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

random walks on networks cumulative distribution of cover time we derive an exact closed form analytical expression for the distribution of the cover time for a random walk over an arbitrary graph in special case we derive simplified exact expressions for the distributions of cover time for a complete graph a cycle graph and a path graph an accurate approximation for the cover time distribution with computational complexity of o2n is also presented the approximation is numerically tested only for graphs with n1000 nodes the random walk is a fundamental dynamic process which can be used to model random processes inherent to many important applications such as transport in disordered media 1 neuron firing dynamics 2 spreading of diseases 3 or transport and search processes 4 8 in this paper we investigate random walks on graphs 9 and derive exact expressions for the cumulative distribution functions for three quantities of a random walk that play the most important role in the theory of random walks 1 hit ting time hij or first passage time which is the number of steps before node j is visited starting from node i 2 com mute time ij hij hji and cover time that is the number of steps to reach every node average hitting time average commute time and average cover time have been recently studied in several papers in 10 the authors investigated random walks on complex net works and derived an exact expression for the mean first passage time between two nodes for each node the random walk centrality is introduced which determines the relative speed by which a node can receive and spread information over the network in a random process using both numerical simulations and scaling arguments the behavior of a random walker on a one dimensional small world network is studied in 11 the average number of distinct sites visited by the random walker the mean square displacement of the walker and the distribution of first return times obey a characteristic scaling form the expected time for a random walk to traverse between two arbitrary sites of the erdos renyi ran dom graph is studied in 12 the properties of random walks on complex trees are studied in 13 both the vertex discov ery rate and the mean topological displacement from the ori gin present a considerable slowing down in the tree case moreover the mean first passage time displays a logarithmic degree dependence in contrast to the inverse degree shape exhibited in looped networks 13 the random walk on net works has also much relevance to algorithmic applications the expected time taken to visit every vertex of connected graphs has recently been extensively studied in a series of papers cooper and frieze studied the average cover time of various models of a random graph see for example 14 this is an outline of the paper in sec ii we derive closed formulas of the cumulative distribution function for hitting time commute time and cover time we also present a simple example of a graph with four nodes and derive closed formulas of the cumulative distribution function for cover time of complete graphs cycle and path graphs an approximation of the cumulative distribution function for cover time is proposed in the sec iii we also present some numerical results of the cumulative distribution function for cover time of different graphs sec iv we finish the paper with conclusions

on the coveting radius of extremal self dual codes it is known that every self dual binary code which is not doubly even is a child of a doubly even parent it will be shown that an n 2 n 2 2 child of an n n 2 d doubly even parent has covering radius geq d 1 every extremal doubly even 32 16 8 code has covering radius 6 and every extremal doubly even 48 24 12 code has covering radius 8 the complete coset weight distribution of the 32 16 8 quadratic residue code is given as well as bounds or exact values for the covering radii of all extremai doubly even codes of length less than or equal to 96

a new approach to the general minimum distance decoding problem the zero neighbors algorithm minimum distance decoding mdd for a general error correcting linear code is a hard computational problem that recently has been shown to be np hard the complexity of known decoding algorithms is determined by min 2 k 2 n k where n is the code length and k is the number of information digits two new algorithms are suggested that reduce substantially the complexity of mdd the algorithms use a new concept of zero neighbors a special set of codewords only these codewords which can be computed in advance should be stored and used in the decoding procedure the number of zero neighbors is shown to be very small compared with min 2 k 2 n k for n gg 1 and a wide range of code rates r k n for example for r approx 0 5 this number grows approximately as a square root of the number of codewords

self avoiding walks on scale free networks several kinds of walks on complex networks are currently used to analyze search and navigation in different systems many analytical and computational results are known for random walks on such networks self avoiding walks saw s are expected to be more suitable than unrestricted random walks to explore various kinds of real life networks here we study long range properties of random saw s on scale free networks characterized by a degree distribution p k ensuremath sim k ensuremath ensuremath gamma in the limit of large networks system size n ensuremath rightarrow ensuremath infty the average number s n of saw s starting from a generic site increases as ensuremath mu n with ensuremath mu k 2 k ensuremath 1 for finite n s n is reduced due to the presence of loops in the network which causes the emergence of attrition of the paths for kinetic growth walks the average maximum length l increases as a power of the system size l ensuremath sim n ensuremath alpha with an exponent ensuremath alpha increasing as the parameter ensuremath gamma is raised we discuss the dependence of ensuremath alpha on the minimum allowed degree in the network a similar power law dependence is found for the mean self intersection length of nonreversal random walks simulation results support our approximate analytical calculations

