Discovering communities in complex networks means grouping nodes similar to each other, to uncover latent information about them. There are hundreds of different algorithms to solve the community detection task, each with its own understanding and definition of what a "community" is. Dozens of review works attempt to order such a diverse landscape -- classifying community discovery algorithms by the process they employ to detect communities, by their explicitly stated definition of community, or by their performance on a standardized task. In this paper, we classify community discovery algorithms according to a fourth criterion: the similarity of their results. We create an Algorithm Similarity Network (ASN), whose nodes are the community detection approaches, connected if they return similar groupings. We then perform community detection on this network, grouping algorithms that consistently return the same partitions or overlapping coverage over a span of more than one thousand synthetic and real world networks. This paper is an attempt to create a similarity-based classification of community detection algorithms based on empirical data. It improves over the state of the art by comparing more than seventy approaches, discovering that the ASN contains well-separated groups, making it a sensible tool for practitioners, aiding their choice of algorithms fitting their analytic needs.
The first -- most popular -- category includes works classifying algorithms by the techniques they employ to divide the graph into groups of nodes, i.e. by their . Examples in this category are @cite_21 , @cite_24 , @cite_33 , @cite_9 , @cite_8 , @cite_26 ; @cite_12 -- focusing on multilayer networks; and @cite_10 -- whose attention narrows down to genetic algorithms. Here, we are agnostic about how an algorithm works, as we are focused on figuring out which algorithm returns similar partitions to which other. This is influenced by how they work, but even algorithms based on the philosophy of modularity maximization might end up in different categories.
@cite_26 @cite_33 @cite_8 @cite_9 @cite_21 @cite_24 @cite_10 @cite_12
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This work addresses graph-based semi-supervised classification and betweenness computation in large, sparse, networks (several millions of nodes). The objective of semi-supervised classification is to assign a label to unlabeled nodes using the whole topology of the graph and the labeling at our disposal. Two approaches are developed to avoid explicit computation of pairwise proximity between the nodes of the graph, which would be impractical for graphs containing millions of nodes. The first approach directly computes, for each class, the sum of the similarities between the nodes to classify and the labeled nodes of the class, as suggested initially in [1,2]. Along this approach, two algorithms exploiting different state-of-the-art kernels on a graph are developed. The same strategy can also be used in order to compute a betweenness measure. The second approach works on a trellis structure built from biased random walks on the graph, extending an idea introduced in [3]. These random walks allow to define a biased bounded betweenness for the nodes of interest, defined separately for each class. All the proposed algorithms have a linear computing time in the number of edges while providing good results, and hence are applicable to large sparse networks. They are empirically validated on medium-size standard data sets and are shown to be competitive with state-of-the-art techniques. Finally, we processed a novel data set, which is made available for benchmarking, for multi-class classification in a large network: the U.S. patents citation network containing 3M nodes (of six different classes) and 38M edges. The three proposed algorithms achieve competitive results (around 85 classification rate) on this large network-they classify the unlabeled nodes within a few minutes on a standard workstation. While much of the work in the design of convolutional networks over the last five years has revolved around the empirical investigation of the importance of depth, filter sizes, and number of feature channels, recent studies have shown that branching, i.e., splitting the computation along parallel but distinct threads and then aggregating their outputs, represents a new promising dimension for significant improvements in performance. To combat the complexity of design choices in multi-branch architectures, prior work has adopted simple strategies, such as a fixed branching factor, the same input being fed to all parallel branches, and an additive combination of the outputs produced by all branches at aggregation points. In this work we remove these predefined choices and propose an algorithm to learn the connections between branches in the network. Instead of being chosen a priori by the human designer, the multi-branch connectivity is learned simultaneously with the weights of the network by optimizing a single loss function defined with respect to the end task. We demonstrate our approach on the problem of multi-class image classification using four different datasets where it yields