models of social networks based on social distance attachment we propose a class of models of social network formation based on a mathematical abstraction of the concept of social distance social distance attachment is represented by the tendency of peers to establish acquaintances via a decreasing function of the relative distance in a representative social space we derive analytical results corroborated by extensive numerical simulations showing that the model reproduces the main statistical characteristics of real social networks large clustering coefficient positive degree correlations and the emergence of a hierarchy of communities the model is confronted with the social network formed by people that shares confidential information using the pretty good privacy pgp encryption algorithm the so called web of trust of pgp

exponential error rates of sdp for block models beyond grothendieck s inequality in this paper we consider the cluster estimation problem under the stochastic block model we show that the semidefinite programming sdp formulation for this problem achieves an error rate that decays exponentially in the signal to noise ratio the error bound implies weak recovery in the sparse graph regime with bounded expected degrees as well as exact recovery in the dense regime an immediate corollary of our results yields error bounds under the censored block model moreover these error bounds are robust continuing to hold under heterogeneous edge probabilities and a form of the so called monotone attack significantly this error rate is achieved by the sdp solution itself without any further pre or post processing and improves upon existing polynomially decaying error bounds proved using the grothendieck s inequality our analysis builds on two key ingredients 1 showing that the graph has a well behaved spectrum even in the sparse regime after discounting an exponentially small number of edges and 2 an order statistics argument that governs the final error rate both arguments highlight the implicit regularization effect of the sdp formulation

minimax rates of community detection in stochastic block models in statistics as well as in computer science probability and applied mathematics community detection for the stochastic block model sbm is probably the most studied topic in network analysis many methodologies have been proposed some beautiful and signicant phase transition results are obtained in various settings in this paper we provide a general minimax theory for community detection it gives minimax rates of the mis match ratio for a wide rage of settings including homogeneous and inhomogeneous sbms dense and sparse networks nite and growing number of communities the minimax rates are exponential dierent from polynomial rates we often see in statistical literature an immediate consequence of the result is to establish threshold phenomenon for strong consistency exact recovery as well as weak consistency partial recovery we obtain the upper bound by a range of penalized likelihood type approaches the lower bound is achieved by a novel reduction from a global mis match ratio to a local clustering problem for one node through an exchangeability property

low distortion embeddings for edit distance we show that 0 1 d endowed with edit distance embeds into e1 with distortion 2o slog d log log d we further show efficient implementation of the embedding that yield solutions to various computational problems involving edit distance these include sketching communication complexity nearest neighbor search for all these problems we improve upon previous bounds

exploring networks with traceroute like probes theory and simulations mapping the internet generally consists in sampling the network from a limited set of sources by using traceroute like probes this methodology akin to the merging of different spanning trees to a set of destination has been argued to introduce uncontrolled sampling biases that might produce statistical properties of the sampled graph which sharply differ from the original ones in this paper we explore these biases and provide a statistical analysis of their origin we derive an analytical approximation for the probability of edge and vertex detection that exploits the role of the number of sources and targets and allows us to relate the global topological properties of the underlying network with the statistical accuracy of the sampled graph in particular we find that the edge and vertex detection probability depends on the betweenness centrality of each element this allows us to show that shortest path routed sampling provides a better characterization of underlying graphs with broad distributions of connectivity we complement the analytical discussion with a throughout numerical investigation of simulated mapping strategies in network models with different topologies we show that sampled graphs provide a fair qualitative characterization of the statistical properties of the original networks in a fair range of different strategies and exploration parameters moreover we characterize the level of redundancy and completeness of the exploration process as a function of the topological properties of the network finally we study numerically how the fraction of vertices and edges discovered in the sampled graph depends on the particular deployements of probing sources the results might hint the steps toward more efficient mapping strategies

