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

unwinding the hairball graph pruning algorithms for weighted complex networks empirical networks of weighted dyadic relations often contain noisy edges that alter the global characteristics of the network and obfuscate the most important structures therein graph pruning is the process of identifying the most significant edges according to a generative null model and extracting the subgraph consisting of those edges here we focus on integer weighted graphs commonly arising when weights count the occurrences of an event relating the nodes we introduce a simple and intuitive null model related to the configuration model of network generation and derive two significance filters from it the marginal likelihood filter mlf and the global likelihood filter glf the former is a fast algorithm assigning a significance score to each edge based on the marginal distribution of edge weights whereas the latter is an ensemble approach which takes into account the correlations among edges we apply these filters to the network of air traffic volume between us airports and recover a geographically faithful representation of the graph furthermore compared with thresholding based on edge weight we show that our filters extract a larger and significantly sparser giant component

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

generalized van der waerden identities the idea of generalized van der waerden identities is introduced to facilitate the polynomial expansion of multi spin functions of the type that are frequently encountered in theoretical studies of ising spin systems

on a question of bhatia and kittaneh abstract we settle in the affirmative a question of bhatia and kittaneh for p and q positive semidefinite n n matrices the inequality r pq 1 2 r p q holds for r 1 2 n

a singular value inequality for heinz means we prove a matrix inequality for matrix monotone functions and apply it to prove a singular value inequality for heinz means recently conjectured by zhan

measuring the dissimilarity of multiplex networks an empirical study of international trade networks abstract in recent years multiplex networks are becoming a research focus in the domain of complex networks discovering significant correlations between layers in multiplex networks can provide an insight to their structures in this study we propose some methods to measure the dissimilarities of different layers in directed and weighted multiplex networks the dissimilarity is defined on two levels node level and layer level the node dissimilarity is computed based on the distance of the probability distribution of its link weights vectors in different layers and the layer level dissimilarity is the weighted sum of the nodes dissimilarities furthermore the dissimilarity is disintegrated into the connection based dissimilarity and the weight based dissimilarity which represent the topological structure changes and the link weight changes respectively the proposed methods are applied to international trade networks

the spin 3 2 blume emery griffiths model abstract the spin 3 2 blume emery griffiths model is studied using the effective field theory based on the use of a probability distribution that correctly accounts for the single site kinematic relations the phase diagrams of the model on a square lattice are obtained

extracting the multiscale backbone of complex weighted networks a large number of complex systems find a natural abstraction in the form of weighted networks whose nodes represent the elements of the system and the weighted edges identify the presence of an interaction and its relative strength in recent years the study of an increasing number of large scale networks has highlighted the statistical heterogeneity of their interaction pattern with degree and weight distributions that vary over many orders of magnitude these features along with the large number of elements and links make the extraction of the truly relevant connections forming the network s backbone a very challenging problem more specifically coarse graining approaches and filtering techniques come into conflict with the multiscale nature of large scale systems here we define a filtering method that offers a practical procedure to extract the relevant connection backbone in complex multiscale networks preserving the edges that represent statistically significant deviations with respect to a null model for the local assignment of weights to edges an important aspect of the method is that it does not belittle small scale interactions and operates at all scales defined by the weight distribution we apply our method to real world network instances and compare the obtained results with alternative backbone extraction techniques

nonlocal evolution of weighted scale free networks we introduce the notion of globally updating evolution for a class of weighted networks in which the weight of a link is characterized by the amount of data packet transport flowing through it by noting that the packet transport over the network is determined nonlocally this approach can explain the generic nonlinear scaling between the strength and the degree of a node we demonstrate by a simple model that the strength driven evolution scheme recently introduced can be generalized to a nonlinear preferential attachment rule generating the power law behaviors in degree and in strength simultaneously

