models of social networks based on social distance attachment we propose a class of models of social network formation based on a mathematical abstraction of the concept of social distance social distance attachment is represented by the tendency of peers to establish acquaintances via a decreasing function of the relative distance in a representative social space we derive analytical results corroborated by extensive numerical simulations showing that the model reproduces the main statistical characteristics of real social networks large clustering coefficient positive degree correlations and the emergence of a hierarchy of communities the model is confronted with the social network formed by people that shares confidential information using the pretty good privacy pgp encryption algorithm the so called web of trust of pgp

seed size strongly affects cascades on random networks the average avalanche size in the watts model of threshold dynamics on random networks of arbitrary degree distribution is determined analytically existence criteria for global cascades are shown to depend sensitively on the size of the initial seed disturbance the dependence of cascade size upon the mean degree z of the network is known to exhibit several transitions these are typically continuous at low z and discontinuous at high z here it is demonstrated that the low z transition may in fact be discontinuous in certain parameter regimes connections between these results and the zero temperature random field ising model on random graphs are discussed

clustering in complex networks i general formalism we develop a full theoretical approach to clustering in complex networks a key concept is introduced the edge multiplicity that measures the number of triangles passing through an edge this quantity extends the clustering coefficient in that it involves the properties of two and not just one vertices the formalism is completed with the definition of a three vertex correlation function which is the fundamental quantity describing the properties of clustered networks the formalism suggests different metrics that are able to thoroughly characterize transitive relations a rigorous analysis of several real networks which makes use of this formalism and the metrics is also provided it is also found that clustered networks can be classified into two main groups the weak and the strong transitivity classes in the first class edge multiplicity is small with triangles being disjoint in the second class edge multiplicity is high and so triangles share many edges as we shall see in the following paper the class a network belongs to has strong implications in its percolation properties

computation of the inverse laplace transform based on a collocation method which uses only real values we develop a numerical algorithm for inverting a laplace transform lt based on laguerre polynomial series expansion of the inverse function under the assumption that the lt is known on the real axis only the method belongs to the class of collocation methods c methods and is applicable when the lt function is regular at infinity difficulties associated with these problems are due to their intrinsic ill posedness the main contribution of this paper is to provide computable estimates of truncation discretization conditioning and roundoff errors introduced by numerical computations moreover we introduce the pseudoaccuracy which will be used by the numerical algorithm in order to provide uniform scaled accuracy of the computed approximation for any x with respect to e x these estimates are then employed to dynamically truncate the series expansion in other words the number of the terms of the series acts like the regularization parameter which provides the trade off between errors with the aim to validate the reliability and usability of the algorithm experiments were carried out on several test functions

epidemic spreading in correlated complex networks we study a dynamical model of epidemic spreading on complex networks in which there are explicit correlations among the node s connectivities for the case of markovian complex networks showing only correlations between pairs of nodes we find an epidemic threshold inversely proportional to the largest eigenvalue of the connectivity matrix that gives the average number of links which from a node with connectivity k go to nodes with connectivity k ensuremath numerical simulations on a correlated growing network model provide support for our conclusions

cohomology of infinitesimal algebraic groups in this paper we study the cohomology h br k of a higher frobenius kernel br r 1 of a borel subgroup b of a reductive algebraic group g over an algebraically closed field k of a positive characteristic p for the first frobenius kernel b 1 the cohomology has been completely determined by h h andersen and j c jantzen 3 in case p h the coxeter number of g in low degrees the cohomology is independent of r 1 and their results allow us to compute the first different case h zp i br k for s lt i 3 6 in higher degrees however we have to resort to a different method although h b 2 k has already been computed for sl z also in 3 it is a considerably harder problem for sl 3 we apply algebraic steenrod operations and the transgression theorem to the lyndon hochschildserre or cartan eilenberg spectral sequence arising from a central series of the unipotent radical u of b to find h k for sl 3 5 1 the method we employ here was originally developed by m tezuka and n yagita 21 in order to investigate h u fq k q a power ofp in fact the results of sections 4 and 5 were first conceived from the corresponding results of h u fq k aside from the interests in relation to the cohomology of finite groups the work was also motivated by a close connection of the bl cohomology to g lusztig s conjectural character formula for the irreducible modules of the reductive group g which is described in section 2 hoping that the representation theory may shed more light on some topological problems we intend to take up the steenrod algebra and the a algebras in a subsequent paper

