Daniel ben-Avraham

Resilience of the internet to random breakdowns

A common property of many large networks, including the Internet, is that the connectivity of the various nodes follows a scale-free power-law distribution, P(k) = ck(-alpha). We study the stability of such networks with respect to crashes, such as random removal of sites. Our approach, based on percolation theory, leads to a general condition for the critical fraction of nodes, p(c), that needs to be removed before the network disintegrates. We show analytically and numerically that for alpha</=3 the transition never takes place, unless the network is finite. In the special case of the physical structure of the Internet (alpha approximately 2.5), we find that it is impressively robust, with p(c)>0.99.

Breakdown of the Internet under Intentional Attack

We study the tolerance of random networks to intentional attack, whereby a fraction p of the most connected sites is removed. We focus on scale-free networks, having connectivity distribution P(k) approximately k(-alpha), and use percolation theory to study analytically and numerically the critical fraction p(c) needed for the disintegration of the network, as well as the size of the largest connected cluster. We find that even networks with alpha < or = 3, known to be resilient to random removal of sites, are sensitive to intentional attack. We also argue that, near criticality, the average distance between sites in the spanning (largest) cluster scales with its mass, M, as square root of [M], rather than as log (k)M, as expected for random networks away from criticality.

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. We study analytically the critical threshold for complete immunization. We also study the strategy with respect to the susceptible-infected-removed epidemiological model. We show that the immunization threshold is dramatically reduced with the suggested strategy, for all studied cases.

Normal modes as refinement parameters for the f−actin model

The slow normal modes of G-actin were used as structural parameters to refine the F-actin model against 8-A resolution x-ray fiber diffraction data. The slowest frequency normal modes of G-actin pertain to collective rearrangements of domains, motions that are characterized by correlation lengths on the order of the resolution of the fiber diffraction data. Using a small number of normal mode degrees of freedom (< or = 12) improved the fit to the data significantly. The refined model of F-actin shows that the nucleotide binding cleft has narrowed and that the DNase I binding loop has twisted to a lower radius, consistent with other refinement techniques and electron microscopy data. The methodology of a normal mode refinement is described, and the results, as applied to actin, are detailed.

Percolation critical exponents in scale-free networks

We study the behavior of scale-free networks, having connectivity distribution P(k) approximately k(-lambda), close to the percolation threshold. We show that for networks with 3<lambda<4, known to undergo a transition at a finite threshold of dilution, the critical exponents are different than the expected mean-field values of regular percolation in infinite dimensions. Networks with 2<lambda<3 possess only a percolative phase. Nevertheless, we show that in this case percolation critical exponents are well defined, near the limit of extreme dilution (where all sites are removed), and that also then the exponents bear a strong lambda dependence. The regular mean-field values are recovered only for lambda>4.

Scale-free networks on lattices.

We suggest a method for embedding scale-free networks, with degree distribution Pk approximately k(-lambda), in regular Euclidean lattices accounting for geographical properties. The embedding is driven by a natural constraint of minimization of the total length of the links in the system. We find that all networks with lambda>2 can be successfully embedded up to a (Euclidean) distance xi which can be made as large as desired upon the changing of an external parameter. Clusters of successive chemical shells are found to be compact (the fractal dimension is df=d), while the dimension of the shortest path between any two sites is smaller than 1: dmin=(lambda-2)/(lambda-1-1/d), contrary to all other known examples of fractals and disordered lattices.

Statics and dynamics of a diffusion-limited reaction: Anomalous kinetics, nonequilibrium self-ordering, and a dynamic transition

We solve exactly the one-dimensional diffusion-limited single-species coagulation process (A+A→A) with back reactions (A→A+A) and/or a steady input of particles (B→A). The exact solution yields not only the steady-state concentration of particles, but also the exact time-dependent concentration as well as the time-dependent probability distribution for the distance between neighboring particles, i. e., the interparticle distribution function (IPDF). The concentration for this diffusion-limited reaction process does not obey the classical “mean-field” rate equation. Rather, the kinetics is described by a finite set of ordinary differential equations only in particular cases, with no such description holding in general. The reaction kinetics is linked to the spatial distribution of particles as reflected in the IPDFs. The spatial distribution of particles is totally random, i. e., the maximum entropy distribution, only in the steady state of the strictly reversible process A+A↔A, a true equilibrium state with detailed balance. Away from this equilibrium state the particles display a static or dynamic self-organization imposed by the nonequilibrium reactions. The strictly reversible process also exhibits a sharp transition in its relaxation dynamics when switching between equilibria of different values of the system parameters. When the system parameters are suddenly changed so that the new equilibrium concentration is greater than exactly twice the old equilibrium concentration, the exponential relaxation time depends on the initial concentration.

Fractal and transfractal recursive scale-free nets

We explore the concepts of self-similarity, dimensionality, and (multi)scaling in a new family of recursive scale-free nets that yield themselves to exact analysis through renormalization techniques. All nets in this family are self-similar and some are fractals—possessing a finite fractal dimension—while others are small-world (their diameter grows logarithmically with their size) and are infinite-dimensional. We show how a useful measure of transfinite dimension may be defined and applied to the small-world nets. Concerning multiscaling, we show how first-passage time for diffusion and resistance between hubs (the most connected nodes) scale differently than for other nodes. Despite the different scalings, the Einstein relation between diffusion and conductivity holds separately for hubs and nodes. The transfinite exponents of small-world nets obey Einstein relations analogous to those in fractal nets.

What is special about diffusion on scale-free nets?

We study diffusion (random walks) on recursive scale-free graphs and contrast the results to similar studies in other analytically soluble media. This allows us to identify ways in which diffusion in scale-free graphs is special. Most notably, scale-free architecture results in a faster transit time between existing nodes when the network grows in size; and walks emanating from the most connected nodes are recurrent, despite the networks infinite dimension. We also find that other attributes of the graph, besides its scale-free distribution, have a strong influence on the nature of diffusion.

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. We find different types of criticality, illustrating the crucial effect of other structural properties aside from the scale-free degree distribution of the nets.