network growth with arbitrary initial conditions degree dynamics for uniform and preferential attachment this paper provides time dependent expressions for the expected degree distribution of a given network that is subject to growth we consider both uniform attachment where incoming nodes form links to existing nodes selected uniformly at random and preferential attachment where probabilities are assigned proportional to the degrees of the existing nodes we consider the cases of single and multiple links being formed by each newly introduced node the initial conditions are arbitrary that is the solution depends on the degree distribution of the initial graph which is the substrate of the growth previous work in the literature focuses on the asymptotic state that is when the number of nodes added to the initial graph tends to infinity rendering the effect of the initial graph negligible our contribution provides a solution for the expected degree distribution as a function of time for arbitrary initial condition previous results match our results in the asymptotic limit the results are discrete in the degree domain and continuous in the time domain where the addition of new nodes to the graph are approximated by a continuous arrival rate

hyperbolic geometry of complex networks we develop a geometric framework to study the structure and function of complex networks we assume that hyperbolic geometry underlies these networks and we show that with this assumption heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry conversely we show that if a network has some metric structure and if the network degree distribution is heterogeneous then the network has an effective hyperbolic geometry underneath we then establish a mapping between our geometric framework and statistical mechanics of complex networks this mapping interprets edges in a network as noninteracting fermions whose energies are hyperbolic distances between nodes while the auxiliary fields coupled to edges are linear functions of these energies or distances the geometric network ensemble subsumes the standard configuration model and classical random graphs as two limiting cases with degenerate geometric structures finally we show that targeted transport processes without global topology knowledge made possible by our geometric framework are maximally efficient according to all efficiency measures in networks with strongest heterogeneity and clustering and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure

constrained spin dynamics description of random walks on hierarchical scale free networks we study a random walk problem on the hierarchical network which is a scale free network grown deterministically the random walk problem is mapped onto a dynamical ising spin chain system in one dimension with a nonlocal spin update rule which allows an analytic approach we show analytically that the characteristic relaxation time scale grows algebraically with the total number of nodes n as t n z from a scaling argument we also show the power law decay of the autocorrelation function c sigma t t alpha which is the probability to find the ising spins in the initial state sigma after t time steps with the state dependent nonuniversal exponent alpha it turns out that the power law scaling behavior has its origin in a quasiultrametric structure of the configuration space

behaviors of susceptible infected epidemics on scale free networks with identical infectivity in this paper we propose a susceptible infected model with identical infectivity in which at every time step each node can only contact a constant number of neighbors we implemented this model on scale free networks and found that the infected population grows in an exponential form with the time scale proportional to the spreading rate furthermore by numerical simulation we demonstrated that the targeted immunization of the present model is much less efficient than that of the standard susceptible infected model finally we investigate a fast spreading strategy when only local information is available different from the extensively studied path finding strategy the strategy preferring small degree nodes is more efficient than that preferring large degree nodes our results indicate the existence of an essential relationship between network traffic and network epidemic on scale free networks

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

the covering radius of the 2 15 16 reed muller code is at least 16276 two interesting cosets of the first order reed muller code of block length 2 15 are described they provide counterexamples to a conjecture on the covering radius

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

the physical character of information the mathematical theory of communication defines information in syntax without reference to its physical representation and semantic significance however in an everyday context information is tied to its representation and its content is valued the dichotomy between the formal definition and the practical perception of information is examined by the second law of thermodynamics that was recently formulated as an equation of motion thermodynamic entropy shows that the physical representation of information is not inconsequential in generation transmission and processing of information according to the principle of increasing entropy communication by dissipative transformations is a natural process among many other evolutionary phenomena that level energy density differences between components of a communication system and its surroundings in addition information guided processes direct down along descents on free energy landscapes the non integrable equation for irreversible processes reveals that there is no universal analytical algorithm to match source to channel noise infiltration is also regarded by the second law as an inevitable consequence of energy transduction between a communication system and its surroundings communication is invariably associated with misunderstanding because mechanisms and means of information processing at the receiver differ from those at the sender the significance of information is ascribed to the increase in thermodynamic entropy in the receiver system that results from execution of the received message