scale free brain functional networks functional magnetic resonance imaging is used to extract functional networks connecting correlated human brain sites analysis of the resulting networks in different tasks shows that a the distribution of functional connections and the probability of finding a link versus distance are both scale free b the characteristic path length is small and comparable with those of equivalent random networks and c the clustering coefficient is orders of magnitude larger than those of equivalent random networks all these properties typical of scale free small world networks reflect important functional information about brain states

network synchronization diffusion and the paradox of heterogeneity many complex networks display strong heterogeneity in the degree connectivity distribution heterogeneity in the degree distribution often reduces the average distance between nodes but paradoxically may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength here we offer a solution to this apparent paradox our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states we show that for a given degree distribution the maximum synchronizability is achieved when the network of couplings is weighted and directed and the overall cost involved in the couplings is minimum this enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size numerical verification of the main results is provided for representative classes of small world and scale free networks

information filtering in complex weighted networks many systems in nature society and technology can be described as networks where the vertices are the system s elements and edges between vertices indicate the interactions between the corresponding elements edges may be weighted if the interaction strength is measurable however the full network information is often redundant because tools and techniques from network analysis do not work or become very inefficient if the network is too dense and some weights may just reflect measurement errors and need to be be discarded moreover since weight distributions in many complex weighted networks are broad most of the weight is concentrated among a small fraction of all edges it is then crucial to properly detect relevant edges simple thresholding would leave only the largest weights disrupting the multiscale structure of the system which is at the basis of the structure of complex networks and ought to be kept in this paper we propose a weight filtering technique based on a global null model global statistical significance gloss filter keeping both the weight distribution and the full topological structure of the network the method correctly quantifies the statistical significance of weights assigned independently to the edges from a given distribution applications to real networks reveal that the gloss filter is indeed able to identify relevant connections between vertices

topology of whole brain functional mri networks improving the truncated scale free model abstract networks of connections within the human brain have been the subject of intense recent research yet their topology is still only partially understood we analyze weighted networks calculated from functional magnetic resonance imaging fmri data acquired during task performance expanding previous work in the area our analysis retains all of the connections between all of the voxels in the full brain fmri data computing correlations between approximately 200 000 voxels per subject for 10 subjects we evaluate the extent to which this rich dataset can be described by existing models of scale free or exponentially truncated scale free topology comparing results across a large number of more complex topological models as well our results suggest that the novel log quadratic model presented in this paper offers a significantly better fit to networks of functional connections at the voxel level in the human brain

novel matrix singular value inequalities and their applications to uncertain mimo channels novel matrix singular value inequalities are established for a sum product of three matrices their application to the uncertain compound multiple input multiple output mimo channel subject to normed additive uncertainty establishes the saddle point property for a wide range of performance metrics monotonic in the channel singular values including among others the mutual information mmse error exponent and pairwise error probability this in turn implies that the transmission on the eigenmodes of the nominal or worst case channel is also optimal for the whole set of channels under a general power constraint and hence achieves the compound channel capacity the worst case channel turns out to be antiparallel of the nominal one for all these performance metrics an application of these results to beamforming over compound mimo channels is discussed an optimal robust precoder for the uncertain mimo channel is obtained in a closed form under the sum mse criterion and the total power constraint the saddle point property is shown to hold and the optimal strategy is to diagonalize the nominal or worst case channel

transport in weighted networks partition into superhighways and roads transport in weighted networks is dominated by the minimum spanning tree mst the tree connecting all nodes with the minimum total weight we find that the mst can be partitioned into two distinct components having significantly different transport properties characterized by centrality the number of times a node or link is used by transport paths one component superhighways is the infinite incipient percolation cluster for which we find that nodes or links with high centrality dominate for the other component roads which includes the remaining nodes low centrality nodes dominate we find also that the distribution of the centrality for the infinite incipient percolation cluster satisfies a power law with an exponent smaller than that for the entire mst the significance of this finding is that one can improve significantly the global transport by improving a tiny fraction of the network the superhighways

