average path length in random networks analytic solution for the average path length in a large class of uncorrelated random networks with hidden variables is found we apply the approach to classical random graphs of erdos and renyi er evolving networks introduced by barabasi and albert as well as random networks with asymptotic scale free connectivity distributions characterized by an arbitrary scaling exponent alpha 2 our result for 2 infinity there is a saturation effect for the average path length

polya number and first return of bursty random walk rigorous solutions abstract the recurrence properties of random walks can be characterized by polya number i e the probability that the walker has returned to the origin at least once in this paper we investigate polya number and first return for bursty random walk on a line in which the walk has different step size and moving probabilities using the concept of the catalan number we obtain exact results for first return probability the average first return time and polya number for the first time we show that polya number displays two different functional behavior when the walk deviates from the recurrent point by utilizing the lagrange inversion formula we interpret our findings by transferring polya number to the closed form solutions of an inverse function we also calculate polya number using another approach which corroborates our results and conclusions finally we consider the recurrence properties and polya number of two variations of the bursty random walk model

link prediction based on path entropy abstract information theory has been taken as a prospective tool for quantifying the complexity of complex networks in this paper first we study the information entropy or uncertainty of a path using the information theory after that we apply the path entropy to the link prediction problem in real world networks specifically we propose a new similarity index namely path entropy pe index which considers the information entropies of shortest paths between node pairs with penalization to long paths empirical experiments demonstrate that pe index outperforms the mainstream of link predictors

random walks in weighted networks with a perfect trap an application of laplacian spectra trapping processes constitute a primary problem of random walks which characterize various other dynamical processes taking place on networks most previous works focused on the case of binary networks while there is much less related research about weighted networks in this paper we propose a general framework for the trapping problem on a weighted network with a perfect trap fixed at an arbitrary node by utilizing the spectral graph theory we provide an exact formula for mean first passage time mfpt from one node to another based on which we deduce an explicit expression for average trapping time att in terms of the eigenvalues and eigenvectors of the laplacian matrix associated with the weighted graph where att is the average of mfpts to the trap over all source nodes we then further derive a sharp lower bound for the att in terms of only the local information of the trap node which can be obtained in some graphs moreover we deduce the att when the trap is distributed uniformly in the whole network our results show that network weights play a significant role in the trapping process to apply our framework we use the obtained formulas to study random walks on two specific networks trapping in weighted uncorrelated networks with a deep trap the weights of which are characterized by a parameter and levy random walks in a connected binary network with a trap distributed uniformly which can be looked on as random walks on a weighted network for weighted uncorrelated networks we show that the att to any target node depends on the weight parameter that is the att to any node can change drastically by modifying the parameter a phenomenon that is in contrast to that for trapping in binary networks for levy random walks in any connected network by using their equivalence to random walks on a weighted complete network we obtain the optimal exponent characterizing levy random walks which have the minimal average of atts taken over all target nodes

finding community structure in networks using the eigenvectors of matrices we consider the problem of detecting communities or modules in networks groups of vertices with a higher than average density of edges connecting them previous work indicates that a robust approach to this problem is the maximization of the benefit function known as modularity over possible divisions of a network here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix which plays a role in community detection similar to that played by the graph laplacian in graph partitioning calculations this result leads us to a number of possible algorithms for detecting community structure as well as several other results including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong the algorithms and measures proposed are illustrated with applications to a variety of real world complex networks

centrality measures and thermodynamic formalism for complex networks in the study of small and large networks it is customary to perform a simple random walk where the random walker jumps from one node to one of its neighbors with uniform probability the properties of this random walk are intimately related to the combinatorial properties of the network in this paper we propose to use the ruelle bowens random walk instead whose probability transitions are chosen in order to maximize the entropy rate of the walk on an unweighted graph if the graph is weighted then a free energy is optimized instead of the entropy rate specifically we introduce a centrality measure for large networks which is the stationary distribution attained by the ruelle bowens random walk we name it entropy rank we introduce a more general version which is able to deal with disconnected networks under the name of free energy rank we compare the properties of those centrality measures with the classic pagerank and hyperlink induced topic search hits on both toy and real life examples in particular their robustness to small modifications of the network we show that our centrality measures are more discriminating than pagerank since they are able to distinguish clearly pages that pagerank regards as almost equally interesting and are more sensitive to the medium scale details of the graph