consistently higher accuracy compared to the state-of-the-art ResNeXt'' multi-branch network given the same learning capacity. Classifying a set of objects into clusters can be done in numerous ways, producing different results. They can be visually compared using contingency tables [27], mosaicplots [13], fluctuation diagrams [15], tableplots [20] , (modified) parallel coordinates plots [28], Parallel Sets plots [18] or circos diagrams [19]. Unfortunately the interpretability of all these graphical displays decreases rapidly with the numbers of categories and clusterings. In his famous book A Semiology of Graphics [5] Bertin writes “the discovery of an ordered concept appears as the ultimate point in logical simplification since it permits reducing to a single instant the assimilation of series which previously required many instants of study”. Or in more everyday language, if you use good orderings you can see results immediately that with other orderings might take a lot of effort. This is also related to the idea of effect ordering [12], that data should be organised to reflect the effect you want to observe. This paper presents an efficient algorithm based on Bertin's idea and concepts related to Kendall's t [17], which finds informative joint orders for two or more nominal classification variables. We also show how these orderings improve the various displays and how groups of corresponding categories can be detected using a top-down partitioning algorithm. Different clusterings based on data on the environmental performance of cars sold in Germany are used for illustration. All presented methods are available in the R package extracat which is used to compute the optimized orderings for the example dataset. This paper studies the problem of leveraging computationally intensive classification algorithms for large scale text categorization problems. We propose a hierarchical approach which decomposes the classification problem into a coarse level task and a fine level task. A simple yet scalable classifier is applied to perform the coarse level classification while a more sophisticated model is used to separate classes at the fine level. However, instead of relying on a human-defined hierarchy to decompose the problem, we we use a graph algorithm to discover automatically groups of highly similar classes. As an illustrative example, we apply our approach to real-world industrial data from eBay, a major e-commerce site where the goal is to classify live items into a large taxonomy of categories. In such industrial setting, classification is very challenging due to the number of classes, the amount of training data, the size of the feature space and the real-world requirements on the response time. We demonstrate through extensive experimental evaluation that (1) the proposed hierarchical approach is superior to flat models, and (2) the data-driven extraction of latent groups works significantly better than the existing human-defined hierarchy. In this paper we study a paradigm to generalize online classification algorithms for binary classification problems to multiclass problems. The particular hypotheses we investigate maintain one prototype vector per class. Given an input instance, a multiclass hypothesis computes a similarity-score between each prototype and the input instance and sets the predicted label to be the index of the prototype achieving the highest similarity. To design and analyze the learning algorithms in this paper we introduce the notion of ultraconservativeness. Ultraconservative algorithms are algorithms that update only the prototypes attaining similarity-scores which are higher than the score of the correct label's prototype. We start by describing a family of additive ultraconservative algorithms where each algorithm in the family updates its prototypes by finding a feasible solution for a set of linear constraints that depend on the instantaneous similarity-scores. We then discuss a specific online algorithm that seeks a set of prototypes which have a small norm. The resulting algorithm, which we term MIRA (for Margin Infused Relaxed Algorithm) is ultraconservative as well. We derive mistake bounds for all the algorithms and provide further analysis of MIRA using a generalized notion of the margin for multiclass problems. We discuss the form the algorithms take in the binary case and show that all the algorithms from the first family reduce to the Perceptron algorithm while MIRA provides a new Perceptron-like algorithm with a margin-dependent learning rate. We then return to multiclass problems and describe an analogous multiplicative family of algorithms with corresponding mistake bounds. We end the formal part by deriving and analyzing a multiclass version of Li and Long's ROMMA algorithm. We conclude with a discussion of experimental results that demonstrate the merits of our algorithms. Analyzing massive social networks challenges both high-performance computers and human understanding. These massive networks cannot be visualized easily, and their scale makes applying complex analysis methods computationally expensive. We present a region-growing method for finding a smaller, more tractable