approach" (DMS-0507745), we have studied swarming control algorithms as an application of our network approach and to achieve a better understanding of swarms in general. Indeed, the ability to tailor a control algorithm to achieve a given result provides a unique perspective in the analysis of swarming systems. This work is being carried out in two different projects.

1. Self-organized flocking in a mobile robot swarm: As part of this project, I collaborated with and advised Dr. Ali Emre Turgut (Kovan Research Lab, Dept. of Computer Engineering, Middle East Technical U, Ankara, Turkey) in his PhD work on autonomous robots. His lab constructed several "Kobots"simple interacting mobile robots) to use in the study of swarming control algorithms and dynamics. Interestingly, these Kobots communicate their orientation through a mechanism that does not depend on their distance, but that instead saturates at a given number of interactions. These are precisely the conditions under which our network approach provides exact analytic solutions. I computed these solutions for an extension of the models previously considered by us, which adds a term to model the Kobot?s effective inertia, and obtained a good match to the experimental results.

   Picture of one Kobot (left) and of a group of seven Kobots (right). These robots were designed by the KOVAN research lab for use in swarm robotic studies. (Images courtesy of Kovan Research Lab)

2. Consensus algorithms for linear and circular swarming motion with no absolute reference frame: As part of this project, I co-advised (with Prof. M. Silber) the PhD work of applied mathematics graduate student Daniel Kiefer (Northwestern U). We constructed consensus based decentralized control algorithms that allow groups of autonomous agents to move collectively in a desired formation specified by the moments of their position distributions. These can control rectilinear or circularly moving swarms. They are based on the idea that swarming algorithms can be divided into two parts: a consensus one and a proper control one. The algorithms work even if agents have a limited range of communication, and they do not require agents to know their absolute positions in space. They could therefore be used in a variety of engineering contexts to control effectively groups of autonomous robots.

The following papers have resulted from these projects:

D. Kiefer, C. Huepe and M. Silber.
Consensus based decentralized control of linear and rotational motion of swarms.
In preparation (2008).

A. E. Turgut, C. Huepe, H. Celikkanat, and F. Gokce.
A simple model for self-organized flocking in a mobile robot swarm.
Submitted to: "The tenth international conference on the simulation of adaptive behavior (SAB?8)" (2008). [pdf]

This is an area of research for which I am currently developing a new project. The objective is to advance our theoretical understanding of living systems by exploring the effects of multi-scale (m-s) structures over the dynamics of evolving complex systems and relating these to experimental data. M-s structures have arisen as an emerging feature in various networks describing biological interactions (e.g. genetic and metabolic networks), in the form of modularity, motifs, or scale-free connectivity. M-s structures are defined here as those that can be subdivided into relatively autonomous embedded modules (with limited number of interactions among them) and where each of these modules can be subdivided into sub-modules which are made of sub-sub-modules and so on. The proposed project will determine how this simple idealized structural constraint can condition the evolutionary dynamics and its resulting features.

Schematic representation a multi-scale network of biological interactions. Each disc contains embedded discs and interacts through network connections (arrows) at its scale.

An important aspect of the m-s approach is that it contains a form of natural selection requiring no ad-hoc survival criteria or environment. Indeed, for an m-s structure to be viable in living systems, all scales (up to the highest embedding level) must support complex dynamical responses. If a component at any level decouples from the rest (i.e. ceases to affect them, e.g. by only producing noise or a constant signal in its interactions) it will not contribute to the response of any of its embedding levels and could be, in practice, discarded. This provides an evolutionary constraint for modules at each level where the embedding system acts as an environment and remaining coupled as the selection criterion. While in reality the physical world imposes additional constraints (selecting an evolutionary path) this idealized perspective produces closed systems for exploring properties of evolving structures that are independent of external factors.

Schematic representation of the evolution and organization of a multi-scale architecture. Subsystems combine to create systems with higher levels of complexity towards the top of the figure.

To study how evolving m-s structures condition biological systems, the project will: 1. Explore simple m-s network models, focusing on numerically manageable computations such as (a) two or three level Boolean networks where the response of each higher-level node results from its embedded network?s dynamics; (b) ideally structured systems that respond collectively as each of its components; (c) experimentally measurable properties of Boolean m-s systems. 2. Develop analysis and representation tools tailored for m-s hierarchical interaction networks. 3. Compare properties of m-s systems to those of various experimental biological interaction networks.

At very low-temperatures, some fluids can flow without any resistance or energy dissipation. The understanding of this phenomenon, which is due to quantum effects and is similar to electromagnetic superconductivity, is essential for the description of collective quantum states and could lead to future almost frictionless processes or mechanical devices.

Simulation of a two-dimensional superflow past a cylinder. The sequence shows the shedding of a vortex pair, which induces a phase jump on the complex field (displayed at the bottom).

In my thesis work, I used the Nonlinear Schrödinger equation as a simple model for superfluidity. Using analytical and numerical tools, we found the conditions under which dissipation appears through the shedding of vortices as the superflow speed around an obstacle is increased.