a stochastic evolutionary model exhibiting power law behaviour with an exponential cutoff recently several authors have proposed stochastic evolutionary models for the growth of complex networks that give rise to power law distributions these models are based on the notion of preferential attachment leading to the rich get richer phenomenon despite the generality of the proposed stochastic models there are still some unexplained phenomena which may arise due to the limited size of networks such as protein and e mail networks such networks may in fact exhibit an exponential cutoff in the power law scaling although this cutoff may only be observable in the tail of the distribution for extremely large networks we propose a modification of the basic stochastic evolutionary model so that after a node is chosen preferentially say according to the number of its inlinks there is a small probability that this node will be discarded we show that as a result of this modification by viewing the stochastic process in terms of an urn transfer model we obtain a power law distribution with an exponential cutoff unlike many other models the current model can capture instances where the exponent of the distribution is less than or equal to two as a proof of concept we demonstrate the consistency of our model by analysing a yeast protein interaction network the distribution of which is known to follow a power law with an exponential cutoff

modified bounds for covering codes the covering radius of binary codes is studied bounds on k n r the minimum cardinality of any binary code of length n and covering radius r are found modifications of the van wee lower bounds are proved for k n r the minimal number of codewords in any binary code of length n and covering radius r the first of the two van wee bounds is based on studying the hamming spheres of radius 1 centered at the points which have distance r to the code c the points covered by more than one codeword are divided into several classes and better estimates for some of these classes are obtained using a suitable averaging process the lower bound for k n r when r or 2 is improved the second van wee bound studies spheres of radius 2 centered at the points which have distance r 1 or r to the code c these points are divided essentially into two classes the points that are covered by only one codeword of c and the points that are covered by more than one codeword

navigation in a small world with local information it is commonly known that there exist short paths between vertices in a network showing the small world effect yet vertices for example the individuals living in society usually are not able to find the shortest paths due to the very serious limit of information to study this issue theoretically here the navigation process of launching messages toward designated targets is investigated on a variant of the one dimensional small world network swn in the network structure considered the probability of a shortcut falling between a pair of nodes is proportional to r alpha where r is the lattice distance between the nodes when alpha 0 it reduces to the swn model with random shortcuts the system shows the dynamic small world effect which is different from the well studied static sw effect we study the effective network diameter the path length as a function of the lattice distance and the dynamics they are controlled by multiple parameters and we use data collapse to show that the parameters are correlated the central finding is that in the one dimensional network studied the dynamic sw effect exists for 0 alpha 2 for each given value of alpha in this region the point where the dynamic sw effect arises is m l approximately 1 where m is the number of useful shortcuts and l is their average reduced effective length

steady state and mean recurrence time for random walks on stochastic temporal networks random walks are basic diffusion processes on networks and have applications in for example searching navigation ranking and community detection recent recognition of the importance of temporal aspects on networks spurred studies of random walks on temporal networks here we theoretically study two types of event driven random walks on a stochastic temporal network model that produces arbitrary distributions of interevent times in the so called active random walk the interevent time is reinitialized on all links upon each movement of the walker in the so called passive random walk the interevent time is reinitialized only on the link that has been used the last time and it is a type of correlated random walk we find that the steady state is always the uniform density for the passive random walk in contrast for the active random walk it increases or decreases with the node s degree depending on the distribution of interevent times the mean recurrence time of a node is inversely proportional to the degree for both active and passive random walks furthermore the mean recurrence time does or does not depend on the distribution of interevent times for the active and passive random walks respectively

perfect codes hardly exist the main lines of a partial proof of the perfect code theorem are presented the relevant part refers to the nonexistence of unknown t perfect codes over arbitrary alphabets for t not in 1 2 6 8 the details of the proof can be found in the author s ph d dissertation at the university of amsterdam

optimization of random searches on defective lattice networks we study the general problem of how to search efficiently for targets randomly located on defective lattice networks i e regular lattices which have some fraction of its nodes randomly removed we consider large but finite triangular lattices and assume for the search dynamics that the walker chooses steps lengths lj from the power law distribution p lj approximately lj mu with the exponent mu regulating the strategy of the search process at each step lj the searcher moves in straight lines and constantly looks within a detection radius of vision rv for the targets along the way if there is contact with a defect the movement stops and a new step length is chosen hence the presence of defects decreases the efficiency of the overall process we study numerically how three different aspects of the lattice influence the optimization of the search efficiency i the type of boundary conditions ii the concentration of targets and defects and iii the category or class of search destructive nondestructive or regenerative motivated by the results we develop a type of mean field model for the problem and obtain an analytical approximation for the search efficiency function finally we discuss in the context of searches how defective lattices compare with perfect lattices and with continuous environments

