a self optimizing single step algorithm for the numerical integration of parabolic equations abstract an unconditionally stable single step time marching algorithm for parabolic equations is presented in this algorithm time stepping parameters are chosen according to an error condition such that large time step accuracy is achieved and the solution is also free from oscillations it is different from conventional algorithms such as the crank nicolson scheme in that it uses a matrix of time stepping parameters rather than a single scalar value these parameters can be approximated inexpensively this algorithm is easily programmed and comparisons are made with other methods for different cases of a heat conduction problem it is found that the algorithm gives a more accurate solution for all degrees of freedom for the cases considered it is therefore advantageous over the conventional variably weighted euler method which has a constraint that the same value must be used for all degrees of freedom and that values have to be changed by trial and error for different time step sizes and mesh refinements in order to achieve best results

effect of trap position on the efficiency of trapping in treelike scale free networks the conventional wisdom is that the role and impact of nodes on dynamical processes in scale free networks are not homogenous because of the presence of highly connected nodes at the tail of their power law degree distribution in this paper we explore the influence of different nodes as traps on the trapping efficiency of the trapping problem taking place on scale free networks to this end we study in detail the trapping problem in two families of deterministically growing scale free networks with treelike structure one family is non fractal the other is fractal in the first part of this work we attack a special case of random walks on the two network families with a perfect trap located at a hub i e node with the highest degree the second study addresses the case with trap distributed uniformly over all nodes in the networks for these two cases we compute analytically the mean trapping time mtt a quantitative indicator characterizing the trapping efficiency of the trapping process we show that in the non fractal scale free networks the mtt for both cases follows different scalings with the network order number of network nodes implying that trap s position has a significant effect on the trapping efficiency in contrast it is presented that for both cases in the fractal scale free networks the two leading scalings exhibit the same dependence on the network order suggesting that the location of trap has no essential impact on the trapping efficiency we also show that for both cases of the trapping problem the trapping efficiency is more efficient in the non fractal scale free networks than in their fractal counterparts

global mean first passage times of random walks on complex networks we present a general framework applicable to a broad class of random walks on complex networks which provides a rigorous lower bound for the mean first passage time of a random walker to a target site averaged over its starting position the so called global mean first passage time gmfpt this bound is simply expressed in terms of the equilibrium distribution at the target and implies a minimal scaling of the gmfpt with the network size we show that this minimal scaling which can be arbitrarily slow is realized under the simple condition that the random walk is transient at the target site and independently of the small world scale free or fractal properties of the network last we put forward that the gmfpt to a specific target is not a representative property of the network since the target averaged gmfpt satisfies much more restrictive bounds

constrained markovian dynamics of random graphs we introduce a statistical mechanics formalism for the study of constrained graph evolution as a markovian stochastic process in analogy with that available for spin systems deriving its basic properties and highlighting the role of the mobility the number of allowed moves for any given graph as an application of the general theory we analyze the properties of degree preserving markov chains based on elementary edge switchings we give an exact yet simple formula for the mobility in terms of the graph s adjacency matrix and its spectrum this formula allows us to define acceptance probabilities for edge switchings such that the markov chains become controlled glauber type detailed balance processes designed to evolve to any required invariant measure representing the asymptotic frequencies with which the allowed graphs are visited during the process as a corollary we also derive a condition in terms of simple degree statistics sufficient to guarantee that in the limit where the number of nodes diverges even for state independent acceptance probabilities of proposed moves the invariant measure of the process will be uniform we test our theory on synthetic graphs and on realistic larger graphs as studied in cellular biology showing explicitly that for instances where the simple edge swap dynamics fails to converge to the uniform measure a suitably modified markov chain instead generates the correct phase space sampling

a new stabilization of adaptive step trapezoid rule based on finite difference interrupts the adaptive step trapezoid rule tr is a generally effective numerical integrator but it is prone to ringing instability and solution stall we introduce a new stabilization algorithm based on finite difference interrupts fdi that reduces ringing and prevents stall unlike previously reported stabilization schemes our algorithm achieves stability and second order accuracy without incurring significant computational cost or spurious diffusion tr with fdi is at least as stable and accurate as existing methods when solving the prototypical scalar problem we demonstrate that it has better stability accuracy and cost when solving the tightly coupled system describing free surface flows of viscoelastic liquids though we demonstrate tr with fdi in the context of a finite element method of lines it is applicable to any tr based algorithm

continuum limit of critical inhomogeneous random graphs the last few years have witnessed tremendous interest in understanding the structure as well as the behavior of dynamics for inhomogeneous random graph models to gain insight into real world systems in this study we analyze the maximal components at criticality of one famous class of such models the rank one inhomogeneous random graph model norros and reittu adv appl probab 38 1 59 75 2006 bollobas et al random struct algorithms 31 1 3 122 2007 section 16 4 viewing these components as measured random metric spaces under finite moment assumptions for the weight distribution we show that the components in the critical scaling window with distances scaled by n 1 3 converge in the gromov haussdorf prokhorov metric to rescaled versions of the limit objects identified for the erd s renyi random graph components at criticality in addario berry et al probab theory related fields 152 3 4 367 406 2012 a key step is the construction of connected components of the random graph through an appropriate tilt of a fundamental class of random trees called mathbf p trees camarri and pitman electron j probab 5 2 1 18 2000 aldous et al probab theory related fields 129 2 182 218 2004 this is the first step in rigorously understanding the scaling limits of objects such as the minimal spanning tree and other strong disorder models from statistical physics braunstein et al phys rev lett 91 16 168701 2003 for such graph models by asymptotic equivalence janson random struct algorithms 36 1 26 45 2010 the same results are true for the chung lu model chung and lu proc natl acad sci 99 25 15879 15882 2002 chung and lu ann combin 6 2 125 145 2002 chung and lu complex graphs and networks 2006 and the britton deijfen martin lof model britton et al j stat phys 124 6 1377 1397 2006 a crucial ingredient of the proof of independent interest are tail bounds for the height of mathbf p trees the techniques developed in this paper form the main technical bedrock for the general program developed in bhamidi et al scaling limits of random graph models at criticality universality and the basin of attraction of the erd s renyi random graph arxiv preprint 2014 for proving universality of the continuum scaling limits in the critical regime for a wide array of other random graph models including the configuration model and inhomogeneous random graphs with general kernels bollobas et al random struct algorithms 31 1 3 122 2007

spectra of random graphs with community structure and arbitrary degrees using methods from random matrix theory researchers have recently calculated the full spectra of random networks with arbitrary degrees and with community structure both reveal interesting spectral features including deviations from the wigner semicircle distribution and phase transitions in the spectra of community structured networks in this paper we generalize both calculations giving a prescription for calculating the spectrum of a network with both community structure and an arbitrary degree distribution in general the spectrum has two parts a continuous spectral band which can depart strongly from the classic semicircle form and a set of outlying eigenvalues that indicate the presence of communities

identifying effective and or safe doses by stepwise confidence intervals for ratios typical randomized clinical dose finding studies consist of the comparison of several doses of a drug versus a placebo interest lies in estimating relevant doses among those under investigation for efficacy and safety variables such as the minimum effective dose or the maximum safe dose or estimating both doses simultaneously step down procedures have been proposed for comparing the standardized differences of the dose groups against placebo in this paper we consider the ratio of population means and propose stepwise confidence intervals for these ratios these confidence intervals do not require multiplicity adjustments and yield the same decisions as the associated test procedures in addition several power concepts are investigated within the present framework the results allow sample size determination in the design phase of a study for the probability of estimating correctly the dose of interest auxiliary results of a numerical study show the range of application of these methods

