Googling the brain: discovering hierarchical and asymmetric network structures, with applications in neuroscience

Hierarchical organisation is a common feature of many directed networks arising in nature and technology. For example, a well-defined message-passing framework based on managerial status typically exists in a business organisation. However, in many real-world networks such patterns of hierarchy are unlikely to be quite so transparent. Due to the nature in which empirical data is collated the nodes will often be ordered so as to obscure any underlying structure. In addition, the possibility of even a small number of links violating any overall “chain of command” makes the determination of such structures extremely challenging. Here we address the issue of how to reorder a directed network in order to reveal this type of hierarchy. In doing so we also look at the task of quantifying the level of hierarchy, given a particular node ordering. We look at a variety of approaches. Using ideas from the graph Laplacian literature, we show that a relevant discrete optimization problem leads to a natural hierarchical node ranking. We also show that this ranking arises via a maximum likelihood problem associated with a new range-dependent hierarchical random graph model. This random graph insight allows us to compute a likelihood ratio that quantifies the overall tendency for a given network to be hierarchical. We also develop a generalization of this node ordering algorithm based on the combinatorics of directed walks. In passing, we note that Google’s PageRank algorithm tackles a closely related problem, and may also be motivated from a combinatoric, walk-counting viewpoint. We illustrate the performance of the resulting algorithms on synthetic network data, and on a real-world network from neuroscience where results may be validated biologically

A distributed Monte Carlo based linear algebra solver applied to the analysis of large complex networks

Methods based on Monte Carlo for solving linear systems have some interesting properties which make them, in many instances, preferable to classic methods. Namely, these statistical methods allow the computation of individual entries of the output, hence being able to handle problems where the size of the resulting matrix would be too large. In this paper, we propose a distributed linear algebra solver based on Monte Carlo. The proposed method is based on an algorithm that uses random walks over the system’s matrix to calculate powers of this matrix, which can then be used to compute a given matrix function. Distributing the matrix over several nodes enables the handling of even larger problem instances, however it entails a communication penalty as walks may need to jump between computational nodes. We have studied different buffering strategies and provide a solution that minimizes this overhead and maximizes performance. We used our method to compute metrics of complex networks, such as node centrality and resolvent Estrada index. We present results that demonstrate the excellent scalability of our distributed implementation on very large networks, effectively providing a solution to previously unreachable problem instances.info:eu-repo/semantics/acceptedVersio

SoftNet: A Package for the Analysis of Complex Networks

Identifying the most important nodes according to specific centrality indices is an important issue in network analysis. Node metrics based on the computation of functions of the adjacency matrix of a network were defined by Estrada and his collaborators in various papers. This paper describes a MATLAB toolbox for computing such centrality indices using efficient numerical algorithms based on the connection between the Lanczos method and Gauss-type quadrature rules

Advanced Applications Of Big Data Analytics

Human life is progressing with advancements in technology such as laptops, smart phones, high speed communication networks etc., which helps us by reducing load in doing our daily activities. For instance, one can chat, talk, make video calls with his/her friends instantly using social networking platforms such as Facebook, Twitter, Google+, WhatsApp etc. LinkedIn, Indeed, etc., connects employees with potential employers. The number of people using these applications are increasing day-by-day, and so is the amount of data generated from these applications. Processing such vast amounts of data, may require new techniques for gaining valuable insights. Network theory concepts form the core of such techniques that are designed to uncover valuable insights from large social network datasets. Many interesting problems such as ranking top-K nodes and top-K communities that can effectively diffuse any given message into the network, restaurant recommendations, friendship recommendations on social networking websites, etc., can be addressed by using the concepts of network centrality. Network centrality measures such as In-degree centrality, Out-degree centrality, Eigen-vector centrality, Katz Broadcast centrality, Katz Receive centrality, and PageRank centrality etc., comes handy in solving these problems. In this thesis, we propose different formulae for computing the strength for identifying top-K nodes and communities that can spread viral marketing messages into the network. The strength formulae are based on Katz Broadcast centrality, Resolvent matrix measure and Personalized PageRank measure. Moreover, the effects of intercommunity and intracommunity connectivity in ranking top-K communities are studied. Top-K nodes for spreading any message effectively into the network are determined by using Katz Broadcast centrality measure. Results obtained through this technique are compared with the top-K nodes obtained by using Degree centrality measure. We also studied the effects of varying α on the number of nodes in search space. In Algorithms 2 and 3, top-K communities are obtained by using Resolvent matrix and Personalized PageRank measure. Algorithm 2 results were studied by varying the parameter α

Mittag-Leffler functions and their applications in network science

We describe a complete theory for walk-based centrality indices in complex networks defined in terms of Mittag–Leffler functions. This overarching theory includes as special cases well known centrality measures like subgraph centrality and Katz centrality. The indices we introduce are parametrized by two numbers; by letting these vary, we show that Mittag–Leffler centralities interpolate between degree and eigenvector centrality, as well as between resolvent-based and exponential-based indices. We further discuss modelling and computational issues, and provide guidelines on parameter 10 selection. The theory is then extended to the case of networks that evolve over time. Numerical experiments on synthetic and real-world networks are provided

Understanding the spreading power of all nodes in a network: a continuous-time perspective

Centrality measures such as the degree, k-shell, or eigenvalue centrality can identify a network's most influential nodes, but are rarely usefully accurate in quantifying the spreading power of the vast majority of nodes which are not highly influential. The spreading power of all network nodes is better explained by considering, from a continuous-time epidemiological perspective, the distribution of the force of infection each node generates. The resulting metric, the \textit{expected force}, accurately quantifies node spreading power under all primary epidemiological models across a wide range of archetypical human contact networks. When node power is low, influence is a function of neighbor degree. As power increases, a node's own degree becomes more important. The strength of this relationship is modulated by network structure, being more pronounced in narrow, dense networks typical of social networking and weakening in broader, looser association networks such as the Internet. The expected force can be computed independently for individual nodes, making it applicable for networks whose adjacency matrix is dynamic, not well specified, or overwhelmingly large

Dynamic Katz and related network measures

We study walk-based centrality measures for time-ordered network sequences. For the case of standard dynamic walk-counting, we show how to derive and compute centrality measures induced by analytic functions. We also prove that dynamic Katz centrality, based on the resolvent function, has the unique advantage of allowing computations to be performed entirely at the node level. We then consider two distinct types of backtracking and develop a framework for capturing dynamic walk combinatorics when either or both is disallowed