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

structural properties of planar graphs of urban street patterns recent theoretical and empirical studies have focused on the structural properties of complex relational networks in social biological and technological systems here we study the basic properties of twenty 1 square mile samples of street patterns of different world cities samples are turned into spatial valued graphs in such graphs the nodes are embedded in the two dimensional plane and represent street intersections the edges represent streets and the edge values are equal to the street lengths we evaluate the local properties of the graphs by measuring the meshedness coefficient and counting short cycles of three four and five edges and the global properties by measuring global efficiency and cost we also consider as extreme cases minimal spanning trees mst and greedy triangulations gt induced by the same spatial distribution of nodes the measures found in the real and the artificial networks are then compared surprisingly cities of the same class e g grid iron or medieval exhibit roughly similar properties the correlation between a priori known classes and statistical properties is illustrated in a plot of relative efficiency vs cost

sampled data fuzzy control for a class of nonlinear parabolic distributed parameter systems under spatially point measurements abstract in this paper the exponential stabilization problem is addressed for a class of nonlinear parabolic partial differential equation pde systems via sampled data fuzzy control approach initially the nonlinear pde system is accurately represented by the takagi sugeno t s fuzzy pde model then based on the t s fuzzy pde model a novel time dependent lyapunov functional is used to design a sampled data fuzzy controller under spatially point measurements such that the closed loop fuzzy pde system is exponentially stable with a given decay rate the stabilization conditions are presented in terms of a set of linear matrix inequalities lmis finally simulation results on the control of the diffusion equation and the fitzhugh nagumo fhn equation to illustrate the effectiveness of the proposed design method

scaling relations in food webs in the last three decades researchers have tried to identify universal patterns in the structure of food webs it was recently proposed that the exponent characterizing the efficiency of the transport of energy in large and small food webs might have a universal value 1 13 in this work we establish lower and upper bounds for this exponent in a general spanning tree with a fixed number of trophic species and levels when the number of species is large the lower and upper bounds are equal to 1 implying that the result 1 13 is due to finite size effects and that the value of this exponent depends on the size of the web we also evaluate analytically and numerically the exponent for hierarchical and random networks in all cases the exponent depends on the number of trophic species k and when k is large we have that 1 moreover this result holds for any fixed number m of trophic levels

random geometric graphs we analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present the critical connectivity is found numerically by examining the size of the largest cluster we derive an analytical expression for the cluster coefficient which shows that the graphs are distinctly different from standard random graphs even for infinite dimensionality insights relevant for graph bipartitioning are included

geographical effects on the path length and the robustness in complex networks the short paths between any two nodes and the robustness of connectivity are advanced properties of scale free sf networks however they may be affected by geographical constraints in realistic situations we consider geographical networks with the sf structure based on planar triangulation for online routings and suggest scaling relations between the average distance or number of hops on the optimal paths and the network size we also show that the tolerance to random failures and attacks on hubs is weakened in geographical networks and that even then it is possible for the extremely vulnerable ones to be improved by adding with the local exchange of links

lattice scale free networks with weighted linking recently models of the scale free sf networks on lattices were investigated which consider the influence of the embedded space on the networks since a lot of real networks exist on the 2d global surface it is helpful to discuss these models in this paper based on the lattice sf networks model a linking weight is added thus an additional parameter which can control the clustering coefficient is introduced depending on the linking weight the properties of the model change smoothly from the lattice sf model to the sf random graphs

self similar disk packings as model spatial scale free networks the network of contacts in space filling disk packings such as the apollonian packing is examined these networks provide an interesting example of spatial scale free networks where the topology reflects the broad distribution of disk areas a wide variety of topological and spatial properties of these systems is characterized their potential as models for networks of connected minima on energy landscapes is discussed