subgraphs in random networks understanding the subgraph distribution in random networks is important for modeling complex systems in classic erdos networks which exhibit a poissonian degree distribution the number of appearances of a subgraph g with n nodes and g edges scales with network size as approximately n n g however many natural networks have a non poissonian degree distribution here we present approximate equations for the average number of subgraphs in an ensemble of random sparse directed networks characterized by an arbitrary degree sequence we find scaling rules for the commonly occurring case of directed scale free networks in which the outgoing degree distribution scales as p k approximately k gamma considering the power exponent of the degree distribution gamma as a control parameter we show that random networks exhibit transitions between three regimes in each regime the subgraph number of appearances follows a different scaling law approximately nalpha where alpha n g s 1 for gamma gamma c where s is the maximal outdegree in the subgraph and gamma c s 1 we find that certain subgraphs appear much more frequently than in erdos networks these results are in very good agreement with numerical simulations this has implications for detecting network motifs subgraphs that occur in natural networks significantly more than in their randomized counterparts

inverted berezinskii kosterlitz thouless singularity and high temperature algebraic order in an ising model on a scale free hierarchical lattice small world network we have obtained exact results for the ising model on a hierarchical lattice incorporating three key features characterizing many real world networks a scale free degree distribution a high clustering coefficient and the small world effect by varying the probability p of long range bonds the entire spectrum from an unclustered non small world network to a highly clustered small world system is studied using the self similar structure of the network we obtain analytic expressions for the degree distribution p k and clustering coefficient c for all p as well as the average path length ensuremath ell for p 0 and 1 the ferromagnetic ising model on this network is studied through an exact renormalization group transformation of the quenched bond probability distribution using up to 562 phantom rule 0 2em 0ex 500 renormalized probability bins to represent the distribution for pl0 494 we find power law critical behavior of the magnetization and susceptibility with critical exponents continuously varying with p and exponential decay of correlations away from t c for p ensuremath ge 0 494 in fact where the network exhibits small world character the critical behavior radically changes we find a highly unusual phase transition namely an inverted berezinskii kosterlitz thouless singularity between a low temperature phase with nonzero magnetization and finite correlation length and a high temperature phase with zero magnetization and infinite correlation length with power law decay of correlations throughout the phase approaching t c from below the magnetization and the susceptibility respectively exhibit the singularities of mathrm exp ensuremath c sqrt t c ensuremath t and mathrm exp d sqrt t c ensuremath t with c and d positive constants with long range bond strengths decaying with distance we see a phase transition with power law critical singularities for all p and evaluate an unusually narrow critical region and important corrections to power law behavior that depend on the exponent characterizing the decay of long range interactions

the average distances in random graphs with given expected degrees abstract random graph theory is used to examine the small world phenomenon any two strangers are connected through a short chain of mutual acquaintances we will show that for certain families of random graphs with given expected degrees the average distance is almost surely of order log n log d where d is the weighted average of the sum of squares of the expected degrees of particular interest are power law random graphs in which the number of vertices of degree k is proportional to 1 k for some fixed exponent for the case of 3 we prove that the average distance of the power law graphs is almost surely of order log n log d however many internet social and citation networks are power law graphs with exponents in the range 2 3 for which the power law random graphs have average distance almost surely of order log log n but have diameter of order log n provided having some mild constraints for the average distance and maximum degree in particular these graphs contain a dense subgraph which we call the core having nc log log n vertices almost all vertices are within distance log log n of the core although there are vertices at distance log n from the core

contact processes on random graphs with power law degree distributions have critical value 0 if we consider the contact process with infection rate on a random graph on n vertices with power law degree distributions mean field calculations suggest that the critical value c of the infection rate is positive if the power 3 physicists seem to regard this as an established fact since the result has recently been generalized to bipartite graphs by gomez gardenes et al proc natl acad sci usa 105 2008 1399 1404 here we show that the critical value c is zero for any value of 3 and the contact process starting from all vertices infected with a probability tending to 1 as n maintains a positive density of infected sites for time at least exp n 1 for any 0 using the last result together with the contact process duality we can establish the existence of a quasi stationary distribution in which a randomly chosen vertex is occupied with probability it is expected that c as 0 here we show that 1 2 3 and so 2 for 3 thus even though the graph is locally tree like does not take the mean field critical value 1

random walks in modular scale free networks with multiple traps extensive empirical investigation has shown that a plethora of real networks synchronously exhibit scale free and modular structure and it is thus of great importance to uncover the effects of these two striking properties on various dynamical processes occurring on such networks in this paper we examine two cases of random walks performed on a class of modular scale free networks with multiple traps located at several given nodes we first derive a formula of the mean first passage time mfpt for a general network which is the mean of the expected time to absorption originating from a specific node averaged over all nontrap starting nodes although the computation is complex the expression of the formula is exact moreover the computational approach and procedure are independent of the number and position of the traps we then determine analytically the mfpt for the two random walks being considered the obtained analytical results are in complete agreement with the numerical ones our results show that the number and location of traps play an important role in the behavior of the mfpt since for both cases the mfpt grows as a power law function of the number of nodes but their exponents are quite different we demonstrate that the root of the difference in the behavior of mfpt is attributed to the modular and scale free topologies of the networks this work can deepen the understanding of diffusion on networks with modular and scale free architecture and motivate relevant studies for random walks running on complex random networks with multiple traps

minimum vertex cover in generalized random graphs with power law degree distribution in this paper we study the approximability of the minimum vertex cover problem in power law graphs in particular we investigate the behavior of a standard 2 approximation algorithm together with a simple pre processing step when the input is a random sample from a generalized random graph model with power law degree distribution more precisely if the probability of a vertex of degree i to be present in the graph is c i where 2 and c is a normalizing constant the expected approximation ratio is 1 1 l i e where is the riemann zeta function of li is the polylogarithmic special function of and li 2 1 e 1

swarming on random graphs ii we consider an individual based model where agents interact over a random network via first order dynamics that involve both attraction and repulsion in the case of all to all coupling of agents in mathbb r d this system has a lowest energy state in which an equal number of agents occupy the vertices of the d dimensional simplex the purpose of this paper is to sharpen and extend a line of work initiated in 56 which studies the behavior of this model when the interaction between the n agents occurs according to an erd s renyi random graph mathcal g n p instead of all to all coupling in particular we study the effect of randomness on the stability of these simplicial solutions and provide rigorous results to demonstrate that stability of these solutions persists for probabilities greater than np o log n in other words only a relatively small number of interactions are required to maintain stability of the state the results rely on basic probability arguments together with spectral properties of random graphs

simultaneous confidence intervals for ratios with application to the gold standard design with more than one experimental treatment considering a study design with two experimental treatments a reference treatment and a placebo we extend a previous approach considering the ratios of effects to a procedure for analyzing multiple ratios the technical framework for constructing tests and compatible simultaneous confidence intervals is set in a general manner besides a single step procedure and its extension to a stepdown procedure also an informative stepwise procedure in the spirit of our previous work is developed the latter is especially interesting because noninferiority studies require informative confidence intervals to infer more information than just noninferiority at the prespecified margin results from a simulation study for the three methods are shown we also argue that an extension to more than two experimental treatments is straightforward

