topological fractal networks introduced by mixed degree distribution abstract several fundamental properties of real complex networks such as the small world effect the scale free degree distribution and recently discovered topological fractal structure have presented the possibility of a unique growth mechanism and allow for uncovering universal origins of collective behaviors however highly clustered scale free network with power law degree distribution or small world network models with exponential degree distribution are not self similarity we investigate networks growth mechanism of the branching deactivated geographical attachment preference that learned from certain empirical evidence of social behaviors it yields high clustering and spectrums of degree distribution ranging from algebraic to exponential average shortest path length ranging from linear to logarithmic we observe that the present networks fit well with small world graphs and scale free networks in both limit cases exponential and algebraic degree distribution respectively obviously lacking self similar property under a length scale transformation interestingly we find perfect topological fractal structure emerges by a mixture of both algebraic and exponential degree distributions in a wide range of parameter values the results present a reliable connection among small world graphs scale free networks and topological fractal networks and promise a natural way to investigate universal origins of collective behaviors

map estimation via agreement on trees message passing and linear programming we develop and analyze methods for computing provably optimal maximum a posteriori probability map configurations for a subclass of markov random fields defined on graphs with cycles by decomposing the original distribution into a convex combination of tree structured distributions we obtain an upper bound on the optimal value of the original problem i e the log probability of the map assignment in terms of the combined optimal values of the tree problems we prove that this upper bound is tight if and only if all the tree distributions share an optimal configuration in common an important implication is that any such shared configuration must also be a map configuration for the original distribution next we develop two approaches to attempting to obtain tight upper bounds a a tree relaxed linear program lp which is derived from the lagrangian dual of the upper bounds and b a tree reweighted max product message passing algorithm that is related to but distinct from the max product algorithm in this way we establish a connection between a certain lp relaxation of the mode finding problem and a reweighted form of the max product min sum message passing algorithm

statistical analysis of airport network of china through the study of airport network of china anc composed of 128 airports nodes and 1165 flights edges we show the topological structure of anc conveys two characteristics of small worlds a short average path length 2 067 and a high degree of clustering 0 733 the cumulative degree distributions of both directed and undirected anc obey two regime power laws with different exponents i e the so called double pareto law in degrees and out degrees of each airport have positive correlations whereas the undirected degrees of adjacent airports have significant linear anticorrelations it is demonstrated both weekly and daily cumulative distributions of flight weights frequencies of anc have power law tails besides the weight of any given flight is proportional to the degrees of both airports at the two ends of that flight it is also shown the diameter of each subcluster consisting of an airport and all those airports to which it is linked is inversely proportional to its density of connectivity efficiency of anc and of its subclusters is measured through a simple definition in terms of that the efficiency of anc s subclusters increases as the density of connectivity does anc is found to have an efficiency of 0 484

understanding the role of noise in stochastic local search analysis and experiments stochastic local search sls algorithms have recently been proven to be among the best approaches to solving computationally hard problems sls algorithms typically have a number of parameters optimized empirically that characterize and determine their performance in this article we focus on the noise parameter the theoretical foundation of sls including an understanding of how to the optimal noise varies with problem difficulty is lagging compared to the strong empirical results obtained using these algorithms a purely empirical approach to understanding and optimizing sls noise as problem instances vary can be very computationally intensive to complement existing experimental results we formulate and analyze several markov chain models of sls in this article in particular we compute expected hitting times and show that they are rational functions for individual problem instances as well as their mixtures expected hitting time curves are analytical counterparts to noise response curves reported in the experimental literature hitting time analysis using polynomials and convex functions is also discussed in addition we present examples and experimental results illustrating the impact of varying noise probability on sls run time in experiments where most probable explanations in bayesian networks are computed we use synthetic problem instances as well as problem instances from applications we believe that our results provide an improved theoretical understanding of the role of noise in stochastic local search thereby providing a foundation for further progress in this area

controlled generation of hard and easy bayesian networks impact on maximal clique size in tree clustering this article presents and analyzes algorithms that systematically generate random bayesian networks of varying difficulty levels with respect to inference using tree clustering the results are relevant to research on efficient bayesian network inference such as computing a most probable explanation or belief updating since they allow controlled experimentation to determine the impact of improvements to inference algorithms the results are also relevant to research on machine learning of bayesian networks since they support controlled generation of a large number of data sets at a given difficulty level our generation algorithms called bpart and mpart support controlled but random construction of bipartite and multipartite bayesian networks the bayesian network parameters that we vary are the total number of nodes degree of connectivity the ratio of the number of non root nodes to the number of root nodes regularity of the underlying graph and characteristics of the conditional probability tables the main dependent parameter is the size of the maximal clique as generated by tree clustering this article presents extensive empirical analysis using the hugin tree clustering approach as well as theoretical analysis related to the random generation of bayesian networks using bpart and mpart

