Learning multifractal structure in large networks

In this paper, three new algorithms, a greedy algorithm, a KL-like algorithm, and an add-all algorithm, are proposed to find local optimal community structures in large networks starting from a given source vertex. The time complexity for finding a local community of all these algorithms is O(K 2d), where K is the number of vertices to be explored in the sub-graph and d is the average degree of the vertices in the sub-graph. A JAVA tool is developed based on these algorithms to identify local community structures in large networks. The results of using this tool to analyze a co-purchase network from Amazon.com show that local community structures exist in this large-scale co-purchase network. Further analyses of the identified local communities show that purchases of media items form more compact local communities than purchases of book items do, indicating that recommending digital media items to customers based on co-purchasing information in the online store is more efficient than recommending books.

Community structure is a fundamental feature for many networks. The problem of discovering communities in those networks thus has been attracting a lot of research. However, due to the rapid increase of networks’ scale and the availability of real communities in many networks, the task of detecting communities in large real networks remains a challenging problem. In this paper, we study the structure of various large real networks and their ground-truth community structures and observe an interesting phenomenon: the difference of degrees (abbreviated as dod) of connected nodes follows a heavy-tailed distribution with an approximate power-law tail for large dod, in both original network and community structure but to different extents. With the aim to explore the effect of this observation on identifying communities in real large networks, we propose a weighting strategy and further embed it into two prominent community detection algorithms. Comparisons against the state of the arts demonstrate a very promising performance of the proposed weighting strategy.

Intelligence needs connectivity and organization with some agents' intention in networks. The networks with intelligence produce some structure spontaneously. Therefore, it is important to focus on the structure in large networks like the Internet or World Wide Web that are constructed by human beings with their preferences or concepts. Especially, it has been found that even these artificial networks have scale-free structures, which are seen also in natural networks. It indicates that universal laws or phenomena exist in the properties and behaviors on scale-free networks. We investigate them from the view point of complex systems, in particular self-organization. From our proposed model and simulation experiments, it is found that growing scale-free networks have self-organized criticality. The phenomena would lead to the intelligence emerged from network structures.

Large graphs arise in a number of contexts and understanding their structure and extracting information from them is an important research area. Early algorithms for mining communities have focused on global graph structure, and often run in time proportional to the size of the entire graph. As we explore networks with millions of vertices and find communities of size in the hundreds, it becomes important to shift our attention from macroscopic structure to microscopic structure in large networks. A growing body of work has been adopting local expansion methods in order to identify communities from a few exemplary seed members. In this article, we propose a novel approach for finding overlapping communities called L emon ( L ocal E xpansion via M inimum O ne N orm). Provided with a few known seeds , the algorithm finds the community by performing a local spectral diffusion. The core idea of L emon is to use short random walks to approximate an invariant subspace near a seed set, which we refer to as local spectra . Local spectra can be viewed as the low-dimensional embedding that captures the nodes’ closeness in the local network structure. We show that L emon ’s performance in detecting communities is competitive with state-of-the-art methods. Moreover, the running time scales with the size of the community rather than that of the entire graph. The algorithm is easy to implement and is highly parallelizable. We further provide theoretical analysis of the local spectral properties, bounding the measure of tightness of extracted community using the eigenvalues of graph Laplacian. We thoroughly evaluate our approach using both synthetic and real-world datasets across different domains, and analyze the empirical variations when applying our method to inherently different networks in practice. In addition, the heuristics on how the seed set quality and quantity would affect the performance are provided.

VLSI designs contain very large resistive networks consisting of hundreds of millions (10e11) of resistors. Accurate parasitic extraction and analysis of such large networks is essential in many phases of the VLSI design flow. Existing techniques to analyze large resistive networks using linear solvers, despite recent optimizations, still take prohibitive computation time. In this paper a new technique based on low-distortion embedding to estimate point-to-point effective resistance is presented. Our proposed method employs recently discovered techniques from theoretical computer science to compute an e-approximate resistance embedding matrix from which effective resistances of all node pairs can be estimated as easily as taking the Euclidean norm of column differences. The proposed method runs in almost linear time (linear in the number of resistors), and the accuracy (e) is user specified. The method has been implemented and experimental results on large networks containing upto 10e11 nodes are presented. Compared to existing method using sparse linear solvers, our methods are more than 10 times faster on mesh networks and more importantly given a network of n nodes, allow computation of effective resistance between arbitrary node pairs in O(lg(n)) time (lg denotes logarithm to base 2).

