link prediction based on local information considering preferential attachment abstract link prediction in complex networks has attracted much attention in many fields in this paper a common neighbors plus preferential attachment index is presented to estimate the likelihood of the existence of a link between two nodes based on local information of the nearest neighbors numerical experiments on six real networks demonstrated the high effectiveness and efficiency of the new index compared with five well known and widely accepted indices the common neighbors resource allocation index preferential attachment index local path index and katz index the new index provides competitively accurate prediction with local path index and katz index while has less computational complexity and is more accurate than the other two indices

abnormal cascading on complex networks in the study of cascading failures on complex networks a key issue is to define capacities of edges and nodes as realistically as possible this leads to the consideration of intrinsic edge capacity associated with laws governing flows on networks which goes beyond the existing definitions of capacity based on the initial load as quantified by the betweenness centrality limited edge capacity or bandwidth and high flux or attack can trigger cascading processes which we find as characteristically different from those reported in the literature in particular there can be an abnormal parameter regime where incrementally augmenting the edge capacity can counterintuitively increase the severeness of the cascading process another striking finding is that heterogeneous flow distribution tends to suppress the cascading process in contrast to the current understanding that heterogeneity can make the network more vulnerable to cascading we provide numerical computations and analysis to substantiate these findings

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

finite bruck loops bruck loops are bol loops satisfying the automorphic inverse property we prove a structure theorem for finite bruck loops x showing that x is essentially the direct product of a bruck loop of odd order with a 2 element bruck loop the former class of loops is well understood we identify the minimal obstructions to the conjecture that all finite 2 element bruck loops are 2 loops leaving open the question of whether such obstructions actually exist

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

radial structure of the internet the structure of the internet at the autonomous system as level has been studied by the physics mathematics and computer science communities we extend this work to include features of the core and the periphery taking a radial perspective on as network structure new methods for plotting as data are described and they are used to analyse datasets that have been extended to contain edges missing from earlier collections the average distance from one vertex to the rest of the network is used as the baseline metric for investigating radial structure common vertex specific quantities are plotted against this metric to reveal distinctive characteristics of central and peripheral vertices two datasets are analysed

percolation theory and fragmentation measures in social networks abstract we study the statistical properties of a recently proposed social networks measure of fragmentation f after removal of a fraction q of nodes or links from the network the measure f is defined as the ratio of the number of pairs of nodes that are not connected in the fragmented network to the total number of pairs in the original fully connected network we compare this measure with the one traditionally used in percolation theory p the fraction of nodes in the largest cluster relative to the total number of nodes using both analytical and numerical methods we study erd s renyi er and scale free sf networks under various node removal strategies we find that for a network obtained after removal of a fraction q of nodes above criticality p 1 f 1 2 for fixed p and close to criticality we show that 1 f better reflects the actual fragmentation for a given p 1 f has a broad distribution and thus one can improve significantly the fragmentation of the network we also study and compare the fragmentation measure f and the percolation measure p for a real national social network of workplaces linked by the households of the employees and find similar results

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

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

characteristic fixed point sets of semifree actions on spheres a group action is semifree if it is free away from its fixed point set p a smith showed that when a finite group of orderqacts semifreely on a sphere the fixed set is a modqhomology sphere conversely given a modqhomology sphere as a subset of a sphere one may try to construct a group action on the sphere fixing the subset the converse question was first systematically studied by jones and then by many others in this note we give new numerical congruences satisfied by the homology of the fixed sets and give a definitive solution to the problem for characteristic fixed point sets c 1999 john wiley sons inc