social inertia in collaboration networks this work is a study of the properties of collaboration networks employing the formalism of weighted graphs to represent their one mode projection the weight of the edges is directly the number of times that a partnership has been repeated this representation allows us to define the concept of social inertia that measures the tendency of authors to keep on collaborating with previous partners we use a collection of empirical datasets to analyze several aspects of the social inertia 1 its probability distribution 2 its correlation with other properties and 3 the correlations of the inertia between neighbors in the network we also contrast these empirical results with the predictions of a recently proposed theoretical model for the growth of collaboration networks

modeling the evolution of weighted networks we present a general model for the growth of weighted networks in which the structural growth is coupled with the edges weight dynamical evolution the model is based on a simple weight driven dynamics and a weights reinforcement mechanism coupled to the local network growth that coupling can be generalized in order to include the effect of additional randomness and nonlinearities which can be present in real world networks the model generates weighted graphs exhibiting the statistical properties observed in several real world systems in particular the model yields a nontrivial time evolution of vertices properties and scale free behavior with exponents depending on the microscopic parameters characterizing the coupling rules very interestingly the generated graphs spontaneously achieve a complex hierarchical architecture characterized by clustering and connectivity correlations varying as a function of the vertices degree

transport on weighted networks when the correlations are independent of the degree most real world networks are weighted graphs with the weight of the edges reflecting the relative importance of the connections in this work we study nondegree dependent correlations between edge weights generalizing thus the correlations beyond the degree dependent case we propose a simple method to introduce weight weight correlations in topologically uncorrelated graphs this allows us to test different measures to discriminate between the different correlation types and to quantify their intensity we also discuss here the effect of weight correlations on the transport properties of the networks showing that positive correlations dramatically improve transport finally we give two examples of real world networks social and transport graphs in which weight weight correlations are present

a new technique in the effective field theory of general spin s dilute ising models abstract the effective field equations for diluted spin s ising models are derived using a probability distribution method based on the use of generalized van der waerden identities the approach is employed within the framework of a single site cluster theory based on the use of exact ising spin identities the effective field equations for ising models having all possible single ion and nearest neighbour pair interactions are reported for all spin values from s 1 2 to s 5 2

load distribution in weighted complex networks we study the load distribution in weighted networks by measuring the effective number of optimal paths passing through a given vertex the optimal path along which the total cost is minimum crucially depends on the cost distribution function p c c in the strong disorder limit where p c c ensuremath sim c ensuremath 1 the load distribution follows a power law both in the erd ifmmode mbox h o else h o fi s r enyi er random graphs and in the scale free sf networks and its characteristics are determined by the structure of the minimum spanning tree the distribution of loads at vertices with a given vertex degree also follows the sf nature similar to the whole load distribution implying that the global transport property is not correlated to the local structural information finally we measure the effect of disorder by the correlation coefficient between vertex degree and load finding that it is larger for er networks than for sf networks

explosive percolation in thresholded networks abstract explosive percolation in a network is a phase transition where a large portion of nodes becomes connected with an addition of a small number of edges although extensively studied in random network models and reconstructed real networks explosive percolation has not been observed in a more realistic scenario where a network is generated by thresholding a similarity matrix describing between node associations in this report i examine construction schemes of such thresholded networks and demonstrate that explosive percolation can be observed by introducing edges in a particular order

weight driven growing networks we study growing networks in which each link carries a certain weight randomly assigned at birth and fixed thereafter the weight of a node is defined as the sum of the weights of the links attached to the node and the network grows via the simplest weight driven rule a newly added node is connected to an already existing node with the probability which is proportional to the weight of that node we show that the node weight distribution n w has a universal tail that is it is independent of the link weight distribution n w approximately w 3 as w infinity results are particularly neat for the exponential link weight distribution when n w is algebraic over the entire weight range