In this paper we present a novel strategy to discover the community structure\nof (possibly, large) networks. This approach is based on the well-know concept\nof network modularity optimization. To do so, our algorithm exploits a novel\nmeasure of edge centrality, based on the k-paths. This technique allows to\nefficiently compute a edge ranking in large networks in near linear time. Once\nthe centrality ranking is calculated, the algorithm computes the pairwise\nproximity between nodes of the network. Finally, it discovers the community\nstructure adopting a strategy inspired by the well-known state-of-the-art\nLouvain method (henceforth, LM), efficiently maximizing the network modularity.\nThe experiments we carried out show that our algorithm outperforms other\ntechniques and slightly improves results of the original LM, providing reliable\nresults. Another advantage is that its adoption is naturally extended even to\nunweighted networks, differently with respect to the LM.\n

Sampling from large graphs is an area of great interest, especially since the emergence of huge structures such as Online Social Networks and the World Wide Web (WWW). The large scale properties of a network can be summarized in terms of parameters of the underlying graph, such as the total number of vertices, edges and triangles. However, the large si ze of these networks makes it computationally expensive to obtain such structural properties of the underlying graph by exhaustive search. If we can estimate these properties by taking small but representative samples from the network, then size is no longer such a problem. In this paper we present a general framework to estimate network properties using random walks. These methods work under the assumption we are able to obtain local characteristics of a vertex during each step of the random walk, for example the number of its neighbours, and their labels. As examples of this approach, we present practical methods to estimate the total number of edges/links m, number of vertices/nodes n and number of connected triads of vertices (triangles) t in graphs. We also give a general method to count any type of small connected subgraph, of which vertices, edges and triangles are specific examples. Additionally we present experimental estimates for n, m, t we obtained using our methods on real or synthetic networks. The synthetic networks were random graphs with power-law degree distributions and designed to have a large number of triangles. We used these graphs as they tend to correspond to the structure of large online networks. The real networks were samples of the WWW and social networks obtained from the SNAP database. In order to test that the methods are indeed practical, the total number of steps made by the walk was limited to at most the size n of the network. In fact the estimates appear to converge to the correct value at a lower number of steps, indicating that our proposed methods are feasible in practice.

We consider the random graphs modelling the structure of large data transmission networks including Internet. We investigate the subset of such graphs consisting of N vertices under the condition that the number of edges is equal to n. We obtain the limit distributions of the maximum degree of vertices and the number of vertices of a given degree as N, n → ∞ so that n/N → λ, where λ is a positive constant.

Conventional Vector Autoregressive (VAR) modelling methods applied to high dimensional neural time series data result in noisy solutions that are dense or have a large number of spurious coefficients. This reduces the speed and accuracy of auxiliary computations downstream and inflates the time required to compute functional connectivity networks by a factor that is at least inversely proportional to the true network density. As these noisy solutions have distorted coefficients, thresholding them as per some criterion, statistical or otherwise, does not alleviate the problem. Thus obtaining a sparse representation of such data is important since it provides an efficient representation of the data and facilitates its further analysis. We propose a fast Sparse Vector Autoregressive Greedy Search (SVARGS) method that works well for high dimensional data, even when the number of time points is relatively low, by incorporating only statistically significant coefficients. In numerical experiments, our methods show high accuracy in recovering the true sparse model. The relative absence of spurious coefficients permits accurate, stable and fast evaluation of derived quantities such as power spectrum, coherence and Granger causality. Consequently, sparse functional connectivity networks can be computed, in a reasonable time, from data comprising tens of thousands of channels/voxels. This enables a much higher resolution analysis of functional connectivity patterns and community structures in such large networks than is possible using existing time series methods. We apply our method to EEG data where computed network measures and community structures are used to distinguish emotional states as well as to ADHD fMRI data where it is used to distinguish children with ADHD from typically developing children.