subgroups of maximal rank in finite exceptional groups of lie type the purpose of this paper is to investigate a large and natural class of maximal subgroups of the finite exceptional groups of lie type which we call subgroups of maximal rank these subgroups play a prominent role both in the classification of local maximal subgroups in 9 theorem 1 and in the reduction theorem for subgroups in 28 theorem 2 in 9 it is shown that any local maximal subgroup of a finite exceptional group of lie type is either a subgroup of maximal rank or one of a small list of exceptions and in 28 using this result it is proved that any maximal subgroup is either of maximal rank or almost simple or in a known list to describe these subgroups we require the following notation let g be a simple adjoint algebraic group over an algebraically closed field k of characteristic p 0 and let o be an endomorphism of g such that l ga is a finite exceptional simple group of lie type over fq where q p let a be a group such that f x l the group aut l is generated by ga together with field and graph automorphisms see 5 38 all of which extend to morphisms of the abstract group g commuting with o thus there is a subgroup x of caut c a such that x x o and so x acts on the set of a stable subsets of g for a a stable subset y we write nx y for the stabilizer in x of y if d is a a stable closed connected reductive subgroup of g containing a maximal torus t of g and m nx d we call m a subgroup of maximal rank in x in this paper we determine the structure and conjugacy classes of those subgroups of maximal rank which are maximal in x

spin glass model for the c dismantling problem c dismantling cd problem aims at finding the minimum vertex set d of a graph g v e after removing which the remaining graph will break into connected components with the size not larger than c in this paper we introduce a spin glass model with c 1 integer value states into the cd problem and then study the properties of this spin glass model by the belief propagation bp equations under the replica symmetry ansatz we give the lower bound rho c of the relative size of d with finite c on regular random graphs and erdos renyi random graphs we find rho c will decrease gradually with growing c and it converges to rho infty as c to infty the cd problem is called dismantling problem when c is a small finite fraction of v therefore rho infty is also the lower bound of the dismantling problem when v to infty to reduce the computation complexity of the bp equations taking the knowledge of the probability of a random selected vertex belonging to a remaining connected component with the size a the original bp equations can be simplified to one with only three states when c to infty the simplified bp equations are very similar to the bp equations of the feedback vertex set spin glass model h j zhou eur phys j b 86 455 2013 at last we develop two practical belief propagation guide decimation algorithms based on the original bp equations cd bpd and the simplified bp equations scd bpd to solve the cd problem on a certain graph our bpd algorithms and two other state of art heuristic algorithms are applied on various random graphs and some real world networks computation results show that the cd bpd is the best in all tested algorithms in the case of small c but considering the performance and computation consumption we recommend using scd bpd for the network with small clustering coefficient when c is large

network structure classification and features of water distribution systems the network connectivity structure of water distribution systems wdss represents the domain where hydraulic processes occur driving the emerging behavior of such systems for example with respect to robustness and vulnerability in complex network theory cnt a common way of classifying the network structure and connectivity is the association of the nodal degree distribution to specific probability distribution models and during the last decades researchers classified many real networks using the poisson or pareto distributions in spite of the fact that degree based network classification could play a crucial role to assess wds vulnerability this task is not easy because the network structure of wdss is strongly constrained by spatial characteristics of the environment where they are constructed the consequence of these spatial constraints is that the nodal degree spans very small ranges in wdss hindering a reliable classification by the standard approach based on the nodal degree distribution this work investigates the classification of the network structure of twenty two real wdss built in different environments demonstrating that the poisson distribution generally models the degree distributions very well in order to overcome the problem of the reliable classification based on the standard nodal degree we define the neighborhood degree equal to the sum of the nodal degrees of the nearest topological neighbors i e the adjacent nodes this definition of neighborhood degree is consistent with the fact that the degree of a single node is not significant for analysis of wdss this article is protected by copyright all rights reserved

coauthorship networks and patterns of scientific collaboration by using data from three bibliographic databases in biology physics and mathematics respectively networks are constructed in which the nodes are scientists and two scientists are connected if they have coauthored a paper we use these networks to answer a broad variety of questions about collaboration patterns such as the numbers of papers authors write how many people they write them with what the typical distance between scientists is through the network and how patterns of collaboration vary between subjects and over time we also summarize a number of recent results by other authors on coauthorship patterns