mixing patterns in networks we study assortative mixing in networks the tendency for vertices in networks to be connected to other vertices that are like or unlike them in some way we consider mixing according to discrete characteristics such as language or race in social networks and scalar characteristics such as age as a special example of the latter we consider mixing according to vertex degree i e according to the number of connections vertices have to other vertices do gregarious people tend to associate with other gregarious people we propose a number of measures of assortative mixing appropriate to the various mixing types and apply them to a variety of real world networks showing that assortative mixing is a pervasive phenomenon found in many networks we also propose several models of assortatively mixed networks both analytic ones based on generating function methods and numerical ones based on monte carlo graph generation techniques we use these models to probe the properties of networks as their level of assortativity is varied in the particular case of mixing by degree we find strong variation with assortativity in the connectivity of the network and in the resilience of the network to the removal of vertices

localized activity profiles and storage capacity of rate based autoassociative networks we study analytically the effect of metrically structured connectivity on the behavior of autoassociative networks the steady state equations are derived for a generic input output function and then we focus on their solutions in the case of networks composed of three alternative simple rate based model neurons threshold linear binary or smoothly saturating units for a connectivity which is short range enough the threshold linear network shows localized retrieval states the saturating and binary models also exhibit spatially modulated retrieval states if the highest activity level that they can achieve is above the maximum activity of the units in the stored patterns we show that this saturation level together with the linear gain of the transfer function are important parameters that determine the possibility of localized retrieval if the ratio of the number of stored patterns to the number of connections per unit goes to zero while the latter goes to infinity it is possible to derive an analytical formula for the critical value of the connectivity width below which one observes spatially nonuniform retrieval states the formula is also shown to offer a good first approximation for higher storage loads we show that even in the case of localized retrieval the storage capacity remains proportional to the number of connections per neurons with the proportionality constant lower by a factor of 3 char21 4 compared to uniform retrieval the approach that we present here is generic in the sense that there are no specific assumptions on the single unit input output function nor on the exact connectivity structure

geographical embedding of scale free networks a method for embedding graphs in euclidean space is suggested the method connects nodes to their geographically closest neighbors and economizes on the total physical length of links the topological and geometrical properties of scalefree networks embedded by the suggested algorithm are studied both analytically and through simulations our findings indicate dramatic changes in the embedded networks in comparison to their off lattice counterparts and call into question the applicability of off lattice scale free models to realistic everyday life networks

connectivity distribution of spatial networks we study spatial networks constructed by randomly placing nodes on a manifold and joining two nodes with an edge whenever their distance is less than a certain cutoff we derive the general expression for the connectivity distribution of such networks as a functional of the distribution of the nodes we show that for regular spatial densities the corresponding spatial network has a connectivity distribution decreasing faster than an exponential in contrast we also show that scale free networks with a power law decreasing connectivity distribution are obtained when a certain information measure of the node distribution integral of higher powers of the distribution diverges we illustrate our results on a simple example for which we present simulation results finally we speculate on the role played by the limiting case p k proportional variant k 1 which appears empirically to be relevant to spatial networks of biological origin such as the ones constructed from gene expression data

scale free network on euclidean space optimized by rewiring of links a barabasi albert scale free network is constructed whose nodes are the poisson distributed random points within a unit square and links are the straight line connections among the nodes the cost function which is the total wiring length associated with such a network defined on a two dimensional plane is optimized the optimization process consists of random selection of a pair of links and rewiring them to reduce the total length of the pair but with the constraint that the degree as well as the out degree and in degree of each node are precisely maintained the resulting optimized network has a small diameter as well as high clustering and the link length distribution has a stretched exponential tail

a spatial weighted network model based on optimal expected traffic abstract we propose a spatial weighted network model based on the optimal expected traffic the expected traffic represents the prediction of the flow created by two vertices and is calculated by the improved gravity equation the model maximizes the total expected traffic of the network by changing two parameters which control the fitness and the geographical constraints the model can vary its topology and give rise to a variety of statistical properties observed in the real world network notably our study shows that a linear and a nonlinear strength degree correlation can emerge when considering and neglecting the transport effect respectively