community extraction for social networks analysis of networks and in particular discovering communities within networks has been a focus of recent work in several fields and has diverse applications most community detection methods focus on partitioning the entire network into communities with the expectation of many ties within communities and few ties between however many networks contain nodes that do not fit in with any of the communities and forcing every node into a community can distort results here we propose a new framework that extracts one community at a time allowing for arbitrary structure in the remainder of the network which can include weakly connected nodes the main idea is that the strength of a community should depend on ties between its members and ties to the outside world but not on ties between nonmembers the proposed extraction criterion has a natural probabilistic interpretation in a wide class of models and performs well on simulated and real networks for the case of the block model we establish asymptotic consistency of estimated node labels and propose a hypothesis test for determining the number of communities

nearly optimal classification for semimetrics we initiate the rigorous study of classification in semimetric spaces which are point sets with a distance function that is non negative and symmetric but need not satisfy the triangle inequality we define the density dimension dens and discover that it plays a central role in the statistical and algorithmic feasibility of learning in semimetric spaces we compute this quantity for several widely used semimetrics and present nearly optimal sample compression algorithms which are then used to obtain generalization guarantees including fast rates our claim of near optimality holds in both computational and statistical senses when the sample has radius r and margin we show that it can be compressed down to roughly d r dens points and further that finding a significantly better compression is algorithmically intractable unless p np this compression implies generalization via standard occam type arguments to which we provide a nearly matching lower bound

statistical computational tradeoffs in planted problems and submatrix localization with a growing number of clusters and submatrices we consider two closely related problems planted clustering and submatrix localization in the planted clustering problem a random graph is generated based on an underlying cluster structure of the nodes the task is to recover these clusters given the graph the submatrix localization problem concerns locating hidden submatrices with elevated means inside a large real valued random matrix of particular interest is the setting where the number of clusters submatrices is allowed to grow unbounded with the problem size these formulations cover several classical models such as planted clique planted densest subgraph planted partition planted coloring and the stochastic block model which are widely used for studying community detection graph clustering and bi clustering for both problems we show that the space of the model parameters cluster submatrix size edge probabilities and the mean of the submatrices can be partitioned into four disjoint regions corresponding to decreasing statistical and computational complexities 1 the impossible regime where all algorithms fail 2 the hard regime where the computationally expensive maximum likelihood estimator mle succeeds 3 the easy regime where the polynomial time convexified mle succeeds 4 the simple regime where a local counting thresholding procedure succeeds moreover we show that each of these algorithms provably fails in the harder regimes our results establish the minimax recovery limits which are tight up to universal constants and hold even with a growing number of clusters submatrices and provide order wise stronger performance guarantees for polynomial time algorithms than previously known our study demonstrates the tradeoffs between statistical and computational considerations and suggests that the minimax limits may not be achievable by polynomial time algorithms

sampling properties of directed networks for many real world networks only a small sampled version of the original network may be investigated those results are then used to draw conclusions about the actual system variants of breadth first search bfs sampling which are based on epidemic processes are widely used although it is well established that bfs sampling fails in most cases to capture the in component s of directed networks a description of the effects of bfs sampling on other topological properties is all but absent from the literature to systematically study the effects of sampling biases on directed networks we compare bfs sampling to random sampling on complete large scale directed networks we present new results and a thorough analysis of the topological properties of seven complete directed networks prior to sampling including three versions of wikipedia three different sources of sampled world wide web data and an internet based social network we detail the differences that sampling method and coverage can make to the structural properties of sampled versions of these seven networks most notably we find that sampling method and coverage affect both the bow tie structure and the number and structure of strongly connected components in sampled networks in addition at a low sampling coverage i e less than 40 the values of average degree variance of out degree degree autocorrelation and link reciprocity are overestimated by 30 or more in bfs sampled networks and only attain values within 10 of the corresponding values in the complete networks when sampling coverage is in excess of 65 these results may cause us to rethink what we know about the structure function and evolution of real world directed networks

asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications in this paper we extend our previous work on the stochastic block model a commonly used generative model for social and biological networks and the problem of inferring functional groups or communities from the topology of the network we use the cavity method of statistical physics to obtain an asymptotically exact analysis of the phase diagram we describe in detail properties of the detectability undetectability phase transition and the easy hard phase transition for the community detection problem our analysis translates naturally into a belief propagation algorithm for inferring the group memberships of the nodes in an optimal way i e that maximizes the overlap with the underlying group memberships and learning the underlying parameters of the block model finally we apply the algorithm to two examples of real world networks and discuss its performance

statistical properties of sampled networks by random walks we study the statistical properties of the sampled networks by a random walker we compare topological properties of the sampled networks such as degree distribution degree degree correlation and clustering coefficient with those of the original networks from the numerical results we find that most of topological properties of the sampled networks are almost the same as those of the original networks for gamma les approximately 3 somewhat deviates from that of the original networks when the ratio of the sampled network size to the original network size becomes smaller we also apply the sampling method to various real networks such as collaboration of movie actor worldwide web and peer to peer networks all topological properties of the sampled networks are essentially the same as those of the original real networks

achieving the bayes error rate in synchronization and block models by sdp robustly we study the statistical performance of semidefinite programming sdp relaxations for clustering under random graph models under the mathbb z 2 synchronization model censored block model cbm and stochastic block model sbm we show that sdp achieves an error rate of the form exp 1 o 1 bar n i here bar n is an appropriate multiple of the number of nodes and i is an information theoretic measure of the signal to noise ratio we provide matching lower bounds on the bayes error for each model and therefore demonstrate that the sdp approach is bayes optimal as a corollary our results imply that sdp achieves the optimal exact recovery threshold under each model furthermore we show that sdp is robust the above bound remains valid under semirandom versions of the models in which the observed graph is modified by a monotone adversary our proof is based on a novel primal dual analysis of sdp under a unified framework for all three models and the analysis shows that sdp tightly approximates a joint majority voting procedure

what is the real size of a sampled network the case of the internet internet mapping projects generally consist in sampling the network from a limited set of sources by usingtraceroute probes this methodology akin to the merging of spanning trees from the different sources to a set of destinations leads necessarily to a partial incomplete map of the internet accordingly determination of internet topology characteristics from such sampled maps is in part a problem of statistical inference our contribution begins with the observation that the inference of many of the most basic topological quantities including network size and degree characteristics fromtraceroute measurements is in fact a version of the so called species problem in statistics this obse rvation has important implications since species problems are often quite challenging we focus here on the most fundamental example of atraceroute internet species the number of nodes in a network specifica lly

statistical properties of sampled networks we study the statistical properties of the sampled scale free networks deeply related to the proper identification of various real world networks we exploit three methods of sampling and investigate the topological properties such as degree and betweenness centrality distribution average path length assortativity and clustering coefficient of sampled networks compared with those of original networks it is found that the quantities related to those properties in sampled networks appear to be estimated quite differently for each sampling method we explain why such a biased estimation of quantities would emerge from the sampling procedure and give appropriate criteria for each sampling method to prevent the quantities from being overestimated or underestimated

fast and compact regular expression matching we study 4 problems in string matching namely regular expression matching approximate regular expression matching string edit distance and subsequence indexing on a standard word ram model of computation that allows logarithmic sized words to be manipulated in constant time we show how to improve the space and or remove a dependency on the alphabet size for each problem using either an improved tabulation technique of an existing algorithm or by combining known algorithms in a new way

convex optimization for the planted k disjoint clique problem we consider the k disjoint clique problem the input is an undirected graph g in which the nodes represent data items and edges indicate a similarity between the corresponding items the problem is to find within the graph k disjoint cliques that cover the maximum number of nodes of g this problem may be understood as a general way to pose the classical clustering problem in clustering one is given data items and a distance function and one wishes to partition the data into disjoint clusters of data items such that the items in each cluster are close to each other our formulation additionally allows noise nodes to be present in the input data that are not part of any of the cliques the k disjoint clique problem is np hard but we show that a convex relaxation can solve it in polynomial time for input instances constructed in a certain way the input instances for which our algorithm finds the optimal solution consist of k disjoint large cliques called planted cliques that are then obscured by noise edges inserted either at random or by an adversary as well as additional nodes not belonging to any of the k planted cliques