numerical evaluation of the upper critical dimension of percolation in scale free networks we propose numerical methods to evaluate the upper critical dimension d c of random percolation clusters in erd ifmmode mbox h o else h o fi s r enyi networks and in scale free networks with degree distribution mathcal p k ensuremath sim k ensuremath ensuremath lambda where k is the degree of a node and ensuremath lambda is the broadness of the degree distribution our results support the theoretical prediction d c 2 ensuremath lambda ensuremath 1 ensuremath lambda ensuremath 3 for scale free networks with 3l ensuremath lambda l4 and d c 6 for erd ifmmode mbox h o else h o fi s r enyi networks and scale free networks with ensuremath lambda g4 when the removal of nodes is not random but targeted on removing the highest degree nodes we obtain d c 6 for all ensuremath lambda g2 our method also yields a better numerical evaluation of the critical percolation threshold p c for scale free networks our results suggest that the finite size effects increases when ensuremath lambda approaches 3 from above

structure and evolution of the world trade network the world trade web wtw the network defined by the international import export trade relationships has been recently shown to display some important topological properties which are tightly related to the gross domestic product of world countries while our previous analysis focused on the static undirected version of the wtw here we address its full evolving directed description this is accomplished by exploiting the peculiar reciprocity structure of the wtw to recover the directed nature of international trade channels and by studying the temporal dependence of the parameters describing the wtw topology

the architecture of complex weighted networks networked structures arise in a wide array of different contexts such as technological and transportation infrastructures social phenomena and biological systems these highly interconnected systems have recently been the focus of a great deal of attention that has uncovered and characterized their topological complexity along with a complex topological structure real networks display a large heterogeneity in the capacity and intensity of the connections these features however have mainly not been considered in past studies where links are usually represented as binary states i e either present or absent here we study the scientific collaboration network and the world wide air transportation network which are representative examples of social and large infrastructure systems respectively in both cases it is possible to assign to each edge of the graph a weight proportional to the intensity or capacity of the connections among the various elements of the network we define appropriate metrics combining weighted and topological observables that enable us to characterize the complex statistical properties and heterogeneity of the actual strength of edges and vertices this information allows us to investigate the correlations among weighted quantities and the underlying topological structure of the network these results provide a better description of the hierarchies and organizational principles at the basis of the architecture of weighted networks

universal behavior of optimal paths in weighted networks with general disorder we study the statistics of the optimal path in both random and scale free networks where weights are taken from a general distribution p w we find that different types of disorder lead to the same universal behavior specifically we find that a single parameter s defined as al 1 v for d dimensional lattices and s defined as an 1 3 for random networks determines the distributions of the optimal path length including both strong and weak disorder regimes here v is the percolation connectivity exponent and a depends on the percolation threshold and p w we show that for a uniform p w poisson or gaussian the crossover from weak to strong does not occur and only weak disorder exists

consistency based thresholding of the human connectome densely seeded probabilistic tractography yields weighted networks that are nearly fully connected hence containing many spurious fibers it is thus necessary to prune spurious connections from probabilistically derived networks to obtain a more reliable overall estimate of the connectivity a standard method is to threshold by weight keeping only the strongest edges here by measuring the consistency of edge weights across subjects we propose a new thresholding method that aims to reduce the rate of false positives in group averaged connectivity matrices close inspection of the relationship between consistency weight and distance suggests that the most consistent edges are in fact those that are strong for their length rather than simply strong overall hence retaining the most consistent edges preserves more long distance connections than traditional weight based thresholding which penalizes long connections for being weak regardless of anatomy by comparing our thresholded networks to mouse and macaque tracer data we also show that consistency based thresholding exhibits the species invariant exponential decay of connection weights with distance while weight based thresholding does not we also show that consistency based thresholding can be used to identify highly consistent and highly inconsistent subnetworks across subjects enabling more nuanced analyses of group level connectivity than just the mean connectivity

an arithmetic geometric mean inequality for singular values and its applications abstract in this short note we give a new equivalent form of the arithmetic geometric mean inequality for singular values as applications of our result we give a new proof of an inequality due to bhatia and davis 1993 4 and we obtain a singular value inequality for matrix means which is similar to one proved by drury 2012 9 finally we present a log majorization inequality for singular values

statistical properties of randomly broken objects and of multivalley structures in disordered systems the statistical properties of the multivalley structure of disordered systems and of randomly broken objects have many features in common for all these problems if w denotes the weight of the sth piece we show that the probability distributions pi w of the largest piece w p2 w of the second largest piece and n y of y 1 wi always have singularities at w l n w2 l n and y ijn n 1 2 3

