Analysis of the impact degree distribution in metabolic networks using branching process approximation

Theoretical frameworks to estimate the tolerance of metabolic networks to various failures are important to evaluate the robustness of biological complex systems in systems biology. In this paper, we focus on a measure for robustness in metabolic networks, namely, the impact degree, and propose an approximation method to predict the probability distribution of impact degrees from metabolic network structures using the theory of branching process. We demonstrate the relevance of this method by testing it on real-world metabolic networks. Although the approximation method possesses a few limitations, it may be a powerful tool for evaluating metabolic robustness.     

Phase transitions for information diffusion in random clustered networks

We study the conditions for the phase transitions of information diffusion in complexnetworks. Using the random clustered network model, a generalisation of the Chung-Lurandom network model incorporating clustering, we examine the effect of clustering underthe Susceptible-Infected-Recovered (SIR) epidemic diffusion model with heterogeneouscontact rates. For this purpose, we exploit the branching process to analyse informationdiffusion in random unclustered networks with arbitrary contact rates, and provide noveliterative algorithms for estimating the conditions and sizes of global cascades,respectively. Showing that a random clustered network can be mapped into a factor graph,which is a locally tree-like structure, we successfully extend our analysis to randomclustered networks with heterogeneous contact rates. We then identify the conditions forphase transitions of information diffusion using our method. Interestingly, for variouscontact rates, we prove that random clustered networks with higher clustering coefficientshave strictly lower phase transition points for any given degree sequence. Finally, weconfirm our analytical results with numerical simulations of both synthetically-generatedand real-world networks.     

Growth and shape of branched polymers,aggregates and percolating clusters

Fractals are described as branching chains, growing stochastically in “time” t. This relates critical branching to conductivity and leads to a generalized Flory approximation describing radius and mass versus t, for branched polymers, percolating clusters and random aggregates.     

Decoupling potentials for the three-body final state Schrödinger equation of knockout reactions

In the conventional distorted wave impulse approximation (DWIA) approach the three-body final state of a knockout reaction is decoupled by assuming a plane wave form for the coupling term. The influence of this decoupling approximation on the analyses of cluster knockout reactions has been investigated for a test case where the exact solution is obtainable. A proper treatment of the coupling term causes large oscillations in the effective distorting optical potentials for the decoupled Schrödinger equation. These decoupling potentials depend strongly not only on the partial wave angular momentum,l but also on their azimuthal projection,m.     

Electron scattering on two-neutron halo nuclei: The case of 6He

The formalism to describe electron scattering reactions on two-neutron halo nuclei is developed. The halo nucleus is described as a three-body system (core +n + n), and the wave function is obtained by solving the Faddeev equations in coordinate space. We discuss elastic and quasielastic scattering using the impulse approximation to describe the reaction mechanism. We apply the method to investigate the case of electron scattering on ⁶He. Spectral functions, response functions, and differential cross sections are calculated for both neutron knockout and α knockout by the electron.     

Stochastic Approximation to Understand Simple Simulation Models

This paper illustrates how a deterministic approximation of a stochastic process can be usefully applied to analyse the dynamics of many simple simulation models. To demonstrate the type of results that can be obtained using this approximation, we present two illustrative examples which are meant to serve as methodological references for researchers exploring this area. Finally, we prove some convergence results for simulations of a family of evolutionary games, namely, intra-population imitation models in n-player games with arbitrary payoffs.     

Exact multiple scattering of waves from random rough surfaces: Speckle contrast

An exact formula for the mean multiple scattered intensity is presented by using normal statistics, as well as a gaussian correlation function, for the random rough surface in the limit of large correlation distance, T → ∞. Comparisons with the Kirchoff-Beckman approximation are also done. Further, we obtain an exact expression for the speckle contrast which explains experimental data and draws conclusions on the physical properties of the surface.     

Statistical physics in QCD evolution towards high energies

The concepts and methods used for the study of disordered systems have proven useful in the analysis of the evolution equations of quantum chromodynamics in the high-energy regime: Indeed, parton branching in the semi-classical approximation relevant at high energies and at a fixed impact parameter is a peculiar branching-diffusion process, and parton branching supplemented by saturation effects(such as gluon recombination) is a reaction-diffusion process. In this review article, we first introduce the basic concepts in the context of simple toy models, we study the properties of the latter, and show how the results obtained for the simple models may be taken over to quantum chromodynamics.     

