community detection algorithms a comparative analysis uncovering the community structure exhibited by real networks is a crucial step toward an understanding of complex systems that goes beyond the local organization of their constituents many algorithms have been proposed so far but none of them has been subjected to strict tests to evaluate their performance most of the sporadic tests performed so far involved small networks with known community structure and or artificial graphs with a simplified structure which is very uncommon in real systems here we test several methods against a recently introduced class of benchmark graphs with heterogeneous distributions of degree and community size the methods are also tested against the benchmark by girvan and newman proc natl acad sci u s a 99 7821 2002 and on random graphs as a result of our analysis three recent algorithms introduced by rosvall and bergstrom proc natl acad sci u s a 104 7327 2007 proc natl acad sci u s a 105 1118 2008 blondel et al j stat mech theory exp 2008 p10008 and ronhovde and nussinov phys rev e 80 016109 2009 have an excellent performance with the additional advantage of low computational complexity which enables one to analyze large systems

robust reconstruction on trees is determined by the second eigenvalue consider a markov chain on an infinite tree t v e rooted at in such a chain once the initial root state is chosen each vertex iteratively chooses its state from the one of its parent by an application of a markov transition rule and all such applications are independent let j denote the resulting measure for j j is defined on configurations x x2v 2 a v where a is some finite set let n denote the restriction of to the sigma algebra generated by the variables x where x is at distance exactly n from letting n maxi j2a dtv n n where dtv denotes total variation distance we say that the reconstruction problem is solvable if liminfn 1 n 0 reconstruction solvability roughly means that the n th level of the tree contains a non vanishing amount of information on the root of the tree as n 1 in this paper we study the problem of robust reconstruction let be a non degenerate distribution on a and 0 let be chosen according to n and 0 be obtained from by letting for each node independently v 0 v with probability 1 and 0 v to be an independent sample from otherwise we denote by n the resulting measure on 0 the measure n is a perturbation of the measure n letting n maxi j2a dtv n n we say that the reconstruction problem is robust solvable if liminfn 1 n 0 for all 0 1 where 2 m is the second largest eigenvalue of m in absolute value then for all non degenerate the reconstruction problem is robust solvable we prove a converse and show that the reconstruction problem is not robust solvable if b 2 m 2 1 this proves a conjecture by the second author and peres we also consider other models of noise and general trees

ergodicity in stationary graph processes a weak law of large numbers for stationary signals in time the weak law of large numbers wlln states that ensemble and realization averages are within epsilon of each other with a probability of order mathcal o 1 n epsilon 2 when considering n signal components the graph wlln introduced in this paper shows that the same is essentially true for signals supported on graphs however the notions of stationarity ensemble mean and realization mean are different recent papers have defined graph stationary signals as those that satisfy a form of invariance with respect to graph diffusion the ensemble mean of a graph stationary signal is not a constant but a node varying signal whose structure depends on the spectral properties of the graph the realization average of a graph signal is defined here as an average of successive weighted averages of local signal values with signal values of neighboring nodes the graph wlln shows that these two node varying signals are within epsilon of each other with probability of order mathcal o 1 n epsilon 2 in at least some nodes in stationary time signals the realization average is not only a consistent estimator of the ensemble mean but also optimal in terms of mean squared error mse this is not true for graph signals optimal mse graph filter designs are also presented an example problem concerning the estimation of the mean of a gaussian random field is presented

performance of modularity maximization in practical contexts although widely used in practice the behavior and accuracy of the popular module identification technique called modularity maximization is not well understood in practical contexts here we present a broad characterization of its performance in such situations first we revisit and clarify the resolution limit phenomenon for modularity maximization second we show that the modularity function q exhibits extreme degeneracies it typically admits an exponential number of distinct high scoring solutions and typically lacks a clear global maximum third we derive the limiting behavior of the maximum modularity q max for one model of infinitely modular networks showing that it depends strongly both on the size of the network and on the number of modules it contains finally using three real world metabolic networks as examples we show that the degenerate solutions can fundamentally disagree on many but not all partition properties such as the composition of the largest modules and the distribution of module sizes these results imply that the output of any modularity maximization procedure should be interpreted cautiously in scientific contexts they also explain why many heuristics are often successful at finding high scoring partitions in practice and why different heuristics can disagree on the modular structure of the same network we conclude by discussing avenues for mitigating some of these behaviors such as combining information from many degenerate solutions or using generative models

