a maximum entropy framework for nonexponential distributions probability distributions having power law tails are observed in a broad range of social economic and biological systems we describe here a potentially useful common framework we derive distribution functions for situations in which a joiner particle k pays some form of price to enter a community of size where costs are subject to economies of scale maximizing the boltzmann gibbs shannon entropy subject to this energy like constraint predicts a distribution having a power law tail it reduces to the boltzmann distribution in the absence of economies of scale we show that the predicted function gives excellent fits to 13 different distribution functions ranging from friendship links in social networks to protein protein interactions to the severity of terrorist attacks this approach may give useful insights into when to expect power law distributions in the natural and social sciences

optimal network configuration for maximal coherence resonance in excitable systems we analyze the coherence resonance phenomenon in an ensemble of noise driven excitable neurons giving special attention to the role of the interaction topology the neural architecture is modeled using a spatially embedded network in which we can tune the network organization between scale free like with dominating long range connections and a network with mostly adjacent neurons connected we found that besides an optimal noise intensity also an optimal network configuration exists at which the largest average coherence of noise induced spikes is achieved furthermore we show that long as well as short range interactions between neurons should exist in order to achieve the optimal response of the neuronal network

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

weighted scale free networks with variable power law exponents abstract we present a weighted scale free network model in which the power law exponents can be controlled by the model parameters the network is generated through the weight driven preferential attachment of new nodes to existing nodes and the growth of the weights of existing links the simplicity of the model enables us to derive analytically the various statistical properties such as the distributions of degree strength and weight the degree strength and degree weight relationship and the dependencies of these power law exponents on the model parameters finally we demonstrate that networks of words coauthorship of researchers and collaboration of actor actresses are quantitatively well described by this model

statistical mechanics of the international trade network analyzing real data on international trade covering the time interval 1950 2000 we show that in each year over the analyzed period the network is a typical representative of the ensemble of maximally random weighted networks whose directed connections bilateral trade volumes are only characterized by the product of the trading countries gdps it means that time evolution of this network may be considered as a continuous sequence of equilibrium states i e a quasistatic process this in turn allows one to apply the linear response theory to make and also verify simple predictions about the network in particular we show that bilateral trade fulfills a fluctuation response theorem which states that the average relative change in imports exports between two countries is a sum of the relative changes in their gdps yearly changes in trade volumes prove that the theorem is valid

scale free property of local world networks and their community structures abstract the scale free property and community structures of complex networks formed by local events have been studied theoretically numerically and empirically we showed analytically and numerically that the degree distribution function p k of the local world evolving network exhibits a crossover from an exponential to power law form by increasing the local world size m for m much larger than the crossover local world size m co the distribution function p k has a power law form for any degree k 1 below m co however p k obeys a power law for 1 k k co and decays exponentially for k k co the crossover size m co and the crossover degree k co have been also elucidated in addition we constructed the drug prescription network dpn as a real local world network in which the local world subsets are definitely specified to reveal how the local world nature affects properties of real world networks we found that the community structure of the dpn strongly correlates with the local worlds

hamiltonian dynamics of preferential attachment prediction and control of network dynamics are grand challenge problems in network science the lack of understanding of fundamental laws driving the dynamics of networks is among the reasons why many practical problems of great significance remain unsolved for decades here we study the dynamics of networks evolving according to preferential attachment pa known to approximate well the large scale growth dynamics of a variety of real networks we show that this dynamics is hamiltonian thus casting the study of complex networks dynamics to the powerful canonical formalism in which the time evolution of a dynamical system is described by hamilton s equations we derive the explicit form of the hamiltonian that governs network growth in pa this hamiltonian turns out to be nearly identical to graph energy in the configuration model which shows that the ensemble of random graphs generated by pa is nearly identical to the ensemble of random graphs with scale free degree distributions in other words pa generates nothing but random graphs with power law degree distribution the extension of the developed canonical formalism for network analysis to richer geometric network models with non degenerate groups of symmetries may eventually lead to a system of equations describing network dynamics at small scales

on the regularity of weak solutions of a one dimensional two phase degenerate stefan problem with convection we shall be concerned with the following one dimensional two phase stefan problem let t 0 and q 0 1 x 0 t find a pair of functions u x t s t which are defined on qt and 0 t respectively and which satisfy al u uxx f u u in x t o x s l o t t l la az ub4 u xx in x t s t x l o tdt l lb u s t r 0 o t t l lc i t u s t t u s t t o t t lid u 0 t 1 06t t l le u 1 t 1 obt t l lf u x 0 u x o x l 1 k s 0 so e 0 1 l lh here so is a given number and the ar a f and u are known functions of their arguments problem 1 1 may be viewed as a model of heat conduction in a region involving phase change an example situation is an ice water mix in which circulation of the water is present then the term f u u represents the contribution of the field velocity generated by the water flow in the liquid phase a more detailed discussion on the formulation of 1 1 can be found elsewhere see e g s