calculation of fibonacci polynomials for gfsr sequences with low discrepancies fibonacci polynomials are defined in the context of the two dimensional discrepancy of tausworthe pseudorandom sequences as an analogue to fibonacci numbers which give the best figure of merit for the two dimensional discrepancy of linear congruential sequences we conduct an exhaustive search for the fibonacci polynomials of degree less than 32 whose associated tausworthe sequences can be easily implemented and very quickly generated

levy walk in complex networks an efficient way of mobility abstract to obtain an efficient diffusion process is an intriguing and important issue in the study of dynamical behaviors on real networks most previous studies are mainly focused on the analysis based on the random walk strategy whose entropy rate is bounded by the logarithm of the largest node degree of a given graph in this paper we take into account a novel strategy named levy walk and derive the general expression of entropy rate of levy walk on networks we present numerical evidences for how the levy walk strategy delivers an efficient diffusion process on networks and significantly increases the entropy rate compared with the random walk strategy it is further demonstrated that the capability of levy walk heavily relies on the network topology as well as the amount of information available regarding the network structure specifically the behavior of levy walk is highly sensitive to the distribution of shortest distances of the network and its variation to address this finding we thereby give a theoretical explanation of the relationship between the variation of shortest distances and the entropy rate of levy walk this work may help to enrich our understanding of the behavior of levy walk and further guide us to find an efficient diffusion process on networks

distance distribution in random graphs and application to network exploration we consider the problem of determining the proportion of edges that are discovered in an erdos renyi graph when one constructs all shortest paths from a given source node to all other nodes this problem is equivalent to the one of determining the proportion of edges connecting nodes that are at identical distance from the source node the evolution of this quantity with the probability of existence of the edges exhibits intriguing oscillatory behavior in order to perform our analysis we introduce a different way of computing the distribution of distances between nodes our method outperforms previous similar analyses and leads to estimates that coincide remarkably well with numerical simulations it allows us to characterize the phase transitions appearing when the connectivity probability varies

combinatorial structure and capacity of the permuting relay channel blackwell s trap door channel is an interesting example of a finite state channel its deterministic version that is the permuting channel has been studied by ahlswede and kaspi ina multiterminal information theoretic framework they determined the capacities of permuting jammer channels and relay channels for some special cases the capacity problem for permuting relay channels is completely solved more specifically when a is the cardinality of alphabet and beta is the number of available storage locations in the channel the capacity c r alpha beta of the permuting relay channel is given by log lambda where lambda denotes the maximum eigenvalue of a matrix q derived from the state transition mechanism associated with the channel

general formalism for inhomogeneous random graphs we present and investigate an extension of the classical random graph to a general class of inhomogeneous random graph models where vertices come in different types and the probability of realizing an edge depends on the types of its terminal vertices this approach provides a general framework for the analysis of a large class of models the generic phase structure is derived using generating function techniques and relations to other classes of models are pointed out

capacity of the trapdoor channel with feedback we establish that the feedback capacity of the trapdoor channel is the logarithm of the golden ratio and provide a simple communication scheme that achieves capacity as part of the analysis we formulate a class of dynamic programs that characterize capacities of unifilar finite state channels the trapdoor channel is an instance that admits a simple closed form solution

long range navigation on complex networks using levy random walks we introduce a strategy of navigation in undirected networks including regular random and complex networks that is inspired by l evy random walks generalizing previous navigation rules we obtained exact expressions for the stationary probability distribution the occupation probability the mean first passage time and the average time to reach a node on the network we found that the long range navigation using the l evy random walk strategy compared with the normal random walk strategy is more efficient at reducing the time to cover the network the dynamical effect of using the l evy walk strategy is to transform a large world network into a small world our exact results provide a general framework that connects two important fields l evy navigation strategies and dynamics on complex networks

on the discrepancy of gfsr pseudorandom numbers a new summation formula based on the orthogonal property of walsh functions is devised using this formula the k dimensional discrepancy of the generalized feedback shift register gfsr pseudorandom numbers is derived the relation between the discrepancy and k distribution of gfsr sequences is also obtained finally the definition of optimal gpsr pseudorandom number generators is introduced

organization of modular networks we examine the global organization of heterogeneous equilibrium networks consisting of a number of well distinguished interconnected parts char22 communities or modules we develop an analytical approach allowing us to obtain the statistics of connected components and the intervertex distance distribution in these modular networks and to describe their global organization and structure in particular we study the evolution of the intervertex distance distribution with an increasing number of interlinks connecting two infinitely large uncorrelated networks we demonstrate that even a relatively small number of shortcuts unite the networks into one in more precise terms if the number of interlinks is any finite fraction of the total number of connections then the intervertex distance distribution approaches a ensuremath delta function peaked form and so the network is united