network cross validation for determining the number of communities in network data abstractthe stochastic block model sbm and its variants have been a popular tool for analyzing large network data with community structures in this article we develop an efficient network cross validation ncv approach to determine the number of communities as well as to choose between the regular stochastic block model and the degree corrected block model dcbm the proposed ncv method is based on a block wise node pair splitting technique combined with an integrated step of community recovery using sub blocks of the adjacency matrix we prove that the probability of under selection vanishes as the number of nodes increases under mild conditions satisfied by a wide range of popular community recovery algorithms the solid performance of our method is also demonstrated in extensive simulations and two data examples supplementary materials for this article are available online

branch and price column generation for solving huge integer programs we discuss formulations of integer programs with a huge number of variables and their solution by column generation methods i e implicit pricing of nonbasic variables to generate new columns or to prove lp optimality at a node of the branch and bound tree we present classes of models for which this approach decomposes the problem provides tighter lp relaxations and eliminates symmetry we then discuss computational issues and implementation of column generation branch and bound algorithms including special branching rules and efficient ways to solve the lp relaxation we also discuss the relationship with lagrangian duality

stochastic blockmodels and community structure in networks stochastic blockmodels have been proposed as a tool for detecting community structure in networks as well as for generating synthetic networks for use as benchmarks most blockmodels however ignore variation in vertex degree making them unsuitable for applications to real world networks which typically display broad degree distributions that can significantly affect the results here we demonstrate how the generalization of blockmodels to incorporate this missing element leads to an improved objective function for community detection in complex networks we also propose a heuristic algorithm for community detection using this objective function or its non degree corrected counterpart and show that the degree corrected version dramatically outperforms the uncorrected one in both real world and synthetic networks

pseudo likelihood methods for community detection in large sparse networks many algorithms have been proposed for fitting network models with communities but most of them do not scale well to large networks and often fail on sparse networks here we propose a new fast pseudo likelihood method for fitting the stochastic block model for networks as well as a variant that allows for an arbitrary degree distribution by conditioning on degrees we show that the algorithms perform well under a range of settings including on very sparse networks and illustrate on the example of a network of political blogs we also propose spectral clustering with perturbations a method of independent interest which works well on sparse networks where regular spectral clustering fails and use it to provide an initial value for pseudo likelihood we prove that pseudo likelihood provides consistent estimates of the communities under a mild condition on the starting value for the case of a block model with two communities

community detection in complex networks using extremal optimization we propose a method to find the community structure in complex networks based on an extremal optimization of the value of modularity the method outperforms the optimal modularity found by the existing algorithms in the literature giving a better understanding of the community structure we present the results of the algorithm for computer simulated and real networks and compare them with other approaches the efficiency and accuracy of the method make it feasible to be used for the accurate identification of community structure in large complex 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

benchmark graphs for testing community detection algorithms community structure is one of the most important features of real networks and reveals the internal organization of the nodes many algorithms have been proposed but the crucial issue of testing i e the question of how good an algorithm is with respect to others is still open standard tests include the analysis of simple artificial graphs with a built in community structure that the algorithm has to recover however the special graphs adopted in actual tests have a structure that does not reflect the real properties of nodes and communities found in real networks here we introduce a class of benchmark graphs that account for the heterogeneity in the distributions of node degrees and of community sizes we use this benchmark to test two popular methods of community detection modularity optimization and potts model clustering the results show that the benchmark poses a much more severe test to algorithms than standard benchmarks revealing limits that may not be apparent at a first analysis

proof of a conjecture on the infinite dimension limit of a unifying model for random matrix theory we study the large n limit of a sparse random block matrix ensemble it depends on two parameters the average connectivity z and the size of the blocks d which is the dimension of an euclidean space in the limit of large d with frac z d fixed we prove the conjecture that the spectral distribution of the sparse random block matrix converges in the case of the adjacency block matrix to the one of the effective medium approximation in the case of the laplacian block matrix to the marchenko pastur distribution we extend previous analytical computations of the moments of the spectral density of the adjacency block matrix and the lagrangian block matrix valid for all values of z and d