compatibility as underlying mechanism behind the evolution of networks abstract in this work we propose a dynamic network model based on an evolutive rule called compatibility the presence of common topological properties observed in two real complex networks are well reproduced by the proposed model these results suggest that the compatibility between nodes is sufficient to generate the complexity observed in real systems even considering that the new added nodes do not have the ability to know global information of the system

generating multi scaling networks with two types of vertices abstract a variety of scale free networks have been created since the pioneer work by barabasi and albert science 286 1999 509 most of these models are homogeneous since they are composed of the same kind of nodes in the realistic world however elements nodes or vertices in the network may play different roles or have different functions in this work we develop an alternative way of vertex classification other than the ordinary modularity method by introducing two types of vertices the interaction between two neighbor vertices is dependent on their types it is found that the vertex degree exhibits a multi scaling law distribution with the scaling exponent of each types of vertex adjustable this network model may exhibit some interesting properties concerning the dynamical processes on it

time cumulative scale free networks without both growth and preferential attachment abstract a model of nongrowing networks with time cumulative scale free sf property based on the apply reply process is proposed in this paper this model is not only without growth but also without a direct expression of preferential attachment the network structure is evolved by the change of nodes active probability which is mathematically equivalent to a special diffusion process this model only allows zero or one edge connecting on a node at any moment and the power law degree distributions can be exhibited from the statistic of the network after a long time accumulation our results imply that both growth and direct preferential attachments are not necessary in the generation of the sf property

evolution of a modified binomial random graph by agglomeration in the classical erd s renyi random graph g n p there are n vertices and each of the possible edges is independently present with probability p the random graph g n p is homogeneous in the sense that all vertices have the same characteristics on the other hand numerous real world networks are inhomogeneous in this respect such an inhomogeneity of vertices may influence the connection probability between pairs of vertices the purpose of this paper is to propose a new inhomogeneous random graph model which is obtained in a constructive way from the erd s renyi random graph g n p given a configuration of n vertices arranged in n subsets of vertices we call each subset a super vertex we define a random graph with n super vertices by letting two super vertices be connected if and only if there is at least one edge between them in g n p our main result concerns the threshold for connectedness we also analyze the phase transition for the emergence of the giant component and the degree distribution even though our model begins with g n p it assumes the existence of some community structure encoded in the configuration furthermore under certain conditions it exhibits a power law degree distribution both properties are important for real world applications

regularity results for a one dimensional two phase degenerate stefan problem l lh 40 30 l li where a u 0 and ai 0 for i 1 2 in the classical stefan problem the specific heat e u is required to be strictly positive where a u ml u if u is in q and 11 if u is in q however in more general free boundary problems various types of degeneracy in a can occur in the electro chemical machining process the specific heat is zero in both phases resulting in elliptic differential equations but with a stefan type boundary condition see 3 for a description of this model in the study of saturated unsaturated flow in

exploring complex networks by walking on them we carry out a comparative study of the problem of a walker searching several typical complex networks the search efficiency is evaluated for various strategies having no knowledge of the global properties of the underlying networks and the optimal path between any two given nodes it is found that the best search strategy is the self avoiding random walk the preferentially self avoiding random walk does not help in improving the search efficiency further in return topological information of the underlying networks may be drawn by comparing the results of the different search strategies

riemannian geometric entropy for measuring network complexity a central issue in the science of complex systems is the quantitative characterization of complexity in the present work we address this issue by resorting to information geometry actually we propose a constructive way to associate with a in principle any network a differentiable object a riemannian manifold whose volume is used to define the entropy the effectiveness of the latter in measuring network complexity is successfully proved through its capability of detecting a classical phase transition occurring in both random graphs and scale free networks as well as of characterizing small exponential random graphs configuration models and real 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

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

social aggregation as a cooperative game abstract a new approach for the description of phenomena of social aggregation is suggested on the basis of psychological concepts as for instance social norms and cultural coordinates we deduce a general mechanism for social aggregation in which different clusters of individuals can merge according to cooperation among the agents in their turn the agents can cooperate or defect according to the clusters distribution inside the system the fitness of an individual increases with the size of its cluster but decreases with the work the individual had to do in order to join it in order to test the reliability of such a new approach we introduce a couple of simple toy models with the features illustrated above we see from this preliminary study how cooperation is the most convenient strategy only in the presence of very large clusters while on the other hand it is not necessary to have one hundred percent of cooperators for reaching a totally ordered configuration with only one megacluster filling the whole system