the complexity of counting cuts and of computing the probability that a graph is connected several enumeration and reliability problems are shown to be p complete and hence at least as hard as np complete problems included are important problems in network reliability analysis namely computing the probability that a graph is connected and counting the number of minimum cardinality s t cuts or directed network cuts also shown to be p complete are counting vertex covers in a bipartite graph counting antichains in a partial order and approximating the probability that a graph is connected and the probability that a pair of vertices is connected

a topological phase transition between small worlds and fractal scaling in urban railway transportation networks abstract fractal and small worlds scaling laws are applied to study the growth of urban railway transportation networks using total length and total population as observational parameters in spite of the variety of populations and urban structures the variation of the total length of the railway network with the total population of conurbations follows similar patterns for large and middle metropolis diachronous analysis of data for urban transportation networks suggests that there is second order phase transition from small worlds behaviour to fractal scaling during their early stages of development

approximating maps for belief networks is np hard and other theorems abstract finding maximum a posteriori map assignments also called most probable explanations is an important problem on bayesian belief networks shimony has shown that finding maps is np hard in this paper we show that approximating maps with a constant ratio bound is also np hard in addition we examine the complexity of two related problems which have been mentioned in the literature we show that given the map for a belief network and evidence set or the family of maps if the optimal is not unique it remains np hard to find or approximate alternative next best explanations furthermore we show that given the map or maps for a belief network and an initial evidence set it is also np hard to find or approximate the map assignment for the same belief network with a modified evidence set that differs from the initial set by the addition or removal of even a single node assignment finally we show that our main result applies to networks with constrained in degree and out degree applies to randomized approximation and even still applies if the ratio bound instead of being constant is allowed to be a polynomial function of various aspects of the network topology

a differential approach to inference in bayesian networks we present a new approach to inference in bayesian networks which is based on representing the network using a polynomial and then retrieving answers to probabilistic queries by evaluating and differentiating the polynomial the network polynomial itself is exponential in size but we show how it can be computed efficiently using an arithmetic circuit that can be evaluated and differentiated in time and space linear in the circuit size the proposed framework for inference subsumes one of the most influential methods for inference in bayesian networks known as the tree clustering or jointree method which provides a deeper understanding of this classical method and lifts its desirable characteristics to a much more general setting we discuss some theoretical and practical implications of this subsumption

maximum order of periodic outer automorphisms of a free group abstract let fn be a free group with rank n and denote by out fn its outer automorphism group for arbitrary n consider the orders of periodic elements in out fn or equivalently the orders of finite cyclic subgroups of out fn by considering group actions on finite connected graphs we obtained the number theoretical characterization of these orders comparing the results with those for cyclic subgroups of finite symmetric groups asymptotic estimation for the maximum order cn is derived

approximating probabilistic inference in bayesian belief networks is np hard abstract it is known that exact computation of conditional probabilities in belief networks is np hard many investigators in the ai community have tacitly assumed that algorithms for performing approximate inference with belief networks are of polynomial complexity indeed special cases of approximate inference can be performed in time polynomial in the input size however we have discovered that the general problem of approximating conditional probabilities with belief networks like exact inference resides in the np hard complexity class we develop a complexity analysis to elucidate the difficulty of approximate probabilistic inference more specifically we show that the existence of a polynomial time relative approximation algorithm for major classes of problem instances implies that np p we present our proof and explore the implications of the result

automorphisms of automorphism groups of free groups abstract if n 3 and f n is free of rank n then out aut f n out out f n 1

the computational complexity of probabilistic inference using bayesian belief networks research note abstract bayesian belief networks provide a natural efficient method for representing probabilistic dependencies among a set of variables for these reasons numerous researchers are exploring the use of belief networks as a knowledge representation in artificial intelligence algorithms have been developed previously for efficient probabilistic inference using special classes of belief networks more general classes of belief networks however have eluded efforts to develop efficient inference algorithms we show that probabilistic inference using belief networks is np hard therefore it seems unlikely that an exact algorithm can be developed to perform probabilistic inference efficiently over all classes of belief networks this result suggests that research should be directed away from the search for a general efficient probabilistic inference algorithm and toward the design of efficient special case average case and approximation algorithms