topology optimization of beam cross sections abstract perhaps this paper reports the first successful applications of the topology optimization in the design of thin walled beam sections in particular topologically different thin walled beam cross sections can be obtained by the present approach which is very useful in identifying the direction and location of stiffeners in formulating the topology optimization problems a simple power law is used for the relation between the density of an element with a hole and the mechanical properties of the element the sensitivity of the torsional rigidity is obtained by developing a finite element model of a st venant torsion problem and the euler beam theory is used for the sensitivity analysis of the bending rigidities

inferring overlapping community structure with degree corrected block model abstract recent research has shown great interest in statistical inference methods for community detection not only in models and algorithms but also in the detectability in this paper we propose a fast community detection algorithm based on the degree corrected block model by introducing a parameter to select the candidate solutions our algorithm is able to detect overlapping communities experiments on a range of networks have achieved state of the art results moreover we show that the algorithm based on the degree corrected block model also suffers the detectability limitation which is in accord with the most recent research on the detectability threshold

estimation and prediction for stochastic blockstructures a statistical approach to a posteriori blockmodeling for digraphs and valued digraphs is proposed the probability model assumes that the vertices of the digraph are partitioned into several unobserved latent classes and that the probability distribution of the relation between two vertices depends only on the classes to which they belong a bayesian estimator based on gibbs sampling is proposed the basic model is not identified because class labels are arbitrary the resulting identifiability problems are solved by restricting inference to the posterior distributions of invariant functions of the parameters and the vertex class membership in addition models are considered where class labels are identified by prior distributions for the class membership of some of the vertices the model is illustrated by an example from the social networks literature kapferer s tailor shop

computational and statistical boundaries for submatrix localization in a large noisy matrix we study in this paper computational and statistical boundaries for submatrix localization given one observation of one or multiple nonoverlapping signal submatrix of magnitude lambda and size k m times k n embedded in a large noise matrix of size m times n the goal is to optimal identify the support of the signal submatrix computationally and statistically two transition thresholds for the signal to noise ratio lambda sigma are established in terms of m n k m and k n the first threshold sf snr c corresponds to the computational boundary we introduce a new linear time spectral algorithm that identifies the submatrix with high probability when the signal strength is above the threshold sf snr c below this threshold it is shown that no polynomial time algorithm can succeed in identifying the submatrix under the hidden clique hypothesis the second threshold sf snr s captures the statistical boundary below which no method can succeed in localization with probability going to one in the minimax sense the exhaustive search method successfully finds the submatrix above this threshold in marked contrast to submatrix detection and sparse pca the results show an interesting phenomenon that sf snr c is always significantly larger than sf snr s under the sub gaussian error model which implies an essential gap between statistical optimality and computational efficiency for submatrix localization

information theoretic bounds and phase transitions in clustering sparse pca and submatrix localization we study the problem of detecting a structured low rank signal matrix corrupted with additive gaussian noise this includes clustering in a gaussian mixture model sparse pca and submatrix localization each of these problems is conjectured to exhibit a sharp information theoretic threshold below which the signal is too weak for any algorithm to detect we derive upper and lower bounds on these thresholds by applying the first and second moment methods to the likelihood ratio between these planted models and null models where the signal matrix is zero for sparse pca and submatrix localization we determine this threshold exactly in the limit where the number of blocks is large or the signal matrix is very sparse for the clustering problem our bounds differ by a factor of sqrt 2 when the number of clusters is large moreover our upper bounds show that for each of these problems there is a significant regime where reliable detection is information theoretically possible but where known algorithms such as pca fail completely since the spectrum of the observed matrix is uninformative this regime is analogous to the conjectured hard but detectable regime for community detection in sparse graphs

large deviations of connected components in the stochastic block model we study the stochastic block model which is often used to model community structures and study community detection algorithms we consider the case of two blocks in regard to its largest connected component and largest biconnected component respectively we are especially interested in the distributions of their sizes including the tails down to probabilities smaller than 10 ensuremath 800 for this purpose we use sophisticated markov chain monte carlo simulations to sample graphs from the stochastic block model ensemble we use these data to study the large deviation rate function and conjecture that the large deviation principle holds further we compare the distribution to the well known erd ifmmode mbox h o else h o fi s r enyi ensemble where we notice subtle differences at and above the percolation threshold

