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We present a link-by-link rule-based method for constructing all members of the ensemble of spanning trees for any recursively generated, finitely articulated graph, such as the Dorogovtsev-Goltsev-Mendes (DGM) net. The recursions allow for many large-scale properties of the ensemble of spanning trees to be analytically solved exactly. We show how a judicious application of the prescribed growth rules selects for certain subsets of the spanning trees with particular desired properties (small world, extended diameter, degree distribution, etc.), and thus approximates and/or provides solutions to several optimization problems on undirected and unweighted networks. The analysis of spanning trees enhances the usefulness of recursive graphs as sophisticated models for everyday life complex networks.

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Stochasticity is introduced to a well studied class of recursively grown graphs: (u,v)-flower nets, which have power-law degree distributions as well as small-world properties (when u=1). The stochastic variant interpolates between different (deterministic) flower graphs thus adding flexibility to the model. The random multiplicative growth process involved, however, leads to a spread ensemble of networks with finite variance for the number of links, nodes, and loops. Nevertheless, the degree exponent and loopiness exponent attain unique values in the thermodynamic limit of infinitely large graphs. We also study a class of mixed flower networks, closely related to the stochastic flowers, but which are grown recursively in a deterministic way. The deterministic growth of mixed flower-nets eliminates ensemble spreads, and their recursive growth allows for exact analysis of their (uniquely defined) mixed properties.

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The Protein data bank (PDB) (Berman et al 2000 Nucl. Acids Res. 28 235-42) contains the atomic structures of over 105 biomolecules with better than 2.8 Å resolution. The listing of the identities and coordinates of the atoms comprising each macromolecule permits an analysis of the slow-time vibrational response of these large systems to minor perturbations. 3D video animations of individual modes of oscillation demonstrate how regions interdigitate to create cohesive collective motions, providing a comprehensive framework for and familiarity with the overall 3D architecture. Furthermore, the isolation and representation of the softest, slowest deformation coordinates provide opportunities for the development of mechanical models of enzyme function. The eigenvector decomposition, therefore, must be accurate, reliable as well as rapid to be generally reported upon. We obtain the eigenmodes of a 1.2 Å 34 kDa PDB entry using either exclusively heavy atoms or partly or fully reduced atomic sets; Cartesian or internal coordinates; interatomic force fields derived either from a full Cartesian potential, a reduced atomic potential or a Gaussian distance-dependent potential; and independently developed software. These varied technologies are similar in that each maintains proper stereochemistry either by use of dihedral degrees of freedom which freezes bond lengths and bond angles, or by use of a full atomic potential that includes realistic bond length and angle restraints. We find that the shapes of the slowest eigenvectors are nearly identical, not merely similar.

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We study the ordering statistics of four random walkers on the line, obtaining a much improved estimate for the long-time decay exponent of the probability that a particle leads to time t, P_{lead}(t)∼t^{-0.91287850}, and that a particle lags to time t (never assumes the lead), P_{lag}(t)∼t^{-0.30763604}. Exponents of several other ordering statistics for N=4 walkers are obtained to eight-digit accuracy as well. The subtle correlations between n walkers that lag jointly, out of a field of N, are discussed: for N=3 there are no correlations and P_{lead}(t)∼P_{lag}(t)^{2}. In contrast, our results rule out the possibility that P_{lead}(t)∼P_{lag}(t)^{3} for N=4, although the correlations in this borderline case are tiny.

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We have extended our analytically derived PDB-NMA formulation, Atomic Torsional Modal Analysis or ATMAN (Tirion and ben-Avraham 2015 Phys. Rev. E 91 032712), to include protein dimers using mixed internal and Cartesian coordinates. A test case on a 1.3 [Formula: see text] resolution model of a small homodimer, ActVA-ORF6, consisting of two 112-residue subunits identically folded in a compact 50 [Formula: see text] sphere, reproduces the distinct experimental Debye-Waller motility asymmetry for the two chains, demonstrating that structure sensitively selects vibrational signatures. The vibrational analysis of this PDB entry, together with biochemical and crystallographic data, demonstrates the cooperative nature of the dimeric interaction of the two subunits and suggests a mechanical model for subunit interconversion during the catalytic cycle.

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It is shown that the density of modes of the vibrational spectrum of globular proteins is universal, i.e. regardless of the protein in question, it closely follows one universal curve. The present study, including 135 proteins analyzed with a full atomic empirical potential (CHARMM22) and using the full complement of all atoms Cartesian degrees of freedom, goes far beyond previous claims of universality, confirming that universality holds even in the frequency range that is well above 100 cm(-1) (300-4000 cm(-1)), where peaks and turns in the density of states are faithfully reproduced from one protein to the next. We also characterize fluctuations of the spectral density from the average, paving the way to a meaningful discussion of rare, unusual spectra and the structural reasons for the deviations in such 'outlier' proteins. Since the method used for the derivation of the vibrational modes (potential energy formulation, set of degrees of freedom employed, etc) has a dramatic effect on the spectral density, another significant implication of our findings is that the universality can provide an exquisite tool for assessing and improving the quality of potential functions and the quality of various models used for NMA computations. Finally, we show that the input configuration also affects the density of modes, thus emphasizing the importance of simplified potential energy formulations that are minimized at the outset. In summary, our findings call for a serious two-way dialogue between theory and experiment: experimental spectra of proteins could now guide the fine tuning of theoretical empirical potentials, and the various features and peaks observed in theoretical studies--being universal, and hence now rising in importance--would hopefully spur experimental confirmation.

