some bounds for total communicability of graphs abstract in a network or a graph the total communicability tc has been defined as the sum of the entries in the exponential of the adjacency matrix of the network this quantity offers a good measure of how easily information spreads across the network and can be useful in the design of networks having certain desirable properties in this paper we obtain some bounds for total communicability of a graph g t c g in terms of spectral radius of the adjacency matrix number of vertices number of edges minimum degree and the maximum degree of g moreover we find some upper bounds for t c g when g is the cartesian product tensor product or the strong product of two graphs in addition nordhaus gaddum type results for the total communicability of a graph g are established

subgraph centrality in complex networks we introduce a new centrality measure that characterizes the participation of each node in all subgraphs in a network smaller subgraphs are given more weight than larger ones which makes this measure appropriate for characterizing network motifs we show that the subgraph centrality c s i can be obtained mathematically from the spectra of the adjacency matrix of the network this measure is better able to discriminate the nodes of a network than alternate measures such as degree closeness betweenness and eigenvector centralities we study eight real world networks for which c s i displays useful and desirable properties such as clear ranking of nodes and scale free characteristics compared with the number of links per node the ranking introduced by c s i for the nodes in the protein interaction network of s cereviciae is more highly correlated with the lethality of individual proteins removed from the proteome

developments in the theory of randomized shortest paths with a comparison of graph node distances there have lately been several suggestions for parametrized distances on a graph that generalize the shortest path distance and the commute time or resistance distance the need for developing such distances has risen from the observation that the above mentioned common distances in many situations fail to take into account the global structure of the graph in this article we develop the theory of one family of graph node distances known as the randomized shortest path dissimilarity which has its foundation in statistical physics we show that the randomized shortest path dissimilarity can be easily computed in closed form for all pairs of nodes of a graph moreover we come up with a new definition of a distance measure that we call the free energy distance the free energy distance can be seen as an upgrade of the randomized shortest path dissimilarity as it defines a metric in addition to which it satisfies the graph geodetic property the derivation and computation of the free energy distance are also straightforward we then make a comparison between a set of generalized distances that interpolate between the shortest path distance and the commute time or resistance distance this comparison focuses on the applicability of the distances in graph node clustering and classification the comparison in general shows that the parametrized distances perform well in the tasks in particular we see that the results obtained with the free energy distance are among the best in all the experiments

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

communicability angle and the spatial efficiency of networks we introduce the concept of communicability angle between a pair of nodes in a graph we provide strong analytical and empirical evidence that the average communicability angle for a given network accounts for its spatial efficiency on the basis of the communications among the nodes in a network we determine characteristics of the spatial efficiency of more than a hundred real world networks that represent complex systems arising in a diverse set of scenarios in particular we find that the communicability angle correlates very well with the experimentally measured value of the relative packing efficiency of proteins that are represented as residue networks we finally show how we can modulate the spatial efficiency of a network by tuning the weights of the edges of the networks this allows us to predict the effects of external stresses on the spatial efficiency of a network as well as to design strategies to improve important parameters in real world systems

analysis of four parameter fractional derivative model of real solid materials the introduction of fractional derivatives into the constitutive equation of the differential operator type of linear solid materials has led to the development of the so called fractional derivative models one of these models characterized by four parameters has been found usable for describing the variation of dynamics elastic and damping properties in a wide frequency range provided that there is only one loss peak in this paper this four parameter model is theoretically analyzed the effect of the parameters on the frequency curves is demonstrated and it is shown that there is a strict relation between the dispersion of the dynamic modulus the loss peak and the slope of the frequency curves the half value bandwidth of the loss modulus frequency curve is investigated and conditions are developed to establish the applicability of the model for a class of materials moreover it is shown that the model can be used to predict the frequency dependences of dynamic properties for a wide range even if measurements are made in only a narrow frequency range around the loss peak

analysis and reduction of the spurious current in a class of multiphase lattice boltzmann models we show that the spurious current present near a curved interface in a class of multiphase lattice boltzmann lb models is due to the insufficient isotropy of the discrete gradient operator a method of obtaining highly isotropic gradient operators on a lattice is given numerical simulations show that both the magnitude and the spatial extent of the spurious current are significantly reduced as gradient operators of increasingly higher order of isotropy is adopted in multiphase lb models