typology of phase transitions in bayesian inference problems many inference problems undergo phase transitions as a function of the signal to noise ratio a prominent example of this phenomenon being found in the stochastic block model sbm that generates a random graph with a hidden community structure some of these phase transitions affect the information theoretic optimal but possibly computationally expensive estimation procedure others concern the behavior of efficient iterative algorithms a computational gap opens when the phase transitions for these two aspects do not coincide leading to a hard phase in which optimal inference is computationally challenging in this paper we refine this description in two ways from a qualitative perspective we emphasize the existence of more generic phase diagrams with a hybrid hard phase in which it is computationally easy to reach a nontrivial inference accuracy but computationally hard to match the information theoretic optimal one we support this discussion by quantitative expansions of the functional cavity equations that describe inference problems on sparse graphs around their trivial solution these expansions shed light on the existence of hybrid hard phases for a large class of planted constraint satisfaction problems and on the question of the tightness of the kesten stigum ks bound for the associated tree reconstruction problem our results show that the instability of the trivial fixed point is not generic evidence for the bayes optimality of the message passing algorithms we clarify in particular the status of the symmetric sbm with four communities and of the tree reconstruction of the associated potts model in the assortative ferromagnetic case the ks bound is always tight whereas in the disassortative antiferromagnetic case we exhibit an explicit criterion involving the degree distribution that separates a large degree regime where the ks bound is tight and a low degree regime where it is not we also investigate the sbm with two communities of different sizes also known as the asymmetric ising model and describe quantitatively its computational gap as a function of its asymmetry as well as a version of the sbm with two groups of q 1 and q 2 communities we complement this study with numerical simulations of the belief propagation iterative algorithm confirming that its behavior on large samples is well described by the cavity method

consistency of community detection in networks under degree corrected stochastic block models community detection is a fundamental problem in network analysis with applications in many diverse areas the stochastic block model is a common tool for model based community detection and asymptotic tools for checking consistency of community detection under the block model have been recently developed however the block model is limited by its assumption that all nodes within a community are stochastically equivalent and provides a poor fit to networks with hubs or highly varying node degrees within communities which are common in practice the degree corrected stochastic block model was proposed to address this shortcoming and allows variation in node degrees within a community while preserving the overall block community structure in this paper we establish general theory for checking consistency of community detection under the degree corrected stochastic block model and compare several community detection criteria under both the standard and the degree corrected models we show which criteria are consistent under which models and constraints as well as compare their relative performance in practice we find that methods based on the degree corrected block model which includes the standard block model as a special case are consistent under a wider class of models and that modularity type methods require parameter constraints for consistency whereas likelihood based methods do not on the other hand in practice the degree correction involves estimating many more parameters and empirically we find it is only worth doing if the node degrees within communities are indeed highly variable we illustrate the methods on simulated networks and on a network of political blogs

gibbs states and the set of solutions of random constraint satisfaction problems abstract an instance of a random constraint satisfaction problem defines a random subset the set of solutions of a large product space x n the set of assignments we consider two prototypical problem ensembles random k satisfiability and q coloring of random regular graphs and study the uniform measure with support on s as the number of constraints per variable increases this measure first decomposes into an exponential number of pure states clusters and subsequently condensates over the largest such states above the condensation point the mass carried by the n largest states follows a poisson dirichlet process for typical large instances the two transitions are sharp we determine their precise location further we provide a formal definition of each phase transition in terms of different notions of correlation between distinct variables in the problem the degree of correlation naturally affects the performances of many search sampling algorithms empirical evidence suggests that local monte carlo markov chain strategies are effective up to the clustering phase transition and belief propagation up to the condensation point finally refined message passing techniques such as survey propagation may also beat this threshold message passing algorithms phase transitions random graphs

statistical mechanical approach to image processing the basic frameworks and techniques of the bayesian approach to image restoration are reviewed from the statistical mechanical point of view first a few basic notions in digital image processing are explained to convince the reader that statistical mechanics has a close formal similarity to this problem second the basic formulation of the statistical estimation from the observed degraded image by using the bayes formula is demonstrated the relationship between bayesian statistics and statistical mechanics is also listed particularly it is explained that some correlation inequalities on the nishimori line of the random spin model also play an important role in bayesian image restoration third the framework of bayesian image restoration for binary images by means of the ising model is reviewed some practical algorithms for binary image restoration are given by employing the mean field and the bethe approximations finally bayesian image restoration for a grey level image using the gaussian model is reviewed and the gaussian model is extended to a more practical probabilistic model by introducing the line state to treat the effects of edges the line state is also extended to quantized values