subgraph, a community, given a few example seed vertices. Unlike existing work, we focus on a small number of seed vertices, from two to a few dozen. We also present the first comparison between five algorithms for expanding a small seed set into a community. Our comparison applies these algorithms to an R-MAT generated graph component with 240 thousand vertices and 32 million edges and evaluates the community size, modularity, Kullback-Leibler divergence, conductance, and clustering coefficient. We find that our new algorithm with a local modularity maximizing heuristic based on Clauset, Newman, and Moore performs very well when the output is limited to 100 or 1 000 vertices. When run without a vertex size limit, a heuristic from McCloskey and Bader generates communities containing around 60 of the graph’s vertices and having a small conductance and modularity appropriate to the result size. A personalized PageRank algorithm based on Andersen, Lang, and Chung also performs well with respect to our metrics. Many methods have been proposed to solve the image classification problem for a large number of categories. Among them, methods based on tree-based representations achieve good trade-off between accuracy and test time efficiency. While focusing on learning a tree-shaped hierarchy and the corresponding set of classifiers, most of them [11, 2, 14] use a greedy prediction algorithm for test time efficiency. We argue that the dramatic decrease in accuracy at high efficiency is caused by the specific design choice of the learning and greedy prediction algorithms. In this work, we propose a classifier which achieves a better trade-off between efficiency and accuracy with a given tree-shaped hierarchy. First, we convert the classification problem as finding the best path in the hierarchy, and a novel branch-and-bound-like algorithm is introduced to efficiently search for the best path. Second, we jointly train the classifiers using a novel Structured SVM (SSVM) formulation with additional bound constraints. As a result, our method achieves a significant 4.65 , 5.43 , and 4.07 (relative 24.82 , 41.64 , and 109.79 ) improvement in accuracy at high efficiency compared to state-of-the-art greedy "tree-based" methods [14] on Caltech-256 [15], SUN [32] and Image Net 1K [9] dataset, respectively. Finally, we show that our branch-and-bound-like algorithm naturally ranks the paths in the hierarchy (Fig. 8) so that users can further process them. We consider the enumeration of maximal bipartite cliques (bicliques) from a large graph, a task central to many practical data mining problems in social network analysis and bioinformatics. We present novel parallel algorithms for the MapReduce platform, and an experimental evaluation using Hadoop MapReduce. Our algorithm is based on clustering the input graph into smaller sized subgraphs, followed by processing different subgraphs in parallel. Our algorithm uses two ideas that enable it to scale to large graphs: (1) the redundancy in work between different subgraph explorations is minimized through a careful pruning of the search space, and (2) the load on different reducers is balanced through the use of an appropriate total order among the vertices. Our evaluation shows that the algorithm scales to large graphs with millions of edges and tens of millions of maximal bicliques. To our knowledge, this is the first work on maximal biclique enumeration for graphs of this scale. Nowadays, in the world of limited attention, the techniques that maximize the spread of social influence are more than welcomed. Companies try to maximize their profits on sales by providing customers with free samples believing in the power of word-of-mouth marketing, governments and non-governmental organizations often want to introduce positive changes in the society by appropriately selecting individuals or election candidates want to spend least budget yet still win the election. In this work we propose the use of evolutionary algorithm as a mean for selecting seeds in social networks. By framing the problem as genetic algorithm challenge we show that it is possible to outperform well-known greedy algorithm in the problem of influence maximization for the linear threshold model in both: quality (up to 16 better) and efficiency (up to 35 times faster). We implemented these two algorithms by using GPGPU approach showing that also the evolutionary algorithm can benefit from GPU acceleration making it efficient and scaling better than the greedy algorithm. As the experiments conducted by using three real world datasets reveal, the evolutionary approach proposed in this paper outperforms the greedy algorithm in terms of the outcome and it also scales much better than the greedy algorithm when the network size is increasing. The only drawback in the GPGPU approach so far is the maximum size of the network that can be processed - it is limited by the memory of the GPU card. We believe that by showing the superiority of the evolutionary approach over the greedy algorithm, we will motivate the scientific community to look for an idea to overcome this limitation of the GPU approach - we also suggest one of the possible paths to