resilience to damage of graphs with degree correlations the existence or nonexistence of a percolation threshold on power law correlated graphs is a fundamental question for which a general criterion is lacking in this work we investigate the problems of site and bond percolation on graphs with degree correlations and their connection with spreading phenomena we obtain some general expressions that allow the computation of the transition thresholds or their bounds using these results we study the effects of assortative and disassortative correlations on the resilience to damage of networks

on the covering radius of codes the covering radius r of a code is the maximal distance of any vector from the code this work gives a number of new results concerning t n k the minimal covering radius of any binary code of length n and dimension k for example t n 4 and t n 5 are determined exactly and reasonably tight bounds on t n k are obtained for any k when n is large these results are found by using several new constructions for codes with small covering radius one construction the amalgamated direct sum involves a quantity called the norm of a code codes with norm leq 2 r 1 are called normal and may be combined efficiently the paper concludes with a table giving bounds on t n k for n leq 64

the second order reed muller code of length 64 has covering radius 18 corresp the assertion of the title is proved by first showing that the covering radius of r 2 6 cannot exceed 18 and then constructing a weight 18 coset leader

new lower bounds for binary covering codes g d chen et al ibid vol it 32 p 680 94 1986 presented two new lower bounds for k n r where k n r denotes the minimum cardinality of a binary code of length n and covering radius r the author shows that a slight modification gives further improvements and some examples are given to confirm the argument codes that have a certain partitioning property are considered

constructions of nonlinear covering codes constructions of nonlinear covering codes are given using any nonlinear starting code of covering radius r spl ges 2 these constructions form an infinite family of codes with the same covering radius a nonlinear code is treated as a union of cosets of a linear code new infinite families of nonlinear covering codes are obtained concepts of r l objects r l partitions and r l length are described for nonlinear codes

mixing search on complex networks abstract in this article we derive the first passage time fpt distribution and the mean first passage time mfpt of random walks from multiple sources on networks on the basis of analysis and simulation we find that the mfpt drops substantially when particle number increases at the first stage and converges to the shortest distance between the sources and the destination when particle number tends to infinite given the fact that a brownian particle from a high degree node often needs a large number of steps to reach an expected low degree node which is the bottleneck for a single random walk we propose a mixing search model to improve the efficiency of search processes by using random walks from multiple sources to continue the searches from high degree nodes to destinations we compare our model with the mixing navigation model proposed by zhou on complex networks and find that our model converges much faster with lower hardware cost than zhou s model moreover simulations on scale free networks show that the search efficiency of our model is much higher than that of a single random walk and comparable to that of multiple random walks which have much higher hardware cost than our model finally we discuss the traffic cost of our model and propose an absorption strategy for our model to recover the additional walkers in networks simulations indicate that this strategy reduces the traffic cost of our model effectively

a note on the thin finite simple groups in a previous paper 11 the author described a new approach to nonabelian simple groups of characteristic 2 type in which every 2 local subgroup of odd index is solvable and has cyclic sylow subgroups for all odd primes in fact we imposed these conditions only on certain special 2 local subgroups of odd index the approach was made possible by certain new general results concerning groups of characteristic 2 type which divide into the following three categories

anomalous biased diffusion in networks we study diffusion with a bias toward a target node in networks this problem is relevant to efficient routing strategies in emerging communication networks like optical networks bias is represented by a probability p of the packet or particle to travel at every hop toward a site that is along the shortest path to the target node we investigate the scaling of the mean first passage time mfpt with the size of the network we find by using theoretical analysis and computer simulations that for random regular rr and erd ifmmode mbox h o else h o fi s r enyi networks there exists a threshold probability p mathrm th such that for pl p mathrm th the mfpt scales anomalously as n ensuremath alpha where n is the number of nodes and ensuremath alpha depends on p for pg p mathrm th the mfpt scales logarithmically with n the threshold value p mathrm th of the bias parameter for which the regime transition occurs is found to depend only on the mean degree of the nodes an exact solution for every value of p is given for the scaling of the mfpt in rr networks the regime transition is also observed for the second moment of the probability distribution function the standard deviation for the case of scale free sf networks we present analytical bounds and simulations results showing that the mfpt scales at most as mathrm ln n to a positive power for any finite bias which means that in sf networks even a very small bias is considerably more efficient in comparison to unbiased walk

