Session Y14: Focus Session: Statistical Mechanics of Complex Networks II

Universal features of dynamic small-world networks

In a dynamic small-world contact network, an individual has fixed short range links within its local neighborhood and time-varying stochastic long range links outside of that neighborhood. The probability of a long range link occurring ($p$, in analogy with the standard small-world rewiring parameter) is a measure of the mobility of the population. In this study we investigate the epidemic to non-epidemic phase transition that occurs in a susceptible-infected-recovered (SIR) disease spreading model located on this type of dynamic network. We first derive the finite-valued critical mobility p$_{c}$ and find excellent agreement with numerical simulations. Close to p$_{c}$ the outbreak size scales as (p-p$_{c})^{\beta }$ since it is a continuous transition; we find that $\beta $ is near 2, but varies as a function of the infectivity and recovery rates. At the critical point our study shows that the distribution of outbreak sizes scales as $\sim $ N$^{-\alpha }$ with $\alpha $ = 1.5$\pm $0.03. We compare these critical exponents to those found in related small-world and dynamic small-world networks and comment on potential universality.   

Robustness and dynamics of networks of coupled modules

Many systems, from power grids and the internet, to the brain and society, can be modeled using networks of coupled overlapping modules. The elements of these networks perform individual and collective tasks such as generating and consuming electrical load or transmitting data. We study the robustness of these systems using percolation theory: a random fraction of the elements fail which may cause the network to lose global connectivity. We show that the modules themselves can become isolated or uncoupled (non-overlapping) well before the network falls apart. This has important structural and dynamical consequences for these networks and may explain how missing information hides pervasive overlap between communities in real networks.   

Explosive Percolation in the Human Protein Homology Network

We study the explosive character of the percolation transition in a real-world network. We show that the emergence of a spanning cluster in the Human Protein Homology Network (H-PHN) exhibits similar features to an Achlioptas-type process and is markedly different from regular random percolation. The underlying mechanism of this transition can be described by slow-growing clusters that remain isolated until the later stages of the process, when the addition of a small number of links leads to the rapid interconnection of these modules into a giant cluster. Our results indicate that the evolutionary-based process that shapes the topology of the H-PHN through duplication-divergence events may occur in sudden steps.   

Power Spectrum of the Finite Kuramoto Model

We study the synchronization of oscillators in the finite Kuramoto model, a simple model for coupled phase oscillators that exhibits a phase transition. The usual self-consistent approach used in studying the Kuramoto model gives a prediction for the distribution of modified frequencies that includes a Dirac delta at the synchronized frequency and a depletion of nearby frequencies. For finite systems, the prediction adequately describes the distribution of frequencies averaged over very long durations, but the accompanying power spectrum of the order parameter looks very different. The sharp peak at the synchronization frequency has a finite width and oscillators that are otherwise entrained manage to occasionally escape. The resulting harmonics of these escaped oscillators leads to a power spectrum with an exponential drop-off from the peak, rather than the originally predicted depletion.   

Recruitment dynamics in adaptive social networks

We model recruitment in social networks in the presence of birth and death processes. The recruitment is characterized by nodes changing their status to that of the recruiting class as a result of contact with recruiting nodes. The recruiting nodes may adapt their connections in order to improve recruitment capabilities, thus changing the network structure. We develop a mean-field theory describing the system dynamics. Using mean-field theory we characterize the dependence of the growth threshold of the recruiting class on the adaptation parameter. Furthermore, we investigate the effect of adaptation on the recruitment dynamics, as well as on network topology. The theoretical predictions are confirmed by the direct simulations of the full system.   

The spread of opinion on co-evolving networks

We discuss a model of opinion formation in co-evolving networks. In realistic scenarios, the network constantly changes structure favoring connections between similar individuals (homophily). Here we allow the opinions to co-evolve with the reorganization of links in the network. This dynamical nature of the network impedes the spreading of opinions. We study how this resistance to the spread can be overcome and consensus can be achieved by randomly distributing a few committed agents (-nodes that are not influenceable in their opinions). In this model adjacent nodes influence each other if they are similar on at least Q attributes, where Q is the influence threshold. Nodes will rewire their existing links if they are not similar enough. We demonstrate through simulations that in the absence of committed agents, time to reach consensus in opinions diverges exponentially with system size N. However, as committed agents are added, beyond a small value of committed fraction, the consensus time becomes a slowly varying function of N. (Ref- F. Vazquez et al. - Phys. Rev. E76, 046120 -2007)   

Network resilience to real-world disasters: Eyjafjallaj\"okull and 9/11

We investigate the resilience of the the world-wide air transportation network (WAN) to the 9/11 terrorist attacks and the recent eruption of the volcano Eyjafjallaj\"okull. Although both disasters caused wide-spread disruption, the number of airports that were closed and the volume of interrupted traffic were well below the percolation threshold predicted by the classical theory. In order to quantify and visualize network deformation before breakdown, we introduce a framework based on the increase in shortest-path distance and homogenization of shortest-path structure. These real-world disasters are a new type of disruption because the removal of all vertices (airports) is geographically compact. Our framework incorporates the dual perspective of individual airports and geopolitical regions to capture how the impact interacts with the sub-network structure.We find that real-world events have an impact signature which is qualitatively different from that of random or high-centrality attacks. Furthermore, we find that the network is more resilient to the 9/11 disaster, although it removed more airports and traffic than the volcanic ash-cloud. This is due to the network roles of Europe and North America. We discuss how regional roles influence resilience to a region's removal.   