explore. Since the proposed approach is based only on topological features of the network, not on the attributes of nodes, the applications of it are broader than the ones that are dataset-specific. A prominent tool in many problems involving metric spaces is a notion of randomized low-diameter decomposition. Loosely speaking, ( )-decomposition refers to a probability distribution over partitions of the metric into sets of low diameter, such that nearby points (parameterized by ( >0 )) are likely to be “clustered” together. Applying this notion to the shortest-path metric in edge-weighted graphs, it is known that n-vertex graphs admit an (O( n) )-padded decomposition (Bartal, 37th annual symposium on foundations of computer science. IEEE, pp 184–193, 1996), and that excluded-minor graphs admit O(1)-padded decomposition (, 25th annual ACM symposium on theory of computing, pp 682–690, 1993; Fakcharoenphol and Talwar, J Comput Syst Sci 69(3), 485–497, 2004; , Proceedings of the 46th annual ACM symposium on theory of computing. STOC ’14, pp 79–88. ACM, New York, NY, USA, 2014). We design decompositions to the family of p-path-separable graphs, which was defined by Abraham and Gavoille (Proceedings of the twenty-fifth annual acm symposium on principles of distributed computing, PODC ’06, pp 188–197, 2006) and refers to graphs that admit vertex-separators consisting of at most p shortest paths in the graph. Our main result is that every p-path-separable n-vertex graph admits an (O( (p n)) )-decomposition, which refines the (O( n) ) bound for general graphs, and provides new bounds for families like bounded-treewidth graphs. Technically, our clustering process differs from previous ones by working in (the shortest-path metric of) carefully chosen subgraphs. We formulate a spectral graph-partitioning algorithm that uses the two leading eigenvectors of the matrix corresponding to a selected quality function to split a network into three communities in a single step. In so doing, we extend the recursive bipartitioning methods developed by Newman [M. E. J. Newman, Proc. Natl. Acad. Sci. U.S.A. 103, 8577 (2006); Phys. Rev. E 74, 036104 (2006)] to allow one to consider the best available two-way and three-way divisions at each recursive step. We illustrate the method using simple "bucket brigade" examples and then apply the algorithm to examine the community structures of the coauthorship graph of network scientists and of U. S. Congressional networks inferred from roll call voting similarities. Chapter 36 Decomposing Graphs into Regions of Small Diameter* Nathan Linialt Michael .lss A decomposition of a graph G = (V, E) is a partition of the vertex set into subsets (called lhks). The diameter of a decomposition is the least. d such that any two vertices belonging to the same connected component of a block are at distance < d. In this paper we prove (nearly best possible) statements of the form: .4ny n–vertex graph has a decomposition into a small number of blocks each having small diameter. Such decompositions provide a tool for efficiently decentralizing distributed computations. In [AGLP1 it was shown that every graph has a decomposition into at most s(n) blocks of diameter at most s(n) for s(n) = o( loglog d h n). usinga,technique of Awerbuch [A] and Awerbuch and Peleg [AP], we improve this result by showing that every graph has a decomposition of diameter ()(log n) into O(log n) blocks. In addition, we give a randomized distributed algorithm that produces such a decomposition and runs in time 0(log2 n). The construction can be parametrized to provide decompositions that trade-off between the number of blocks and the diameter. We show that this trade-off is nearly best possible for two families of graphs the first consists of skeletons of certain triangulations of a simplex and the second consists of grid graphs with added diagonals. The proofs in both cases rely on basic results in combinatorial topology, Sperner’s lemma for the first class and Tucker’s lemma for the second. *This work was supported in part by NSF contracts DMS87-03541 and CCR-8911388 tDepartment of Computer Science, Hebrew University, Jerusalem, Israel. + ‘Department of Computer Science and Engineering, Mail Code C-014, University of California, San Diego, La Jolla, CA 92093-0114. Graph classification is a problem with practical applications in many different domains. To solve this problem, one usually calculates certain graph statistics (i.e., graph features) that help discriminate between graphs of different classes. When calculating such features, most existing approaches process the entire graph. In a graphlet-based approach, for instance, the entire graph is processed to get the total count of different graphlets or subgraphs. In many real-world applications, however, graphs can be noisy with discriminative patterns confined to certain regions in the graph only. In this work, we study the problem of attention-based graph classification. The use of attention allows us to focus on small but informative parts of the graph, avoiding noise in the rest of the graph. We present a novel RNN model, called the Graph Attention Model (GAM), that processes only a portion of the