robustness of a network formed of spatially embedded networks we present analytic and numeric results for percolation in a network formed of interdependent spatially embedded networks we show results for a treelike and a random regular network of networks each with i unconstrained dependency links and ii dependency links restricted to a maximum euclidean length r analytic results are given for each network of networks with spatially unconstrained dependency links and compared to simulations for the case of two fully interdependent spatially embedded networks it was found li et al phys rev lett 108 228702 2012 that the system undergoes a first order phase transition only for r r c 8 we find here that for treelike networks of networks composed of n networks r c significantly decreases as n increases and rapidly n 11 reaches its limiting value of 1 for cases where the dependencies form loops such as in random regular networks we show analytically and confirm through simulations that there is a certain fraction of dependent nodes q max above which the entire network structure collapses even if a single node is removed the value of q max decreases quickly with m the degree of the random regular network of networks our results show the extreme sensitivity of coupled spatial networks and emphasize the susceptibility of these networks to sudden collapse the theory proposed here requires only numerical knowledge about the percolation behavior of a single network and therefore can be used to find the robustness of any network of networks where the profile of percolation of a singe network is known numerically

monopoles and maps from s 2 to s 2 the topology of the configuration space the configuration space for the su 2 yang mills higgs equations on 3 is shown to be homotopic to the space of smooth maps froms 2 tos 2 this configuration space indexes a family of twisted dirac operators the dirac family is used to prove that the configuration space does not retract onto any subspace on which the su 2 yang mills higgs functional is bounded

optimal interdependence enhances the dynamical robustness of complex systems although interdependent systems have usually been associated with increased fragility we show that strengthening the interdependence between dynamical processes on different networks can make them more likely to survive over long times by coupling the dynamics of networks that in isolation exhibit catastrophic collapse with extinction of nodal activity we demonstrate system wide persistence of activity for an optimal range of interdependence between the networks this is related to the appearance of attractors of the global dynamics comprising disjoint sets islands of stable activity

min max theory for the yang mills higgs equations in each monopole sector there exist an infinite number of finite energy solutions to the prasad sommerfield limit of the su 2 yang mills higgs equations on 3 whose energy is greater than any finite number

coupled effects of local movement and global interaction on contagion abstract by incorporating segregated spatial domain and individual based linkage into the sis susceptible infected susceptible model we propose a generalized epidemic model which can change from the territorial epidemic model to the networked epidemic model the role of the individual based linkage between different spatial domains is investigated as we adjust the timescale parameter from 0 to unity which represents the degree of activation of the individual based linkage three regions are found within the region of 0 0 02 the epidemic is determined by local movement and is sensitive to the timescale within the region of 0 02 0 5 the epidemic is insensitive to the timescale within the region of 0 5 1 the outbreak of the epidemic is determined by the structure of the individual based linkage as we keep an eye on the first region the role of activating the individual based linkage in the present model is similar to the role of the shortcuts in the two dimensional small world network only activating a small number of the individual based linkage can prompt the outbreak of the epidemic globally the role of narrowing segregated spatial domain and reducing mobility in epidemic control is checked these two measures are found to be conducive to curbing the spread of infectious disease only when the global interaction is suppressed a log log relation between the change in the number of infected individuals and the timescale is found by calculating the epidemic threshold and the mean first encounter time we heuristically analyze the microscopic characteristics of the propagation of the epidemic in the present model

interdependent networks with identical degrees of mutually dependent nodes we study a problem of failure of two interdependent networks in the case of identical degrees of mutually dependent nodes we assume that both networks a and b have the same number of nodes n connected by the bidirectional dependency links establishing a one to one correspondence between the nodes of the two networks in a such a way that the mutually dependent nodes have the same number of connectivity links i e their degrees coincide this implies that both networks have the same degree distribution p k we call such networks correspondently coupled networks ccns we assume that the nodes in each network are randomly connected we define the mutually connected clusters and the mutual giant component as in earlier works on randomly coupled interdependent networks and assume that only the nodes that belong to the mutual giant component remain functional we assume that initially a 1 p fraction of nodes are randomly removed because of an attack or failure and find analytically for an arbitrary p k the fraction of nodes p that belong to the mutual giant component we find that the system undergoes a percolation transition at a certain fraction p p c which is always smaller than p c for randomly coupled networks with the same p k we also find that the system undergoes a first order transition at p c 0 if p k has a finite second moment for the case of scale free networks with 2 0 finally we find that the robustness of ccn increases with the broadness of their degree distribution