link scheduling in polynomial time two polynomial time algorithms are given for scheduling conversations in a spread spectrum radio network the constraint on conversations is that each station can converse with only one other station at a time the first algorithm is strongly polynomial and finds a schedule of minimum length that allows each pair of neighboring stations to converse directly for a prescribed length of time the second algorithm is designed for the situation in which messages must be relayed multiple hops the algorithm produces in polynomial time a routing vector and compatible link schedule that jointly meet a prespecified end to end demand so that the schedule has the smallest possible length

community mining with new node similarity by incorporating both global and local topological knowledge in a constrained random walk abstract detection of community is a crucial step to understand the structure and dynamics of complex networks most of conventional community detection methods focus on optimizing a certain objective function or on clustering nodes based on their similarities which leads to a phenomenon that they have preference for specific types of networks but are not general using constrained random walk we exploit global and local topology structures of network to propose a modified transition matrix and further to define a new similarity metric named isim between two nodes in contrast to the existing similarities isim does not work directly on the observed data but in a convergent stable space this feature makes isim robust to the observed noisy data in real world networks isim not only measures node s distance but also captures node s topology structure in network experiments on synthetic and real world networks demonstrate that isim can be successfully applied to community detection in broader types of networks and outperforms other community detection methods

on equivalence of likelihood maximization of stochastic block model and constrained nonnegative matrix factorization abstract community structures detection in complex network is important for understanding not only the topological structures of the network but also the functions of it stochastic block model and nonnegative matrix factorization are two widely used methods for community detection which are proposed from different perspectives in this paper the relations between them are studied the logarithm of likelihood function for stochastic block model can be reformulated under the framework of nonnegative matrix factorization despite model equivalence the algorithms employed by the two methods are different preliminary numerical experiments are carried out to compare the behaviors of the algorithms demonstrating that the multiplicative update rules for nmf are more effective

hiding quiet solutions in random constraint satisfaction problems we study constraint satisfaction problems on the so called planted random ensemble we show that for a certain class of problems e g graph coloring many of the properties of the usual random ensemble are quantitatively identical in the planted random ensemble we study the structural phase transitions and the easy hard easy pattern in the average computational complexity we also discuss the finite temperature phase diagram finding a close connection with the liquid glass solid phenomenology

efficient and principled method for detecting communities in networks a fundamental problem in the analysis of network data is the detection of network communities groups of densely interconnected nodes which may be overlapping or disjoint here we describe a method for finding overlapping communities based on a principled statistical approach using generative network models we show how the method can be implemented using a fast closed form expectation maximization algorithm that allows us to analyze networks of millions of nodes in reasonable running times we test the method both on real world networks and on synthetic benchmarks and find that it gives results competitive with previous methods we also show that the same approach can be used to extract nonoverlapping community divisions via a relaxation method and demonstrate that the algorithm is competitively fast and accurate for the nonoverlapping problem

a sequential approach for optimal broadcast scheduling in packet radio networks an important problem that arises in the design of packet radio networks is that of scheduling access to the high speed communications channel in such a way as to avoid interference while keeping the frame length to a minimum the broadcast scheduling problem is known to be np hard and to date this problem has been formulated as a nonlinear discrete optimization problem for a given frame length and solved via heuristic approaches by parametrically varying the length of the frame this paper presents a linear integer programming formulation for the composite problem of maximizing channel utilization while minimizing the length of the frame it then introduces a solution approach based on solving two relatively easier though still np complete integer programs in succession computational experiments are conducted on a set of benchmark cases and additional randomly generated problem instances results show that this sequential integer programming approach is very effective solving all the problems optimally within a few seconds these results imply that optimal solutions can be identified in very little time for problems of realistic size and that heuristic approaches will be needed only when problems get much larger than those considered in the literature to date

the nishimori line and bayesian statistics the nishimori line is a line or hypersurface in the parameter space of systems with quenched disorder where simple expressions of the averages of physical quantities over the quenched random variables are obtained it has been playing an important role in the theoretical studies of the random frustrated systems since its discovery in around 1980 in this paper an interpretation of the nishimori line from the viewpoint of statistical information processing is developed our main aim is the reconstruction of the whole theory of the nishimori line from the viewpoint of bayesian statistics or almost equivalently from the viewpoint of the theory of error correcting codes as a byproduct of the interpretation counterparts of the nishimori line in models without gauge invariance are given we also discussed the issues on the finite temperature decoding of error correcting codes and clarify the role of gauge invariance in this topic