distributed random walks performing random walks in networks is a fundamental primitive that has found applications in many areas of computer science including distributed computing in this article we focus on the problem of sampling random walks efficiently in a distributed network and its applications given bandwidth constraints the goal is to minimize the number of rounds required to obtain random walk samples all previous algorithms that compute a random walk sample of length e as a subroutine always do so naively that is in o e rounds the main contribution of this article is a fast distributed algorithm for performing random walks we present a sublinear time distributed algorithm for performing random walks whose time complexity is sublinear in the length of the walk our algorithm performs a random walk of length e in o ed rounds o hides polylog n factors where n is the number of nodes in the network with high probability on an undirected network where d is the diameter of the network for small diameter graphs this is a significant improvement over the naive o e bound furthermore our algorithm is optimal within a poly logarithmic factor as there exists a matching lower bound nanongkai et al 2011 we further extend our algorithms to efficiently perform k independent random walks in o ked k rounds we also show that our algorithm can be applied to speedup the more general metropolis hastings sampling our random walk algorithms can be used to speed up distributed algorithms in applications that use random walks as a subroutine we present two main applications first we give a fast distributed algorithm for computing a random spanning tree rst in an arbitrary undirected unweighted network which runs in o md rounds with high probability m is the number of edges our second application is a fast decentralized algorithm for estimating mixing time and related parameters of the underlying network our algorithm is fully decentralized and can serve as a building block in the design of topologically aware networks

weight distribution of translates covering radius and perfect codes correcting errors of given weights let v bed binary linear n k code defined by a check matrix h and let h x be the characteristic function for the set of columns of h connections between the walsh transform of h x and the weight distributions of all translates of the code are obtained explicit formulas for the weight distributions of translates are given for small weights i i the computation of the weight distribution of all translates including the code itself for i requires at most 7 n k 2 n k additions and subtractions 6 cdot 2 n k multiplications and 2 n k l memory cells this method may be very effective if there is an analytic expression for h x a simple method for computing the covering radius of the code by the walsh transform of h x is described the implementation of this method requires for large n at most 2 n k n k log 2 n k arithmetical operations and 2 n k 1 memory cells we define the concept l perfect for codes where l is a set of weights after describing several linear and nonlinear l perfect codes necessary and sufficient conditions for a code to be l perfect in terms of the walsh transform of h x are established an analog of the lloyd theorem for such codes is proved

dynamic model of time dependent complex networks the characterization of the most connected nodes in static or slowly evolving complex networks has helped in understanding and predicting the behavior of social biological and technological networked systems including their robustness against failures vulnerability to deliberate attacks and diffusion properties however recent empirical research of large dynamic networks characterized by irregular connections that evolve rapidly has demonstrated that there is little continuity in degree centrality of nodes over time even when their degree distributions follow a power law this unexpected dynamic centrality suggests that the connections in these systems are not driven by preferential attachment or other known mechanisms we present an approach to explain real world dynamic networks and qualitatively reproduce these dynamic centrality phenomena this approach is based on a dynamic preferential attachment mechanism which exhibits a sharp transition from a base pure random walk scheme

the covering radius of the leech lattice we investigate the points in 24 dimensional space that are at maximal distance from the leech lattice i e the deep holes in that lattice the distance of such a point from the leech lattice is 1 sqrt 2 times the minimal distance between the lattice points furthermore there are 23 inequivalent types of deep hole one for each of the 23 even unimodular 24 dimensional lattices found by niemeier

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

computational complexity arising from degree correlations in networks we apply a bethe peierls approach to statistical mechanics models defined on random networks of arbitrary degree distribution and arbitrary correlations between the degrees of neighboring vertices using the nondeterministic polynomial time hard optimization problem of finding minimal vertex covers on these graphs we show that such correlations may lead to a qualitatively different solution structure as compared to uncorrelated networks this results in a higher complexity of the network in a computational sense simple heuristic algorithms fail to find a minimal vertex cover in the highly correlated case whereas uncorrelated networks seem to be simple from the point of view of combinatorial optimization