search in power law networks many communication and social networks have power law link distributions containing a few nodes that have a very high degree and many with low degree the high connectivity nodes play the important role of hubs in communication and networking a fact that can be exploited when designing efficient search algorithms we introduce a number of local search strategies that utilize high degree nodes in power law graphs and that have costs scaling sublinearly with the size of the graph we also demonstrate the utility of these strategies on the gnutella peer to peer network

entropy measures for networks toward an information theory of complex topologies the quantification of the complexity of networks is today a fundamental problem in the physics of complex systems a possible roadmap to solve the problem is via extending key concepts of information theory to networks in this rapid communication we propose how to define the shannon entropy of a network ensemble and how it relates to the gibbs and von neumann entropies of network ensembles the quantities we introduce here will play a crucial role for the formulation of null models of networks through maximum entropy arguments and will contribute to inference problems emerging in the field of complex networks

an empirical study of an agglomeration network recently researchers have reported many models mimicking real network evolution growth among which some are based on network aggregation growth however until now relatively few experiments have been reported accordingly in this paper photomicrographs of real materials the agglomeration in the filtrate of slurry formed by a gap nanoparticle conglomerate dispersed in water are analyzed within the framework of complex network theory by data mapping from photomicrographs we generate undirected networks and as a definition of degree we adopt the number of pixel s nearest neighbors while adjacent pixels define a connection or an edge we study the topological structure of these networks including degree distribution clustering coefficient and average path length in addition we discuss the self similarity and synchronizability of the networks we find that the synchronizability of high concentration agglomeration is better than that of low concentration agglomeration we also find that agglomeration networks possess good self similar features

origin of degree correlations in the internet and other networks it has been argued that the observed anticorrelation between the degrees of adjacent vertices in the network representation of the internet has its origin in the restriction that no two vertices have more than one edge connecting them here we propose a formalism for modeling ensembles of graphs with single edges only and derive values for the exponents and correlation coefficients characterizing them our results confirm that the conjectured mechanism does indeed give rise to correlations of the kind seen in the internet although only a part of the measured correlation can be accounted for in this way

approximate entropy of network parameters we study the notion of approximate entropy within the framework of network theory approximate entropy is an uncertainty measure originally proposed in the context of dynamical systems and time series we first define a purely structural entropy obtained by computing the approximate entropy of the so called slide sequence this is a surrogate of the degree sequence and it is suggested by the frequency partition of a graph we examine this quantity for standard scale free and erdos renyi networks by using classical results of pincus we show that our entropy measure often converges with network size to a certain binary shannon entropy as a second step with specific attention to networks generated by dynamical processes we investigate approximate entropy of horizontal visibility graphs visibility graphs allow us to naturally associate with a network the notion of temporal correlations therefore providing the measure a dynamical garment we show that approximate entropy distinguishes visibility graphs generated by processes with different complexity the result probes to a greater extent these networks for the study of dynamical systems applications to certain biological data arising in cancer genomics are finally considered in the light of both approaches

highly clustered scale free networks we propose a model for growing networks based on a finite memory of the nodes the model shows stylized features of real world networks power law distribution of degree linear preferential attachment of new links and a negative correlation between the age of a node and its link attachment rate notably the degree distribution is conserved even though only the most recently grown part of the network is considered as the network grows the clustering reaches an asymptotic value larger than that for regular lattices of the same average connectivity and similar to the one observed in the networks of movie actors coauthorship in science and word synonyms these highly clustered scale free networks indicate that memory effects are crucial for a correct description of the dynamics of growing networks

organization of growing random networks the organizational development of growing random networks is investigated these growing networks are built by adding nodes successively and linking each to an earlier node of degree k with an attachment probability a k when a k grows more slowly than linearly with k the number of nodes with k links n k t decays faster than a power law in k while for a k growing faster than linearly in k a single node emerges which connects to nearly all other nodes when a k is asymptotically linear n k t ensuremath sim mathrm tk ensuremath ensuremath nu with ensuremath nu dependent on details of the attachment probability but in the range 2l ensuremath nu l ensuremath infty the combined age and degree distribution of nodes shows that old nodes typically have a large degree there is also a significant correlation in the degrees of neighboring nodes so that nodes of similar degree are more likely to be connected the size distributions of the in and out components of the network with respect to a given node namely its descendants and ancestors are also determined the in component exhibits a robust s ensuremath 2 power law tail where s is the component size the out component has a typical size of order mathrm ln t and it provides basic insights into the genealogy of the network