scaling of average sending time on weighted koch networks random walks on weighted complex networks especially scale free networks have attracted considerable interest in the past but the efficiency of a hub sending information on scale free small world networks has been addressed less in this paper we study random walks on a class of weighted koch networks with scaling factor 0 r 1 we derive some basic properties for random walks on the weighted koch networks based on which we calculate analytically the average sending time ast defined as the average of mean first passage times mfpts from a hub node to all other nodes excluding the hub itself the obtained result displays that for 0 r 1 in large networks the ast grows as a power law function of the network order with the exponent represented by log43r 1r and for r 1 in large networks the ast grows with network order as n ln n which is larger than the linear scaling of the average receiving time defined as the average of mfpts for random walks to a given hub node averaged over all starti

localization transition of biased random walks on random networks we study random walks on large random graphs that are biased towards a randomly chosen but fixed target node we show that a critical bias strength bc exists such that most walks find the target within a finite time when b bc for b bc a finite fraction of walks drift off to infinity before hitting the target the phase transition at b bc is a critical point in the sense that quantities such as the return probability p t show power laws but finite size behavior is complex and does not obey the usual finite size scaling ansatz by extending rigorous results for biased walks on galton watson trees we give the exact analytical value for bc and verify it by large scale simulations

a note on multiple testing procedures in dose finding it is shown that in some recently proposed multiple test procedures for dose finding tamhane hochberg and dunnett 1996 biometrics 52 21 37 strong control of the familywise error is only guaranteed if order relations are assumed to hold among the means procedures that can be applied safely without assuming any restrictions on the means can be based on the classical many one comparisons of doses with the zero dose

cluster tails for critical power law inhomogeneous random graphs recently the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank 1 type having finite variance but infinite third moment degrees was obtained in bhamidi et al ann probab 40 2299 2361 2012 it was proved that when the degrees obey a power law with exponent tau in 3 4 the sequence of clusters ordered in decreasing size and multiplied through by n tau 2 tau 1 converges as n rightarrow infty to a sequence of decreasing non degenerate random variables here we study the tails of the limit of the rescaled largest cluster i e the probability that the scaling limit of the largest cluster takes a large value u as a function of u this extends a related result of pittel j combin theory ser b 82 2 237 269 2001 for the erd s renyi random graph to the setting of rank 1 inhomogeneous random graphs with infinite third moment degrees we make use of delicate large deviations and weak convergence arguments

community structure and scale free collections of erdős renyi graphs community structure plays a significant role in the analysis of social networks and similar graphs yet this structure is little understood and not well captured by most models we formally define a community to be a subgraph that is internally highly connected and has no deeper substructure we use tools of combinatorics to show that any such community must contain a dense erd ifmmode mbox h o else h o fi s r enyi er subgraph based on mathematical arguments we hypothesize that any graph with a heavy tailed degree distribution and community structure must contain a scale free collection of dense er subgraphs these theoretical observations corroborate well with empirical evidence from this we propose the block two level erd ifmmode mbox h o else h o fi s r enyi bter model and demonstrate that it accurately captures the observable properties of many real world social networks

network cross validation by edge sampling summary while many statistical models and methods are now available for network analysis resampling of network data remains a challenging problem cross validation is a useful general tool for model selection and parameter tuning but it is not directly applicable to networks since splitting network nodes into groups requires deleting edges and destroys some of the network structure in this paper we propose a new network resampling strategy based on splitting node pairs rather than nodes that is applicable to cross validation for a wide range of network model selection tasks we provide theoretical justification for our method in a general setting and examples of how the method can be used in specific network model selection and parameter tuning tasks numerical results on simulated networks and on a statisticians citation network show that the proposed cross validation approach works well for model selection

on small sample confidence intervals for parameters in discrete distributions summary the traditional definition of a confidence interval requires the coverage probability at any value of the parameter to be at least the nominal confidence level in constructing such intervals for parameters in discrete distributions less conservative behavior results from inverting a single two sided test than inverting two separate one sided tests of half the nominal level each we illustrate for a variety of discrete problems including interval estimation of a binomial parameter the difference and the ratio of two binomial parameters for independent samples and the odds ratio

scale free networks with a large to hypersmall world transition abstract recently there has been a tremendous interest in models of networks with a power law distribution of degree so called scale free networks it has been observed that such networks normally have extremely short path lengths scaling logarithmically or slower with system size as an exotic and counterintuitive example we propose a simple stochastic model capable of generating scale free networks with linearly scaling distances furthermore by tuning a parameter the model undergoes a phase transition to a regime with extremely short average distances apparently slower than log log n which we call a hypersmall world regime we characterize the degree degree correlation and clustering properties of this class of networks

influences of degree inhomogeneity on average path length and random walks in disassortative scale free networks various real life networks exhibit degree correlations and heterogeneous structure with the latter being characterized by power law degree distribution p k k where the degree exponent describes the extent of heterogeneity in this paper we study analytically the average path length apl of and random walks rws on a family of deterministic networks recursive scale free trees rsfts with negative degree correlations and various 2 1 ln 3 ln 2 with an aim to explore the impacts of structure heterogeneity on the apl and rws we show that the degree exponent has no effect on the apl d of rsfts in the full range of d behaves as a logarithmic scaling with the number of network nodes n i e d ln n which is in sharp contrast to the well known double logarithmic scaling d ln ln n previously obtained for uncorrelated scale free networks with 2 3 in addition we present that some scaling efficiency exponents of random walks are reliant on the degree exponent

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

epidemic processes on complex networks abstract we analyse the dynamics of the sis and reed frost models on complex networks in contrast to most of the previous analytical work which relied on mean field approximations and simulations we present a probabilistic approach whereby general results are derived and illustrated on various families of graphs

sparse maximum entropy random graphs with a given power law degree distribution even though power law or close to power law degree distributions are ubiquitously observed in a great variety of large real networks the mathematically satisfactory treatment of random power law graphs satisfying basic statistical requirements of realism is still lacking these requirements are sparsity exchangeability projectivity and unbiasedness the last requirement states that entropy of the graph ensemble must be maximized under the degree distribution constraints here we prove that the hypersoft configuration model belonging to the class of random graphs with latent hyperparameters also known as inhomogeneous random graphs or w random graphs is an ensemble of random power law graphs that are sparse unbiased and either exchangeable or projective the proof of their unbiasedness relies on generalized graphons and on mapping the problem of maximization of the normalized gibbs entropy of a random graph ensemble to the graphon entropy maximization problem showing that the two entropies converge to each other in the large graph limit

universality for the distance in finite variance random graphs we generalize the asymptotic behavior of the graph distance between two uniformly chosen nodes in the configuration model to a wide class of random graphs among others this class contains the poissonian random graph the expected degree random graph and the generalized random graph including the classical erdos renyi graph in the paper we assign to each node a deterministic capacity and the probability that there exists an edge between a pair of nodes is equal to a function of the product of the capacities of the pair divided by the total capacity of all the nodes we consider capacities which are such that the degrees of a node have uniformly bounded moments of order strictly larger than two so that in particular the degrees have finite variance we prove that the graph distance grows like log nu n where the nu depends on the capacities and n denotes the size of the graph in addition the random fluctuations around this asymptotic mean log nu n are shown to be tight we also consider the case where the capacities are independent copies of a positive random lambda with p lambda x 3 againg resulting in graphs where the degrees have finite variance the method of proof of these results is to couple each member of the class to the poissonian random graph for which we then give the complete proof by adapting the arguments of van der hofstad et al