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 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

effective graph resistance this paper studies an interesting graph measure that we call the effective graph resistance the notion of effective graph resistance is derived from the field of electric circuit analysis where it is defined as the accumulated effective resistance between all pairs of vertices the objective of the paper is twofold first we survey known formulae of the effective graph resistance and derive other representations as well the derivation of new expressions is based on the analysis of the associated random walk on the graph and applies tools from markov chain theory this approach results in a new method to approximate the effective graph resistance a second objective of this paper concerns the optimisation of the effective graph resistance for graphs with given number of vertices and diameter and for optimal edge addition a set of analytical results is described as well as results obtained by exhaustive search one of the foremost applications of the effective graph resistance we have in mind is the analysis of robustness related problems however with our discussion of this informative graph measure we hope to open up a wealth of possibilities of applying the effective graph resistance to all kinds of networks problems 2011 elsevier inc all rights reserved

correlations in complex networks under attack for any initially correlated network after any kind of attack where either nodes or edges are removed we obtain general expressions for the degree degree probability matrix and degree distribution we show that the proposed analytical approach predicts the correct topological changes after the attack by comparing the evolution of the assortativity coefficient for different attack strategies and intensities in theory and simulations we find that it is possible to turn an initially assortative network into a disassortative one and vice versa by fine tuning removal of either nodes or edges for an initially uncorrelated network on the other hand we discover that only a targeted edge removal attack can induce such correlations

bridges in complex networks a bridge in a graph is an edge whose removal disconnects the graph and increases the number of connected components we calculate the fraction of bridges in a wide range of real world networks and their randomized counterparts we find that real networks typically have more bridges than their completely randomized counterparts but they have a fraction of bridges that is very similar to their degree preserving randomizations we define an edge centrality measure called bridgeness to quantify the importance of a bridge in damaging a network we find that certain real networks have a very large average and variance of bridgeness compared to their degree preserving randomizations and other real networks finally we offer an analytical framework to calculate the bridge fraction and the average and variance of bridgeness for uncorrelated random networks with arbitrary degree distributions

on bruck loops of 2 power exponent the goal of this paper is two fold first we provide the information needed to study bol ar or bruck loops by applying group theoretic methods this information is used in this paper as well as in baumeister and stein 2010 bs3 and in stein 2009 s moreover we determine the groups associated to bruck loops of 2 power exponent under the assumption that every non abelian simple group s is either passive or isomorphic to psl2 q q 1 4 a 2 power in a separate paper it is proven that indeed every non abelian simple group s is either passive or isomorphic to psl2 q q 1 4 a 2 power stein 2009 s the results obtained here are used in baumeister and stein 2010 bs3 where we determine the structure of the groups associated to the finite bruck loops

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

graph partitioning induced phase transitions we study the percolation properties of graph partitioning on random regular graphs with n vertices of degree k optimal graph partitioning is directly related to optimal attack and immunization of complex networks we find that for any partitioning process even if nonoptimal that partitions the graph into essentially equal sized connected components clusters the system undergoes a percolation phase

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

attack vulnerability of complex networks we study the response of complex networks subject to attacks on vertices and edges several existing complex network models as well as real world networks of scientific collaborations and internet traffic are numerically investigated and the network performance is quantitatively measured by the average inverse geodesic length and the size of the largest connected subgraph for each case of attacks on vertices and edges four different attacking strategies are used removals by the descending order of the degree and the betweenness centrality calculated for either the initial network or the current network during the removal procedure it is found that the removals by the recalculated degrees and betweenness centralities are often more harmful than the attack strategies based on the initial network suggesting that the network structure changes as important vertices or edges are removed furthermore the correlation between the betweenness centrality and the degree in complex networks is studied

predicting percolation thresholds in networks we consider different methods which do not rely on numerical simulations of the percolation process to approximate percolation thresholds in networks we perform a systematic analysis on synthetic graphs and a collection of 109 real networks to quantify their effectiveness and reliability as prediction tools our study reveals that the inverse of the largest eigenvalue of the nonbacktracking matrix of the graph often provides a tight lower bound for true percolation threshold however in more than 40 of the cases this indicator is less predictive than the naive expectation value based solely on the moments of the degree distribution we find that the performance of all indicators becomes worse as the value of the true percolation threshold grows thus none of them represents a good proxy for the robustness of extremely fragile networks