classification of extremal and s extremal binary self dual codes of length 38 in this paper we classify all extremal and s extremal binary self dual codes of length 38 there are exactly 2744 extremal self dual codes two s extremal codes and 1730 s extremal codes we obtain our results from the use of a recursive algorithm used in the recent classification of all extremal self dual codes of length 36 and from a generalization of this recursive algorithm for the shadow the classification of extremal codes permits to achieve the classification of all extremal codes with

the complexity of computing the covering radius of a code the problem of finding the covering radius of a binary linear code is shown to be nondeterministic polynomial np hard in fact a problem that is complete for the class prod 2 p in the polynomial hierarchy is shown to be reducible to the covering radius problem so that finding the covering radius is strictly harder than any np complete problem unless the polynomial hierarchy collapses with np prod 2 p

partition of networks into basins of attraction we study partition of networks into basins of attraction based on a steepest ascent search for the node of highest degree each node is associated with or attracted to its neighbor of maximal degree as long as the degree is increasing a node that has no neighbors of higher degree is a peak attracting all the nodes in its basin maximally random scale free networks exhibit different behavior based on their degree distribution exponent gamma for small gamma broad distribution networks are dominated by a giant basin whereas for large gamma narrow distribution there are numerous basins with peaks attracting mainly their nearest neighbors we derive expressions for the first two moments of the number of basins we also obtain the complete distribution of basin sizes for a class of hierarchical deterministic scale free networks that resemble random nets finally we generalize the problem to regular networks and lattices where all degrees are equal and thus the attractiveness of a node must be determined by an assigned weight rather than the degree we derive the complete distribution of basins of attraction resulting from randomly assigned weights in one dimensional chains

kinetic growth walks on complex networks kinetically grown self avoiding walks on various types of generalized random networks have been studied networks with short and long tailed degree distributions p k were considered k degree or connectivity including scale free networks with p k k the long range behaviour of self avoiding walks on random networks is found to be determined by finite size effects the mean self intersection length of non reversal random walks l scales as a power of the system size n l n with an exponent 0 5 for short tailed degree distributions and 0 5 for scale free networks with 3 the mean attrition length of kinetic growth walks l scales as l n with an exponent which depends on the lowest degree in the network results of approximate probabilistic calculations are supported by those derived from simulations of various kinds of networks the efficiency of kinetic growth walks to explore networks is largely reduced by inhomogeneity in the degree distribution as happens for scale free networks

voronoi regions of lattices second moments of polytopes and quantization if a point is picked at random inside a regular simplex octahedron 600 cell or other polytope what is its average squared distance from the centroid in n dimensional space what is the average squared distance of a random point from the closest point of the lattice a n or d n e n a n ast or d n ast the answers are given here together with a description of the voronoi or nearest neighbor regions of these lattices the results have applications to quantization and to the design of signals for the gaussian channel for example a quantizer based on the eight dimensional lattice e8 has a mean squared error per symbol of 0 0717 cdots when applied to uniformly distributed data compared with 0 08333 cdots for the best one dimensional quantizer

on the string consensus problem and the manhattan sequence consensus problem abstract we study the manhattan sequence consensus problem msc problem in which we are given k integer sequences each of length l and we are to find an integer sequence x of length l called a consensus sequence such that the maximum manhattan distance of x from each of the input sequences is minimized a related problem with hamming distance instead of manhattan distance is called hamming string consensus hsc also known under the names of string center problem or closest string problem for binary sequences manhattan distance coincides with hamming distance hence in this case hsc is a special case of msc we design a practically efficient o l time algorithm solving msc for k 5 sequences it improves upon the quadratic algorithm by amir et al 2012 1 for hsc for k 5 binary strings similarly as in the algorithm of amir et al we use a column based framework we replace the implied general integer linear programming by its easy special cases due to combinatorial properties of msc for k 5 practicality of our algorithms has been verified experimentally we also show that for a general parameter k any instance can be reduced in linear time to a kernel of size k so the problem is fixed parameter tractable nevertheless for k 4 this is still too large for any naive solution to be feasible in practice this is a full version of an article published at spire 2014 15

some new upper bounds on the covering radius of binary linear codes a griesmer like upper bound on the covering radius r is given to the author s knowledge this is the only upper bound which explicitly depends on all three parameters n k and d an upper bound on r for cyclic codes is then given which depends on the generator polynomial of the cyclic code and which in many cases leads to an improvement of the previous bound an upper bound on the irreducible generator polynomial cyclic codes is also given new interpretations and applications of the so called norse bounds and necessary and sufficient conditions to attain one of these bounds are provided generalizations of most of the results for codes over gf q are outlined