sampling promotes community structure in social and information networks abstract any network studied in the literature is inevitably just a sampled representative of its real world analogue additionally network sampling is lately often applied to large networks to allow for their faster and more efficient analysis nevertheless the changes in network structure introduced by sampling are still far from understood in this paper we study the presence of characteristic groups of nodes in sampled social and information networks we consider different network sampling techniques including random node and link selection network exploration and expansion we first observe that the structure of social networks reveals densely linked groups like communities while the structure of information networks is better described by modules of structurally equivalent nodes however despite these notable differences the structure of sampled networks exhibits stronger characterization by community like groups than the original networks irrespective of their type and consistently across various sampling techniques hence rich community structure commonly observed in social and information networks is to some extent merely an artifact of sampling

phase transitions in semidefinite relaxations statistical inference problems arising within signal processing data mining and machine learning naturally give rise to hard combinatorial optimization problems these problems become intractable when the dimensionality of the data is large as is often the case for modern datasets a popular idea is to construct convex relaxations of these combinatorial problems which can be solved efficiently for large scale datasets semidefinite programming sdp relaxations are among the most powerful methods in this family and are surprisingly well suited for a broad range of problems where data take the form of matrices or graphs it has been observed several times that when the statistical noise is small enough sdp relaxations correctly detect the underlying combinatorial structures in this paper we develop asymptotic predictions for several detection thresholds as well as for the estimation error above these thresholds we study some classical sdp relaxations for statistical problems motivated by graph synchronization and community detection in networks we map these optimization problems to statistical mechanics models with vector spins and use nonrigorous techniques from statistical mechanics to characterize the corresponding phase transitions our results clarify the effectiveness of sdp relaxations in solving high dimensional statistical problems

reconstruction and estimation in the planted partition model the planted partition model also known as the stochastic blockmodel is a classical cluster exhibiting random graph model that has been extensively studied in statistics physics and computer science in its simplest form the planted partition model is a model for random graphs on n nodes with two equal sized clusters with an between class edge probability of q and a within class edge probability of p although most of the literature on this model has focused on the case of increasing degrees ie pn qn rightarrow infty as n rightarrow infty the sparse case p q o 1 n is interesting both from a mathematical and an applied point of view a striking conjecture of decelle krzkala moore and zdeborova based on deep non rigorous ideas from statistical physics gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model in particular if p a n and q b n then decelle et al conjectured that it is possible to cluster in a way correlated with the true partition if a b 2 2 a b and impossible if a b 2 2 a b by comparison the best known rigorous result is that of coja oghlan who showed that clustering is possible if a b 2 c a b for some sufficiently large c we prove half of their prediction showing that it is indeed impossible to cluster if a b 2 2 a b furthermore we show that it is impossible even to estimate the model parameters from the graph when a b 2 2 a b on the other hand we provide a simple and efficient algorithm for estimating a and b when a b 2 2 a b following decelle et al our work establishes a rigorous connection between the clustering problem spin glass models on the bethe lattice and the so called reconstruction problem this connection points to fascinating applications and open problems

sharp uniform convexity and smoothness inequalities for trace norms we prove several sharp inequalities specifying the uniform convexity and uniform smoothness properties of the schatten trace ideals c p which are the analogs of the lebesgue spaces l p in non commutative integration the inequalities are all precise analogs of results which had been known in l p but were only known in c p for special values of p in the course of our treatment of uniform convexity and smoothness inequalities for c p we obtain new and simple proofs of the known inequalities for l p