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We introduce a formulation for normal mode analyses of globular proteins that significantly improves on an earlier one-parameter formulation [M. M. Tirion, Phys. Rev. Lett. 77, 1905 (1996)] that characterized the slow modes associated with protein data bank structures. Here we develop that empirical potential function that is minimized at the outset to include two features essential to reproduce the eigenspectra and associated density of states in the 0 to 300cm-1 frequency range, not merely the slow modes. First, introduction of preferred dihedral-angle configurations via use of torsional stiffness constants eliminates anomalous dispersion characteristics due to insufficiently bound surface side chains and helps fix the spectrum thin tail frequencies (100-300cm-1). Second, we take into account the atomic identities and the distance of separation of all pairwise interactions, improving the spectrum distribution in the 20 to 300cm-1 range. With these modifications, not only does the spectrum reproduce that of full atomic potentials, but we obtain stable reliable eigenmodes for the slow modes and over a wide range of frequencies.

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We study the Krapivsky-Redner (KR) network growth model, but where new nodes can connect to any number of existing nodes, m, picked from a power-law distribution p(m)∼m^{-α}. Each of the m new connections is still carried out as in the KR model with probability redirection r (corresponding to degree exponent γ_{KR}=1+1/r in the original KR model). The possibility to connect to any number of nodes resembles a more realistic type of growth in several settings, such as social networks, routers networks, and networks of citations. Here we focus on the in-, out-, and total-degree distributions and on the potential tension between the degree exponent α, characterizing new connections (outgoing links), and the degree exponent γ_{KR}(r) dictated by the redirection mechanism.

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We study a model for a random walk of two classes of particles (A and B). Where both species are present in the same site, the motion of A's takes precedence over that of B's. The model was originally proposed and analyzed in Maragakis et al. [Phys. Rev. E 77, 020103(R) (2008)]; here we provide additional results. We solve analytically the diffusion coefficients of the two species in lattices for a number of protocols. In networks, we find that the probability of a B particle to be free decreases exponentially with the node degree. In scale-free networks, this leads to localization of the B's at the hubs and arrest of their motion. To remedy this, we investigate several strategies to avoid trapping of the B's, including moving an A instead of the hindered B, allowing a trapped B to hop with a small probability, biased walk toward non-hub nodes, and limiting the capacity of nodes. We obtain analytic results for lattices and networks, and we discuss the advantages and shortcomings of the possible strategies.

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We introduce a "water retention" model for liquids captured on a random surface with open boundaries and investigate the model for both continuous and discrete surface heights 0,1,…,n-1 on a square lattice with a square boundary. The model is found to have several intriguing features, including a nonmonotonic dependence of the retention on the number of levels: for many n, the retention is counterintuitively greater than that of an (n+1)-level system. The behavior is explained using percolation theory, by mapping it to a 2-level system with variable probability. Results in one dimension are also found.

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We discuss entropy production in nonequilibrium steady states by focusing on paths obtained by sampling at regular (small) intervals, instead of sampling on each change of the system's state. This allows us to directly study entropy production in systems with microscopic irreversibility. The two sampling methods are equivalent otherwise, and the fluctuation theorem also holds for the different paths. We focus on a fully irreversible three-state loop, as a canonical model of microscopic irreversibility, finding its entropy distribution, rate of entropy production, and large deviation function in closed analytical form, and showing that the observed kink in the large deviation function arises solely from microscopic irreversibility.

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The nonconsensus opinion model (NCO) introduced recently by Shao et al. [Phys. Rev. Lett. 103, 018701 (2009)] is solved exactly on the line. Although, as expected, the model exhibits no phase transition in one dimension, its study is interesting because of the possible connection with invasion percolation with trapping. The system evolves exponentially fast to the steady state, rapidly developing long-range correlations: The average cluster size in the steady state scales as the square of the initial cluster size, of the (uncorrelated) initial state. We also discuss briefly the NCO model on Bethe lattices, arguing that its phase transition diagram is different than that of regular percolation.