ring structures and mean first passage time in networks in this paper we address the problem of the calculation of the mean first passage time on generic graphs we focus in particular on the mean first passage time on a node s for a random walker starting from a generic unknown node x we introduce an approximate scheme of calculation which maps the original process in a markov process in the space of the so called rings described by a transition matrix of size o ln n ln k ln n ln k where n is the size of the graph and k the average degree in the graph in this way one has a drastic reduction of degrees of freedom with respect to the size n of the transition matrix of the original process corresponding to an extremely low computational cost we first apply the method to the erdos renyi random graphs for which the method allows for almost perfect agreement with numerical simulations then we extend the approach to the barabasi albert graph as an example of scale free graph for which one obtains excellent results finally we test the method with two real world graphs internet and a network of the brain for which we obtain accurate results

optimization of random searches on regular lattices we investigate random searches on isotropic and topologically regular square and triangular lattices with periodic boundary conditions and study the efficiency of search strategies based on a power law distribution p ensuremath ell ensuremath sim ensuremath ell ensuremath ensuremath mu of step lengths ensuremath ell we consider both destructive searches in which a target can be visited only once and nondestructive searches when a target site is always available for future visits we discuss i the dependence of the search efficiency on the choice of the lattice topology ii the relevance of the periodic boundary conditions iii the behavior of the optimal power law exponent ensuremath mu mathrm opt as a function of target site density iv the differences between destructive and nondestructive environments and finally v how the results for the discrete searches differ from the continuous cases previously studied

the multicovering radii of codes the covering radius of a code is the least r such that the set of balls of radius r around codewords covers the entire ambient space we introduce a generalization of the notion of covering radius the m covering radius of a code is the least radius such that the set of balls of that radius covers all m tuples of elements in the ambient space we investigate basic properties of m covering radii we investigate whether codes exist with given m covering radii not always we derive bounds on the size of the smallest code with a given m covering radius based on generalizations of the sphere bound and the method of counting excesses

scaling in weighted networks and complex systems abstract network systems evolving by preferential attachment of new vertices and by link loading are considered the probability of node attaching is assumed to depend linearly on degrees of existing nodes to which new ones are connected the rule of the network loading allows both deterministic and random processes of distributing weights among links specifically the rule is determined by assigning to each link a weight dependent on randomly modified intervals between times at which linked vertices are connected to the system this enables the investigation of the influence of local and global loading processes on statistical properties of weighted networks it is demonstrated that for different values of parameters controlling these processes the distribution of weights concentrated at vertices of such networks displays a large variety of scale free and truncated scale free behaviors it is also shown that statistical properties of the studied weighted networks and statistical properties of some complex evolving spatial systems exhibit a close relationship

covering radius survey and recent results all known results on covering radius are presented as well as some new results there are a number of upper and lower bounds including asymptotic results a few exact determinations of covering radius some extensive relations with other aspects of coding theory through the reed muller codes and new results on the least covering radius of any linear n k code there is also a recent result on the complexity of computing the covering radius

levy flights and random searches in this work we discuss some recent contributions to the random search problem our analysis includes superdiffusive levy processes and correlated random walks in several regimes of target site density mobility and revisitability we present results in the context of mean field like and closed form average calculations as well as numerical simulations we then consider random searches performed in regular lattices and lattices with defects and we discuss a necessary criterion for distinguishing true superdiffusion from correlated random walk processes we invoke energy considerations in relation to critical survival states on the edge of extinction and we analyze the emergence of levy behavior in deterministic search walks finally we comment on the random search problem in the context of biological foraging

explicit codes with low covering radius a series of explicit low rate binary linear codes that have relatively low covering radius and can be rapidly decoded is exhibited these codes can be derived from higher dimensional analogs of the gale berlekamp switching game conjectures of independent interest involving hadamard matrices are given that could yield semiexplicit covering codes of very low density

another upper bound on covering radius a new upper bound on the covering radius of binary linear codes improves on previously known bounds for some codes the bound is w 2 rho 2 where w is one of the weights occurring among the codewords and rho 2 is the covering radius of an associated code some comparisons with other bounds are made

pathlength scaling in graphs with incomplete navigational information the graph navigability problem concerns how one can find as short paths as possible between a pair of vertices given an incomplete picture of a graph we study the navigability of graphs where the vertices are tagged by a number between 1 and the total number of vertices in a way to aid navigation this information is too little to ensure errorfree navigation but enough as we will show for the agents to do significantly better than a random walk in our setup given a graph we first assign information to the vertices that agents can utilize for their navigation to evaluate the navigation we calculate the average distance traveled over random pairs of source and target and different graph realizations we show that this type of embedding can be made quite efficiently the more information is embedded the more efficient it gets we also investigate the embedded navigational information in a standard graph layout algorithm and find that although this information does not make algorithms as efficient as the above mentioned schemes it is significantly helpful