Thermalization and temperature reached in heavy-ion collision using the isospin-dependent quantum molecular dynamics model

One-neutron knockout reactions in a ⁹Be target have been investigated at relativistic energies, near 700 MeV/u, for a set of sd-shell, neutron-rich nuclei. The experiment was performed in the FRS spectrometer, at GSI. ??-ray measurements were carried out by means of the MINIBALL ??-ray spectrometer and allowed the determination of partial cross-sections and branching ratios corresponding to the final states of the emerging knockout fragments. Experimental results are presented for ¹⁷C, ¹⁹N, ²¹O and ²⁵F projectiles. The role of excited states of the N ? 1 fragments in the composition of the ground state of these neutron-rich projectiles is outlined in this work.     

On the construction of complex networks with optimal Tsallis entropy

In this work, we first formulate the Tsallis entropy in the context of complex networks. We then propose a network construction whose topology maximizes the Tsallis entropy. The growing network model has two main ingredients: copy process and random attachment mechanism (C-R model). We show that the resulting degree distribution exactly agrees with the required degree distribution that maximizes the Tsallis entropy. We also provide another example of network model using a combination of preferential and random attachment mechanisms (P-R model) and compare it with the distribution of the Tsallis entropy. In this case, we show that by adequately identifying the exponent factor q, the degree distribution can also be written in the q-exponential form. Taken together, our findings suggest that both mechanisms, copy process and preferential attachment, play a key role for the realization of networks with maximum Tsallis entropy. Finally, we discuss the interpretation of q parameter of the Tsallis entropy in the context of complex networks.     

Computational complexity of impact size estimation for spreading processes on networks

Spreading processes on networks are often analyzed to understand how the outcome of the process (e.g. the number of affected nodes) depends on structural properties of the underlying network. Most available results are ensemble averages over certain interesting graph classes such as random graphs or graphs with a particular degree distributions. In this paper, we focus instead on determining the expected spreading size and the probability of large spreadings for a single (but arbitrary) given network and study the computational complexity of these problems using reductions from well-known network reliability problems. We show that computing both quantities exactly is intractable, but that the expected spreading size can be efficiently approximated with Monte Carlo sampling. When nodes are weighted to reflect their importance, the problem becomes as hard as the s-t reliability problem, which is not known to yield an efficient randomized approximation scheme up to now. Finally, we give a formal complexity-theoretic argument why there is most likely no randomized constant-factor approximation for the probability of large spreadings, even for the unweighted case. A hybrid Monte Carlo sampling algorithm is proposed that resorts to specialized s-t reliability algorithms for accurately estimating the infection probability of those nodes that are rarely affected by the spreading process.     

The Dynamics of Power laws: Fitness and Aging in Preferential Attachment Trees

Continuous-time branching processes describe the evolution of a population whose individuals generate a random number of children according to a birth process. Such branching processes can be used to understand preferential attachment models in which the birth rates are linear functions. We are motivated by citation networks, where power-law citation counts are observed as well as aging in the citation patterns. To model this, we introduce fitness and age-dependence in these birth processes. The multiplicative fitness moderates the rate at which children are born, while the aging is integrable, so that individuals receives a finite number of children in their lifetime. We show the existence of a limiting degree distribution for such processes. In the preferential attachment case, where fitness and aging are absent, this limiting degree distribution is known to have power-law tails. We show that the limiting degree distribution has exponential tails for bounded fitnesses in the presence of integrable aging, while the power-law tail is restored when integrable aging is combined with fitness with unbounded support with at most exponential tails. In the absence of integrable aging, such processes are explosive.     

Scaling Limits and Generic Bounds for Exploration Processes

We consider exploration algorithms of the random sequential adsorption type both for homogeneous random graphs and random geometric graphs based on spatial Poisson processes. At each step, a vertex of the graph becomes active and its neighboring nodes become blocked. Given an initial number of vertices N growing to infinity, we study statistical properties of the proportion of explored (active or blocked) nodes in time using scaling limits. We obtain exact limits for homogeneous graphs and prove an explicit central limit theorem for the final proportion of active nodes, known as the jamming constant, through a diffusion approximation for the exploration process which can be described as a unidimensional process. We then focus on bounding the trajectories of such exploration processes on random geometric graphs, i.e., random sequential adsorption. As opposed to exploration processes on homogeneous random graphs, these do not allow for such a dimensional reduction. Instead we derive a fundamental relationship between the number of explored nodes and the discovered volume in the spatial process, and we obtain generic bounds for the fluid limit and jamming constant: bounds that are independent of the dimension of space and the detailed shape of the volume associated to the discovered node. Lastly, using coupling techinques, we give trajectorial interpretations of the generic bounds.     