community detection as an inference problem we express community detection as an inference problem of determining the most likely arrangement of communities we then apply belief propagation and mean field theory to this problem and show that this leads to fast accurate algorithms for community detection

achieving exact cluster recovery threshold via semidefinite programming extensions resolving a conjecture of abbe bandeira and hall the authors have recently shown that the semidefinite programming sdp relaxation of the maximum likelihood estimator achieves the sharp threshold for exactly recovering the community structure under the binary stochastic block model sbm of two equal sized clusters the same was shown for the case of a single cluster and outliers extending the proof techniques in this paper it is shown that sdp relaxations also achieve the sharp recovery threshold in the following cases 1 binary sbm with two clusters of sizes proportional to network size but not necessarily equal 2 sbm with a fixed number of equal sized clusters and 3 binary censored block model with the background graph being erd s renyi furthermore a sufficient condition is given for an sdp procedure to achieve exact recovery for the general case of a fixed number of clusters plus outliers these results demonstrate the versatility of sdp relaxation as a simple general purpose computationally feasible methodology for community detection

fundamental limits of symmetric low rank matrix estimation we consider the high dimensional inference problem where the signal is a low rank symmetric matrix which is corrupted by an additive gaussian noise given a probabilistic model for the low rank matrix we compute the limit in the large dimension setting for the mutual information between the signal and the observations as well as the matrix minimum mean squared error while the rank of the signal remains constant we also show that our model extends beyond the particular case of additive gaussian noise and we prove an universality result connecting the community detection problem to our gaussian framework we unify and generalize a number of recent works on pca sparse pca submatrix localization or community detection by computing the information theoretic limits for these problems in the high noise regime in addition we show that the posterior distribution of the signal given the observations is characterized by a parameter of the same dimension as the square of the rank of the signal i e scalar in the case of rank one this allows to locate precisely the information theoretic thresholds for the above mentioned problems finally we connect our work with the hard but detectable conjecture in statistical physics

phase transitions in the coloring of random graphs we consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color using the cavity method we present a detailed and systematic analytical study of the space of proper colorings solutions we show that for a fixed number of colors and as the average vertex degree number of constraints increases the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses first at the clustering transition the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard afterward the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state eventually above the coloring threshold no more solutions are available

exact recovery in the hypergraph stochastic block model a spectral algorithm abstract we consider the exact recovery problem in the hypergraph stochastic block model hsbm with k blocks of equal size more precisely we consider a random d uniform hypergraph h with n vertices partitioned into k clusters of size s n k hyperedges e are added independently with probability p if e is contained within a single cluster and q otherwise where 0 q p 1 we present a spectral algorithm which recovers the clusters exactly with high probability given mild conditions on n k p q and d our algorithm is based on the adjacency matrix of h which is a symmetric n n matrix whose u v th entry is the number of hyperedges containing both u and v to the best of our knowledge our algorithm is the first to guarantee exact recovery when the number of clusters k n

block models and personalized pagerank methods for ranking the importance of nodes in a network have a rich history in machine learning and across domains that analyze structured data recent work has evaluated these methods through the seed set expansion problem given a subset formula see text of nodes from a community of interest in an underlying graph can we reliably identify the rest of the community we start from the observation that the most widely used techniques for this problem personalized pagerank and heat kernel methods operate in the space of landing probabilities of a random walk rooted at the seed set ranking nodes according to weighted sums of landing probabilities of different length walks both schemes however lack an a priori relationship to the seed set objective in this work we develop a principled framework for evaluating ranking methods by studying seed set expansion applied to the stochastic block model we derive the optimal gradient for separating the landing probabilities of two classes in a stochastic block model and find surprisingly that under reasonable assumptions the gradient is asymptotically equivalent to personalized pagerank for a specific choice of the pagerank parameter formula see text that depends on the block model parameters this connection provides a formal motivation for the success of personalized pagerank in seed set expansion and node ranking generally we use this connection to propose more advanced techniques incorporating higher moments of landing probabilities our advanced methods exhibit greatly improved performance despite being simple linear classification rules and are even competitive with belief propagation

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