structure of shells in complex networks we define shell l in a network as the set of nodes at distance l with respect to a given node and define rl as the fraction of nodes outside shell l in a transport process information or disease usually diffuses from a random node and reach nodes shell after shell thus understanding the shell structure is crucial for the study of the transport property of networks we study the statistical properties of the shells of a randomly chosen node for a randomly connected network with given degree distribution we derive analytically the degree distribution and average degree of the nodes residing outside shell l as a function of rl further we find that rl follows an iterative functional form rl phi rl 1 where phi is expressed in terms of the generating function of the original degree distribution of the network our results can explain the power law distribution of the number of nodes bl found in shells with l larger than the network diameter d which is the average distance between all pairs of nodes for real world networks the theoretical prediction of rl deviates from the empirical rl we introduce a network correlation function c rl identical with rl phi rl 1 to characterize the correlations in the network where rl is the empirical value and phi rl 1 is the theoretical prediction c rl 1 indicates perfect agreement between empirical results and theory we apply c rl to several model and real world networks we find that the networks fall into two distinct classes i a class of poorly connected networks with c rl 1 where a larger smaller fraction of nodes resides outside inside distance l from a given node than in randomly connected networks with the same degree distributions examples include the watts strogatz model and networks characterizing human collaborations such as citation networks and the actor collaboration network ii a class of well connected networks with c rl 1 examples include the barabasi albert model and the autonomous system internet network

integrality of the monopole number in su 2 yang mills higgs theory on ℝ3 we prove that in classical su 2 yang mills higgs theories on 3 with a higgs field in the adjoint representation an integer valued monopole number magnetic charge is canonically defined for any finite actionl1 loc2 configuration in particular the result is true for smooth configurations the monopole number is shown to decompose the configuration space into path components

the existence of multi monopole solutions to the non abelian yang mills higgs equations for arbitrary simple gauge groups we prove that for arbitrary simple gauge groups the non abelian yang mills higgs equations on 3 in the prasad sommerfield limit have at least a countably infinite set of distinct solutions these solutions may be interpreted physically as configurations of widely spaced non interacting fundamental monopoles the solutions are generically not spherically symmetric

robustness of a network formed by n interdependent networks with a one to one correspondence of dependent nodes many real world networks interact with and depend upon other networks we develop an analytical framework for studying a network formed by n fully interdependent randomly connected networks each composed of the same number of nodes n the dependency links connecting nodes from different networks establish a unique one to one correspondence between the nodes of one network and the nodes of the other network we study the dynamics of the cascades of failures in such a network of networks non caused by a random initial attack on one of the networks after which a fraction p of its nodes survives we find for the fully interdependent loopless non that the final state of the non does not depend on the dynamics of the cascades but is determined by a uniquely defined mutual giant component of the non which generalizes both the giant component of regular percolation of a single network n 1 and the recently studied case of the mutual giant component of two interdependent networks n 2 we also find that the mutual giant component does not depend on the topology of the non and express it in terms of generating functions of the degree distributions of the network our results show that for any n 2 there exists a critical p p c 0 below which the mutual giant component abruptly collapses from a finite nonzero value for p p c to zero for p 2 a rr non is stable for any n with p c 1 this results arises from the critical role played by singly connected nodes which exist in an er non and enhance the cascading failures but do not exist in a rr non

percolation of partially interdependent networks under targeted attack we study a system composed of two partially interdependent networks when nodes in one network fail they cause dependent nodes in the other network to also fail in this paper the percolation of partially interdependent networks under targeted attack is analyzed we apply a general technique that maps a targeted attack problem in interdependent networks to a random attack problem in a transformed pair of interdependent networks we illustrate our analytical solutions for two examples i the probability for each node to fail is proportional to its degree and ii each node has the same probability to fail in the initial time we find the following i for any targeted attack problem for the case of weak coupling the system shows a second order phase transition and for the strong coupling the system shows a first order phase transition ii for any coupling strength when the high degree nodes have higher probability to fail the system becomes more vulnerable iii there exists a critical coupling strength and when the coupling strength is greater than the critical coupling strength the system shows a first order transition otherwise the system shows a second order transition

method for estimating critical exponents in percolation processes with low sampling in phase transition phenomena the estimation of the critical point is crucial for the calculation of the various critical exponents and the determination of the universality class they belong to however this is not an easy task since a large amount of realizations is needed to eliminate the noise in the data in this paper we introduce a novel method for the simultaneous estimation of the critical point p c and the critical exponent applied for the case of explosive bond percolation on two dimensional square lattices and erdos renyi networks the results show that with only a few hundred realizations it is possible to acquire accurate values for these quantities guidelines are given at the end for the applicability of the method to other cases as well

