Spanning trees of recursive scale-free graphs.

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.

Stochastic and mixed flower graphs.

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.

Slow normal modes of proteins are accurately reproduced across different platforms.

The Protein data bank (PDB) (Berman et al 2000 Nucl. Acids Res. 28 235-42) contains the atomic structures of over 10 biomolecules with better than 2.

Ordering statistics of four random walkers on a line.

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}.

PDB-NMA of a protein homodimer reproduces distinct experimental motility asymmetry.

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.

Universality of vibrational spectra of globular proteins.

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.

Atomic torsional modal analysis for high-resolution proteins.

We introduce a formulation for normal mode analyses of globular proteins that significantly improves on an earlier one-parameter formulation [M. M. Tirion, Phys.

Growing networks with superjoiners.

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.

Random walk with priorities in communicationlike networks.

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.

Retention capacity of random surfaces.

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.

Entropy production in nonequilibrium steady states: a different approach and an exactly solvable canonical model.

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.

Exact solution of the nonconsensus opinion model on the line.

The nonconsensus opinion model (NCO) introduced recently by Shao et al. [Phys. Rev.

Realm of validity of the Crooks relation.

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.

Greedy connectivity of geographically embedded graphs.

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.

Asymptotic behavior of the Kleinberg model.

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.

Partition of networks into basins of attraction.

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.

Sequence nets.

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.

Priority diffusion model in lattices and complex networks.

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.

An artificial neural network model of orienting attention toward threatening somatosensory stimuli.

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.

Percolation in hierarchical scale-free nets.

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.

Kleinberg navigation in fractal small-world networks.

We study the Kleinberg problem of navigation in small-world networks when the underlying lattice is a fractal consisting of N>>1 nodes. Our extensive numerical simulations confirm the prediction that the most efficient navigation is attained when the length r of long-range links is taken from the distribution P(r) approximately r(-alpha), where alpha=d(f) is the fractal dimension of the underlying lattice. We find finite-size corrections to the exponent alpha, proportional to 1/(ln N)2.

Synchronous and asynchronous recursive random scale-free nets.

We investigate the differences between scale-free recursive nets constructed by a synchronous, deterministic updating rule (e.g., Apollonian nets), versus an asynchronous, random sequential updating rule (e.

Velocity distribution in a viscous granular gas.

We investigate the velocity relaxation of a viscous one-dimensional granular gas in which neither energy nor momentum is conserved in a collision. Of interest is the distribution of velocities in the gas as it cools, and the time dependence of the relaxation behavior. A Boltzmann equation of instantaneous binary collisions leads to a two-peaked distribution, as do numerical simulations of grains on a line.

Designer nets from local strategies.

We propose a local strategy for constructing scale-free networks of arbitrary degree distributions, based on the redirection method of Krapivsky and Redner [Phys. Rev. E 63, 066123 (2001)].

Efficient immunization strategies for computer networks and populations.

We present an effective immunization strategy for computer networks and populations with broad and, in particular, scale-free degree distributions. The proposed strategy, acquaintance immunization, calls for the immunization of random acquaintances of random nodes (individuals). The strategy requires no knowledge of the node degrees or any other global knowledge, as do targeted immunization strategies.