Paul L. Krapivsky

A Statistical Physics Perspective on Web Growth

Abstract Approaches from statistical physics are applied to investigate the structure of network models whose growth rules mimic aspects of the evolution of the World Wide Web. We first determine the degree distribution of a growing network in which nodes are introduced one at a time and attach to an earlier node of degree k with rate Ak∼kγ. Very different behaviors arise for γ 1. We also analyze the degree distribution of a heterogeneous network, the joint age-degree distribution, the correlation between degrees of neighboring nodes, as well as global network properties. An extension to directed networks is then presented. By tuning model parameters to reasonable values, we obtain distinct power-law forms for the in-degree and out-degree distributions with exponents that are in good agreement with current data for the web. Finally, a general growth process with independent introduction of nodes and links is investigated. This leads to independently growing sub-networks that may coalesce with other sub-networks. General results for both the size distribution of sub-networks and the degree distribution are obtained.

Transitional aggregation kinetics in dry and damp environments

We investigate the kinetics of constant-kernel aggregation which is augmented by ~a! evaporation of monomers from clusters, which is termed aggregation in a ‘‘dry’’ environment, and ~b! continuous cluster growth or condensation, termed aggregation in a ‘‘damp’’ environment. The rate equations for these two processes are analyzed using both exact and asymptotic methods. In dry aggregation, mass conserving evaporation is treated, in which the monomers which evaporate remain in the system and continue to be reactive. For this reaction process, the competition between evaporation and aggregation leads to several asymptotic outcomes. When the evaporation is weak, the kinetics is similar to that of aggregation with no evaporation, while a steady state is quickly reached in the opposite case. At a critical evaporation rate, a steady state is slowly reached in which the cluster mass distribution decays as k 25/2 , where k is the mass, while the typical cluster mass, or upper cutoff in the mass distribution, grows with time as t 2/3 . For damp aggregation, several cases are considered for the dependence of the cluster growth rate Lk on k. ~i! For Lk independent of k, the mass distribution attains a conventional scaling form, but with the typical cluster mass growing as t ln t. ~ii! When Lk}k, the typical mass grows exponentially in time, while the mass distribution again scales. ~iii! In the intermediate case of Lk}k m , scaling generally applies, with the typical mass growing as t 1/~12m! . The scaling approach is also adapted to treat diffusion-limited damp aggregation for spatial dimension d<2. @S1063-651X~96!10210-5#

Reinforcement-driven spread of innovations and fads

We investigate how social reinforcement drives the spread of permanent innovations and transient fads. We account for social reinforcement by endowing each individual with M + 1 possible awareness states 0, 1, 2,..., M, with state M corresponding to adopting an innovation. An individual with awareness k 1. When individuals can abandon the innovation at rate λ, the population fraction that remains clueless about the fad undergoes a phase transition at a critical rate λc; this transition is second order for M = 1 and first order for M > 1, with macroscopic fluctuations accompanying the latter. The time for the fad to disappear has an intriguing non-monotonic dependence on λ.

Extreme value statistics and traveling fronts: application to computer science.

We study the statistics of height and balanced height in the binary search tree problem in computer science. The search tree problem is first mapped to a fragmentation problem that is then further mapped to a modified directed polymer problem on a Cayley tree. We employ the techniques of traveling fronts to solve the polymer problem and translate back to derive exact asymptotic properties in the original search tree problem. The second mapping allows us not only to again derive the already known results for random binary trees but to obtain exact results for search trees where the entries arrive according to an arbitrary distribution, not necessarily randomly. Besides it allows us to derive the asymptotic shape of the full probability distribution of height and not just its moments. Our results are then generalized to m-ary search trees with arbitrary distribution.

The Inelastic Maxwell Model

Dynamics of inelastic gases are studied within the framework of random collision processes. The corresponding Boltzmann equation with uniform collision rates is solved analytically for gases, impurities, and mixtures. Generally, the energy dissipation leads to a significant departure from the elastic case. Specifically, the velocity distributions have overpopulated high energy tails and different velocity components are correlated. In the freely cooling case, the velocity distribution develops an algebraic high-energy tail, with an exponent that depends sensitively on the dimension and the degree of dissipation. Moments of the velocity distribution exhibit multiscaling asymptotic behavior, and the autocorrelation function decays algebraically with time. In the forced case, the steady state velocity distribution decays exponentially at large velocities. An impurity immersed in a uniform inelastic gas may or may not mimic the behavior of the background, and the departure from the background behavior is characterized by a series of phase transitions.