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

search in weighted complex networks we study trade offs presented by local search algorithms in complex networks which are heterogeneous in edge weights and node degree we show that search based on a network measure local betweenness centrality lbc utilizes the heterogeneity of both node degrees and edge weights to perform the best in scale free weighted networks the search based on lbc is universal and performs well in a large class of complex networks

multiple scale analysis of complex networks using the empirical mode decomposition method abstract empirical mode decomposition emd method can decompose any complicated data into finite intrinsic mode functions imfs in this paper we use emd method to analyze and discuss the structural properties of complex networks a random walk method is used to collect the data series of network systems utilizing the emd method we decompose the obtained data into finite imfs under different spatial scales the analysis results show that emd method is an effective tool for capturing the topological properties of network systems under different spatial scales such as the modular structures of network systems and their energy densities

diffusive capture process on complex networks we study the dynamical properties of a diffusing lamb captured by a diffusing lion on the complex networks with various sizes of n we find that the lifetime t of a lamb scales as t approximately n and the survival probability s n infinity t becomes finite on scale free networks with degree exponent gamma 3 however s n t for gamma 3 has a long living tail on tree structured scale free networks and decays exponentially on looped scale free networks this suggests that the second moment of degree distribution k2 is the relevant factor for the dynamical properties in the diffusive capture process we numerically find that the normalized number of capture events at a node with degree k n k decreases as n k approximately k sigma when gamma 3 n k still increases anomalously for k approximately kmax where kmax is the maximum value of k of given networks with size n we analytically show that n k satisfies the relation n k approximately k2 p k for any degree distribution p k and the total number of capture events ntot is proportional to k2 which causes the gamma dependent behavior of s n t and t

dynamical properties of model communication networks we study the dynamical properties of a collection of models for communication processes characterized by a single parameter j representing the relation between information load of the nodes and its ability to deliver this information the critical transition to congestion reported so far occurs only for j51 this case is well analyzed for different network topologies we focus on the properties of the order parameter the susceptibility and the time correlations when approaching the critical point for j 1 no transition to congestion is observed but it remains a crossover from a low density to a high density state for j 1 the transition to congestion is discontinuous and congestion nuclei arise

greedy connectivity of geographically embedded graphs we introduce a measure of greedy connectivity for geographical networks graphs embedded in space and where the search for connecting paths relies only on local information such as a node s location and that of its neighbors constraints of this type are common in everyday life applications greedy connectivity accounts also for imperfect transmission across established links and is larger the higher the proportion of nodes that can be reached from other nodes with a high probability greedy connectivity can be used as a criterion for optimal network design

lower bounds on the covering radius of the non binary and binary irreducible goppa codes new lower bounds on the covering radius for different subclasses of irreducible goppa codes are obtained

fast quantizing and decoding and algorithms for lattice quantizers and codes for each of the lattices a n n geq 1 d n n geq 2 e 6 e 7 e 8 and their duals a very fast algorithm is given for finding the closest lattice point to an arbitrary point if these lattices are used for vector quantizing of uniformly distributed data the algorithm finds the minimum distortion lattice point if the lattices are used as codes for a gaussian channel the algorithm performs maximum likelihood decoding

mean field diffusive dynamics on weighted networks diffusion is a key element of a large set of phenomena occurring on natural and social systems modeled in terms of complex weighted networks here we introduce a general formalism that allows to easily write down mean field equations for any diffusive dynamics on weighted networks we also propose the concept of annealed weighted networks in which such equations become exact we show the validity of our approach addressing the problem of the random walk process pointing out a strong departure of the behavior observed in quenched real scale free networks from the mean field predictions additionally we show how to employ our formalism for more complex dynamics our work sheds light on mean field theory on weighted networks and on its range of validity and warns about the reliability of mean field results for complex dynamics