Binding corrections for reactions on the deuteron in the 3-3 resonance region

Binding corrections evaluated to all orders in a phenomenological nucleon-nucleon potential are compared with the leading term and with a double scattering correction in the Watson expansion for breakup and for coherent reactions on the deuteron at intermediate pion or photon energies. As principal result, we establish that for charged pion photoproduction, and for certain pion induced “spectator” reactions, the binding correction remains sufficiently small that truncation of the Watson expansion at terms of second order in the projectile-nucleon amplitudes is justifiable, but sufficiently large that it cannot be neglected in comparison with multiple scattering corrections. For the reactions π⁺d → π⁺p + spectator neutron and π^?d → π^?n + spectator proton, our theory predicts deviations from the simplest impulse approximation which become very large in well-defined kinematical regions, and should be easily accessible to experimental investigation.     

Distinct Clusterings and Characteristic Path Lengths in Dynamic Small-World Networks with Identical Limit Degree Distribution

Many real-world networks belong to a particular class of structures, known as small-world networks, that display short distance between pair of nodes. In this paper, we introduce a simple family of growing small-world networks where both addition and deletion of edges are possible. By tuning the deletion probability q _t , the model undergoes a transition from large worlds to small worlds. By making use of analytical or numerical means we determine the degree distribution, clustering coefficient and average path length of our networks. Surprisingly, we find that two similar evolving mechanisms, which provide identical degree distribution under a reciprocal scaling as t goes to infinity, can lead to quite different clustering behaviors and characteristic path lengths. It is also worth noting that Farey graphs constitute the extreme case q _t ??0 of our random construction.     

Modeling Final-State Interactions with a Relativistic Multiple-Scattering Approximation

We address the issue of nuclear attenuation in nucleon and pion knockout reactions. A selection of results from a model based on a relativistic multiple-scattering approximation is presented. We show transparency calculations for pion electroproduction on several nuclei, where data are in very good agreement with calculations including color transparency. Secondly, we discuss the density dependence of reactions involving one or double proton knockout. The latter reaction succeeds in probing the high density regions in the deep interior of the nucleus.     

Cluster knockout reactions

Cluster knockout reactions are expected to reveal the amount of clustering (such as that of α, d and even of heavier clusters such as¹²C, ¹⁶O etc.) in the target nucleus. In simple terms, incident medium high-energy nuclear projectile interacts strongly with the cluster (present in the target nucleus) as if it were existing as a free entity. Theoretically, the relatively softer interactions of the two outgoing particles with the residual nucleus lead to optical distortions and are treated in terms of distorted wave (DW) formalism. The long-range projectile–cluster interaction is accounted for, in terms of the finite range (FR) direct reaction formalism, as against the more commonly adopted zero-range (ZR) distorted wave impulse approximation (DWIA) formalism. Comparison of the DWIA calculations with the observed data provide information about the momentum distribution and the clustering spectroscopic factor of the target nucleus. Interesting results and some recent advancements in the area of (α, 2 α) reactions and heavy cluster knockout reactions are discussed. Importance of the finite-range vertex and the final-state interactions are brought out.     

Systematic approximations in disordered systems

We discuss two systematic approximation schemes for disordered systems (random lattices) by extending the single site CPA. In the first scheme all self energy diagrams up to those includingν scattering centers are summed up. Thisν-center approximation has been discussed before and is here derived using a straightforward projection operator formalism. The second scheme is a “true” cluster-CPA in which the scattering from an arbitrarily large cluster of neighbouring atoms (ν-cluster) embedded in a random crystal is treated just in the same way as in the single site CPA. In contrast to other cluster formulations our theory preserves translational invariance and determines the self energy in a natural way. The two schemes are compared with each other concerning their practical applicability.     

Survival,Extinction and Approximation of Discrete-time Branching Random Walks

We consider a general discrete-time branching random walk on a countable set X. We relate local, strong local and global survival with suitable inequalities involving the first-moment matrix M of the process. In particular we prove that, while the local behavior is characterized by M, the global behavior cannot be completely described in terms of properties involving M alone. Moreover we show that locally surviving branching random walks can be approximated by sequences of spatially confined and stochastically dominated branching random walks which eventually survive locally if the (possibly finite) state space is large enough. An analogous result can be achieved by approximating a branching random walk by a sequence of multitype contact processes and allowing a sufficiently large number of particles per site. We compare these results with the ones obtained in the continuous-time case and we give some examples and counterexamples.