The power of choice in growing trees

Abstract.The “power of choice” has been shown to radically alter the behavior of a number of randomized algorithms. Here we explore the effects of choice on models of random tree growth. In our models each new node has k randomly chosen contacts, where k > 1 is a constant. It then attaches to whichever one of these contacts is most desirable in some sense, such as its distance from the root or its degree. Even when the new node has just two choices, i.e., when k = 2, the resulting tree can be very different from a random graph or tree. For instance, if the new node attaches to the contact which is closest to the root of the tree, the distribution of depths changes from Poisson to a traveling wave solution. If the new node attaches to the contact with the smallest degree, the degree distribution is closer to uniform than in a random graph, so that with high probability there are no nodes in the tree with degree greater than O(log log N). Finally, if the new node attaches to the contact with the largest degree, we find that the degree distribution is a power law with exponent -1 up to degrees roughly equal to k, with an exponential cutoff beyond that; thus, in this case, we need k ≫ 1 to see a power law over a wide range of degrees.

A model of ballistic aggregation and fragmentation

A simple model of ballistic aggregation and fragmentation is proposed. The model is characterized by two energy thresholds, Eagg and Efrag, which demarcate different types of impacts: if the kinetic energy of the relative motion of a colliding pair is smaller than Eagg or larger than Efrag, particles respectively merge or break; otherwise they rebound. We assume that particles are formed from monomers which cannot split any further and that in a collision-induced fragmentation the larger particle splits into two fragments. We start from the Boltzmann equation for the mass–velocity distribution function and derive Smoluchowski-like equations for concentrations of particles of different mass. We analyze these equations analytically, solve them numerically and perform Monte Carlo simulations. When aggregation and fragmentation energy thresholds do not depend on the masses of the colliding particles, the model becomes analytically tractable. In this case we show the emergence of the two types of behavior: the regime of unlimited cluster growth arises when fragmentation is (relatively) weak and the relaxation towards a steady state occurs when fragmentation prevails. In a model with mass-dependent Eagg and Efrag the evolution with a crossover from one of the regimes to another has been detected.

Random Fibonacci sequences

Solutions to the random Fibonacci recurrence xn + 1 = xn??xn-1 decrease (increase) exponentially, xn~exp (?n), for sufficiently small (large) ?. In the limits ??0 and ???, we expand the Lyapunov exponent ?(?) in powers of ? and ?-1, respectively. For the classical case of ? = 1 we obtain exact non-perturbative results. In particular, an invariant measure associated with Ricatti variable rn = xn + 1/xn is shown to exhibit plateaux around all rational r.

Signatures of Arithmetic Simplicity in Metabolic Network Architecture

Metabolic networks perform some of the most fundamental functions in living cells, including energy transduction and building block biosynthesis. While these are the best characterized networks in living systems, understanding their evolutionary history and complex wiring constitutes one of the most fascinating open questions in biology, intimately related to the enigma of lifes origin itself. Is the evolution of metabolism subject to general principles, beyond the unpredictable accumulation of multiple historical accidents? Here we search for such principles by applying to an artificial chemical universe some of the methodologies developed for the study of genome scale models of cellular metabolism. In particular, we use metabolic flux constraint-based models to exhaustively search for artificial chemistry pathways that can optimally perform an array of elementary metabolic functions. Despite the simplicity of the model employed, we find that the ensuing pathways display a surprisingly rich set of properties, including the existence of autocatalytic cycles and hierarchical modules, the appearance of universally preferable metabolites and reactions, and a logarithmic trend of pathway length as a function of input/output molecule size. Some of these properties can be derived analytically, borrowing methods previously used in cryptography. In addition, by mapping biochemical networks onto a simplified carbon atom reaction backbone, we find that properties similar to those predicted for the artificial chemistry hold also for real metabolic networks. These findings suggest that optimality principles and arithmetic simplicity might lie beneath some aspects of biochemical complexity.

Scale-free networks as preasymptotic regimes of superlinear preferential attachment.

We study the following paradox associated with networks growing according to superlinear preferential attachment: superlinear preference cannot produce scale-free networks in the thermodynamic limit, but there are superlinearly growing network models that perfectly match the structure of some real scale-free networks, such as the Internet. We obtain an analytic solution, supported by extensive simulations, for the degree distribution in superlinearly growing networks with arbitrary average degree, and confirm that in the true thermodynamic limit these networks are indeed degenerate, i.e., almost all nodes have low degrees. We then show that superlinear growth has vast preasymptotic regimes whose depths depend both on the average degree in the network and on how superlinear the preference kernel is. We demonstrate that a superlinearly growing network model can reproduce, in its preasymptotic regime, the structure of a real network, if the model captures some sufficiently strong structural constraints-rich-club connectivity, for example. These findings suggest that real scale-free networks of finite size may exist in pre-asymptotic regimes of network evolution processes that lead to degenerate network formations in the thermodynamic limit.