The Google Earth satellite image software is adopted as the data source and analysis tool in this study. Some indices of river networks and channel profiles have been calculated in order to investigate their quantitive relationship. The results indicated that: For small basins local conditions may affect the topology structure of river networks, however, the topology structure of large river networks will tend to be the same regardless of any factors. The longitudinal profiles of incisive channels were formed under the influence of discharge and incision depth. The river section slope is larger when incision depth is also larger and when discharge is smaller.

The eigenvalues of matrices representing the structure of large-scale complex networks present a wide range of applications, from the analysis of dynamical processes taking place in the network to spectral techniques aiming to rank the importance of nodes in the network. A common approach to study the relationship between the structure of a network and its eigenvalues is to use synthetic random networks in which structural properties of interest, such as degree distributions, are prescribed. Although very common, synthetic models present two major flaws: 1) These models are only suitable to study a very limited range of structural properties; and 2) they implicitly induce structural properties that are not directly controlled and can deceivingly influence the network eigenvalue spectrum. In this paper, we propose an alternative approach to overcome these limitations. Our approach is not based on synthetic models. Instead, we use algebraic graph theory and convex optimization to study how structural properties influence the spectrum of eigenvalues of the network. Using our approach, we can compute, with low computational overhead, global spectral properties of a network from its local structural properties. We illustrate our approach by studying how structural properties of online social networks influence their eigenvalue spectra.

<p>Deep neural networks have achieved tremendous success in many domains (e.g., computer vision~\cite{Alexnet12,vggnet15,fastrcnn15}, speech recognition~\cite{hinton2012deep,dahl2012context}, natural language processing~\cite{dahl2012context,collobert2011natural}, games~\cite{silver2017mastering,silver2016mastering}), however, there are still many challenges in deep learning comunity such as how to speed up training large deep neural networks, how to compress large nerual networks for mobile/embed device without performance loss, how to automatically design the optimal network structures for a certain task, and how to further design the optimal networks with improved performance and certain model size with reduced computation cost.</p> <p>To speed up training large neural networks, we propose to use multinomial sampling for dropout, i.e., sampling features or neurons according to a multinomial distribution with different probabilities for different features/neurons. To exhibit the optimal dropout probabilities, we analyze the shallow learning with multinomial dropout and establish the risk bound for stochastic optimization. By minimizing a sampling dependent factor in the risk bound, we obtain a distribution-dependent dropout with sampling probabilities dependent on the second order statistics of the data distribution. To tackle the issue of evolving distribution of neurons in deep learning, we propose an efficient adaptive dropout (named evolutional dropout) that computes the sampling probabilities on-the-fly from a mini-batch of examples.</p> <p>To compress large neural network structures, we propose a simple yet powerful method for compressing the size of deep Convolutional Neural Networks (CNNs) based on parameter binarization. The striking difference from most previous work on parameter binarization/quantization lies at different treatments of $1\times 1$ convolutions and $k\times k$ convolutions ($k>1$), where we only binarize $k\times k$ convolutions into binary patterns. By doing this, we show that previous deep CNNs such as GoogLeNet and Inception-type Nets can be compressed dramatically with marginal drop in performance. Second, in light of the different functionalities of $1\times 1$ (data projection/transformation) and $k\times k$ convolutions (pattern extraction), we propose a new block structure codenamed the pattern residual block that adds transformed feature maps generated by $1\times 1$ convolutions to the pattern feature maps generated by $k\times k$ convolutions, based on which we design a small network with $\sim 1$ million parameters. Combining with our parameter binarization, we achieve better performance on ImageNet than using similar sized networks including recently released Google MobileNets.</p> <p>To automatically design neural networks, we study how to design a genetic programming approach for optimizing the structure of a CNN for a given task under limited computational resources yet without imposing strong restrictions on the search space. To reduce the computational costs, we propose two general strategies that are observed to be helpful: (i) aggressively selecting strongest individuals for survival and reproduction, and killing weaker individuals at a very early age; (ii) increasing mutation frequency to encourage diversity and faster evolution. The combined strategy with additional optimization techniques allows us to explore a large search space but with affordable computational costs.</p> <p>To further design the optimal networks with improved performance and certain model size under reduced computation cost, we propose an ecologically inspired genetic approach for neural network structure search , that includes two types of succession: primary and secondary succession as well as accelerated extinction. Specifically, we first use primary succession to rapidly evolve a community of poor initialized neural network structures into a more diverse community, followed by a secondary succession stage for fine-grained searching based on the networks from the primary succession. Accelerated extinction is applied in both stages to reduce computational cost. In addition, we also introduce the gene duplication to further utilize the novel block of layers that appeared in the discovered network structure.</p>