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

on the nullspace the rangespace and the characteristic polynomial of euclidean distance matrices we present a characterization of the nullspace and the rangespace of a euclidean distance matrix edm d in terms of the vector of all ones and in terms of the gale subspace g d and the realization matrix p corresponding to d this characterization is then used to compute the characteristic polynomial of d we also present some results concerning edms generated by regular figures and edms generated by centrally symmetric points

the communicability distance in graphs let g be a simple connected graph with adjacency matrix a the communicabilitygpq between two nodes p and q of the graph is defined as the pq entry of g exp a we prove here that p q gpp gqq 2gpq 1 2 is a euclidean distance and give expressions for it in paths cycles stars and complete graphs with n nodes the sum of all communicabilitydistances in a graph is introduced as a new graph invariant g we compare this index with the wiener and kirchhoff indices of graphs and conjecture about the graphs with maximum and minimum values of this index

spectral measures of bipartivity in complex networks we introduce a quantitative measure of network bipartivity as a proportion of even to total number of closed walks in the network spectral graph theory is used to quantify how close to bipartite a network is and the extent to which individual nodes and edges contribute to the global network bipartivity it is shown that the bipartivity characterizes the network structure and can be related to the efficiency of semantic or communication networks trophic interactions in food webs construction principles in metabolic networks or communities in social networks

communicability in complex networks we propose a new measure of the communicability of a complex network which is a broad generalization of the concept of the shortest path according to the new measure most of the real world networks display the largest communicability between the most connected popular nodes of the network assortative communicability there are also several networks with the disassortative communicability where the most popular nodes communicate very poorly to each other using this information we classify a diverse set of real world complex systems into a small number of universality classes based on their structure dynamic correlation in addition the new communicability measure is able to distinguish finer structures of networks such as communities into which a network is divided a community is unambiguously defined here as a set of nodes displaying larger communicability among them than to the rest of the nodes in the network

numerical simulation of moving contact line problems using a volume of fluid method abstract moving contact lines are implemented in a volume of fluid scheme with piecewise linear interface construction interfacial tension is treated as a continuous body force computed from numerical derivatives of a smoothed volume of fluid function two methods for implementing the contact angle condition are investigated the first extrapolates the volume of fluid function beyond the flow domain on the basis of the condition that its gradient is perpendicular to the interface and that the normal to the interface at the wall is determined by the contact angle the second method treats the problem as a three phase situation and mimics the classical argument of young it is found that the latter approach introduces an artificial localized flow and the extrapolation method is preferable slip is a crucial factor in the spreading of contact lines the numerical method introduces slip at the discrete level effectively introducing a slip length on the order of the mesh size

minimizing effective resistance of a graph the effective resistance between two nodes of a weighted graph is the electrical resistance seen between the nodes of a resistor network with branch conductances given by the edge weights the effective resistance comes up in many applications and fields in addition to electrical network analysis including for example markov chains and continuous time averaging networks in this paper we study the problem of allocating edge weights on a given graph in order to minimize the total effective resistance i e the sum of the resistances between all pairs of nodes we show that this is a convex optimization problem and can be solved efficiently either numerically or in some cases analytically we show that optimal allocation of the edge weights can reduce the total effective resistance of the graph compared to uniform weights by a factor that grows unboundedly with the size of the graph we show that among all graphs with n nodes the path has the largest value of optimal total effective resistance and the complete graph has the least