robustness of network controllability in cascading failure controllability of networks widely existing in real life systems have been a critical and attractive research subject for both network science and control systems communities research in network controllability has mostly focused on the effects of the network structure on its controllability and some studies have begun to investigate the controllability robustness of complex networks cascading failure is common phenomenon in many infrastructure networks which largely affect normal operation of networks and sometimes even lead to collapse resulting in considerable economic losses the robustness of network controllability against the cascading failure is studied by a linear load capacity model with a breakdown probability in this paper the controllability of canonical model networks under different node attack strategies is investigated random failure and malicious attack it is shown by numerical simulations that the tolerant parameter of load capacity model has an important role in the emergence of cascading failure independent to the types of network the networks with moderate average degree are more vulnerable to the cascading failure while these with high average degree are very robust in particular betweenness attack strategy is more harmful to the network controllability than degree attack one especially for the scale free networks

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

the surgery semicharacteristic given a degree one normal map m vm x c t c wall defined the associated surgery obstruction a f f e l jln x this obstruction vanishes if and only if is normally bordant to a simple homotopy equivalence using wall s approach one must perform preliminary surgery to make highly connected before the obstruction can be calculated it is natural to ask for invariants which can be calculated without preliminary surgery in the even dimensional case the multisignature gives such an invariant the purpose of this paper is to present an odd dimensional normal bordism invariant defined without preliminary surgery this invariant is analogous to the semicharacteristic bordism invariant introduced by lee 13 a justification for the normal bordism invariant is that it gives strong restrictions on the homology of a manifold with a free action of a finite group in particular it gives a good explanation of lee s results on free actions of finite groups on spheres let i be a 2n l dimensional poincare complex with fundamental group n suppose a is a semisimple ring with involution a a for a e a then l2n 1 a can be identified as a quotient of a subgroup of 0 2 given a homomorphism zn co a we can regard a as a local coefficient system that is a znmodule the surgery semicharacteristic is

a class of finite simple bol loops of exponent 2 in this paper we give an infinite class of finite simple right bol loops of exponent 2 the right multiplication group of these loops is an extension of an elementary abelian 2 group by s 5 the construction uses the description of the structure of such loops given by m aschbacher 2005 these results answer some questions of m aschbacher

expander graphs in pure and applied mathematics expander graphs are highly connected sparse finite graphs they play an important role in computer science as basic building blocks for network constructions error correcting codes algorithms and more in recent years they have started to play an increasing role also in pure mathematics number theory group theory geometry and more this expository article describes their constructions and various applications in pure and applied mathematics

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

enhancing robustness and immunization in geographical networks we find that different geographical structures of networks lead to varied percolation thresholds although these networks may have similar abstract topological structures thus strategies for enhancing robustness and immunization of a geographical network are proposed using the generating function formalism we obtain an explicit form of the percolation threshold q c for networks containing arbitrary order cycles for three cycles the dependence of q c on the clustering coefficients is ascertained the analysis substantiates the validity of the strategies with analytical evidence

betweenness centrality of an edge in tree like components with finite size using the generating function formalism we propose an exact formula for estimating betweenness centrality of an edge in finite tree like components of random networks with arbitrary degree distributions as a function of the degrees of two vertices interconnected by the edge and the size of the component to which the edge belongs the formula can give an estimation of the mean value of the edge betweenness we confirm the formula by simulations for poisson and power law distributions respectively we also study the asymptotic behavior of the formula and find that there is an asymptotic power law relation with an exponent 1 5 between the edge betweenness and the size of the component to which the edge belongs

nonlinear analogs of linear group actions on spheres introduction this article is based upon a principle which is so standard that it is almost a cliche the first step to understanding a nonlinear phenomenon is to define and study a suitable linear approximation to be more specific we shall describe some applications of this to the symmetry questions of topological transformation groups given a topological space x let homeo denote the set of self homeomorphisms of x this is a group under composition of mappings if g is an arbitrary group then a group action of g on i is a homomorphism homeo x frequently we wish to impose some weak assumptions on p for example if g is a topological group then we might want cp to have suitable continuity properties the usual assumption is that the map