cascade of failures in coupled network systems with multiple support dependence relations we study both analytically and numerically the cascade of failures in two coupled network systems a and b where multiple support dependence relations are randomly built between nodes of networks a and b in our model we assume that each node in one network can function only if it has at least a single support link connecting it to a functional node in the other network we assume that networks a and b have i sizes n a and n b ii degree distributions of connectivity links p a k and p b k iii degree distributions of support links p a k and p b k and iv random attack removes 1 r a n a and 1 r b n b nodes form the networks a and b respectively we find the fractions of nodes a and b which remain functional giant component at the end of the cascade process in networks a and b in terms of the generating functions of the degree distributions of their connectivity and support links in a special case of erd s renyi networks with average degrees a and b in networks a and b respectively and poisson distributions of support links with average degrees a and b in networks a and b respectively a r a 1 exp a b 1 exp a a and b r b 1 exp b a 1 exp b b in the limit of a and b both networks become independent and our model becomes equivalent to a random attack on a single erd s renyi network we also test our theory on two coupled scale free networks and find good agreement with the simulations

stability in yang mills theories at a solution of the yang mills equations ons4 or the yang mills higgs equation on 3 the hessian of the action functional defines a natural second order elliptic operator the number of negative eigenvalues of this operator is bounded below by a multiple of the relevant topological charge the proof of this assertion requires a relative index theorem for dirac type operators on n n 3

monopoles braid groups and the dirac operator using the relation between the space of rational functions on the space ofsu 2 monopoles on 3 and the classifying space of the braid group see 10 we show how the index bundle of the family of real dirac operators coupled tosu 2 monopoles can be described using permutation representations of artin s braid groups we also show how this implies the existence of a pair consisting of a gauge fielda and a higgs field on 3 whose corresponding dirac equation has an arbitrarily large dimensional space of solutions

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

percolation of a general network of networks percolation theory is an approach to study the vulnerability of a system we develop an analytical framework and analyze the percolation properties of a network composed of interdependent networks netonet typically percolation of a single network shows that the damage in the network due to a failure is a continuous function of the size of the failure i e the fraction of failed nodes in sharp contrast in netonet due to the cascading failures the percolation transition may be discontinuous and even a single node failure may lead to an abrupt collapse of the system we demonstrate our general framework for a netonet composed of n classic erd ifmmode mbox h o else h o fi s r enyi er networks where each network depends on the same number m of other networks i e for a random regular network rr formed of interdependent er networks the dependency between nodes of different networks is taken as one to one correspondence i e a node in one network can depend only on one node in the other network no feedback condition in contrast to a treelike netonet in which the size of the largest connected cluster mutual component depends on n the loops in the rr netonet cause the largest connected cluster to depend only on m and the topology of each network but not on n we also analyzed the extremely vulnerable feedback condition of coupling where the coupling between nodes of different networks is not one to one correspondence in the case of netonet formed of er networks percolation only exhibits two phases a second order phase transition and collapse and no first order percolation transition regime is found in the case of the no feedback condition in the case of netonet composed of rr networks there exists a first order phase transition when the coupling strength q fraction of interdependency links is large and a second order phase transition when q is small our insight on the resilience of coupled networks might help in designing robust interdependent systems

percolation of partially interdependent scale free networks we study the percolation behavior of two interdependent scale free sf networks under random failure of 1 p fraction of nodes our results are based on numerical solutions of analytical expressions and simulations we find that as the coupling strength between the two networks q reduces from 1 fully coupled to 0 no coupling there exist two critical coupling strengths q 1 and q 2 which separate three different regions with different behavior of the giant component as a function of p i for q q 1 an abrupt collapse transition occurs at p p c ii for q 2 3 q 1 decreases with increasing here is the scaling exponent of the degree distribution p k proportionality k b in the hybrid transition at the q 2 0 for 3 thus the known theoretical p c 0 for a single network with 3 is expected to be valid also for strictly partial interdependent networks

a note and erratum concerning min max theory for the yang mills higgs equations an error in an argument which was used to prove the existence of non minimal solutions to thesu 2 yang mills higgs equations has been shown to the author a revised proof is presented here to establish the existence of infinitely many non minimal solutions to the afore mentioned equations

localized versions of function spaces and generic results abstract we consider generalizations of classical function spaces by requiring that a holomorphic in function satisfies some property when we approach from not the whole boundary but only a part of it these spaces endowed with their natural topology are frechet spaces we prove some generic non extendability results in such spaces and generic nowhere differentiability on the corresponding part of