For two-dimensional superflows around a cylinder, we calculated numerically for the first time the saddle-node bifurcation that leads to superfluid dissipation. Through a secondary pitchfork bifurcation, the unstable branch generates one-vortex asymmetric fields that correspond to the nucleation solutions. We found symmetric and non-symmetric nucleation solutions. We also characterized the influence of the ratio of the coherence length to the disc diameter on the properties of the bifurcation diagram.

Simulation of a three-dimensional superflow past a cylinder. The images show the vortex stretching mechanism which is essential in 3D.

Finally, we studied the three-dimensional instabilities that develop in a superflow around a cylinder. We found numerically a mechanism of vortex stretching that is the main responsible for dissipation in three-dimensions.

Here are some of my papers in this area:

C. Nore, C. Huepe, M. E. Brachet.
Subcritical Dissipation in Three-Dimensional Superflows.
Phys. Rev. Lett. 84, 2191 (2000). [link] [pdf]

C. Huepe and M.-E. Brachet.
Scaling laws for vortical nucleation solutions in a model of superflow.
Physica D 140 (2000) 126-140. [link] [pdf]

C. Huepe and M.-E. Brachet.
Vortical Nucleation Solutions in a Model of Super-flow.
C.R. Acad. Sci. Paris, t. 325, Serie IIb, 195-202, (1997). [link] [pdf]

At extremely low temperatures, some gases can undergo a Bose-Einstein condensation where particles coalesce into a collective quantum state. This new state of matter, which was first achieved experimentally in 1995 (more than 70 years after it was theoretically predicted), has reenergized the research in fundamental aspects of quantum mechanics. This may help achieve a better understanding of the very nature of matter and its interactions, and develop new applications relating to quantum communication, quantum information, and quantum computers.

During my thesis, I used the nonlinear Schrödinger equation as a simple model to study attractive Bose-Einstein condensates confined in external potential wells such as the ones first achieved experimentally with lithium-7 in the group of R. G. Hulet.

Energy potential barrier (top) and most unstable eigenvalue (bottom) for an attractive Bose-Einstein condensate as a function of the number of atoms. The universal saddle-node properties are apparent.

For isotropic and non-isotropic three-dimensional potentials, we showed the presence of a Hamiltonian saddle-node bifurcation where the stable (elliptic) and unstable (hyperbolic) solutions meet. This bifurcation determines the maximum size of condensates with attractive interactions. The condensate decay rates corresponding to macroscopic quantum tunneling, thermal fluctuations and inelastic collisions were thus determined based on generic saddle-node characteristics and compared to experimental measurements. The influence of anisotropy on the bifurcation diagram was also characterized, which required large three-dimensional numerical computations.

Here are some of my papers in this area:

C. Huepe, L.S. Tuckerman, S. Metens and M.E. Brachet.
Stability and decay rates in nonisotropic attractive Bose-Einstein condensates.
Phys. Rev. A 68 (2) Art. No. 023609 (2003). [link] [pdf]

M. Abid, C. Huepe, S. Metens, C. Nore, C.T. Pham, L.S. Tuckerman and M.E. Brachet.
Gross-Pitaevskii dynamics of Bose-Einstein condensates and superfluid turbulence.
Fluid Dynamics Research 33 (5-6) 509-544 (2003). [link] [pdf]

C. Huepe, S. Metens, G. Dewel, P. Borckmans and M. E. Brachet.
Decay Rates in Attractive Bose-Einstein Condensates.
Phys. Rev. Lett. 82, 1616 (1999). [link] [pdf]

Cosmological models aim to describe the evolution of the universe as a whole. One ongoing challenge is to develop a theory that includes both its gravitation and quantum properties. Since no theory is yet able to do so, a relevant approach is to use over-simplified toy models.

In my thesis work, I combined a complex-field and a relativistic hydrodynamic approach by analyzing a toy cosmological model where the matter is in a Bose-Einstein condensate-like state. This is described using the nonlinear Klein-Gordon equation, which is a relativistic covariant generalization of the nonlinear Schrödinger equation.

Diagrams displaying the reduced dynamics of a toy cosmological model filled with a complex field that follows a Nonlinear Klein-Gordon Equation. Inflation arises as the fast dynamics towards a central manifold.

We showed that the nonlinear dynamics of our toy-model universe, that includes quantum and gravitational components, has a wide range of inflationary solutions at short time scales, which we characterized. Using adiabatic invariants, we also showed the existence of a conventional cosmological expansion regime at longer time scales.

Here are my papers in this area:

C. Huepe, M.-E. Brachet and F. Debbasch.
Generic inflationary and noninflationary behavior in toy-cosmology.
Physica D 144 (2000) 20-36. [link] [pdf]

C. Huepe, F. Debbasch and M. -E. Brachet.
Hydrodynamical Interpretation of Relativistic Charged Scalar Field Dynamics.
In "Topological defects in cosmology",
M. Signore and F. Melchiorri (Eds.)
World Scientific (1996).