correlations in scale free networks tomography and percolation we discuss three related models of scale free networks with the same degree distribution but different correlation properties starting from the barabasi albert construction based on growth and preferential attachment we discuss two other networks emerging when randomizing it with respect to links or nodes we point out that the barabasi albert model displays dissortative behavior with respect to the nodes degrees while the node randomized network shows assortative mixing these kinds of correlations are visualized by discussing the shell structure of the networks around an arbitrary node in spite of different correlation behaviors all three constructions exhibit similar percolation properties this result for percolation is also detected for a network with finite second moment and its corresponding randomized models

evolutionary construction of geographical networks with nearly optimal robustness and efficient routing properties abstract robust and efficient design of networks on a realistic geographical space is one of the important issues for the realization of dependable communication systems in this paper based on a percolation theory and a geometric graph property we investigate such a design from the following viewpoints 1 network evolution according to a spatially heterogeneous population 2 trimodal low degrees for the tolerant connectivity against both failures and attacks and 3 decentralized routing within short paths furthermore we point out the weakened tolerance by geographical constraints on local cycles and propose a practical strategy by adding a small fraction of shortcut links between randomly chosen nodes in order to improve the robustness to a similar level to that of the optimal bimodal networks with a larger degree o n for the network size n these properties will be useful for constructing future ad hoc networks in wide area communications

evolving networks with distance preferences we study evolving networks where new nodes when attached to the network form links with other nodes of preferred distances a particular case is where always the shortest distances are selected make friends with the friends of your present friends we present simulation results for network parameters like the first eigenvalue of the graph laplacian synchronizability clustering coefficients average distances and degree distributions for different distance preferences and compare them with the parameter values for random and scale free networks we find that for the shortest distance rule we obtain a power law degree distribution as in scale free networks while the other parameters are significantly different especially the clustering coefficient

clustering properties of a generalized critical euclidean network many real world networks exhibit a scale free feature have a small diameter and a high clustering tendency we study the properties of a growing network which has all these features in which an incoming node is connected to its ith predecessor of degree k i with a link of length l using a probability proportional to k beta i l alpha for alpha 0 5 the network is scale free at beta 1 with the degree distribution p k proportional to k gamma and gamma 3 0 as in the barabasi albert model alpha 0 beta 1 we find a phase boundary in the alpha beta plane along which the network is scale free interestingly we find a scale free behavior even for beta 1 for alpha 0 5 where the existence of a different universality class is indicated from the behavior of the degree distribution and the clustering coefficients the network has a small diameter in the entire scale free region the clustering coefficients emulate the behavior of most real networks for increasing negative values of alpha on the phase boundary

evolving networks with disadvantaged long range connections we consider a growing network whose growth algorithm is based on the preferential attachment typical for scale free constructions but where the long range bonds are disadvantaged thus the probability of getting connected to a site at distance d is proportional to d alpha where alpha is a tunable parameter of the model we show that the properties of the networks grown with alpha 1 the structure of the network is quite different thus in this regime the node degree distribution is no longer a power law and it is well represented by a stretched exponential on the other hand the small world property of the growing networks is preserved at all values of alpha

growing networks under geographical constraints inspired by the structure of technological weblike systems we discuss network evolution mechanisms which give rise to topological properties found in real spatial networks thus we suggest that the peculiar structure of transport and distribution networks is fundamentally determined by two factors these are the dependence of the spatial interaction range of vertices on the vertex attractiveness or importance within the network and on the inhomogeneous distribution of vertices in space we propose and analyze numerically a simple model based on these generating mechanisms which seems for instance to be able to reproduce known structural features of the internet

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

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

modulated scale free network in euclidean space a random network is grown by introducing at unit rate randomly selected nodes on the euclidean space a node is randomly connected to its ith predecessor of degree k i with a directed link of length l using a probability proportional to k i l alpha our numerical study indicates that the network is scale free for all values of alpha alpha c and the degree distribution decays stretched exponentially for the other values of alpha the link length distribution follows a power law d l approximately l delta where delta is calculated exactly for the whole range of values of alpha