generation of scale free networks using a simple preferential rewiring dynamics abstract we propose a simple dynamical process for non growing networks where steady states in the long time limit exhibit power law degree distributions with the exponent 2 at each time step two nodes i and j are randomly selected and one incoming link to i is redirected to j with the rewiring probability r determined only by degrees of two nodes k i and k j while higher degree nodes are preferred to get another link this is an application of the general model introduced earlier s ree phys rev e 73 2006 026115 to take the structure of networks into account we also consider three possible distinctions for the model i how we choose a rewiring link out of all incoming links to i three cases ii whether links are directed or not two cases iii types of networks considering the existence of self loops and multiple links two cases as a result we specify the total of 12 different cases of the model we then observe numerically that networks will evolve to steady states with power law degree distributions when parameters of the model satisfy certain conditions this work is from an effort to find a simple model of the network dynamics generating scale free networks and has a potential to become an underlying mechanism for wide range of scale free non growing networks

random networks with tunable degree distribution and clustering we present an algorithm for generating random networks with arbitrary degree distribution and clustering frequency of triadic closure we use this algorithm to generate networks with exponential power law and poisson degree distributions with variable levels of clustering such networks may be used as models of social networks and as a testable null hypothesis about network structure finally we explore the effects of clustering on the point of the phase transition where a giant component forms in a random network and on the size of the giant component some analysis of these effects is presented

interacting individuals leading to zipf s law institut de physique th eorique universit e de fribourg ch 1700 february 1 2008 we present a general approach to explain the zipf s law of city distribution if the simplestinteraction pairwise is assumed individuals tend to form cities in agreement with the well knownstatisticspacs numbers 89 50 r 05 20 y 05 40 j

on the existence of the weak solution and the regularity of the free boundary to a one dimensional two phase degenerate stefan problem existence and regularity of the free boundary s t are demonstrated for the weak solution u u x t and s s t of the degenerate stefan problem 1 u ut uxx 0 0 0 t t u 1 t 2 t 0 0 t t u x 0 u0 x 0 x 1 u s t t 0 0 t t t ux s t 0 t ux s t 0 t 0 t t s 0 s0 where i u 0 and i 0 0 i 1 2

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

general free boundary evolution problems in several space dimensions abstract existence results are given for the weak formulation of two phase free boundary evolution problems with free boundary jump conditions generalizing that of stefan and with possibly vanishing coefficient of time derivative

scale free property of directed networks with two intrinsic node weights abstract this paper proposes a network model to understand the scale free property of directed networks the proposed model assigns two intrinsic variables incoming and outgoing weights to every node a directed link is established from node i to node j if the sum of the outgoing weight of node i and the incoming weight of node j exceeds a predetermined threshold the proposed model allows us to know the exact analytical expressions for degree distributions and clustering we analytically find that the in degree and out degree distributions have power law tails and their scaling exponents are controllable within the range 1 the average clustering coefficient of nodes with out degree or in degree n also has a power law tail as a function of n we also find that the scaling exponent of the clustering coefficient depends on the correlation between incoming and outgoing weights

number of cliques in random scale free network ensembles abstract in this paper we calculate the average number of cliques in random scale free networks in the limit of large network size n 1 we consider first the hidden variable ensemble and subsequently the molloy reed ensemble in both cases we find that cliques i e fully connected subgraphs appear also when the average degree is finite this is in contrast to what happens in erdos and renyi graphs in which diverging average degree is required to observe cliques of size c 3 moreover we show that in random scale free networks the clique number i e the size of the largest clique present in the network diverges with the system size

what big data tells sampling the social network by communication channels big data has become the primary source of understanding the structure and dynamics of the society at large scale the network of social interactions can be considered as a multiplex where each layer corresponds to one communication channel and the aggregate of all of them constitutes the entire social network however usually one has information only about one of the channels or even a part of it which should be considered as a subset or sample of the whole here we introduce a model based on a natural bilateral communication channel selection mechanism which for one channel leads to consistent changes in the network properties for example while it is expected that the degree distribution of the whole social network has a maximum at a value larger than one we get a monotonically decreasing distribution as observed in empirical studies of single channel data we also find that assortativity may occur or get strengthened due to the sampling method we analyze the far reaching consequences of our findings

g2 van der waerden invariant the g2 van der waerden invariant is given it solves the external labeling problem connected with direct products of irreducible representations of g2

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

scale free networks from a hamiltonian dynamics contrary to many recent models of growing networks we present a model with fixed number of nodes and links where a dynamics favoring the formation of links between nodes with degree of connectivity as different as possible is introduced by applying a local rewiring move the network reaches equilibrium states assuming broad degree distributions which have a power law form in an intermediate range of the parameters used interestingly in the same range we find nontrivial hierarchical clustering

numerical solution of inverse heat conduction problem with nonstationary measurements the inverse heat conduction problem involves the calculation of surface heat flux and or temperature histories from transient measured temperatures inside solid in this paper the temperature is not specified at x 0 in an inverse heat conduction problem we choose an additional moving boundary condition this problem may be subdivided into two separate problems one of these problems is a moving boundary problem which is solved exactly by means of fundamental solution of heat equation the second problem is an inverse moving boundary problem which is solved with the residual minimization least squares method by using numerical examples we show the accuracy of our method