path finding strategies in scale free networks we numerically investigate the scale free network model of barab asi and albert a l barab asi and r albert science 286 509 1999 through the use of various path finding strategies in real networks global network information is not accessible to each vertex and the actual path connecting two vertices can sometimes be much longer than the shortest one a generalized diameter depending on the actual path finding strategy is introduced and a simple strategy which utilizes only local information on the connectivity is suggested and shown to yield small world behavior the diameter d of the network increases logarithmically with the network size n the same as is found with global strategy if paths are sought at random d ensuremath sim n 0 5 is found

topologically biased random walk and community finding in networks we present an approach of topology biased random walks for undirected networks we focus on a one parameter family of biases and by using a formal analogy with perturbation theory in quantum mechanics we investigate the features of biased random walks this analogy is extended through the use of parametric equations of motion to study the features of random walks vs parameter values furthermore we show an analysis of the spectral gap maximum associated with the value of the second eigenvalue of the transition matrix related to the relaxation rate to the stationary state applications of these studies allow ad hoc algorithms for the exploration of complex networks and their communities

statistical independence properties of pseudorandom vectors produced by matrix generators abstract the uniformity test and the serial test are standard statistical tests for equidistribution and statistical independence respectively in sequences of uniform pseudorandom numbers we introduce analogues of these tests for sequences of uniform pseudorandom vectors and we apply these tests to matrix generators for uniform pseudorandom vector generation the results show that these generators behave well under these tests provided that the matrix in the generation algorithm is chosen carefully the essential condition is that a certain figure of merit attached to the matrix be large we also prove a general theorem which guarantees the existence of matrices with large figure of merit

shortest paths and load scaling in scale free trees the average node to node distance of scale free graphs depends logarithmically on n the number of nodes while the probability distribution function of the distances may take various forms here we analyze these by considering mean field arguments and by mapping the m 1 case of the barabasi albert model into a tree with a depth dependent branching ratio this shows the origins of the average distance scaling and allows one to demonstrate why the distribution approaches a gaussian in the limit of n large the load the number of the shortest distance paths passing through any node is discussed in the tree presentation

flow graphs interweaving dynamics and structure the behavior of complex systems is determined not only by the topological organization of their interconnections but also by the dynamical processes taking place among their constituents a faithful modeling of the dynamics is essential because different dynamical processes may be affected very differently by network topology a full characterization of such systems thus requires a formalization that encompasses both aspects simultaneously rather than relying only on the topological adjacency matrix to achieve this we introduce the concept of flow graphs namely weighted networks where dynamical flows are embedded into the link weights flow graphs provide an integrated representation of the structure and dynamics of the system which can then be analyzed with standard tools from network theory conversely a structural network feature of our choice can also be used as the basis for the construction of a flow graph that will then encompass a dynamics biased by such a feature we illustrate the ideas by focusing on the mathematical properties of generic linear processes on complex networks that can be represented as biased random walks and their dual consensus dynamics and show how our framework improves our understanding of these processes

scale free networks from varying vertex intrinsic fitness a new mechanism leading to scale free networks is proposed in this letter it is shown that in many cases of interest the connectivity power law behavior is neither related to dynamical properties nor to preferential attachment assigning a quenched fitness value x i to every vertex and drawing links among vertices with a probability depending on the fitnesses of the two involved sites gives rise to what we call a good get richer mechanism in which sites with larger fitness are more likely to become hubs i e to be highly connected

mean field solution of the small world network model the small world network model is a simple model of the structure of social networks which possesses characteristics of both regular lattices and random graphs the model consists of a one dimensional lattice with a low density of shortcuts added between randomly selected pairs of points these shortcuts greatly reduce the typical path length between any two points on the lattice we present a mean field solution for the average path length and for the distribution of path lengths in the model this solution is exact in the limit of large system size and either a large or small number of shortcuts

large deviations of random walks on random graphs we study the rare fluctuations or large deviations of time integrated functionals or observables of an unbiased random walk evolving on erdos renyi random graphs and construct a modified biased random walk that explains how these fluctuations arise in the long time limit two observables are considered the sum of the degrees visited by the random walk and the sum of their logarithm related to the trajectory entropy the modified random walk is used for both quantities to explain how sudden changes in degree fluctuations similar to dynamical phase transitions are related to localization transitions for the second quantity we also establish links between the large deviations of the trajectory entropy and the maximum entropy random walk