percolation on correlated networks we reconsider the problem of percolation on an equilibrium random network with degree degree correlations between nearest neighboring vertices focusing on critical singularities at a percolation threshold we obtain criteria for degree degree correlations to be irrelevant for critical singularities we present examples of networks in which assortative and disassortative mixing leads to unusual percolation properties and new critical exponents

resilience to damage of graphs with degree correlations the existence or nonexistence of a percolation threshold on power law correlated graphs is a fundamental question for which a general criterion is lacking in this work we investigate the problems of site and bond percolation on graphs with degree correlations and their connection with spreading phenomena we obtain some general expressions that allow the computation of the transition thresholds or their bounds using these results we study the effects of assortative and disassortative correlations on the resilience to damage of networks

analytical results for bond percolation and k core sizes on clustered networks an analytical approach to calculating bond percolation thresholds sizes of k cores and sizes of giant connected components on structured random networks with nonzero clustering is presented the networks are generated using a generalization of trapman s p trapman theor popul biol 71 160 2007 model of cliques embedded in treelike random graphs the resulting networks have arbitrary degree distributions and tunable degree dependent clustering the effect of clustering on the bond percolation thresholds for networks of this type is examined and contrasted with some recent results in the literature for very high levels of clustering the percolation threshold in these generalized trapman networks is increased above the value it takes in a randomly wired unclustered network of the same degree distribution in assortative scale free networks where the variance of the degree distribution is infinite this clustering effect can lead to a nonzero percolation epidemic threshold

mean size of avalanches on directed random networks with arbitrary degree distributions the mean size of unordered binary avalanches on infinite directed random networks may be determined using the damage propagation function introduced by b samuelsson and j e s socolar phys rev e 74 036113 2006 the derivation of samuelsson and socolar explicitly assumes a poisson distribution of out degrees it is shown here that the damage propagation function method may be used whenever the in degree and out degree of network nodes are independently distributed in particular it is not necessary that the out degree distribution be poisson the general case of correlated in and out degrees is discussed and numerical simulations on large finite networks are compared with the theoretical predictions for infinite networks

large scale topological and dynamical properties of the internet we study the large scale topological and dynamical properties of real internet maps at the autonomous system level collected in a 3 yr time interval we find that the connectivity structure of the internet presents statistical distributions settled in a well defined stationary state the large scale properties are characterized by a scale free topology consistent with previous observations correlation functions and clustering coefficients exhibit a remarkable structure due to the underlying hierarchical organization of the internet the study of the internet time evolution shows a growth dynamics with aging features typical of recently proposed growing network models we compare the properties of growing network models with the present real internet data analysis

geography in a scale free network model we offer an example of a network model with a power law degree distribution p k approximately k alpha for nodes but which nevertheless has a well defined geography and a nonzero threshold percolation probability for alpha 2 the range of real world contact networks this is different from p c 0 for alpha 3 results for the original well mixed scale free networks in our lattice based scale free network individuals link to nearby neighbors on a lattice even considerable additional small world links do not change our conclusion of nonzero thresholds when applied to disease propagation these results suggest that random immunization may be more successful in controlling human epidemics than previously suggested if there is geographical clustering

perturbing general uncorrelated networks this paper is a direct continuation of an earlier work where we studied erd os r enyi random graphs perturbed by an interaction hamiltonian favouring the formation of short cycles here we generalize these results we keep the same interaction hamiltonian but let it act on general graphs with uncorrelated nodes and an arbitrary given degree distribution it is shown that the results obtained for erd os r enyi graphs are generic at the qualitative level however scale free graphs are an exception to this general rule and exhibit a singular behaviour studied thoroughly in this paper both analytically and numerically