geographical threshold graphs with small world and scale free properties many real networks are equipped with short diameters high clustering and power law degree distributions with preferential attachment and network growth the model by barabasi and albert simultaneously reproduces these properties and geographical versions of growing networks have also been analyzed however nongrowing networks with intrinsic vertex weights often explain these features more plausibly since not all networks are really growing we propose a geographical nongrowing network model with vertex weights edges are assumed to form when a pair of vertices are spatially close and or have large summed weights our model generalizes a variety of models as well as the original nongeographical counterpart such as the unit disk graph the boolean model and the gravity model which appear in the contexts of percolation wire communication mechanical and solid physics sociology economy and marketing in appropriate configurations our model produces small world networks with power law degree distributions we also discuss the relation between geography power laws in networks and power laws in general quantities serving as vertex weights

vertex similarity in networks we consider methods for quantifying the similarity of vertices in networks we propose a measure of similarity based on the concept that two vertices are similar if their immediate neighbors in the network are themselves similar this leads to a self consistent matrix formulation of similarity that can be evaluated iteratively using only a knowledge of the adjacency matrix of the network we test our similarity measure on computer generated networks for which the expected results are known and on a number of real world networks

hyperbolic geometry of complex networks we develop a geometric framework to study the structure and function of complex networks we assume that hyperbolic geometry underlies these networks and we show that with this assumption heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry conversely we show that if a network has some metric structure and if the network degree distribution is heterogeneous then the network has an effective hyperbolic geometry underneath we then establish a mapping between our geometric framework and statistical mechanics of complex networks this mapping interprets edges in a network as noninteracting fermions whose energies are hyperbolic distances between nodes while the auxiliary fields coupled to edges are linear functions of these energies or distances the geometric network ensemble subsumes the standard configuration model and classical random graphs as two limiting cases with degenerate geometric structures finally we show that targeted transport processes without global topology knowledge made possible by our geometric framework are maximally efficient according to all efficiency measures in networks with strongest heterogeneity and clustering and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure

multiple test procedures for identifying the maximum safe dose we consider dose response studies for safety assessment of crop protection compounds and drugs and we offer a hypothesis testing approach for identifying the maximum dose level that is guaranteed to be safe with preassigned confidence the focus is on step down sd multiple test procedures for identifying the maximum safe dose we propose two classes of contrasts among the dose means as test statistics for these procedures pairwise contrasts pc and helmert contrasts hc the first procedure sd2pc consists of a sequence of ordinary t tests and is thus easy to apply but it can have very low power for certain step response functions the second procedure sd1hc does not suffer from this drawback but it requires a weak monotonicity assumption for its mathematical validity the powers of sd2pc and sd1hc are studied via monte carlo simulation we recommend sd2pc for linear response functions and sd1hc for step response functions the procedures are illustrated by applying them to data from an aquatic

generating simple random graphs with prescribed degree distribution let f be a probability distribution with support on the non negative integers four methods for generating a simple undirected graph with approximate degree distribution f are described and compared two methods are based on the so called configuration model with modifications ensuring a simple graph one method is an extension of the classical erdos renyi graph where the edge probabilities are random variables and the last method starts with a directed random graph which is then modified to a simple undirected graph all methods are shown to give the correct distribution in the limit of large graph size but under different assumptions on the degree distribution f and also using different order of operations

spectra of random graphs with given expected degrees in the study of the spectra of power law graphs there are basically two competing approaches one is to prove analogues of wigner s semicircle law whereas the other predicts that the eigenvalues follow a power law distribution although the semicircle law and the power law have nothing in common we will show that both approaches are essentially correct if one considers the appropriate matrices we will prove that under certain mild conditions the eigenvalues of the normalized laplacian of a random power law graph follow the semicircle law whereas the spectrum of the adjacency matrix of a power law graph obeys the power law our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants of interest are a number of new values for the exponent where phase transitions for eigenvalue distributions occur the spectrum distributions have direct implications to numerous graph algorithms such as for example randomized algorithms that involve rapidly mixing markov chains

laplacian spectra as a diagnostic tool for network structure and dynamics we examine numerically the three way relationships among structure laplacian spectra and frequency synchronization dynamics on complex networks we study the effects of clustering degree distribution and a particular type of coupling asymmetry input normalization all of which are known to have effects on the synchronizability of oscillator networks we find that these topological factors produce marked signatures in the laplacian eigenvalue distribution and in the localization properties of individual eigenvectors using a set of coordinates based on the laplacian eigenvectors as a diagnostic tool for synchronization dynamics we find that the process of frequency synchronization can be visualized as a series of quasi independent transitions involving different normal modes particular features of the partially synchronized state can be understood in terms of the behavior of particular modes or groups of modes for example there are important partially synchronized states in which a set of low lying modes remain unlocked while those in the main spectral peak are locked we find therefore that spectra are correlated with dynamics in ways that go beyond results relating a single threshold to a single extremal eigenvalue

navigation in a small world with local information it is commonly known that there exist short paths between vertices in a network showing the small world effect yet vertices for example the individuals living in society usually are not able to find the shortest paths due to the very serious limit of information to study this issue theoretically here the navigation process of launching messages toward designated targets is investigated on a variant of the one dimensional small world network swn in the network structure considered the probability of a shortcut falling between a pair of nodes is proportional to r alpha where r is the lattice distance between the nodes when alpha 0 it reduces to the swn model with random shortcuts the system shows the dynamic small world effect which is different from the well studied static sw effect we study the effective network diameter the path length as a function of the lattice distance and the dynamics they are controlled by multiple parameters and we use data collapse to show that the parameters are correlated the central finding is that in the one dimensional network studied the dynamic sw effect exists for 0 alpha 2 for each given value of alpha in this region the point where the dynamic sw effect arises is m l approximately 1 where m is the number of useful shortcuts and l is their average reduced effective length

core periphery organization of complex networks networks may or may not be wired to have a core that is both itself densely connected and central in terms of graph distance in this study we propose a coefficient to measure if the network has such a clear cut core periphery dichotomy we measure this coefficient for a number of real world and model networks and find that different classes of networks have their characteristic values among other things we conclude that geographically embedded transportation networks have a strong core periphery structure we proceed to study radial statistics of the core i e properties of the n neighborhoods of the core vertices for increasing n we find that almost all networks have unexpectedly many edges within n neighborhoods at a certain distance from the core suggesting an effective radius for nontrivial network processes

the exact solution of the mean geodesic distance for vicsek fractals vicsek fractals are one of the most interesting classes of fractals and the study of their structural properties is important in this paper the exact formula for the mean geodesic distance of vicsek fractals is found the quantity is computed precisely through the recurrence relations derived from the self similar structure of the fractals considered the obtained exact solution exhibits that the mean geodesic distance approximately increases as a power law function of the number of nodes with the exponent equal to the reciprocal of the fractal dimension the closed form solution is confirmed by extensive numerical calculations