understanding the scalability of bayesian network inference using clique tree growth curves one of the main approaches to performing computation in bayesian networks bns is clique tree clustering and propagation the clique tree approach consists of propagation in a clique tree compiled from a bn and while it was introduced in the 1980s there is still a lack of understanding of how clique tree computation time depends on variations in bn size and structure in this article we improve this understanding by developing an approach to characterizing clique tree growth as a function of parameters that can be computed in polynomial time from bns specifically i the ratio of the number of a bn s non root nodes to the number of root nodes and ii the expected number of moral edges in their moral graphs analytically we partition the set of cliques in a clique tree into different sets and introduce a growth curve for the total size of each set for the special case of bipartite bns there are two sets and two growth curves a mixed clique growth curve and a root clique growth curve in experiments where random bipartite bns generated using the bpart algorithm are studied we systematically increase the out degree of the root nodes in bipartite bayesian networks by increasing the number of leaf nodes surprisingly root clique growth is well approximated by gompertz growth curves an s shaped family of curves that has previously been used to describe growth processes in biology medicine and neuroscience we believe that this research improves the understanding of the scaling behavior of clique tree clustering for a certain class of bayesian networks presents an aid for trade off studies of clique tree clustering using growth curves and ultimately provides a foundation for benchmarking and developing improved bn inference and machine learning algorithms

the maximum order of finite groups of outer automorphisms of free groups let fr be free group of rank r l and o u t f r a u t fr innfr the outer au tomorph ism group automorphisms modulo inner automorphisms we shall prove the following theorem the maximum order of a f ini te subgroup g of out f is 12 for r 2 and 2 r r for r 2 furthermore the f inite subgroup of out f realizing the maximum order is unique up to conjugacy for r 3 the proof of the above theorem is based on the following geometric realization result which was observed first in z p 478 see also cu proposition 0 let g be a finite subgroup of out f then there exist a f inite connected graph f with 1 f f and an action of g on f realizing the given action on f remarks and definitions let f be a subgraph obtained by deleting all free edges of f i e edges with vertex of valence 1 note that the g action restricts on the new graph f and the fundamental g roup and the induced action on it do not change so we may assume in proposi t ion 0 that f contains no vertex of valence 1 we use sym f to denote the symmetry group of f in a combina tor ial sense i e mapping vertices to vertices and oriented edges to oriented edges or equivalently homeomorph i sms acting linearly on edges in general we allow inversions of edges i e reflections in the midpoints of edges which is not considered as the identity then in proposi t ion 0 we may also assume that f has no vertices of valence 2 by amalgamat ing the two adjacent edges containing the same vertex of valence 2 into one edge again g acts on the new graph with the same induced action on the fundamenta l group we say that a g roup g acts effectively on f if it can be embedded into sym f or equivalently no non trivial element of g is the identity on f for a connected graph f its rank denoted by rank f is the rank of its fundamental group lemma 1 let f be a connected graph of rank r 1 without vertices of valence 1 and g be a f ini te group acting on f then g acts effectively on f if and only if the induced action on f injects into out f

on the hardness of approximate reasoning abstract many ai problems when formalized reduce to evaluating the probability that a propositional expression is true in this paper we show that this problem is computationally intractable even in surprisingly restricted cases and even if we settle for an approximation to this probability we consider various methods used in approximate reasoning such as computing degree of belief and bayesian belief networks as well as reasoning techniques such as constraint satisfaction and knowledge compilation that use approximation to avoid computational difficulties and reduce them to model counting problems over a propositional domain we prove that counting satisfying assignments of propositional languages is intractable even for horn and monotone formulae and even when the size of clauses and number of occurrences of the variables are extremely limited this should be contrasted with the case of deductive reasoning where horn theories and theories with binary clauses are distinguished by the existence of linear time satisfiability algorithms what is even more surprising is that as we show even approximating the number of satisfying assignments i e approximating approximate reasoning is intractable for most of these restricted theories we also identify some restricted classes of propositional formulae for which efficient algorithms for counting satisfying assignments can be given

rise and fall of historic tram networks logistic approximation and discontinuous events abstract a logistic approximation was used to describe in terms of total length l and population h variables the growth and decay of historic transportation systems three successive stages separated for sharp discontinuities were detected for several european tramway and metro systems corresponding to a fast initial growth followed by an intermediate step of slow growth and a final stage of rapid decay a common generalized behaviour was obtained in the l h vs h variations relative to critical values of l and h parameters defined from the maximum in the l h ratio

