Sergey V. Buldyrev

Revisiting Levy flight search patterns of wandering albatrosses, bumblebees and deer

The study of animal foraging behaviour is of practical ecological importance, and exemplifies the wider scientific problem of optimizing search strategies. Lévy flights are random walks, the step lengths of which come from probability distributions with heavy power-law tails, such that clusters of short steps are connected by rare long steps. Lévy flights display fractal properties, have no typical scale, and occur in physical and chemical systems. An attempt to demonstrate their existence in a natural biological system presented evidence that wandering albatrosses perform Lévy flights when searching for prey on the ocean surface. This well known finding was followed by similar inferences about the search strategies of deer and bumblebees. These pioneering studies have triggered much theoretical work in physics (for example, refs 11, 12), as well as empirical ecological analyses regarding reindeer, microzooplankton, grey seals, spider monkeys and fishing boats. Here we analyse a new, high-resolution data set of wandering albatross flights, and find no evidence for Lévy flight behaviour. Instead we find that flight times are gamma distributed, with an exponential decay for the longest flights. We re-analyse the original albatross data using additional information, and conclude that the extremely long flights, essential for demonstrating Lévy flight behaviour, were spurious. Furthermore, we propose a widely applicable method to test for power-law distributions using likelihood and Akaike weights. We apply this to the four original deer and bumblebee data sets, finding that none exhibits evidence of Lévy flights, and that the original graphical approach is insufficient. Such a graphical approach has been adopted to conclude Lévy flight movement for other organisms, and to propose Lévy flight analysis as a potential real-time ecosystem monitoring tool. Our results question the strength of the empirical evidence for biological Lévy flights.

Cascade of failures in coupled network systems with multiple support-dependence relations

We study, both analytically and numerically, the cascade of failures in two coupled network systems A and B, where multiple support-dependence relations are randomly built between nodes of networks A and B. In our model we assume that each node in one network can function only if it has at least a single support link connecting it to a functional node in the other network. We assume that networks A and B have (i) sizes N{A} and N{B}, (ii) degree distributions of connectivity links P{A}(k) and P{B}(k), (iii) degree distributions of support links P̃{A}(k) and P̃{B}(k), and (iv) random attack removes (1-R{A})N{A} and (1-R{B})N{B} nodes form the networks A and B, respectively. We find the fractions of nodes μ{∞}{A} and μ{∞}{B} which remain functional (giant component) at the end of the cascade process in networks A and B in terms of the generating functions of the degree distributions of their connectivity and support links. In a special case of Erdős-Rényi networks with average degrees a and b in networks A and B, respectively, and Poisson distributions of support links with average degrees ã and b̃ in networks A and B, respectively, μ{∞}{A}=R{A}[1-exp(-ãμ{∞}{B})][1-exp(-aμ{∞}{A})] and μ{∞}{B}=R{B}[1-exp(-b̃μ{∞}{A})][1-exp(-bμ{∞}{B})]. In the limit of ã→∞ and b̃→∞, both networks become independent, and our model becomes equivalent to a random attack on a single Erdős-Rényi network. We also test our theory on two coupled scale-free networks, and find good agreement with the simulations.

Interdependent networks with identical degrees of mutually dependent nodes.

We study a problem of failure of two interdependent networks in the case of identical degrees of mutually dependent nodes. We assume that both networks (A and B) have the same number of nodes N connected by the bidirectional dependency links establishing a one-to-one correspondence between the nodes of the two networks in a such a way that the mutually dependent nodes have the same number of connectivity links; i.e., their degrees coincide. This implies that both networks have the same degree distribution P(k). We call such networks correspondently coupled networks (CCNs). We assume that the nodes in each network are randomly connected. We define the mutually connected clusters and the mutual giant component as in earlier works on randomly coupled interdependent networks and assume that only the nodes that belong to the mutual giant component remain functional. We assume that initially a 1-p fraction of nodes are randomly removed because of an attack or failure and find analytically, for an arbitrary P(k), the fraction of nodes μ(p) that belong to the mutual giant component. We find that the system undergoes a percolation transition at a certain fraction p=p(c), which is always smaller than p(c) for randomly coupled networks with the same P(k). We also find that the system undergoes a first-order transition at p(c)>0 if P(k) has a finite second moment. For the case of scale-free networks with 2<λ≤3, the transition becomes a second-order transition. Moreover, if λ<3, we find p(c)=0, as in percolation of a single network. For λ=3 we find an exact analytical expression for p(c)>0. Finally, we find that the robustness of CCN increases with the broadness of their degree distribution.

Thermodynamics and dynamics of the two-scale spherically symmetric Jagla ramp model of anomalous liquids

Using molecular dynamics simulations, we study the Jagla model of a liquid which consists of particles interacting via a spherically symmetric two-scale potential with both repulsive and attractive ramps. This potential displays anomalies similar to those found in liquid water, namely expansion upon cooling and an increase of diffusivity upon compression, as well as a liquid-liquid (LL) phase transition in the region of the phase diagram accessible to simulations. The LL coexistence line, unlike in tetrahedrally coordinated liquids, has a positive slope, because of the Clapeyron relation, corresponding to the fact that the high density phase (HDL) is more ordered than low density phase (LDL). When we cool the system at constant pressure above the critical pressure, the thermodynamic properties rapidly change from those of LDL-like to those of HDL-like upon crossing the Widom line. The temperature dependence of the diffusivity also changes rapidly in the vicinity of the Widom line, namely the slope of the Arrhenius plot sharply increases upon entering the HDL domain. The properties of the glass transition are different in the two phases, suggesting that the less ordered phase is fragile, while the more ordered phase is strong, which is consistent with the behavior of tetrahedrally coordinated liquids such as water silica, silicon, and BeF2.