sampling properties of random graphs the degree distribution we discuss two sampling schemes for selecting random subnets from a network random sampling and connectivity dependent sampling and investigate how the degree distribution of a node in the network is affected by the two types of sampling here we derive a necessary and sufficient condition that guarantees that the degree distributions of the subnet and the true network belong to the same family of probability distributions for completely random sampling of nodes we find that this condition is satisfied by classical random graphs for the vast majority of networks this condition will however not be met we furthermore discuss the case where the probability of sampling a node depends on the degree of a node and we find that even classical random graphs are no longer closed under this sampling regime we conclude by relating the results to real eschericia coli protein interaction network data

exploring complex networks through random walks most real complex networks such as protein interactions social contacts and the internet are only partially known and available to us while the process of exploring such networks in many cases resembles a random walk it becomes a key issue to investigate and characterize how effectively the nodes and edges of such networks can be covered by different strategies at the same time it is critically important to infer how well can topological measurements such as the average node degree and average clustering coefficient be estimated during such network explorations the present article addresses these problems by considering random barabasi albert ba and geographical network models with varying connectivity explored by three types of random walks traditional preferential to untracked edges and preferential to unvisited nodes a series of relevant results are obtained including the fact that networks of the three studied models with the same size and average node degree allow similar node and edge coverage efficiency the identification of linear scaling with the size of the network of the random walk step at which a given percentage of the nodes edges is covered and the critical result that the estimation of the averaged node degree and clustering coefficient by random walks on ba networks often leads to heavily biased results many are the theoretical and practical implications of such results

euclidean quotients of finite metric spaces this paper is devoted to the study of quotients of finite metric spaces the basic type of question we ask is given a finite metric space m and 1 what is the largest quotient of a subset of m which well embeds into hilbert space we obtain asymptotically tight bounds for these questions and prove that they exhibit phase transitions we also study the analogous problem for embeddings into lp and the particular case of the hypercube

improved graph clustering graph clustering involves the task of dividing nodes into clusters so that the edge density is higher within clusters as opposed to across clusters a natural classic and popular statistical setting for evaluating solutions to this problem is the stochastic block model also referred to as the planted partition model in this paper we present a new algorithm a convexified version of maximum likelihood for graph clustering we show that in the classic stochastic block model setting it outperforms existing methods by polynomial factors when the cluster size is allowed to have general scalings in fact it is within logarithmic factors of known lower bounds for spectral methods and there is evidence suggesting that no polynomial time algorithm would do significantly better we then show that this guarantee carries over to a more general extension of the stochastic block model our method can handle the settings of semirandom graphs heterogeneous degree distributions unequal cluster sizes unaffiliated nodes partially observed graphs planted clique coloring and so on in particular our results provide the best exact recovery guarantees to date for the planted partition planted k disjoint cliques and planted noisy coloring models with general cluster sizes in other settings we match the best existing results up to logarithmic factors

community detection in sparse networks via grothendieck s inequality we present a simple and flexible method to prove consistency of semidefinite optimization problems on random graphs the method is based on grothendieck s inequality unlike the previous uses of this inequality that lead to constant relative accuracy we achieve any given relative accuracy by leveraging randomness we illustrate the method with the problem of community detection in sparse networks those with bounded average degrees we demonstrate that even in this regime various simple and natural semidefinite programs can be used to recover the community structure up to an arbitrarily small fraction of misclassified vertices the method is general it can be applied to a variety of stochastic models of networks and semidefinite programs

adaptive metric dimensionality reduction we study data adaptive dimensionality reduction in the context of supervised learning in general metric spaces our main statistical contribution is a generalization bound for lipschitz functions in metric spaces that are doubling or nearly doubling which yields a new theoretical explanation for empirically reported improvements gained by preprocessing euclidean data by pca principal components analysis prior to constructing a linear classifier on the algorithmic front we describe an analogue of pca for metric spaces namely an efficient procedure that approximates the data s intrinsic dimension which is often much lower than the ambient dimension our approach thus leverages the dual benefits of low dimensionality 1 more efficient algorithms e g for proximity search and 2 more optimistic generalization bounds