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We consider the distribution P(Φ) of the Hatano-Sasa entropy, Φ, in reversible and irreversible processes, finding that the Crooks relation for the ratio of the probability density functions of the forward and backward processes, P(F)(Φ)/P(R)(-Φ)=e(Φ), is satisfied not only for reversible, but also for irreversible processes, in general, in the adiabatic limit of "slow processes." Focusing on systems with a finite set of discrete states (and no absorbing states), we observe that two-state systems always fulfill detailed balance, and obey the Crooks relation. We also identify a wide class of systems, with more than two states, that can be "coarse grained" into two-state systems and obey the Crooks relation despite their irreversibility and violation of detailed balance. We verify these results in selected cases numerically.

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We introduce a measure of greedy connectivity for geographical networks (graphs embedded in space) and where the search for connecting paths relies only on local information, such as a node's location and that of its neighbors. Constraints of this type are common in everyday life applications. Greedy connectivity accounts also for imperfect transmission across established links and is larger the higher the proportion of nodes that can be reached from other nodes with a high probability. Greedy connectivity can be used as a criterion for optimal network design.

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We study Kleinberg navigation (the search of a target in a d-dimensional lattice, where each site is connected to one other random site at distance r, with probability approximately r(-alpha) by means of an exact master equation for the process. We show that the asymptotic scaling behavior for the delivery time T to a target at distance L scales as T approximately ln(2)L when alpha=d, and otherwise as T approximately L(x), with x=(d-alpha)/(d+1-alpha) for alphad+1. These values of x exceed the rigorous lower bounds established by Kleinberg. We also address the situation where there is a finite probability for the message to get lost along its way and find short delivery times (conditioned upon arrival) for a wide range of alpha's.

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We study a class of networks generated by sequences of letters taken from a finite alphabet consisting of m letters (corresponding to m types of nodes) and a fixed set of connectivity rules. Recently, it was shown how a binary alphabet might generate threshold nets in a similar fashion [A. Hagberg, Phys. Rev. E 74, 056116 (2006)]. Just like threshold nets, sequence nets in general possess a modular structure reminiscent of everyday-life nets and are easy to handle analytically (i.e., calculate degree distribution, shortest paths, betweenness centrality, etc.). Exploiting symmetry, we make a full classification of two- and three-letter sequence nets, discovering two classes of two-letter sequence nets. These sequence nets retain many of the desirable analytical properties of threshold nets while yielding richer possibilities for the modeling of everyday-life complex networks more faithfully.

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We introduce a model for diffusion of two classes of particles (A and B ) with priority: where both species are present in the same site the motion of A's takes precedence over that of B's. This describes realistic situations in wireless and communication networks. In regular lattices the diffusion of the two species is normal, but the B particles are significantly slower due to the presence of the A particles. From the fraction of sites where the B particles can move freely, which we compute analytically, we derive the diffusion coefficients of the two species. In heterogeneous networks the fraction of sites where B's are free decreases exponentially with the degree of the sites. This, coupled with accumulation of particles in high-degree nodes, leads to trapping of the low priority particles in scale-free networks.

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An artificial neural network model was designed to test the threat detection hypothesis developed in our experimental studies, where threat detector activity in the somatosensory association areas is monitored by the medial prefrontal cortex, which signals the lateral prefrontal cortex to redirect attention to the threat. As in our experimental studies, simulated threat-evoked activations of all three brain areas were larger when the somatosensory target stimulus was unattended than attended, and the increase in behavioral reaction times when the target stimulus was unattended was smaller for threatening than nonthreatening stimuli. The model also generated a number of novel predictions, for example, the effect of threat on reaction time only occurs when the target stimulus is unattended, and the P3a indexes prefrontal cortex activity involved in redirecting attention toward response processes on that trial and sensory processes on subsequent trials.

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We study partition of networks into basins of attraction based on a steepest ascent search for the node of highest degree. Each node is associated with, or "attracted" to its neighbor of maximal degree, as long as the degree is increasing. A node that has no neighbors of higher degree is a peak, attracting all the nodes in its basin. Maximally random scale-free networks exhibit different behavior based on their degree distribution exponent gamma : For small gamma (broad distribution) networks are dominated by a giant basin, whereas for large gamma (narrow distribution) there are numerous basins, with peaks attracting mainly their nearest neighbors. We derive expressions for the first two moments of the number of basins. We also obtain the complete distribution of basin sizes for a class of hierarchical deterministic scale-free networks that resemble random nets. Finally, we generalize the problem to regular networks and lattices where all degrees are equal, and thus the attractiveness of a node must be determined by an assigned weight, rather than the degree. We derive the complete distribution of basins of attraction resulting from randomly assigned weights in one-dimensional chains.

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We study the percolation phase transition in hierarchical scale-free nets. Depending on the method of construction, the nets can be fractal or small world (the diameter grows either algebraically or logarithmically with the net size), assortative or disassortative (a measure of the tendency of like-degree nodes to be connected to one another), or possess various degrees of clustering. The percolation phase transition can be analyzed exactly in all these cases, due to the self-similar structure of the hierarchical nets. We find different types of criticality, illustrating the crucial effect of other structural properties aside from the scale-free degree distribution of the nets.