Finite-Time Singularities

The dynamics of various physical and biological systems can lead to mathematically ill-defined singularities where quantities at one particular point either diverge to infinity or vanish to zero at a given time.

Eigenfunctions of the perturbations of self-similar solutions as they evolve towards a finite-time singularity.

A common example of this phenomenon occurs when a drop separates from a fluid through pinch-off. At the point where this happens, hydrodynamic equations behave in a singular way, which presents a problem for standard analytical and numerical techniques. Close to this singularity, however, equations can become self-similar, following simple relations. This property could be used to achieve a simplified rigorous pinch-off description that would be helpful for engineering applications such as inkjet printer design and large-scale hydrodynamic simulations.

During my work at the James Franck Institute of the University of Chicago, I studied numerically the stability of trajectories leading to finite-time singularities in a simplified model of aggregation. This model has been used describe chemotactic bacteria aggregation and dissipative gravitational collapse. It has the advantage of having analytic self-similar solutions that lead to the singularity. I analyzed the perturbations that can change this path to singularity numerically and found their approximate analytical solutions. I also developed and tested a new simple adaptive mesh method for solving this and other similar problems.

Here is a paper I wrote in this area:

C. Huepe.
Dynamics of the convergence towards a self-similar blowup solution in a simplified model of aggregation.
Nonlinearity 15(5), 1699-1715 (2002). [link] [pdf]

Faraday waves are the perturbations that appear on the surface of a fluid when its container is rapidly oscillated vertically. It is one of the best studied pattern forming systems, but the effect of the oscillation details over the resulting surface patterns is not well understood. Progress in this front would clarify the relation between the forcing and its resulting resonant excitations in various systems and could help develop applications such as achieving microscopic or macroscopic patterns in certain industrial processes by simply shaking the system adequately.

Pictures of the patterns arising on the liquid surface in a vertically oscillated container (Faraday experiment). The bi-stable solutions achieved through a complex oscillation function are displayed.

In my work with Prof. M. Silber at the Applied Math Department of Northwestern University, I analyzed two aspects of this problem:

1. Faraday patterns generated by impulsive oscillations: When the oscillations correspond to instant accelerations (impulses or kicks), many exact analytical results can be obtained. We found the complete impulsive solution for an arbitrary number of impulses on a finite depth viscous fluid layer, showing analytically that the problem is ill-posed for the high viscosity case. We also computed numerically the continuous-to-impulsive limit.
2. Faraday patterns generated by complex multi-frequency oscillations: Very little is known about the effect of oscillations that are not described by a simple sinusoidal motion. We implemented a semi-analytic code in Mathematica to compute the Faraday excitations for any forcing function and used it to study the effect of various multi-frequency oscillation shapes. We then used a WKB approximation as a new analytical tool to compute Faraday waves in shallow viscous fluids under arbitrary oscillation patterns.

Here is one of my papers in this area:

C. Huepe, Y. Ding, P. Umbanhowar and M. Silber.
Forcing function control of Faraday wave instabilities in viscous shallow fluids.
Phys. Rev. E 73 (1) 016310 (2006). [link] [pdf]

Many nonequilibrium pattern-forming systems exhibit transitions from spatially ordered and temporally regular patterns to spatio-temporal chaos. Although much progress has been made, the identification and classification of spatio-temporal chaotic states remains a challenge. Topological defects, on the other side, appear to play a prominent role in disrupting the pattern order in many systems. For example, in certain two-dimensional equilibrium systems the Kosterlitz-Thouless transition from disordered to ordered states involves the unbinding of defects.

In my work with Prof H. Riecke at the Applied Math Department of Northwestern University, we analyzed the statistics of defect trajectories (SDT) on a Complex Ginzburg-Landau Equation (CGLE) to study its different chaotic states. The CGLE is a generic amplitude equation used to describe various systems. Its defect-chaos state is a prototype for spatio-temporal chaos. By studying its SDT we obtained a reduced variable description that could be applied to other chaotic dynamics found, for example, in the atmosphere, oceans, or reactive processes. In addition, we used the SDT to analyze the Kosterlitz-Thouless transition in an XY-model (commonly used to describe ferromagnetic materials and similar systems).

Picture of the undulation chaos and defects observed an inclined-layer convection experiment. The lines correspond to the convection rolls.

By combining our work with experiments by Profs E. Bodenschatz and K. Daniels at Cornell University, we also used the SDT to study the defect chaos states arising from the convection motion in an inclined thin fluid layer. We discovered a scale transformation that produces a universal SDT behavior that matches our experimental and theoretical data for a wide range of parameters. In other specific parameter regions, however, our SDT results show significant deviations from the previously reported finding that the defect dynamics are consistent with those of random walkers that are created with fixed probability and annihilated through random collisions.

This is one of our papers in this area:

C. Huepe, H. Riecke, K. Daniels and E. Bodenschatz.
Statistics of Defect Trajectories in Spatio-Temporal Chaos in Inclined Layer Convection and the Complex Ginzburg-Landau Equation.
Chaos 14 (3) 864-874 (2004). [link] [pdf]