dempster s rule of combination is p complete research note abstract we consider the complexity of combining bodies of evidence according to the rules of the dempster shafer theory of evidence we prove that given as input a set of tables representing basic probability assignments m 1 m n over a frame of discernment and a set a the problem of computing the combined basic probability value m 1 m n a is p complete as a corollary we obtain that while the simple belief plausibility and commonality values bel a pl a and q a can be computed in polynomial time the problems of computing the combinations bel 1 bel n a pl 1 pl n a and q 1 q n a are p complete

fractal small world dichotomy in real world networks we draw attention to a clear dichotomy between small world networks exhibiting exponential neighborhood growth and fractal like networks where neighborhoods grow according to a power law this distinction is observed in a number of real world networks and is related to the degree correlations and geographical constraints we conclude by pointing out that the status of human social networks in this dichotomy is far from clear

topological parameters for time space tradeoff in this paper we propose a family of algorithms combining tree clustering with conditioning that trade space for time such algorithms are useful for reasoning in probabilistic and deterministic networks as well as for accomplishing optimization tasks by analyzing the problem structure the user can select from a spectrum of algorithms the one that best meets a given time space specification to determine the potential of this approach we analyze the structural properties of problems coming from the circuit diagnosis domain the analysis demonstrates how the tradeoffs associated with various hybrids can be used for each problem instance 2001 elsevier science b v all rights reserved

on the maximum order of torsion elements ingl n z and aut fn we study the maximum order of torsion elements ingl n z and aut fn denotedg n andh n respectively we prove a landau type estimate log and we show thath n g n if and only ifn 2 6 12

growing networks under geographical constraints inspired by the structure of technological weblike systems we discuss network evolution mechanisms which give rise to topological properties found in real spatial networks thus we suggest that the peculiar structure of transport and distribution networks is fundamentally determined by two factors these are the dependence of the spatial interaction range of vertices on the vertex attractiveness or importance within the network and on the inhomogeneous distribution of vertices in space we propose and analyze numerically a simple model based on these generating mechanisms which seems for instance to be able to reproduce known structural features of the internet

finding maps for belief networks is np hard given a probabilistic world model an important problem is to find the maximum a posteriori probability map instantiation of all the random variables given the evidence numerous researchers using such models employ some graph representation for the distributions such as a bayesian belief network this representation simplifies the complexity of specifying the distributions from exponential in n the number of variables in the model to linear in n in many interesting cases we show however that finding the map is np hard in the general case when these representations are used even if the size of the representation happens to be linear in n furthermore minor modifications to the proof show that the problem remains np hard for various restrictions of the topology of the graphs the same technique can be applied to the results of a related paper by cooper to further restrict belief network topology in the proof that probabilistic inference is np hard

maximum order of finite abelian subgroups in the outer automorphism group of a rank n free group let fn be a free group of rank n denote by out fn its outer automorphism group that is its automorphism group modulo its inner automorphism group for arbitrary n by considering group actions on finite connected graphs we derived the maximum order of finite abelian subgroups in out fn moreover it is shown that the subgroups reaching this maximum order can be determined up to isomorphisms

a differential semantics for jointree algorithms a new approach to inference in belief networks has been recently proposed which is based on an algebraic representation of belief networks using multi linear functions according to this approach belief network inference reduces to a simple process of evaluating and differentiating multilinear functions we show here that mainstream inference algorithms based on jointrees are a special case of the approach based on multi linear functions in a very precise sense we use this result to prove new properties of jointree algorithms we also discuss some practical and theoretical implications of this new finding

change in order of phase transitions on fractal lattices abstract we re examine a population model which exhibits a continuous absorbing phase transition belonging to directed percolation in 1d and a first order transition in 2d and above studying the model on sierpinski carpets of varying fractal dimensions we examine at what fractal dimension 1 d f 2 the change in order occurs as well as commenting on the order of the transitions we produce estimates for the critical points and for continuous transitions some critical exponents

structure of a large social network we study a social network consisting of over 10 4 individuals with a degree distribution exhibiting two power scaling regimes separated by a critical degree k mathrm crit and a power law relation between degree and local clustering we introduce a growing random model based on a local interaction mechanism that reproduces the observed scaling features and their exponents we suggest that the double power law originates from two very different kinds of networks that are simultaneously present in the human social network