gyrogroup actions a generalization of group actions abstract this article explores the novel notion of gyrogroup actions which is a natural generalization of the usual notion of group actions as a first step toward the study of gyrogroup actions from the algebraic viewpoint we prove three well known theorems in group theory for gyrogroups the orbit stabilizer theorem the orbit decomposition theorem and the burnside lemma or the cauchy frobenius lemma we then prove that under a certain condition a gyrogroup g acts transitively on the set g h of left cosets of a subgyrogroup h in g in a natural way from this we prove the structure theorem that every transitive action of a gyrogroup can be realized as a gyrogroup action by left gyroaddition we also exhibit concrete examples of gyrogroup actions from the mobius and einstein gyrogroups

effect of edge removal on topological and functional robustness of complex networks abstract we study the robustness of several network models subject to edge removal the robustness is measured by the statistics of network breakdowns where a breakdown is defined as the destroying of the total connectedness of a network rather than the disappearance of the giant component we introduce a simple traffic dynamics as the function of a network topology and the total connectedness can be destroyed in the sense of either the topology or the function the overall effect of the topological breakdown and the functional breakdown as well as the relative importance of the topological robustness and the functional robustness are studied under two edge removal strategies

a vibrational approach to node centrality and vulnerability in complex networks we propose a new measure of vulnerability of a node in a complex network the measure is based on the analogy in which the nodes of the network are represented by balls and the links are identified with springs we define the measure as the node displacement or the amplitude of vibration of each node under fluctuation due to the thermal bath in which the network is supposed to be submerged we prove exact relations among the thus defined node displacement the information centrality and the kirchhoff index the relation between the first two suggests that the node displacement has a better resolution of the vulnerability than the information centrality because the latter is the sum of the local node displacement and the node displacement averaged over the entire network

asymptotic estimates for efficiency vulnerability and cost for random networks an investigation both analytical and numerical of the values of efficiency vulnerability and cost functions is provided several comparisons among these functions are made following a pairwise strategy and taking into account different efficiency vulnerability and cost functions it is shown in all cases that these magnitudes display strong correlations which are typically piecewise linear in log space as well as universal i e independent of the number of nodes n to a great extent the analytical part of the work develops explicit bounds linking all pairs of such parameters these results are numerically compared with the result of a random testing for erdos renyi networks

local surgery obstructions and space forms the space form problem concerning the existence of free actions of finite groups on spheres has now been substantially solved except in dimension three i 3 4 8 11 13 14 this paper gives an effective method for the uniqueness question which homotopy types are realized by such actions our results are fairly complete for 2 hyperelementary periodic groups and give new necessary conditions for actions even on s 3 in particular corollary c below eliminates certain homotopically non linear actions which presented an obstacle to knowing precise dimensional bounds see 4 conjecture d the methods also apply to the euclidean space form problem of semi free actions on i k fixing i k 2 where the linear model is a free representation direct sum with a trivial representation a free action on s 1 yields a semi free topological action on l by coning the method involves the calculation of a certain local surgery obstructions w167 2 3 the basic cases are the 2 hyperelementary type i groups extensions of the form l z m z 2k l where m is odd we write o for a sylow 2 subgroup of z and identify it with z 2k let t be the twisting defining the extension and let t cr im t z 2 k aut z m 7 2 z

the finite bruck loops abstract we continue the work of aschbacher kinyon and phillips 2006 akp06 as well as of glauberman 1964 1968 g64 g68 by describing the structure of the finite bruck loops we show that a finite bruck loop x is the direct product of a bruck loop of odd order with either a soluble bruck loop of 2 power order or a product of loops related to the groups psl 2 q q 9 or q 5 a fermat prime the latter possibility does occur as is shown in nagy 2008 n08 and baumeister and stein 2010 bs10 as corollaries we obtain versions of sylow s lagrange s and hall s theorems for loops