Stochastic Moment Equations - Case Closed

Reaction networks frequently appear in many natural systems, such as chemistry, biology and ecology. The modelling of these networks is commonly based on rate equations models, incorporating the law of mass action kinetics. However, when the system is microscopic, it becomes governed by fluctuations, the law of mass action kinetics no longer applies, and the rate equations fail. To obtain an accurate description of microscopic reaction networks, one must refer to stochastic methods based on the master equation. The problem is that the number of equations rises exponentially with the number of species, rendering the treatment of the master equation infeasible. Moment equations are known to be more efficient, however the equations are not closed, and become prohibitively complicated when moments of high order are included. In this talk we present the binomial moment equations. The binomial moments are linear combinations of the ordinary moments related to the population size of the reactive species. They capture the essential combinatorics of the reaction processes reflecting their stoichiometric structure. This leads to a simple and transparent form of the equations, allows a highly efficient and surprisingly simple truncation scheme and enables the inclusion of moments up to any desired order. The result is a set of equations that enables an equation-based stochastic analysis of reaction networks under a very broad range of conditions.   

Using a Projector to Control BZ Drops: Attractor Selection by Pattern Entrainment

An emulsion consisting of drops in the 100$\mu$m diameter range containing the Belousov-Zhabotinsky (BZ) oscillatory chemicals can interact via diffusive inhibition and can be thought of as coupled phase oscillators. For weak coupling, a 2-D hexagonal lattice of these drops naturally develop regions of attractor states of sequential oscillations with phase differences of plus/minus $2\pi/3$ much like the 2D anti-ferromagnetic Heisenberg spin model. An untrained system of these oscillators will develop unstable regions of both attractors that grow and compete. We use photo-initiated inhibition to optically entrain the system with a projected $+2\pi/3$ pattern in an attempt to force the system into the $+2\pi/3$ attractor state. However, both the left and right handed variants of the $2\pi/3$ attractor are present in the entrained system. Defining an order parameter $e^{i 3 \phi}$ allows for a quantitation of the purity of the $2\pi/3$ attractor state in the final system.   

Complexity facilitates perturbation of a coherent dynamical process

We discuss the influence of perturbation on networks of globally coupled three state stochastic oscillators. When coupled, the system shows intermittent behavior characterized by a waiting time distribution which reveals both inverse power-law and coherent dynamical properties. Specifically, we compare the results of perturbation realized with a periodic signal to those obtained using perturbation provided by a matching system. We find that the SNR (signal-to-noise ratio) does not depend on the frequency of the perturbing signal. We also observe that the second approach results in higher values of SNR. We discuss how those findings cannot be explained by either classical or statistical resonance theory. With the help of the fluctuation-dissipation theorem [1] we determine the role of the scaling dynamics in the system under investigation. \\[4pt] [1] Aquino G., Bologna M., Grigolioni P., West B.J., PRL 105, 040601 (2010)   

Coupled Oscillations in a 1D Emulsion of Belousov-Zhabotinsky Droplets

We experimentally and computationally study the dynamics of interacting oscillating Belousov-Zhabotinsky (BZ) droplets of $\sim $120 $\mu $m diameter separated by perfluorinated oil and arranged in a one-dimensional array (1D). The coupling between BZ droplets is dominated by inhibition and is strongest at low concentrations of malonic acid (MA) and small droplet separations. A microfluidic chip is used for mixing the BZ reactants, forming monodisperse droplets by flow-focusing and directing them into a hydrophobized 100 $\mu $m diameter capillary. For samples composed of many drops and in the absence of well defined initial conditions, the anti-phase attractor, in which adjacent droplets oscillate 180\r{ } out of phase, is observed for strong coupling. When the coupling strength is reduced the initial transients in the phase difference between neighboring droplets persist until the BZ reactants are exhausted. In order to make quantitative comparison with theory, we use photosensitive Ru(bipy)$_{3}^{2+}$-catalyzed BZ droplets and set both boundary and initial conditions of arrays of small numbers of oscillating BZ droplets with a programmable illumination source. In these small collections of droplets, transient patterns decay rapidly and we observe several more complex attractors, including ones in which some adjacent droplets are in-phase.   

Quarantine generated phase transition in epidemic spreading

We study the critical effect of quarantine on the propagation of epidemics on an adaptive network of social contacts. For this purpose, we analyze the susceptible-infected-recovered (SIR) model in the presence of quarantine, where susceptible individuals protect themselves by disconnecting their links to infected neighbors with probability w, and reconnecting them to other susceptible individuals chosen at random. Starting from a single infected individual, we show by an analytical approach and simulations that there is a phase transition at a critical rewiring (quarantine) threshold $w_c$ separating a phase ($w= w_c$) where the disease does not spread out. We find that in our model the topology of the network strongly affects the size of the propagation, and that $w_c$ increases with the mean degree and heterogeneity of the network. We also find that $w_c$ is reduced if we perform a preferential rewiring, in which the rewiring probability is proportional to the degree of infected nodes.