graph by adaptively selecting a sequence of "informative" nodes. Experimental results on multiple real-world datasets show that the proposed method is competitive against various well-known methods in graph classification even though our method is limited to only a portion of the graph. In this paper, we present and study a class of graph partitioning algorithms that reduces the size of the graph by collapsing vertices and edges, we find ak-way partitioning of the smaller graph, and then we uncoarsen and refine it to construct ak-way partitioning for the original graph. These algorithms compute ak-way partitioning of a graphG= (V,E) inO(|E|) time, which is faster by a factor ofO(logk) than previously proposed multilevel recursive bisection algorithms. A key contribution of our work is in finding a high-quality and computationally inexpensive refinement algorithm that can improve upon an initialk-way partitioning. We also study the effectiveness of the overall scheme for a variety of coarsening schemes. We present experimental results on a large number of graphs arising in various domains including finite element methods, linear programming, VLSI, and transportation. Our experiments show that this new scheme produces partitions that are of comparable or better quality than those produced by the multilevel bisection algorithm and requires substantially smaller time. Graphs containing up to 450,000 vertices and 3,300,000 edges can be partitioned in 256 domains in less than 40 s on a workstation such as SGI's Challenge. Compared with the widely used multilevel spectral bisection algorithm, our new algorithm is usually two orders of magnitude faster and produces partitions with substantially smaller edge-cut. Recent algorithms in convolutional neural networks (CNN) considerably advance the fine-grained image classification, which aims to differentiate subtle differences among subordinate classes. However, previous studies have rarely focused on learning a fined-grained and structured feature representation that is able to locate similar images at different levels of relevance, e.g., discovering cars from the same make or the same model, both of which require high precision. In this paper, we propose two main contributions to tackle this problem. 1) A multitask learning framework is designed to effectively learn fine-grained feature representations by jointly optimizing both classification and similarity constraints. 2) To model the multi-level relevance, label structures such as hierarchy or shared attributes are seamlessly embedded into the framework by generalizing the triplet loss. Extensive and thorough experiments have been conducted on three finegrained datasets, i.e., the Stanford car, the Car-333, and the food datasets, which contain either hierarchical labels or shared attributes. Our proposed method has achieved very competitive performance, i.e., among state-of-the-art classification accuracy when not using parts. More importantly, it significantly outperforms previous fine-grained feature representations for image retrieval at different levels of relevance. Graphical models (e.g., Bayesian and constraint networks, influence diagrams, and Markov decision processes) have become a central paradigm for knowledge representation and reasoning in both artificial intelligence and computer science in general. These models are used to perform many reasoning tasks, such as scheduling, planning and learning, diagnosis and prediction, design, hardware and software verification, and bioinformatics. These problems can be stated as the formal tasks of constraint satisfaction and satisfiability, combinatorial optimization, and probabilistic inference. It is well known that the tasks are computationally hard, but research during the past three decades has yielded a variety of principles and techniques that significantly advanced the state of the art. In this book we provide comprehensive coverage of the primary exact algorithms for reasoning with such models. The main feature exploited by the algorithms is the model's graph. We present inference-based, message-passing schemes (e.g., variable-elimination) and search-based, conditioning schemes (e.g., cycle-cutset conditioning and AND OR search). Each class possesses distinguished characteristics and in particular has different time vs. space behavior. We emphasize the dependence of both schemes on few graph parameters such as the treewidth, cycle-cutset, and (the pseudo-tree) height. We believe the principles outlined here would serve well in moving forward to approximation and anytime-based schemes. The target audience of this book is researchers and students in the artificial intelligence and machine learning area, and beyond. Table of Contents: Preface Introduction What are Graphical Models Inference: Bucket Elimination for Deterministic Networks Inference: Bucket Elimination for Probabilistic Networks Tree-Clustering Schemes AND OR Search Spaces and Algorithms for Graphical Models Combining Search and Inference: Trading Space for Time Conclusion Bibliography Author's Biography It has been recently recognized that heterogeneous information networks composed of multiple types of nodes