network properties revealed through matrix functions the emerging field of network science deals with the tasks of modeling comparing and summarizing large data sets that describe complex interactions because pairwise affinity data can be stored in a two dimensional array graph theory and applied linear algebra provide extremely useful tools here we focus on the general concepts of centrality communicability and betweenness each of which quantifies important features in a network some recent work in the mathematical physics literature has shown that the exponential of a network s adjacency matrix can be used as the basis for defining and computing specific versions of these measures we introduce here a general class of measures based on matrix functions and show that a particular case involving a matrix resolvent arises naturally from graph theoretic arguments we also point out connections between these measures and the quantities typically computed when spectral methods are used for data mining tasks such as clustering and ordering we finish with computational examples showing the new matrix resolvent version applied to real networks

dispersion curves of viscoelastic plane waves and rayleigh surface wave in high frequency range with fractional derivatives abstract the moduli of conventional elastic structural materials are extended to one of the viscoelastic materials through a modification whereby the dynamic moduli converge to the static moduli of elasticity as the fractional order approaches zero by plotting phase velocity curves and group velocity curves of plane waves and rayleigh surface wave for a viscoelastic material polyvinyl chloride foam the influence of the fractional order of viscoelasticity is examined the phase and group velocity curves in the high frequency range were derived for longitudinal transverse and rayleigh waves inherent to the viscoelastic material in addition the equation for the phase velocity was mathematically derived on the complex plane too and graphically illustrated a phenomenon was found that at the moment when the fractional order of the time derivative reaches an integer value 1 the curve on the complex plane becomes completely different exhibiting snap through behavior we examined the mechanism of the snap through mathematically numerical calculation examples were solved and good agreement was confirmed between the numerical calculation and the analytical expression mentioned above from the results of the numerical example regularities were derived for the absolute value of the complex phase and group velocities on the complex plane

a note on the estrada hatano communicability algorithm for detecting community structure in complex networks abstract recently estrada and hatano proposed an algorithm for the detection of community structure in complex networks using the concept of network communicability here we amend this algorithm by eliminating the subjectivity of choosing degree of overlapping and including an additional check of the fitness of detected communities we show that this amendment can detect some communities which remain undetected by estrada and hatano s algorithm for instance let g p q be a graph obtained from two cliques k p and k q p q 3 joined by a single edge it is apparent that this graph contains two communities namely the two cliques however estrada and hatano s algorithm detects only k q as a community when p is sufficiently larger than q our algorithm correctly detects both communities also our method also finds the correct community structure in one of the classic benchmark networks the zachary karate club

volume of fluid interface tracking with smoothed surface stress methods for three dimensional flows motivated by the need for three dimensional methods for interface calculations that can deal with topology changes we describe a numerical scheme built from a volume of fluid interface tracking technique that uses a piecewise linear interface calculation in each cell momentum balance is computed using explicit finite volume finite differences on a regular cubic grid surface tension is implemented by the continuous surface stress or continuous surface force method examples and verifications of the method are given by comparing simulations to analytical results and experiments for sedimenting droplet arrays and capillary waves at finite reynolds number in the case of a pinching pendant drop both three dimensional and axisymmetric simulations are compared to experiments agreement is found both before and after the reconnections

boundary conditions for lattice boltzmann simulations with complex geometry flows boundary conditions of lattice boltzmann method to simulate flows embedded with a solid object is proposed the closest nodes adjacent to the boundary in the fluid domain are used as boundary nodes of the flow domain the fluid velocity of the boundary node is obtained by linear interpolation between the velocities of the solid object and the second fluid node further away then distribution functions originating from the solid domain at the boundary nodes are modified using known distribution functions and correctors to satisfy the momentum this boundary condition is an extended form of a method proposed by hou et al c f hou c chang c a lin consistent boundary conditions for 2d and 3d lattice boltzmann simulations submitted for publication for plane wall and regular geometry the technique is examined by simulating decaying vortex transient flow induced by an abruptly rotating ring and flow over an asymmetrically placed cylinder numerical simulations indicate that this method is second order accurate and all the numerical results are compatible with the benchmark solutions