structure identification in relational data this paper presents several investigations into the prospects for identifying meaningful structures in empirical data namely structures permitting effective organization of the data to meet requirements of future queries we propose a general framework whereby the notion of identifiability is given a precise formal definition similar to that of learnability using this framework we then explore if a tractable procedure exists for deciding whether a given relation is decomposable into a constraint network or a cnf theory with desirable topology and if the answer is positive identifying the desired decomposition finally we address the problem of expressing a given relation as a horn theory and if this is impossible finding the best k horn approximation to the given relation we show that both problems can be solved in time polynomial in the length of the data

on approximation algorithms for p the theme of this paper is to investigate to what extent approximation possibly together with randomization can reduce the complexity of problems in valiant s class p in general any function in p can be approximated to within any constant factor by a function in the class delta 3 p of the polynomial time hierarchy relative to a particular oracle delta 3 p cannot be replaced by delta 2 p in this result another part of the paper introduces a model of random sampling where the size of a set x is estimated by checking for various sample sets s whether or not s intersects x for various classes of sample sets upper and lower bounds on the number of samples required to estimate the size of x are discussed this type of sampling is motivated by particular problems in p such as computing the size of a backtrack search tree in the case of backtrack search trees a sample amounts to checking whether a certain path exists in the tree one of the lower bounds suggests that such tests

horn approximations of empirical data abstract formal ai systems traditionally represent knowledge using logical formulas sometimes however a model based representation is more compact and enables faster reasoning than the corresponding formula based representation the central idea behind our work is to represent a large set of models by a subset of characteristic models more specifically we examine model based representations of horn theories and show that there are large horn theories that can be exactly represented by an exponentially smaller set of characteristic models we show that deduction based on a set of characteristic models requires only polynomial time as it does using horn theories more surprisingly abduction can be performed in polynomial time using a set of characteristic models whereas abduction using horn theories is np complete finally we discuss algorithms for generating efficient representations of the horn theory that best approximates a general set of models

scaling and percolation in the small world network model in this paper we study the small world network model of watts and strogatz which mimics some aspects of the structure of networks of social interactions we argue that there is one nontrivial length scale in the model analogous to the correlation length in other systems which is well defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero giving a normal critical point in this limit this length scale governs the crossover from large to small world behavior in the model as well as the number of vertices in a neighborhood of given radius on the network we derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex vertex distance on the network and using series expansion and pad e approximants find an approximate analytic form for the scaling function we calculate the effective dimension of small world graphs and show that this dimension varies as a function of the length scale on which it is measured in a manner reminiscent of multifractals we also study the problem of site percolation on small world networks as a simple model of disease propagation and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms in epidemiological terms the point at which an epidemic occurs the typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph all our analytic results are confirmed by extensive numerical simulations of the model

statistical analysis of 22 public transport networks in poland public transport systems in 22 polish cities have been analyzed the sizes of these networks range from n 152 to 2881 depending on the assumed definition of network topology the degree distribution can follow a power law or can be described by an exponential function distributions of path lengths in all considered networks are given by asymmetric unimodal functions clustering assortativity and betweenness are studied all considered networks exhibit small world behavior and are hierarchically organized a transition between dissortative small networks n approximately or 500 is observed

tree based reparameterization framework for analysis of sum product and related algorithms we present a tree based reparameterization trp framework that provides a new conceptual view of a large class of algorithms for computing approximate marginals in graphs with cycles this class includes the belief propagation bp or sum product algorithm as well as variations and extensions of bp algorithms in this class can be formulated as a sequence of reparameterization updates each of which entails refactorizing a portion of the distribution corresponding to an acyclic subgraph i e a tree or more generally a hypertree the ultimate goal is to obtain an alternative but equivalent factorization using functions that represent exact or approximate marginal distributions on cliques of the graph our framework highlights an important property of the sum product algorithm and the larger class of reparameterization algorithms the original distribution on the graph with cycles is not changed the perspective of tree based updates gives rise to a simple and intuitive characterization of the fixed points in terms of tree consistency we develop interpretations of these results in terms of information geometry the invariance of the distribution in conjunction with the fixed point characterization enables us to derive an exact expression for the difference between the true marginals on an arbitrary graph with cycles and the approximations provided by belief propagation more broadly our analysis applies to any algorithm that minimizes the bethe free energy we also develop bounds on the approximation error which illuminate the conditions that govern their accuracy finally we show how the reparameterization perspective extends naturally to generalizations of bp e g kikuchi 1951 approximations and variants via the notion of hypertree reparameterization