group actions on homology spheres this can be stated in a more symmetric manner let r be any positive integer not equal to 3 then n acts freely and homologically trivially on z r i ff n acts freely and homologically trivially on sl in fact there is a one to one correspondence between such actions on u and such actions on s r the classification of such actions is discussed in w in addition the actions constructed have the property that every group element is isotopic to the identity theorem a holds in the topological or piecewise linear categories well known counterexamples exist in the smooth category even for homotopy spheres theorem a would be false without the assumption of homological triviality or without the assumption of simple connectivity see remark 8 8 in fact there exist groups n and simply connected z in homology spheres z r such that n acts freely on u but not on sl also there are groups n which act freely on s r but do not act freely on certain lens spaces lp with p prime to in if the group n is cyclic a result similar to our main theorem was first proved by l sffler l by different methods the second author independently established the cyclic case in wel by a method that extends to cyclic group actions on more general manifolds see c w1 and via this paper to general finite groups however the difficulties in making this extension have application elsewhere the main theorem is motivated by the philosophy that all of the geometry of a group action is present at the order of the group thus it is only natural to expect that any manifold homologically resembling a sphere admitting a

isolated involutions whose centraliser is soluble abstract we analyse a minimal counterexample to glauberman s z theorem from a local group theoretic point of view one of the main results is a group theoretic proof for the z theorem in the special case where the centraliser of an isolated involution is soluble

when is a scale free graph ultra small in this paper we study typical distances in the configuration model when the degrees have asymptotically infinite variance we assume that the empirical degree distribution follows a power law with exponent tau in 2 3 up to value n beta n for some beta n gg log n gamma and gamma in 0 1 this assumption is satisfied for power law i i d degrees and also includes truncated power law empirical degree distributions where the possibly exponential truncation happens at n beta n these examples are commonly observed in many real life networks we show that the graph distance between two uniformly chosen vertices centers around 2 log log n beta n log tau 2 1 beta n 3 tau with tight fluctuations thus the graph is an ultrasmall world whenever 1 beta n o log log n we determine the distribution of the fluctuations around this value in particular we prove these form a sequence of tight random variables with distributions that show log log periodicity and as a result it is non converging we describe the topology and number of shortest paths we show that the number of shortest paths is of order n f n beta n where f n in 0 1 is a random variable that oscillates with n we decompose shortest paths into three segments two end segments starting at each of the two uniformly chosen vertices and a middle segment the two end segments of any shortest path have length log log n beta n log tau 2 tight and the total degree is increasing towards the middle of the path on these segments the connecting middle segment has length 1 beta n 3 tau tight and it contains only vertices with degree at least of order n 1 f n beta n thus all the degrees on this segment are comparable to the maximal degree our theorems also apply when instead of truncating the degrees we start with a configuration model and we remove every vertex with degree at least n beta n and the edges attached to these vertices this sheds light on the attack vulnerability of the configuration model with infinite variance degrees

quantifying network heterogeneity despite degree distributions give some insights about how heterogeneous a network is they fail in giving a unique quantitative characterization of network heterogeneity this is particularly the case when several different distributions fit for the same network when the number of data points is very scarce due to network size or when we have to compare two networks with completely different degree distributions here we propose a unique characterization of network heterogeneity based on the difference of functions of node degrees for all pairs of linked nodes we show that this heterogeneity index can be expressed as a quadratic form of the laplacian matrix of the network which allows a spectral representation of network heterogeneity we give bounds for this index which is equal to zero for any regular network and equal to one only for star graphs using it we study random networks showing that those generated by the erdos renyi algorithm have zero heterogeneity and those generated by the preferential attachment method of barabasi and albert display only 11 of the heterogeneity of a star graph we finally study 52 real world networks and we found that they display a large variety of heterogeneities we also show that a classification system based on degree distributions does not reflect the heterogeneity properties of real world networks

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

surgery with finite fundamental group the relationship between the topology of manifolds and homotopy theory has been analysed by surgery 1 18 28 31 however the effectiveness of this analysis depends on being able to compute surgery obstructions in this paper we introduce a new technique for evaluating the weakly simple surgery obstructions for degree 1 normal maps of closed manifolds and apply it to oriented manifolds with finite fundamental group our main result is theorem a below let x dx be a finite poincare pair of dimension n and a top bundle over x which is stably fibre homotopy equivalent to the spivak normal fibre space v let m dm be a compact n dimensional manifold and let