a bound on the spectral radius of graphs abstract let g be a simple connected graph with n vertices and m edges let a be the adjacency matrix of g the spectral radius a of a satisfies a 2m n 1 with equality if and only if g is isomorphic to one of the following two graphs a the star k1 n 1 b the complete graph kn

properties of euclidean and non euclidean distance matrices abstract a distance matrix d of order n is symmetric with elements 1 2 d ij 2 where dii 0 d is euclidean when the 1 2 n n 1 quantities dij can be generated as the distances between a set of n points x n p in a euclidean space of dimension p the dimensionality of d is defined as the least value of p rank x of any generating x in general p 1 and p 2 are also acceptable but may include imaginary coordinates even when d is euclidean basic properties of euclidean distance matrices are established in particular when rank d it is shown that depending on whether etd e is not or is zero the generating points lie in either p 1 dimensions in which case they lie on a hypersphere or in p 2 dimensions in which case they do not the notation e is used for a vector all of whose values are one when d is non euclidean its dimensionality p r s will comprise r real and s imaginary columns of x and r s are invariant for all generating x of minimal rank higher ranking representations can arise only from p 1 r 1 s or p 1 r s 1 or p 2 r 1 s 1 so that not only are r s invariant but they are both minimal for all admissible representations x

walks and the spectral radius of graphs abstract given a graph g write g for the largest eigenvalue of its adjacency matrix g for its clique number and wk g for the number of its k walks we prove that the inequalities w q r g w q g r g g 1 g w r g hold for all r 0 and odd q 0 we also generalize a number of other bounds on g and characterize semiregular and pseudo regular graphs in spectral terms

topological structural classes of complex networks we use theoretical principles to study how complex networks are topologically organized at large scale using spectral graph theory we predict the existence of four different topological structural classes of networks these classes correspond respectively to highly homogenous networks lacking structural bottlenecks networks organized into highly interconnected modules with low inter community connectivity networks with a highly connected central core surrounded by a sparser periphery and networks displaying a combination of highly connected groups quasicliques and groups of nodes partitioned into disjoint subsets quasibipartites here we show by means of the spectral scaling method that these classes really exist in real world ecological biological informational technological and social networks we show that neither of three network growth mechanisms random with uniform distribution preferential attachment and random with the same degree sequence as real network is able to reproduce the four structural classes of complex networks these models reproduce two of the network classes as a function of the average degree but completely fail in reproducing the other two classes of networks

analysis of free vibrations of a viscoelastic oscillator via the models involving several fractional parameters and relaxation retardation times free damped vibrations of a linear viscoelastic oscillator based on the fractional derivative model involving more than one different fractional parameters and several relaxation retardation times are investigated the analytical solution is obtained in the form of two terms one of which governs the drift of the system s equilibrium position and is defined by the dynamic relaxation retardation processes occurring in the system and the other term describes damped vibrations around the equilibrium position and is determined by the system s inertia and energy dissipation the drift is governed by an improper integral taken along two sides of the cut of the complex plane while damped vibrations are determined according to the two complex conjugate roots of the characteristic equation which are located on the left half of the complex plane the behaviour of the characteristic equation roots as functions of the system s rheological parameters which enable the control of the dynamic response of the oscillator is shown in the complex plane

dispersion curves for a viscoelastic timoshenko beam with fractional derivatives abstract the kramers kronig dispersion relation often used as a viscoelastic constitutive law for polymeric materials is based on purely mathematical properties of linearity convergence of improper integrals and causality thus it may also be valid as a viscoelastic constitutive law for general structural materials accordingly the motion equation of a timoshenko beam composed of conventional elastic structural materials is extended to one composed of viscoelastic materials from the derived governing equation a dispersive equation is derived for a viscoelastic timoshenko beam by plotting phase velocity curves and group velocity curves for a beam of solid circular cross section composed of a viscoelastic material polyvinyl chloride foam the influence of the fractional order of viscoelasticity is examined as a result it is found that in the high frequency range only the first mode of a timoshenko beam converged to the propagation velocity of the rayleigh wave which takes account of the fractional order of viscoelasticity in addition the phase velocity and the group velocity were found to increase as the fractional order approaches 0 and to decrease as the fractional order approaches 1 furthermore the rate of velocity change becomes greater as the fractional order approaches 0 and becomes smaller as the fractional order approaches 1