stepwise confidence intervals without multiplicity adjustment for dose response and toxicity studies abstract not all simultaneous inferences need multiplicity adjustment if the sequence of individual inferences is predefined and failure to achieve the desired inference at any step renders subsequent inferences unnecessary then multiplicity adjustment is not needed this can be justified using the closed testing principle to test appropriate hypotheses that are nested in sequence starting with the most restrictive one but what hypotheses are appropriate may not be obvious in some problems we give a fundamentally different confidence set based justification by partitioning the parameter space naturally and using the principle that exactly one member of the partition contains the true parameter in dose response studies designed to show superiority of treatments over a placebo negative control or a drug known to be efficacious active control the confidence set approach generates methods with meaningful guarantee against incorrect decision whereas previous applications of the closed testing app

a class of vertex edge growth small world network models having scale free self similar and hierarchical characters abstract the problem of uncovering the internal operating function of network models is intriguing demanded and attractive in researches of complex networks notice that in the past two decades a great number of artificial models are built to try to answer the above mentioned task based on the different growth ways these previous models can be divided into two categories one type possessing the preferential attachment follows a power law p k k 2 3 the other has exponential scaling feature p k k however there are no models containing above two kinds of growth ways to be presented even the study of interconnection between these two growth manners in the same model is lacking hence in this paper we construct a class of planar and self similar graphs motivated from a new attachment way vertex edge growth network operation more precisely the couple of both them we report that this model is sparse small world and hierarchical and then not only is scale free feature in our model but also lies the degree parameter 3 242 out the typical range note that we suggest that the coexistence of multiple vertex growth ways will have a prominent effect on the power law parameter and the preferential attachment plays a dominate role on the development of networks over time at the end of this paper we obtain an exact analytical expression for the total number of spanning trees of models and also capture spanning trees entropy which we have compared with those of their corresponding component elements

universality for first passage percolation on sparse random graphs we consider rst passage percolation on sparse random graphs with prescribed degree distributions and general independent and identically distributed edge weights assumed to have a density assuming that the degree distribution satises a uniform x 2 logx condition we analyze the asymptotic distribution for the minimal weight path between a pair of typical vertices as well the number of edges on this path or hopcount the hopcount satises a central limit theorem where the norming constants are expressible in terms of the parameters of an associated continuous time branching process centered by a multiple of logn where the constant is the inverse of the malthusian rate of growth of the associated branching process the minimal weight converges in distribution the limiting random variable equals the sum of the logarithms of the martingale limits of the branching processes that measure the relative growth of neighborhoods about the two vertices and a gumbel random variable and thus shows a remarkably universal behavior the proofs rely on a rened coupling between the shortest path problems on these graphs and continuous time branching processes and on a poisson point process limit for the potential closing edges of shortest weight paths between the source and destination the results extend to a host of related random graph models ranging from random rregular graphs inhomogeneous random graphs and uniform random graphs with a prescribed degree sequence

a unified set of single step algorithms part 1 general formulation and applications a general algorithm for a single step time marching scheme for use in dynamic or diffusion equations is presented this algorithm is easily programmed in its universal form for all orders of approximation and covers most of the currently used schemes as well as presenting many new possibilities in many cases it presents a computationally advantageous form over conventional procedures this is particularly so when compared with the newmark algorighm and its variants

power and sample size computations in simultaneous tests for non inferiority based on relative margins in this paper we address the problem of calculating power and sample sizes associated with simultaneous tests for non inferiority we consider the case of comparing several experimental treatments with an active control the approach is based on the ratio view where the common non inferiority margin is chosen to be some percentage of the mean of the control treatment two power definitions in multiple hypothesis testing namely complete power and minimal power are used in the computations the sample sizes associated with the ratio based inference are also compared with that of a comparable inference based on the difference of means for various scenarios it is found that the sample size required for ratio based inferences is smaller than that of difference based inferences when the relative non inferiority margin is less than one and when large response values indicate better treatment effects the results are illustrated with examples

the partitioning principle a powerful tool in multiple decision theory a first general principle and nowadays state of the art for the construction of powerful multiple test procedures controlling a multiple level a is the so called closure principle in this article we introduce another powerful tool for the construction of multiple decision procedures especially for the construction of multiple test procedures and selection procedures this tool is based on a partition of the parameter space and will be called partitioning principle pp in the first part of the paper we review basic concepts of multiple hypotheses testing and discuss a slight generalization of the current theory in the second part we present various variants of the pp for the construction of multiple test procedures these are a general pp gpp a weak pp wpp and a strong pp spp it will be shown that depending on the underlying decision problem a pp may lead to more powerful test procedures than a formal application of the closure principle fcp moreover the more complex spp may be more powerful than the wpp based on a duality between testing and selecting pps can also be applied for the construction of more powerful selection procedures in the third part of the paper fcp wpp and spp are applied and compared in some examples

explicit determination of mean first passage time for random walks on deterministic uniform recursive trees the determination of mean first passage time mfpt for random walks in networks is a theoretical challenge and is a topic of considerable recent interest within the physics community in this paper according to the known connections between mfpt effective resistance and the eigenvalues of graph laplacian we first study analytically the mfpt between all node pairs of a class of growing treelike networks which we term deterministic uniform recursive trees durts since one of its particular cases is a deterministic version of the famous uniform recursive tree the interesting quantity is determined exactly through the recursive relation of the laplacian spectra obtained from the special construction of durts the analytical result shows that the mfpt between all couples of nodes in durts varies as n text text ln text n for large networks with node number n second we study trapping on a particular network of durts focusing on a special case with the immobile trap positioned at a node having largest degree we determine exactly the average trapping time att that is defined as the average of fpt from all nodes to the trap in contrast to the scaling of the mfpt the leading behavior of att is a linear function of n interestingly we show that the behavior for att of the trapping problem is related to the trapping location which is in comparison with the phenomenon of trapping on fractal t graph although both networks exhibit tree structure finally we believe that the methods could open the way to exactly calculate the mfpt and att in a wide range of deterministic media

a family of small world network models built by complete graph and iteration function abstract small world networks are popular in real life complex systems in the past few decades researchers presented amounts of small world models in which some are stochastic and the rest are deterministic in comparison with random models it is not only convenient but also interesting to study the topological properties of deterministic models in some fields such as graph theory theorem computer sciences and so on as another concerned darling in current researches community structure modular topology is referred to as an useful statistical parameter to uncover the operating functions of network so building and studying such models with community structure and small world character will be a demanded task hence in this article we build a family of sparse network space n t which is different from those previous deterministic models even though our models are established in the same way as them iterative generation by randomly connecting manner in each time step every resulting member in n t has no absolutely self similar feature widely shared in a large number of previous models this makes our insight not into discussing a class certain model but into investigating a group various ones spanning a network space somewhat surprisingly our results prove all members of n t to possess some similar characters a sparsity b exponential scale feature p k k and c small world property here we must stress a very screming but intriguing phenomenon that the difference of average path length a p l between any two members in n t is quite small which indicates this random connecting way among members has no great effect on a p l at the end of this article as a new topological parameter correlated to reliability synchronization capability and diffusion properties of networks the number of spanning trees on a representative member n b t of n t is studied in detail then an exact analytical solution for its spanning trees entropy is also obtained