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

optimal paths in complex networks with correlated weights the worldwide airport network we study complex networks with weights wij associated with each link connecting node i and j the weights are chosen to be correlated with the network topology in the form found in two real world examples a the worldwide airport network and b the e coli metabolic network here wijxijkikj where ki and kj are the degrees of nodes i and j xij is a random number and represents the strength of the correlations the case 0 represents correlation between weights and degree while 0 represents anticorrelation and the case 0 reduces to the case of no correlations we study the scaling of the lengths of the optimal paths opt with the system size n in strong disorder for scale free networks for different we find two different universality classes for opt in strong disorder depending on i if 0 then for 2 the scaling law optn 1 3 where is the power law exponent of the degree distribution of scale free networks and ii if 0 then opt nopt with opt identical to its value for the uncorrelated case 0 we calculate the robustness of correlated scale free networks with different and find the networks with 0 to be the most robust networks when compared to the other values of we propose an analytical method to study percolation phenomena on networks with this kind of correlation and our numerical results suggest that for scale free networks with 0 the percolation threshold pc is finite for 3 which belongs to the same universality class as 0 we compare our simulation results with the real worldwide airport network and we find good agreement

on singular value inequalities for matrix means abstract for positive semidefinite n n matrices a and b the singular value inequality 2 t s j a r b 2 r a 2 r b r 2 s j a 2 t a b b 2 is shown to hold for r 1 2 1 3 2 and all 2 t 2

structural properties of statistically validated empirical information networks abstract we construct the empirical information network ein of traders using the order flow data of the constituent stocks of szse 100 index in 2013 a statistical validation method is applied to the edges of the network to filter out noises and uncover the intrinsic interaction behaviors of traders we investigate the correlation between topological structures and statistical properties for their largest connected components we find that the statistical validated network shows an assortative mixing pattern while the original network exhibits a disassortative mixing pattern we consider two definitions of edge weight for comparison but there is no significant difference in a same network we also analyze the mutual relationships among node degree edge weight and node strength

generalized percolation in random directed networks we develop a general theory for percolation in directed random networks with arbitrary two point correlations and bidirectional edges that is edges pointing in both directions simultaneously these two ingredients alter the previously known scenario and open new views and perspectives on percolation phenomena equations for the percolation threshold and the sizes of the giant components are derived in the most general case we also present simulation results for a particular example of uncorrelated network with bidirectional edges confirming the theoretical predictions

a theoretical study of the diluted spin 5 2 blume capel model the quenched site diluted spin 52 blume capel model is studied using a new technique in effective field theory that employs a probability distribution within the framework of single site cluster theory based on the use of exact ising spin identities the probability distribution derived with the aid of tucker s generalised van der waerden operators correctly account for the single site kinematic relations and certain configurational averaging identities the concentration dependence of the ferromagnetic phase boundary is studied in particular it is found that a range of single ion field strengths exists over which the critical concentration for ordering varies anomalously results for the honeycomb square and cubic lattices are presented the thermal variation of the magnetization the quadrupolar and higher order magnetization moments are presented for selected values of the concentration

notes on matrix arithmetic geometric mean inequalities abstract for positive semi definite n n matrices the inequality 4 ab a b 2 isshown to hold for every unitarily invariant norm the connection of this with some other matrix arithmetic geometric mean inequalities and trace inequalities is discussed