random neighborhood graphs as models of fracture networks on rocks structural and dynamical analysis abstract we propose a new model to account for the main structural characteristics of rock fracture networks rfns the model is based on a generalization of the random neighborhood graphs to consider fractures embedded into rectangular spaces we study a series of 29 real world rfns and find the best fit with the random rectangular neighborhood graphs rrngs proposed here we show that this model captures most of the structural characteristics of the rfns and allows a distinction between small and more spherical rocks and large and more elongated ones we use a diffusion equation on the graphs in order to model diffusive processes taking place through the channels of the rfns we find a small set of structural parameters that highly correlates with the average diffusion time in the rfns we found analytically some bounds for the diameter and the algebraic connectivity of these graphs that allow to bound the diffusion time in these networks we also show that the rrngs can be used as a suitable model to replace the rfns in the study of diffusion like processes indeed the diffusion time in rfns can be predicted by using structural and dynamical parameters of the rrngs finally we also explore some potential extensions of our model to include variable fracture apertures the possibility of long range hops of the diffusive particles as a way to account for heterogeneities in the medium and possible superdiffusive processes and the extension of the model to 3 dimensional space

estimating the estrada index abstract let g be a graph on n vertices and let 1 2 n be its eigenvalues the estrada index of g is a recently introduced graph invariant defined as ee i 1 n e i we establish lower and upper bounds for ee in terms of the number of vertices and number of edges also some inequalities between ee and the energy of g are obtained

viscous flow and colloid transport near air water interface in a microchannel in order to understand the transport behavior of colloids near an air water interface awi two computational methods are applied to simulate the local water flow field near a moving awi in a 2d microfluidic channel the first method is a mesoscopic multicomponent and multiphase lattice boltzmann lbm model and the second is the macroscopic navier stokes based volume of fluid interface tracking method in the lbm it is possible to predict the dynamic contact angles after the static contact angle is correctly set and the predicted dynamic contact angles are in good agreement with previous observations it is demonstrated that the two methods can yield a similar flow velocity field if they are applied properly the flow field relative to awi depends on the direction of the flow and exhibits curved streamlines that transport fluid between the center of the channel and the wall region using the obtained flow the motion of sub micron colloids in a de ionized water solution is then studied by a lagrangian approach the observed colloid trajectories are in qualitative agreement with our visualizations using a confocal microscope

current flow efficiency of networks abstract many real world networks from infrastructure networks to social and communication networks can be formulated as flow networks how to realistically measure the transport efficiency of these networks is of fundamental importance the shortest path based efficiency measurement has limitations as it assumes that flow travels only along those shortest paths here we propose a new metric named current flow efficiency in which we calculate the average reciprocal effective resistance between all pairs of nodes in the network this metric takes the multipath effect into consideration and is more suitable for measuring the efficiency of many real world flow equilibrium networks moreover this metric can handle a disconnected graph and can thus be used to identify critical nodes and edges from the efficiency loss perspective we further analyze how the topological structure affects the current flow efficiency of networks based on some model and real world networks our results enable a better understanding of flow networks and shed light on the design and improvement of such networks with higher transport efficiency

vulnerability of complex networks under intentional attack with incomplete information we study the vulnerability of complex networks under intentional attack with incomplete information which means that one can only preferentially attack the most important nodes among a local region of a network the known random failure and the intentional attack are two extreme cases of our study using the generating function method we derive the exact value of the critical removal fraction fc of nodes for the disintegration of networks and the size of the giant component to validate our model and method we perform simulations of intentional attack with incomplete information in scale free networks we show that the attack information has an important effect on the vulnerability of scale free networks we also demonstrate that hiding a fraction of the nodes information is a cost efficient strategy for enhancing the robustness of complex networks