and links are prevalent in the real world. Both classification and ranking of the nodes (or data objects) in such networks are essential for network analysis. However, so far these approaches have generally been performed separately. In this paper, we combine ranking and classification in order to perform more accurate analysis of a heterogeneous information network. Our intuition is that highly ranked objects within a class should play more important roles in classification. On the other hand, class membership information is important for determining a quality ranking over a dataset. We believe it is therefore beneficial to integrate classification and ranking in a simultaneous, mutually enhancing process, and to this end, propose a novel ranking-based iterative classification framework, called RankClass. Specifically, we build a graph-based ranking model to iteratively compute the ranking distribution of the objects within each class. At each iteration, according to the current ranking results, the graph structure used in the ranking algorithm is adjusted so that the sub-network corresponding to the specific class is emphasized, while the rest of the network is weakened. As our experiments show, integrating ranking with classification not only generates more accurate classes than the state-of-art classification methods on networked data, but also provides meaningful ranking of objects within each class, serving as a more informative view of the data than traditional classification. We propose a new local, deterministic and parameter-free algorithm that detects fuzzy and crisp overlapping communities in a weighted network and simultaneously reveals their hierarchy. Using a local fitness function, the algorithm greedily expands natural communities of seeds until the whole graph is covered. The hierarchy of communities is obtained analytically by calculating resolution levels at which communities grow rather than numerically by testing different resolution levels. This analytic procedure is not only more exact than its numerical alternatives such as LFM and GCE but also much faster. Critical resolution levels can be identified by searching for intervals in which large changes of the resolution do not lead to growth of communities. We tested our algorithm on benchmark graphs and on a network of 492 papers in information science. Combined with a specific post-processing, the algorithm gives much more precise results on LFR benchmarks with high overlap compared to other algorithms and performs very similarly to GCE. This paper presents an approach to combine rank aggregation techniques using a soft computing technique -- Genetic Programming -- in order to improve the results in Information Retrieval tasks. Previous work shows that by combining rank aggregation techniques in an agglomerative way, it is possible to get better results than with individual methods. However, these works either combine only a small set of lists or are performed in a completely ad-hoc way. Therefore, given a set of ranked lists and a set of rank aggregation techniques, we propose to use a supervised genetic programming approach to search combinations of them that maximize effectiveness in large search spaces. Experimental results conducted using four datasets with different properties show that our proposed approach reaches top performance in most datasets. Moreover, this cross-dataset performance is not matched by any other baseline among the many we experiment with, some being the state-of-the-art in learning-to-rank and in the supervised rank aggregation tasks. We also show that our proposed framework is very efficient, flexible, and scalable. We prove the approximation ratio 8 5 for the metric s,t -path-TSP, and more generally for shortest connected T-joins. The algorithm that achieves this ratio is the simple "Best of Many" version of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists in determining the best Christofides s,t -tour out of those constructed from a family @math of trees having a convex combination dominated by an optimal solution x* of the Held-Karp relaxation. They give the approximation guarantee @math for such an s,t -tour, which is the first improvement after the 5 3 guarantee of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao (2012) extended this result to a 13 8-approximation of shortest connected T-joins, for |T|≥4. The ratio 8 5 is proved by simplifying and improving the approach of An, Kleinberg and Shmoys that consists in completing x* 2 in order to dominate the cost of "parity correction" for spanning trees. We partition the edge-set of each spanning tree in @math into an s,t -path (or more generally, into a T-join) and its complement, which induces a decomposition of x*. This decomposition can be refined and then efficiently used to complete x* 2 without using linear programming or particular properties of T, but by adding to each cut deficient for x* 2 an individually tailored explicitly given vector, inherent in x*. A simple example shows that the Best of Many Christofides algorithm may not find a shorter s,t -tour than 3 2 times the incidentally common optima of the problem and of its fractional relaxation.