scale free networks from varying vertex intrinsic fitness a new mechanism leading to scale free networks is proposed in this letter it is shown that in many cases of interest the connectivity power law behavior is neither related to dynamical properties nor to preferential attachment assigning a quenched fitness value x i to every vertex and drawing links among vertices with a probability depending on the fitnesses of the two involved sites gives rise to what we call a good get richer mechanism in which sites with larger fitness are more likely to become hubs i e to be highly connected

fuzzy extractors how to generate strong keys from biometrics and other noisy data we provide formal definitions and efficient secure techniques for turning noisy information into keys usable for any cryptographic application and in particular reliably and securely authenticating biometric data our techniques apply not just to biometric information but to any keying material that unlike traditional cryptographic keys is 1 not reproducible precisely and 2 not distributed uniformly we propose two primitives a fuzzy extractor reliably extracts nearly uniform randomness r from its input the extraction is error tolerant in the sense that r will be the same even if the input changes as long as it remains reasonably close to the original thus r can be used as a key in a cryptographic application a secure sketch produces public information about its input w that does not reveal w and yet allows exact recovery of w given another value that is close to w thus it can be used to reliably reproduce error prone biometric inputs without incurring the security risk inherent in storing them we define the primitives to be both formally secure and versatile generalizing much prior work in addition we provide nearly optimal constructions of both primitives for various measures of closeness of input data such as hamming distance edit distance and set difference

what big data tells sampling the social network by communication channels big data has become the primary source of understanding the structure and dynamics of the society at large scale the network of social interactions can be considered as a multiplex where each layer corresponds to one communication channel and the aggregate of all of them constitutes the entire social network however usually one has information only about one of the channels or even a part of it which should be considered as a subset or sample of the whole here we introduce a model based on a natural bilateral communication channel selection mechanism which for one channel leads to consistent changes in the network properties for example while it is expected that the degree distribution of the whole social network has a maximum at a value larger than one we get a monotonically decreasing distribution as observed in empirical studies of single channel data we also find that assortativity may occur or get strengthened due to the sampling method we analyze the far reaching consequences of our findings

class of correlated random networks with hidden variables we study a class of models of correlated random networks in which vertices are characterized by hidden variables controlling the establishment of edges between pairs of vertices we find analytical expressions for the main topological properties of these models as a function of the distribution of hidden variables and the probability of connecting vertices the expressions obtained are checked by means of numerical simulations in a particular example the general model is extended to describe a practical algorithm to generate random networks with an a priori specified correlation structure we also present an extension of the class to map nonequilibrium growing networks to networks with hidden variables that represent the time at which each vertex was introduced in the system

nonembeddability theorems via fourier analysis various new nonembeddability results mainly into l 1 are proved via fourier analysis in particular it is shown that the edit distance on 0 1 d has l 1 distortion we also give new lower bounds on the l 1 distortion of flat tori quotients of the discrete hypercube under group actions and the transportation cost earthmover metric

why social networks are different from other types of networks we argue that social networks differ from most other types of networks including technological and biological networks in two important ways first they have nontrivial clustering or network transitivity and second they show positive correlations also called assortative mixing between the degrees of adjacent vertices social networks are often divided into groups or communities and it has recently been suggested that this division could account for the observed clustering we demonstrate that group structure in networks can also account for degree correlations we show using a simple model that we should expect assortative mixing in such networks whenever there is variation in the sizes of the groups and that the predicted level of assortative mixing compares well with that observed in real world networks

achieving exact cluster recovery threshold via semidefinite programming the binary symmetric stochastic block model deals with a random graph of n vertices partitioned into two equal sized clusters such that each pair of vertices is independently connected with probability p within clusters and q across clusters in the asymptotic regime of p a log n n and q b log n n for fixed a b and n to infty we show that the semidefinite programming relaxation of the maximum likelihood estimator achieves the optimal threshold for exactly recovering the partition from the graph with probability tending to one resolving a conjecture of abbe et al furthermore we show that the semidefinite programming relaxation also achieves the optimal recovery threshold in the planted dense subgraph model containing a single cluster of size proportional to n