Model for reversible colloidal gelation

We report a numerical study, covering a wide range of packing fraction Phi and temperature T, for a system of particles interacting via a square well potential supplemented by an additional constraint on the maximum number n(max) of bonded interactions. We show that, when n(max)<6, the liquid-gas coexistence region shrinks, giving access to regions of low Phi where dynamics can be followed down to low T without an intervening phase separation. We characterize these arrested states at low densities (gel states) in terms of structure and dynamical slowing down, pointing out features which are very different from the standard glassy states observed at high Phi values.

Unusual phase behavior of one-component systems with two-scale isotropic interactions.

We study the phase behavior of systems of particles interacting through pair potentials with a hard core plus a soft repulsive component. We consider several different forms of soft repulsion, including a square shoulder, a linear ramp and a quasi-exponential tail. The common feature of these potentials is the presence of two repulsive length scales, which may be the origin of unusual phase behaviors such as polyamorphism both in the equilibrium liquid phase and in the glassy state, water-like anomalies in the liquid state and anomalous melting at very high pressures.

A tetrahedral entropy for water

We introduce the space-dependent correlation function C Q(r) and time-dependent autocorrelation function C Q(t) of the local tetrahedral order parameter Q ≡ Q(r,t). By using computer simulations of 512 waterlike particles interacting through the transferable interaction potential with five points (TIP5 potential), we investigate C Q(r) in a broad region of the phase diagram. We find that at low temperatures C Q(t) exhibits a two-step time-dependent decay similar to the self-intermediate scattering function and that the corresponding correlation time τQ displays a dynamic cross-over from non-Arrhenius behavior for T > T W to Arrhenius behavior for T < T W, where T W denotes the Widom temperature where the correlation length has a maximum as T is decreased along a constant-pressure path. We define a tetrahedral entropy S Q associated with the local tetrahedral order of water molecules and find that it produces a major contribution to the specific heat maximum at the Widom line. Finally, we show that τQ can be extracted from S Q by using an analog of the Adam–Gibbs relation.

Static and dynamic anomalies in a repulsive spherical ramp liquid : Theory and simulation

We compare theoretical and simulation results for static and dynamic properties for a model of particles interacting via a spherically symmetric repulsive ramp potential. The model displays anomalies similar to those found in liquid water, namely, expansion upon cooling and an increase of diffusivity upon compression. In particular, we calculate the state points P (rho,T) from the simulation and successfully compare it with the state points P (rho,T) obtained using the Rogers-Young (RY) closure for the Ornstein-Zernike (OZ) equation. Both the theoretical and the numerical calculations confirm the presence of a line of isobaric density maxima, and lines of compressibility minima and maxima. Indirect evidence of a liquid-liquid critical point is found. Dynamic properties also show anomalies. Along constant temperature paths, as the density increases, the dynamics alternate between slowing down and speeding up, and we associate this behavior with the progressive structuring and destructuring of the liquid. Finally we confirm that mode coupling theory successfully predicts the nonmonotonic behavior of dynamics and the presence of multiple glass phases, providing strong evidence that structure (the only input of mode coupling theory) controls dynamics.

Water-like solvation thermodynamics in a spherically symmetric solvent model with two characteristic lengths

We examine by molecular dynamics simulation the solubility of small apolar solutes in a solvent whose particles interact via the Jagla potential, a spherically symmetric ramp potential with two characteristic lengths: an impenetrable hard core and a penetrable soft core. The Jagla fluid has been recently shown to possess water-like structural, dynamic, and thermodynamic anomalies. We find that the solubility exhibits a minimum with respect to temperature at fixed pressure and thereby show that the Jagla fluid also displays water-like solvation thermodynamics. We further find low-temperature swelling of a hard-sphere chain dissolved in the Jagla fluid and relate this phenomenon to cold unfolding of globular proteins. Our results are consistent with the possibility that the presence of two characteristic lengths in the Jagla potential is a key feature of water-like solvation thermodynamics. The penetrable core becomes increasingly important at low temperatures, which favors the formation of low-density, open structures in the Jagla solvent.

Gel to glass transition in simulation of a valence-limited colloidal system

We numerically study a simple model for thermoreversible colloidal gelation in which particles can form reversible bonds with a predefined maximum number of neighbors. We focus on three and four maximally coordinated particles, since in these two cases the low valency makes it possible to probe, in equilibrium, slow dynamics down to very low temperatures T. By studying a large region of T and packing fraction phi we are able to estimate both the location of the liquid-gas phase separation spinodal and the locus of dynamic arrest, where the system is trapped in a disordered nonergodic state. We find that there are two distinct arrest lines for the system: a glass line at high packing fraction, and a gel line at low phi and T. The former is rather vertical (phi controlled), while the latter is rather horizontal (T controlled) in the phi-T plane. Dynamics on approaching the glass line along isotherms exhibit a power-law dependence on phi, while dynamics along isochores follow an activated (Arrhenius) dependence. The gel has clearly distinct properties from those of both a repulsive and an attractive glass. A gel to glass crossover occurs in a fairly narrow range in phi along low-T isotherms, seen most strikingly in the behavior of the nonergodicity factor. Interestingly, we detect the presence of anomalous dynamics, such as subdiffusive behavior for the mean squared displacement and logarithmic decay for the density correlation functions in the region where the gel dynamics interferes with the glass dynamics.