scaling theory of transport in complex biological networks transport is an important function in many network systems and understanding its behavior on biological social and technological networks is crucial for a wide range of applications however it is a property that is not well understood in these systems probably because of the lack of a general theoretical framework here based on the finding that renormalization can be applied to bionetworks we develop a scaling theory of transport in self similar networks we demonstrate the networks invariance under length scale renormalization and we show that the problem of transport can be characterized in terms of a set of critical exponents the scaling theory allows us to determine the influence of the modular structure on transport in metabolic and protein interaction networks we also generalize our theory by presenting and verifying scaling arguments for the dependence of transport on microscopic features such as the degree of the nodes and the distance between them using transport concepts such as diffusion and resistance we exploit this invariance and we are able to explain based on the topology of the network recent experimental results on the broad flow distribution in metabolic networks

multiple test procedures for dose finding the problem of identifying the lowest dose level for which the mean response differs from that at the zero dose level is considered a general framework for stepwise testing procedures that use contrasts among the dose level means is proposed using this framework several new procedures are derived these and some existing procedures including that of williams 1971 biometrics 27 103 117 1972 biometrics 28 519 531 are compared analytically and by an extensive simulation study for the normal theory balanced one way layout case it is pointed out that the procedures based on the so called step and basin contrasts proposed by ruberg 1989 journal of american statistical association 84 816 822 have excessively high type i familywise error rates fwes and hence they should not be used some findings of the simulation study are as follows for monotone dose mean configurations williams procedure and two step down test procedures based on helmert and linear contrasts offer the best performance for nonmonotone dose mean configurations the performance of williams procedure does degrade somewhat but the other two procedures are still the best for more complex designs a simple step down test procedure that uses any alpha level tests not necessarily t tests to compare each dose level with the zero dose level controls the fwe and is the only alternative available but its power is rather low especially under nonmonotone configurations step up procedures are generally dominated by step down procedures when the same contrasts are used although the differences are not great

geometric fractal growth model for scale free networks we introduce a deterministic model for scale free networks whose degree distribution follows a power law with the exponent ensuremath gamma at each time step each vertex generates its offspring whose number is proportional to the degree of that vertex with proportionality constant m ensuremath 1 mg1 we consider the two cases first each offspring is connected to its parent vertex only forming a tree structure second it is connected to both its parent and grandparent vertices forming a loop structure we find that both models exhibit power law behaviors in their degree distributions with the exponent ensuremath gamma 1 mathrm ln 2m ensuremath 1 mathrm ln m thus by tuning m the degree exponent can be adjusted in the range 2l ensuremath gamma l3 we also solve analytically a mean shortest path distance d between two vertices for the tree structure showing the small world behavior that is d ensuremath sim mathrm ln n mathrm ln overline k where n is system size and overline k is the mean degree finally we consider the case that the number of offspring is the same for all vertices and find that the degree distribution exhibits an exponential decay behavior

exact encounter times for many random walkers on regular and complex networks the exact mean time between encounters of a given particle in a system consisting of many particles undergoing random walks in discrete time is calculated on both regular and complex networks analytical results are obtained both for independent walkers where any number of walkers can occupy the same site and for walkers with an exclusion interaction when no site can contain more than one walker these analytical results are then compared with numerical simulations showing very good agreement

generalized adjacency and laplacian spectra of the weighted corona graphs abstract recently corona product graphs have attracted a great deal of attention from scientific communities such as coding theory frequency assignment mathematical chemistry meanwhile weight factor gradually became a research hotpoint because of closing to reality hence we introduce the weighted corona graphs in this paper the weighted corona product graphs are the in depth and promotion of the corona product graphs linking the spectra between the initial graph and the product graph then we deduce the complete information of generalized adjacency and laplacian eigenvalues of the weighted product graph with different structures and we generalize the results obtained previously into product corona graph with m 1 iterations

counting cliques and cycles in scale free inhomogeneous random graphs scale free networks contain many small cliques and cycles we model such networks as inhomogeneous random graphs with regularly varying infinite variance weights for these models the number of cliques and cycles have exact integral expressions amenable to asymptotic analysis we obtain various asymptotic descriptions for how the average number of cliques and cycles of any size grow with the network size for the cycle asymptotics we invoke the theory of circulant matrices

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

multifractal network generator we introduce a new approach to constructing networks with realistic features our method in spite of its conceptual simplicity it has only two parameters is capable of generating a wide variety of network types with prescribed statistical properties e g with degree or clustering coefficient distributions of various very different forms in turn these graphs can be used to test hypotheses or as models of actual data the method is based on a mapping between suitably chosen singular measures defined on the unit square and sparse infinite networks such a mapping has the great potential of allowing for graph theoretical results for a variety of network topologies the main idea of our approach is to go to the infinite limit of the singular measure and the size of the corresponding graph simultaneously a very unique feature of this construction is that with the increasing system size the generated graphs become topologically more structured we present analytic expressions derived from the parameters of the to be iterated initial generating measure for such major characteristics of graphs as their degree clustering coefficient and assortativity coefficient distributions the optimal parameters of the generating measure are determined from a simple simulated annealing process thus the present work provides a tool for researchers from a variety of fields such as biology computer science biology or complex systems enabling them to create a versatile model of their network data

testing strategies in multi dose experiments including active control inferential test strategies for multi arm trials are adapted or proposed for the special situation when more than one dose of a test treatment placebo and active control s are compared this includes between doses dose placebo and dose active control comparisons the procedures refer to situations when detailed comparisons make sense only if the sensitivity of the trial has been shown for example if a dose response relationship or a difference between active control and placebo has been established split strategies hierarchical assuming an order restriction among doses or linked procedures are introduced in linked procedures equivalence to the active control will be established only if the dose is also shown to be effective as compared to placebo all the inferential procedures control the experimentwise error rate in the strong sense for the respective sets of null hypotheses considered

random graph models for directed acyclic networks we study random graph models for directed acyclic graphs a class of networks that includes citation networks food webs and feed forward neural networks among others we propose two specific models roughly analogous to the fixed edge number and fixed edge probability variants of traditional undirected random graphs we calculate a number of properties of these models including particularly the probability of connection between a given pair of vertices and compare the results with real world acyclic network data finding that theory and measurements agree surprisingly well far better than the often poor agreement of other random graph models with their corresponding real world networks

network reachability of real world contact sequences we use real world contact sequences time ordered lists of contacts from one person to another to study how fast information or disease can spread across network of contacts specifically we measure the reachability time the average shortest time for a series of contacts to spread information between a reachable pair of vertices a pair where a chain of contacts exists leading from one person to the other and the reachability ratio the fraction of reachable vertex pairs these measures are studied using conditional uniform graph tests we conclude among other things that the network reachability depends much on a core where the path lengths are short and communication frequent that clustering of the contacts of an edge in time tends to decrease the reachability and that the order of the contacts really does make sense for dynamical spreading processes