sharp nonasymptotic bounds on the norm of random matrices with independent entries we obtain nonasymptotic bounds on the spectral norm of random matrices with independent entries that improve significantly on earlier results if xx is the n nn n symmetric matrix with xij n 0 b2ij xij n 0 bij2 we show that this bound is optimal in the sense that a matching lower bound holds under mild assumptions and the constants are sufficiently sharp that we can often capture the precise edge of the spectrum analogous results are obtained for rectangular matrices and for more general sub gaussian or heavy tailed distributions of the entries and we derive tail bounds in addition to bounds on the expected norm the proofs are based on a combination of the moment method and geometric functional analysis techniques as an application we show that our bounds immediately yield the correct phase transition behavior of the spectral edge of random band matrices and of sparse wigner matrices we also recover a result of seginer on the norm of rademacher matrices

efficient search for approximate nearest neighbor in high dimensional spaces we address the problem of designing data structures that allow efficient search for approximate nearest neighbors more specifically given a database consisting of a set of vectors in some high dimensional euclidean space we want to construct a space efficient data structure that would allow us to search given a query vector for the closest or nearly closest vector in the database we also address this problem when distances are measured by the l1 norm and in the hamming cube significantly improving and extending recent results of kleinberg we construct data structures whose size is polynomial in the size of the database and search algorithms that run in time nearly linear or nearly quadratic in the dimension depending on the case the extra factors are polylogarithmic in the size of the database

sampling networks by nodal attributes in a social network individuals or nodes connect to other nodes by choosing one of the channels of communication at a time to re establish the existing social links since available data sets are usually restricted to a limited number of channels or layers these autonomous decision making processes by the nodes constitute the sampling of a multiplex network leading to just one though very important example of sampling bias caused by the behavior of the nodes we develop a general setting to get insight and understand the class of network sampling models where the probability of sampling a link in the original network depends on the attributes h of its adjacent nodes assuming that the nodal attributes are independently drawn from an arbitrary distribution h and that the sampling probability r h i h j for a link ij of nodal attributes h i and h j is also arbitrary we derive exact analytic expressions of the sampled network for such network characteristics as the degree distribution degree correlation and clustering spectrum the properties of the sampled network turn out to be sums of quantities for the original network topology weighted by the factors stemming from the sampling based on our analysis we find that the sampled network may have sampling induced network properties that are absent in the original network which implies the potential risk of a naive generalization of the results of the sample to the entire original network we also consider the case when neighboring nodes have correlated attributes to show how to generalize our formalism for such sampling bias and we get good agreement between the analytic results and the numerical simulations

the computational hardness of estimating edit distance we prove the first nontrivial communication complexity lower bound for the problem of estimating the edit distance aka levenshtein distance between two strings to the best of our knowledge this is the first computational setting in which the complexity of estimating the edit distance is provably larger than that of hamming distance our lower bound exhibits a trade off between approximation and communication asserting for example that protocols with o 1 bits of communication can obtain only approximation alpha geq omega log d log log d where d is the length of the input strings this case of o 1 communication is of particular importance since it captures constant size sketches as well as embeddings into spaces like l 1 and squared l 2 two prevailing algorithmic approaches for dealing with edit distance indeed the known nontrivial communication upper bounds are all derived from embeddings into l 1 by excluding low communication protocols for edit distance we rule out a strictly richer class of algorithms than previous results furthermore our lower bound holds not only for strings over a binary alphabet but also for strings that are permutations aka the ulam metric for this case our bound nearly matches an upper bound known via embedding the ulam metric into l 1 our proof uses a new technique that relies on fourier analysis in a rather elementary way