To find interesting structure in networks, community detection algorithms have to take into account not only the network topology, but also dynamics of interactions between nodes. We investigate this claim using the paradigm of synchronization in a network of coupled oscillators. As the network evolves to a global steady state, nodes belonging to the same community synchronize faster than nodes belonging to different communities. Traditionally, nodes in network synchronization models are coupled via one-to-one, or conservative interactions. However, social interactions are often one-to-many, as for example, in social media, where users broadcast messages to all their followers. We formulate a novel model of synchronization in a network of coupled oscillators in which the oscillators are coupled via one-to-many, or non-conservative interactions. We study the dynamics of different interaction models and contrast their spectral properties. To find multi-scale community structure in a network of interacting nodes, we define a similarity function that measures the degree to which nodes are synchronized and use it to hierarchically cluster nodes. We study real-world social networks, including networks of two social media providers. To evaluate the quality of the discovered communities in a social media network we propose a community quality metric based on user activity. We find that conservative and non-conservative interaction models lead to dramatically different views of community structure even within the same network. Our work offers a novel mathematical framework for exploring the relationship between network structure, topology and dynamics.
It can be shown that some of the interaction models described above not only solve certain regularized Semi-Definite Programs but also give fast solutions to these problems @cite_12 .
@cite_12
2672830324 2615873886 1967311985 1971332773
The notions of serializability, linearizability, and sequential consistency are used in the specification of concurrent systems. We show that the model checking problem for each of these properties can be cast in terms of the containment of one regular language in another regular language shuffled using a semicommutative alphabet. The three model checking problems are shown to be, respectively, in Pspace, in Expspace, and undecidable. Recent breakthroughs have been made in the use of semi-definite programming and its application to real polynomial solving. For example, the real radical of a zero dimensional ideal, can be determined by such approaches. Some progress has been made on the determination of the real radical in positive dimension by Ma, Wang and Zhi[5, 4]. Such work involves the determination of maximal rank semidefinite matrices. Existing methods are computationally expensive and have poorer accuracy on larger examples. In previous work we showed that regularity in the form of the Slater constraint qualification (strict feasibility) fails for the moment matrix in the SDP feasibility problem[6]. We used facial reduction to obtain a smaller regularized problem for which strict feasibility holds. However we did not have a theoretical guarantee that our methods, based on facial reduction and Douglas-Rachford iteration ensured the satisfaction of the maximum rank condition. Our work is motivated by the problems above. We discuss how to compute the moment matrix and its kernel using facial reduction techniques where the maximum rank property can be guaranteed by solving the dual problem. The facial reduction algorithms on the primal form is presented. We give examples that exhibit for the first time additional facial reductions beyond the first which are effective in practice with much better accuracy than SeDuMi(CVX). In this paper we study linear-programming based approaches to the maximum matching problem in the semi-streaming model. In this model edges are presented sequentially, possibly in an adversarial order, and we are only allowed to use a small space. The allowed space is near linear in the number of vertices (and sublinear in the number of edges) of the input graph. The semi-streaming model is relevant in the context of processing of very large graphs. In recent years, there have been several new and exciting results in the semi-streaming model. However broad techniques such as linear programming have not been adapted to this model. In this paper we present several techniques to adapt and optimize linear-programming based approaches in the semi-streaming model. We use the maximum matching problem as a foil to demonstrate the effectiveness of adapting such tools in this model. As a consequence we improve almost all previous results on the semi-streaming maximum matching problem. We also prove new results on interesting variants. We address the verification problem of finite-state concurrent programs running under weak memory models. These models capture the reordering of program (read and write) operations done by modern multi-processor architectures for performance. The verification problem we study is crucial for the correctness of concurrency libraries and other performance-critical system services employing lock-free synchronization, as well as for the correctness of compiler backends that generate code targeted to run on such architectures. We consider in this paper combinations of three well-known program order relaxations. We consider first the "write to read" relaxation, which corresponds to the TSO (Total Store Ordering) model. This relaxation is used in most hardware architectures available today. Then, we consider models obtained by adding either (1) the "write to write" relaxation, leading to a model which is essentially PSO (Partial Store Ordering), or (2) the "read to read write" relaxation, or (3) both of them, as it is done in the RMO (Relaxed Memory Ordering) model for instance. We define abstract operational models for these weak memory models based on state machines with (potentially unbounded) FIFO buffers, and we investigate the decidability of their reachability and their repeated reachability problems. We prove that the reachability problem is decidable for the TSO model, as well as for its extension with "write to write" relaxation (PSO). Furthermore, we prove that the reachability problem becomes undecidable when the "read to read write" relaxation is added to either of these two memory models, and we give a condition under which this addition preserves the decidability of the reachability problem. We show also that the repeated reachability problem is undecidable for all the considered memory models.