the multiplicative coalescent inhomogeneous continuum random trees and new universality classes for critical random graphs one major open conjecture in the area of critical random graphs formulated by statistical physicists and supported by a large amount of numerical evidence over the last decade braunstein et al in phys rev lett 91 16 168701 2003 wu et al in phys rev lett 96 14 148702 2006 braunstein et al int j bifurc chaos 17 07 2215 2255 2007 chen et al in phys rev lett 96 6 068702 2006 is as follows for a wide array of random graph models with degree exponent tau in 3 4 distances between typical points both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like n tau 3 tau 1 in this paper we study the metric space structure of maximal components of the multiplicative coalescent in the regime where the sizes converge to excursions of levy processes without replacement aldous and limic electron in j probab 3 3 59 1998 yielding a completely new class of limiting random metric spaces a by product of the analysis yields the continuum scaling limit of one fundamental class of random graph models with degree exponent tau in 3 4 where edges are rescaled by n tau 3 tau 1 yielding the first rigorous proof of the above conjecture the limits in this case are compact tree like random fractals with a dense collection of hubs infinite degree vertices a finite number of which are identified with leaves to form shortcuts in a special case we show that the minkowski dimension of the limiting spaces equal tau 2 tau 3 a s in stark contrast to the erd s renyi scaling limit whose minkowski dimension is 2 a s it is generally believed that dynamic versions of a number of fundamental random graph models as one moves from the barely subcritical to the critical regime can be approximated by the multiplicative coalescent in work in progress the general theory developed in this paper is used to prove analogous limit results for other random graph models with degree exponent tau in 3 4 our proof makes crucial use of inhomogeneous continuum random trees icrt which have previously arisen in the study of the entrance boundary of the additive coalescent we show that tilted versions of the same objects using the associated mass measure describe connectivity properties of the multiplicative coalescent since convergence of height processes of corresponding approximating mathbf p trees is not known we use general methodology in athreya et al 2014 and develop novel techniques relying on first showing convergence in the gromov weak topology and then extending this to gromov hausdorff prokhorov convergence by proving a global lower mass bound

multiple test procedures for identifying the minimum effective and maximum safe doses of a drug we address the problem of determining the therapeutic window of a drug by finding its minimum effective and maximum safe doses mined and maxsd the mined is the lowest dose that exceeds the mean efficacy of the zero dose by a specified threshold and the maxsd is the highest dose that does not exceed the mean toxicity of the zero dose by a specified threshold step down multiple test procedures are proposed to identify the mined and maxsd assuming a bivariate normal model these procedures control the type i familywise error probability of declaring any ineffective dose as effective or any unsafe dose as safe at a prespecified level a new multivariate t distribution is introduced whose critical points are required to implement the exact normal theory procedures because these critical points depend on the unknown correlation coefficient between the efficacy and safety variables the bonferroni method is proposed as an alternative which amounts to separately testing for efficacy and safety each at t

a scalable generative graph model with community structure network data is ubiquitous and growing yet we lack realistic generative network models that can be calibrated to match real world data the recently proposed block two level erdos renyi bter model can be tuned to capture two fundamental properties degree distribution and clustering coefficients the latter is particularly important for reproducing graphs with community structure such as social networks in this paper we compare bter to other scalable models and show that it gives a better fit to real data we provide a scalable implementation that requires only o d rm max storage where d rm max is the maximum number of neighbors for a single node the generator is trivially parallelizable and we show results for a hadoop mapreduce implementation for modeling a real world web graph with over 4 6 billion edges we propose that the bter model can be used as a graph generator for benchmarking purposes and provide idealized degree distributions and clustering coefficient profiles that can b

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 g random apollonian nets we show that the dramatic discrepancies observed recently for the degree exponent in these two cases result from a biased choice of the units to be updated sequentially in the asynchronous version

determining global mean first passage time of random walks on vicsek fractals using eigenvalues of laplacian matrices the family of vicsek fractals is one of the most important and frequently studied regular fractal classes and it is of considerable interest to understand the dynamical processes on this treelike fractal family in this paper we investigate discrete random walks on the vicsek fractals with the aim to obtain the exact solutions to the global mean first passage time gmfpt defined as the average of first passage time fpt between two nodes over the whole family of fractals based on the known connections between fpts effective resistance and the eigenvalues of graph laplacian we determine implicitly the gmfpt of the vicsek fractals which is corroborated by numerical results the obtained closed form solution shows that the gmfpt approximately grows as a power law function with system size number of all nodes with the exponent lies between 1 and 2 we then provide both the upper bound and lower bound for gmfpt of general trees and show that the leading behavior of the upper bound is the square of system size and the dominating scaling of the lower bound varies linearly with system size we also show that the upper bound can be achieved in linear chains and the lower bound can be reached in star graphs this study provides a comprehensive understanding of random walks on the vicsek fractals and general treelike networks

trapping in scale free networks with hierarchical organization of modularity a wide variety of real life networks share two remarkable generic topological properties scale free behavior and modular organization and it is natural and important to study how these two features affect the dynamical processes taking place on such networks in this paper we investigate a simple stochastic process trapping problem a random walk with a perfect trap fixed at a given location performed on a family of hierarchical networks that exhibit simultaneously striking scale free and modular structure we focus on a particular case with the immobile trap positioned at the hub node having the largest degree using a method based on generating functions we determine explicitly the mean first passage time mfpt for the trapping problem which is the mean of the node to trap first passage time over the entire network the exact expression for the mfpt is calculated through the recurrence relations derived from the special construction of the hierarchical networks the obtained rigorous formula corroborated by extensive direct numerical calculations exhibits that the mfpt grows algebraically with the network order concretely the mfpt increases as a power law function of the number of nodes with the exponent much less than 1 we demonstrate that the hierarchical networks under consideration have more efficient structure for transport by diffusion in contrast with other analytically soluble media including some previously studied scale free networks we argue that the scale free and modular topologies are responsible for the high efficiency of the trapping process on the hierarchical networks

exploring the assortativity clustering space of a network s degree sequence nowadays there is a multitude of measures designed to capture different aspects of network structure to be able to say if a measured value is expected or not one needs to compare it with a refere

diameters in preferential attachment models in this paper we investigate the diameter in preferential attachment pa models thus quantifying the statement that these models are small worlds the models studied here are such that edges are attached to older vertices proportional to the degree plus a constant i e we consider affine pa models there is a substantial amount of literature proving that quite generally pa graphs possess power law degree sequences with a power law exponent 2

ring structures and mean first passage time in networks in this paper we address the problem of the calculation of the mean first passage time on generic graphs we focus in particular on the mean first passage time on a node s for a random walker starting from a generic unknown node x we introduce an approximate scheme of calculation which maps the original process in a markov process in the space of the so called rings described by a transition matrix of size o ln n ln k ln n ln k where n is the size of the graph and k the average degree in the graph in this way one has a drastic reduction of degrees of freedom with respect to the size n of the transition matrix of the original process corresponding to an extremely low computational cost we first apply the method to the erdos renyi random graphs for which the method allows for almost perfect agreement with numerical simulations then we extend the approach to the barabasi albert graph as an example of scale free graph for which one obtains excellent results finally we test the method with two real world graphs internet and a network of the brain for which we obtain accurate results

the relations between network operation and topological property in a scale free and small world network with community structure abstract it is always an open demanding and difficult task for generating available model to simulate dynamical functions and reveal inner principles from complex systems and networks in this article due to lots of real life and artificial networks are built from series of simple and small groups components we discuss some interesting and helpful network operation to generate more realistic network models in view of community structure modular topology we present a class of sparse network models n t m at the moment we capture the fact the n t 4 has not only scale free feature which means that the probability that a randomly selected vertex with degree k decays as a power law following p k k where is the degree exponent but also small world property which indicates that the typical distance between two uniform randomly chosen vertices grows proportionally to logarithm of the order of n t 4 namely relatively shorter diameter and lower average path length simultaneously displays higher clustering coefficient next as a new topological parameter correlating to reliability synchronization capability and diffusion properties of networks the number of spanning trees over a network is studied in more detail an exact analytical solution for the number of spanning trees of the n t 4 is obtained based on the network operation part hub vertex linking with each other will be helpful for structuring various network models and investigating the rules related with real life networks