properties of highly clustered networks we propose and solve exactly a model of a network that has both a tunable degree distribution and a tunable clustering coefficient among other things our results indicate that increased clustering leads to a decrease in the size of the giant component of the network we also study susceptible infective recovered type epidemic processes within the model and find that clustering decreases the size of epidemics but also decreases the epidemic threshold making it easier for diseases to spread in addition clustering causes epidemics to saturate sooner meaning that they infect a near maximal fraction of the network for quite low transmission rates

k core bootstrap percolation on complex networks critical phenomena and nonlocal effects we develop the theory of the k core bootstrap percolation on uncorrelated random networks with arbitrary degree distributions we show that the k core percolation is an unusual hybrid phase transition with a jump emergence of the k core as at a first order phase transition but also with a critical singularity as at a continuous transition we describe the properties of the k core explain the meaning of the order parameter for the k core percolation and reveal the origin of the specific critical phenomena we demonstrate that a so called corona of the k core plays a crucial role corona is a subset of vertices in the k core which have exactly k neighbors in the k core it turns out that the k core percolation threshold is at the same time the percolation threshold of finite corona clusters the mean separation of vertices in corona clusters plays the role of the correlation length and diverges at the critical point we show that a random removal of even one vertex from the k core may result in the collapse of a vast region of the k core around the removed vertex the mean size of this region diverges at the critical point we find an exact mapping of the k core percolation to a model of cooperative relaxation this model undergoes critical relaxation with a divergent rate at some critical moment

zero temperature hysteresis in the random field ising model on a bethe lattice we consider the single spin flip dynamics of the random field ising model on a bethe lattice at zero temperature in the presence of a uniform external field we determine the average magnetization as the external field is varied from to by setting up the self consistent field equations which we show are exact in this case the qualitative behaviour of magnetization as a function of the external field unexpectedly depends on the coordination number z of the bethe lattice for z 3 with a gaussian distribution of the quenched random fields we find no jump in magnetization for any non zero strength of disorder for for weak disorder the magnetization shows a jump discontinuity as a function of the external uniform field which disappears for a larger variance of the quenched field we determine exactly the critical point separating smooth hysteresis curves from those with a jump we have checked our results by monte carlo simulations of the model on three and four coordinated random graphs which for large system sizes give the same results as on the bethe lattice but avoid surface effects altogether

cyclic topology in complex networks we introduce a cyclic coefficient r which characterizes the degree of circulation in complex networks if a network has a perfect treelike structure then r becomes zero the larger value of r represents that the network has more cyclic structure we measure both the cyclic coefficients and the distributions of local cyclic coefficients for various networks and discuss the cyclic structures of them

pseudofractal scale free web we find that scale free random networks are excellently modeled by simple deterministic graphs our graph has a discrete degree distribution degree is the number of connections of a vertex which is characterized by a power law with exponent gamma 1 ln 3 ln 2 properties of this compact structure are surprisingly close to those of growing random scale free networks with gamma in the most interesting region between 2 and 3 we succeed to find exactly and numerically with high precision all main characteristics of the graph in particular we obtain the exact shortest path length distribution for a large network ln n 1 the distribution tends to a gaussian of width approximately sqrt ln n centered at l approximately ln n we show that the eigenvalue spectrum of the adjacency matrix of the graph has a power law tail with exponent 2 gamma

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

clustering in complex networks ii percolation properties the percolation properties of clustered networks are analyzed in detail in the case of weak clustering we present an analytical approach that allows us to find the critical threshold and the size of the giant component numerical simulations confirm the accuracy of our results in more general terms we show that weak clustering hinders the onset of the giant component whereas strong clustering favors its appearance this is a direct consequence of the differences in the k core structure of the networks which are found to be totally different depending on the level of clustering an empirical analysis of a real social network confirms our predictions

bond percolation on a class of clustered random networks analytical results are derived for the bond percolation threshold and the size of the giant connected component in a class of random networks with nonzero clustering the network s degree distribution and clustering spectrum may be prescribed and theoretical results match well with numerical simulations on both synthetic and real world networks