The decentralisation of existing district heating systems by network separation is a promising approach for district heating transformation. A district heating network is separated into several smaller district heating network structures that can be individually adapted, e.g. in relation to the existing network structure or concerning the properties of the local buildings supplied, such as the temperature requirements of their heating systems. In this work, an approach for the optimal separation of existing district heating networks is expanded, which enables the decentralisation of a specific district heating system and the partial transformation of the separated network depending on the heat source utilised and the local district heating system conditions. The approach first identifies coherent network areas through community detection and aggregation, then optimises network separation and the installation of heat pumps, and finally evaluates the arising district heating network systems through district heating network simulations. The individual steps of this approach are adapted to be applicable to large district heating network structures, e.g. to handle the great number of separation options. The approach is applied to two existing district heating systems with over four thousand buildings supplied. Two types of utilised waste heat sources are investigated, a sewage plant with fluctuating temperatures and an electrolyser with a dynamic operation schedule. The results show favourable transformation alternatives for existing district heating networks, as the separated network areas are transformed by reducing supply temperatures depending on the heat source potential utilised and the temperature requirements of the buildings supplied. • Separating district heating networks facilitates the utilisation of heat sources. • An optimisation model reveals the optimal separation of district heating networks. • The network separation considers available heat capacity and temperature level. • Heat pumps are installed in the separated network to utilise low-temperature heat. • A subsequent simulation evaluates the separated district heating networks.

Individuals' access to information in a social network depends on its distributed and where in the network individuals position themselves. However, individuals have limited capacity to manage their social connections and process information. In this work, we study how this limited capacity and network structure interact to affect the diversity of information social media users receive. Previous studies of the role of networks in information access were limited in their ability to measure the diversity of information. We address this problem by learning the topics of interest to social media users by observing messages they share online with their followers. We present a probabilistic model that incorporates human cognitive constraints in a generative model of information sharing. We then use the topics learned by the model to measure the diversity of information users receive from their social media contacts.We confirm that users in structurally diverse network positions, which bridge otherwise disconnected regions of the follower graph, are exposed to more diverse information. In addition, we identify user effort as an important variable that mediates access to diverse information in social media.Users who invest more effort into their activity on the site not only place themselves in more structurally diverse positions within the network than the less engaged users, but they also receive more diverse information when located in similar network positions.These findings indicate that the relationship between network structure and access to information in networks is more nuanced than previously thought.

Applying effective methods to identify important nodes in a large and complex network is highly invaluable. Recently, in a large and complex network, finding a powerful node in the community to spread information throughout the network is the concern of many researchers. In this paper, to identify powerful nodes in a large and complex network, community-based important node (CBIN) method, which reflects on the community structure of the network and degree of neighbor nodes is proposed as a metrics. CBIN considers a random walk from a node to each community and degree of neighbor nodes. Then, CBIN describes how many influential friends the node has and how the node is essential to connect two or more than two communities of the network. The performance of CBIN is evaluated by Susceptible-Infected (SI) model. Simulation results on a real network show that the proposed method performs better than the existing methods to spread information in the network.