efficiency of attack strategies on complex model and real world networks we investigated the efficiency of attack strategies to network nodes when targeting several complex model and real world networks we tested 5 attack strategies 3 of which were introduced in this work for the first time to attack 3 model networks erdos and renyi barabasi and albert preferential attachment network and scale free network configuration models and 3 real networks gnutella peer to peer network email network of the university of rovira i virgili and immunoglobulin interaction network nodes were removed sequentially according to the importance criterion defined by the attack strategy and we used the size of the largest connected component lcc as a measure of network damage we found that the efficiency of attack strategies fraction of nodes to be deleted for a given reduction of lcc size depends on the topology of the network although attacks based on either the number of connections of a node or betweenness centrality were often the most efficient strategies sequential deletion of nodes in decreasing order of betweenness centrality was the most efficient attack strategy when targeting real world networks the relative efficiency of attack strategies often changed during the sequential removal of nodes especially for networks with power law degree distribution

stability and continuity of centrality measures in weighted graphs this paper presents a formal definition of stability for node centrality measures in weighted graphs it is shown that the commonly used measures of degree closeness and eigenvector centrality are stable whereas betweenness centrality is not an alternative definition of the latter that preserves the same centrality notion while satisfying the stability criteria is introduced continuity is presented as a less stringent alternative to stability betweenness centrality is shown to be not only unstable but discontinuous numerical experiments in synthetic random networks and real world data show that in practice stability and continuity imply different levels of robustness in the presence of noisy data in particular the stable betweenness centrality is shown to exhibit resilience against noise that is absent in the discontinuous and unstable standard betweenness centrality while preserving a similar notion of centrality

complex networks in the euclidean space of communicability distances we study the properties of complex networks embedded in a euclidean space of communicability distances the communicability distance between two nodes is defined as the difference between the weighted sum of walks self returning to the nodes and the weighted sum of walks going from one node to the other we give some indications that the communicability distance identifies the least crowded routes in networks where simultaneous submission of packages is taking place we define an index q based on communicability and shortest path distances which allows reinterpreting the small world phenomenon as the region of minimum q in the watts strogatz model it also allows the classification and analysis of networks with different efficiency of spatial uses consequently the communicability distance displays unique features for the analysis of complex networks in different scenarios

bol loops and bruck loops of order pq abstract right bol loops are loops satisfying the identity z x y x z x y x and right bruck loops are right bol loops satisfying the identity x y 1 x 1 y 1 let p and q be odd primes such that p q advancing the research program of niederreiter and robinson from 1981 we classify right bol loops of order pq when q does not divide p 2 1 the only right bol loop of order pq is the cyclic group of order pq when q divides p 2 1 there are precisely p q 4 2 right bol loops of order pq up to isomorphism including a unique nonassociative right bruck loop b p q of order pq let q be a nonassociative right bol loop of order pq we prove that the right nucleus of q is trivial the left nucleus of q is normal and is equal to the unique subloop of order p in q and the right multiplication group of q has order p 2 q or p 3 q when q b p q the right multiplication group of q is isomorphic to the semidirect product of z p z p with z q finally we offer computational results as to the number of right bol loops of order pq up to isotopy

a comparative analysis of network robustness against different link attacks abstract recently the study of optimizing network robustness has attracted increasing attentions and the constraint that every node s degree cannot be changed is considered although this constraint maintains the node degree distribution consistently in order to reserve the structure of networks it makes the network structure be lack of flexibility since many network structure always transform in the modern society given this consideration in this paper we analyze the robustness of networks through setting a new constraint that is only the number of edges should be unchanged then we use the link robustness index r l as the measure of the network robustness against either random failures or intentional attacks and make a comparative analysis of network robustness against different types of link attacks moreover we use four types of networks as initial networks namely scale free networks random networks regular networks and small world networks the experimental results show that the values of robustness measures for the optimized networks starting from different initial networks are similar under different link attacks but the network topologies may be different that is to say networks with different topologies may have similar robustness in terms of the robustness measures we also find that the optimized networks obtained by one link attack may not robust against other link attacks sometimes even weaker than the original networks therefore before building networks it is better to study which type of link attacks may happen

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