zero error capacity of a class of timing channels we analyze the problem of zero error communica tion through certain timing channels that can be interpreted as discrete time queues with bounded waiting times these timing channels are partly motivated by settings in which communica tion is done with rather unconventional physical carriers such as particles molecules items etc the channel model includes the following assumptions 1 at most n particles are sent in each time slot 2 every particle is delayed in the channel for a number of slots chosen randomly from the setf0 1 kg and 3 the particles are indistinguishable and hence the information is conveyed via the number of particles in each slot only it is shown that the zero error capacity of this channel is logr where r is the unique positive real root of the polynomialx k 1 x k n it is interesting to note that in the particular case n 1 k 1 the capacity is equal to log where 1 p 5 2 is the golden ratio capacity achieving codes are explicitly constructed index terms error correcting code zero error capacity zero error code timing channel timing capacity discrete time queue fibonacci sequence recurrence relation

diffusion on hierarchical systems of weakly coupled networks abstract we analyzed diffusion dynamics on weakly coupled networks interconnected networks by means of separation of time scales using an adiabatic approximation we reduced the system dynamics to a markov chain with aggregated variables and derived a transport equation that is analogous to fick s first law and includes a driving force entropy production is a sum of microscopic entropy transport which results from the particle s migration between networks of different topologies and macroscopic entropy production of the markov chain equilibrium particles partition between different sub networks depends only on internal sub network parameters by changing structure of networks one can not only modify diffusion constants but can also induce or reverse the direction of the particles flow between different networks our framework confirmed by numerical simulations is also useful for considering diffusion in nested systems corresponding to hierarchical networks with several different time scales thus it can serve to uncover hidden hierarchy levels from observations of diffusion processes

entropy of network ensembles in this paper we generalize the concept of random networks to describe network ensembles with nontrivial features by a statistical mechanics approach this framework is able to describe undirected and directed network ensembles as well as weighted network ensembles these networks might have nontrivial community structure or in the case of networks embedded in a given space they might have a link probability with a nontrivial dependence on the distance between the nodes these ensembles are characterized by their entropy which evaluates the cardinality of networks in the ensemble in particular in this paper we define and evaluate the structural entropy i e the entropy of the ensembles of undirected uncorrelated simple networks with given degree sequence we stress the apparent paradox that scale free degree distributions are characterized by having small structural entropy while they are so widely encountered in natural social and technological complex systems we propose a solution to the paradox by proving that scale free degree distributions are the most likely degree distribution with the corresponding value of the structural entropy finally the general framework we present in this paper is able to describe microcanonical ensembles of networks as well as canonical or hidden variable network ensembles with significant implications for the formulation of network constructing algorithms

research on the search ability of brownian particles on networks with an adaptive mechanism abstract in this paper we focus on the search ability of brownian particles with an adaptive mechanism in the adaptive mechanism nodes are allowed to be able to change their own accepting probability according to their congestion states two searching traffic models the static one in which nodes have fixed accepting probability to the incoming particles and the adaptive one in which nodes have adaptive accepting probability to the incoming particles are presented for testing the adaptive mechanism instead of number of hops we use the traveling time which includes not only the number of hops for a particle to jump from the source node to the destination but also the time that the particle stays in the queues of nodes to evaluate the search ability of brownian particles we apply two models to different networks the experiment results show that the adaptive mechanism can decrease the network congestion and the traveling time of the first arriving particle furthermore we investigate the influence of network topologies on the congestion of networks by addressing several main properties degree distribution average path length and clustering coefficient we show the reason why random topologies are more able to deal with congested traffic states than others we also propose an absorption strategy to deal with the additional brownian particles in networks the experiment results on barabasi albert ba scale free networks show that the absorption strategy can increase the probability of a successful search and decrease the average per node particles overhead for our models

two results on the permuting mailbox channel two problems of information transmission on the permuting mailbox or relay channel are considered such a channel is characterized by the cardinality epsilon of the alphabet and the number b of letters in the mailbox in the case b 1 an upper bound is obtained on the achievable rate in the jamming situation in the case epsilon 2 it is proven that time sharing is not always the best solution when both main users wish to transmit information to a third user