distribution of shortest path lengths in subcritical erdős renyi networks networks that are fragmented into small disconnected components are prevalent in a large variety of systems these include the secure communication networks of commercial enterprises government agencies and illicit organizations as well as networks that suffered multiple failures attacks or epidemics the structural and statistical properties of such networks resemble those of subcritical random networks which consist of finite components whose sizes are nonextensive surprisingly such networks do not exhibit the small world property that is typical in supercritical random networks where the mean distance between pairs of nodes scales logarithmically with the network size unlike supercritical networks whose structure has been studied extensively subcritical networks have attracted relatively little attention a special feature of these networks is that the statistical and geometric properties vary between different components and depend on their sizes and topologies the overall statistics of the network can be obtained by a summation over all the components with suitable weights we use a topological expansion to perform a systematic analysis of the degree distribution and the distribution of shortest path lengths dspl on components of given sizes and topologies in subcritical erd ifmmode mbox h o else h o fi s r enyi er networks from this expansion we obtain an exact analytical expression for the dspl of the entire subcritical network in the asymptotic limit the dspl which accounts for all the pairs of nodes that reside on the same finite component fc is found to follow a geometric distribution of the form p mathrm fc l ensuremath ell ll ensuremath infty 1 ensuremath c c ensuremath ell ensuremath 1 where cl1 is the mean degree using computer simulations we calculate the dspl in subcritical er networks of increasing sizes and confirm the convergence to this asymptotic result we also obtain exact asymptotic results for the mean distance ensuremath langle l ensuremath rangle mathrm fc and for the standard deviation of the dspl ensuremath sigma l mathrm fc and show that the simulation results converge to these asymptotic results using the duality relations between subcritical and supercritical er networks we obtain the dspl on the nongiant components of er networks above the percolation transition

degree dependent intervertex separation in complex networks we study the mean length l k of the shortest paths between a vertex of degree k and other vertices in growing networks where correlations are essential in a number of deterministic scale free networks we observe a power law correction to a logarithmic dependence l k a ln n k gamma 1 2 ck gamma 1 n in a wide range of network sizes here n is the number of vertices in the network gamma is the degree distribution exponent and the coefficients a and c depend on a network we compare this law with a corresponding l k dependence obtained for random scale free networks growing through the preferential attachment mechanism in stochastic and deterministic growing trees with an exponential degree distribution we observe a linear dependence on degree l k approximately a ln n ck we compare our findings for growing networks with those for uncorrelated graphs

random walks on complex trees we study the properties of random walks on complex trees we observe that the absence of loops is reflected in physical observables showing large differences with respect to their looped counterparts first both the vertex discovery rate and the mean topological displacement from the origin present a considerable slowing down in the tree case second the mean first passage time mfpt displays a logarithmic degree dependence in contrast to the inverse degree shape exhibited in looped networks this deviation can be ascribed to the dominance of source target topological distance in trees to show this we study the distance dependence of a symmetrized mfpt and derive its logarithmic profile obtaining good agreement with simulation results these unique properties shed light on the recently reported anomalies observed in diffusive dynamical systems on trees

degree mixing and the enhancement of synchronization in complex weighted networks real networks often consist of local units interacting with each other by means of heterogeneous connections in many cases furthermore such networks feature degree mixing properties i e the tendency of nodes with high degree with low degree to connect with connectivity peers with highly connected nodes such degree degree correlations may have an important influence in the spreading of information or infectious agents on a network we explore the role played by these correlations for the synchronization of networks of coupled dynamical systems using a stochastic optimization technique we find that the value of degree mixing providing optimal conditions for synchronization depends on the weighted coupling scheme we also show that a minimization of the assortative coefficient may induce a strong destabilization of the synchronous state we illustrate our findings for weighted networks with scale free and random topologies

approximate interval estimation of the ratio of binomial parameters a review and corrections for skewness various methods for finding confidence intervals for the ratio of binomial parameters are reviewed and evaluated numerically it is found that the method based on likelihood scores koopman 1984 biometrics 40 513 517 miettinen and nurminen 1985 statistics in medicine 4 213 226 performs best in achieving the nominal confidence coefficient but it may distribute the tail probabilities quite disparately using general theory of bartlett 1953 biometrika 40 306 317 1955 biometrika 42 201 203 we correct this method for asymptotic skewness following gart 1985 biometrika 72 673 677 we extend this correction to the case of estimating the common ratio in a series of two by two tables computing algorithms are given and applied to numerical examples parallel methods for the odds ratio and the ratio of poisson parameters are noted

random walks and diameter of finite scale free networks abstract dynamical scalings for the end to end distance r e e and the number of distinct visited nodes n v of random walks rws on finite scale free networks sfns are studied numerically r e e shows the dynamical scaling behavior r e e l t l n g t l z where l is the average minimum distance between all possible pairs of nodes in the network n is the number of nodes is the degree exponent of the sfn and t is the step number of rws especially r e e l t in the limit t satisfies the relation r e e l d where d is the diameter of network with d l ln n for 3 or d l ln ln n for 3 based on the scaling relation r e e we also find that the scaling behavior of the diameter of networks can be measured very efficiently by using rws

first passage properties of the erdos renyi random graph we study the mean time for a random walk to traverse between two arbitrary sites of the erd s renyi random graph we develop an effective medium approximation that predicts that the mean first passage time between pairs of nodes as well as all moments of this first passage time are insensitive to the fraction p of occupied links this prediction qualitatively agrees with numerical simulations away from the percolation threshold near the percolation threshold the statistically meaningful quantity is the mean transit rate namely the inverse of the first passage time this rate varies non monotonically with p near the percolation transition much of this behaviour can be understood by simple heuristic arguments

universality for distances in power law random graphs we survey the recent work on phase transition and distances in various random graph models with general degree sequences we focus on inhomogeneous random graphs the configuration model and affine preferential attachment models and pay special attention to the setting where these random graphs have a power law degree sequence this means that the proportion of vertices with degree k in large graphs is approximately proportional to k for some 1 since many real networks have been empirically shown to have power law degree sequences these random graphs can be seen as more realistic models for real complex networks than classical random graphs such as the erd s renyi random graph it is often suggested that the behavior of random graphs should have a large amount of universality meaning in this case that random graphs with similar degree sequences share similar behavior we survey the available results on graph distances in power law random graphs that are consistent with this prediction

on the likelihood of forests abstract how complex a network is crucially impacts its function and performance in many modern applications the networks involved have a growth property and sparse structures which pose challenges to physicists and applied mathematicians in this paper we introduce the forest likelihood as a plausible measure to gauge how difficult it is to construct a forest in a non preferential attachment way based on the notions of admittable labeling and path construction we propose algorithms for computing the forest likelihood of a given forest concrete examples as well as the distributions of forest likelihoods for all forests with some fixed numbers of nodes are presented moreover we illustrate the ideas on real life networks including a benzenoid tree a mathematical family tree and a peer to peer network