maximal entropy random walks in complex networks with limited information maximization of the entropy rate is an important issue to design diffusion processes aiming at a well mixed state we demonstrate that it is possible to construct maximal entropy random walks with only local information on the graph structure in particular we show that an almost maximal entropy random walk is obtained when the step probabilities are proportional to a power of the degree of the target node with an exponent that depends on the degree degree correlations and is equal to 1 in uncorrelated graphs

shannon and von neumann entropy of random networks with heterogeneous expected degree entropic measures of complexity are able to quantify the information encoded in complex network structures several entropic measures have been proposed in this respect here we study the relation between the shannon entropy and the von neumann entropy of networks with given expected degree sequence we find in different examples of network topologies that when the degree distribution contains some heterogeneity an intriguing correlation emerges between the two entropic quantities this results seems to suggest that heterogeneity in the expected degree distribution is implying an equivalence between a quantum and a classical description of networks which respectively corresponds to the von neumann and the shannon entropy

statistical ensemble of scale free random graphs a thorough discussion of the statistical ensemble of scale free connected random tree graphs is presented methods borrowed from field theory are used to define the ensemble and to study analytically its properties the ensemble is characterized by two global parameters the fractal and the spectral dimensions which are explicitly calculated it is discussed in detail how the geometry of the graphs varies when the weights of the nodes are modified the stability of the scale free regime is also considered when it breaks down either a scale is spontaneously generated or else a singular node appears and the graphs become crumpled a new computer algorithm to generate these random graphs is proposed possible generalizations are also discussed in particular more general ensembles are defined along the same lines and the computer algorithm is extended to arbitrary degenerate scale free random graphs

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

class of correlated random networks with hidden variables we study a class of models of correlated random networks in which vertices are characterized by hidden variables controlling the establishment of edges between pairs of vertices we find analytical expressions for the main topological properties of these models as a function of the distribution of hidden variables and the probability of connecting vertices the expressions obtained are checked by means of numerical simulations in a particular example the general model is extended to describe a practical algorithm to generate random networks with an a priori specified correlation structure we also present an extension of the class to map nonequilibrium growing networks to networks with hidden variables that represent the time at which each vertex was introduced in the system

identifying the borders of mathematical knowledge based on a divide and conquer approach knowledge about nature has been organized into a set of interrelated facts allowing a natural representation in terms of graphs each chunk of knowledge corresponds to a node while relationships between such chunks are expressed as edges this organization becomes particularly clear in the case of mathematical theorems with their intense cross implications and relationships we have derived a web of mathematical theorems from wikipedia and thanks to the powerful concept of entropy identified its more central and frontier elements our results also suggest that the central nodes are the oldest theorems while the frontier nodes are those recently added to the network the network communities have also been identified allowing further insights about the organization of this network such as its highly modular structure

using mapping entropy to identify node centrality in complex networks abstract the problem of finding the best strategy to attack a network or immunize a population with a minimal number of nodes has attracted much current research interest the assessment of node importance has been a fundamental issue in the research of complex networks in this paper we propose a new concept called mapping entropy me to identify the importance of a node in the complex network the concept is established according to the local information which considers the correlation among all neighbors of a node we evaluate the efficiency of the centrality by static and dynamic attacks on standard network models and real world networks the simulation result shows that the new centrality is more efficient than traditional attack strategies whether it is static or dynamic

entropy rate of diffusion processes on complex networks we introduce the concept of entropy rate to characterize a diffusion process on a complex network the entropy rate represents the minimal amount of information necessary to describe the diffusion on the network and is a quantity extremely sensitive to the network topology and dynamics by opportunely tuning the kind of diffusion the entropy rate allows one to extract different properties of the network structure moreover entropy maximization indicates how to design optimal diffusion processes providing a new theoretical tool with applications to social technological and communication systems

creating order in sequence spaces with simple machines abstract we intend to open a new research field towards say a theory of creating order under various constraints as a prototype of problems guiding our investigations we study models involving sequence spaces by creating order or equivalently organization we mean reducing in size the range of outputs by an organizer via a permuting channel a simple machine when it is fed by a given domain of inputs the creation of order is assumed to come only from the permutation operation in these channels four types of order creation are considered depending on the structure of the knowledge of the organizer limitations on mind about the future input and past output sequences and the kinds of admissible permutations inside the channel limitations on matter in any case the organizer s goal is to produce output spaces of minimal cardinality optimal organization we present some strategies of ordering and some first and seemingly basic optimality results after this more technical part of the paper we present some ideas about a general theory of ordering

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved