Statistical Physics of Hard Optimization Problems

Optimization is fundamental in many areas of science, from computer science and information theory to engineering and statistical physics, as well as to biology or social sciences. It typically involves a large number of variables and a cost function depending on these variables. Optimization problems in the NP-complete class are particularly difficult, it is believed that the number of operations required to minimize the cost function is in the most difficult cases exponential in the system size. However, even in an NP-complete problem the practically arising instances might, in fact, be easy to solve. The principal question we address in this thesis is: How to recognize if an NP-complete constraint satisfaction problem is typically hard and what are the main reasons for this? We adopt approaches from the statistical physics of disordered systems, in particular the cavity method developed originally to describe glassy systems. We describe new properties of the space of solutions in two of the most studied constraint satisfaction problems - random satisfiability and random graph coloring. We suggest a relation between the existence of the so-called frozen variables and the algorithmic hardness of a problem. Based on these insights, we introduce a new class of problems which we named "locked" constraint satisfaction, where the statistical description is easily solvable, but from the algorithmic point of view they are even more challenging than the canonical satisfiability.Comment: PhD thesi

Spectral Theory of Sparse Non-Hermitian Random Matrices

Sparse non-Hermitian random matrices arise in the study of disordered physical systems with asymmetric local interactions, and have applications ranging from neural networks to ecosystem dynamics. The spectral characteristics of these matrices provide crucial information on system stability and susceptibility, however, their study is greatly complicated by the twin challenges of a lack of symmetry and a sparse interaction structure. In this review we provide a concise and systematic introduction to the main tools and results in this field. We show how the spectra of sparse non-Hermitian matrices can be computed via an analogy with infinite dimensional operators obeying certain recursion relations. With reference to three illustrative examples --- adjacency matrices of regular oriented graphs, adjacency matrices of oriented Erd\H{o}s-R\'{e}nyi graphs, and adjacency matrices of weighted oriented Erd\H{o}s-R\'{e}nyi graphs --- we demonstrate the use of these methods to obtain both analytic and numerical results for the spectrum, the spectral distribution, the location of outlier eigenvalues, and the statistical properties of eigenvectors.Comment: 60 pages, 10 figure

An Emergent Architecture for Scaling Decentralized Communication Systems (DCS)

With recent technological advancements now accelerating the mobile and wireless Internet solution space, a ubiquitous computing Internet is well within the research and industrial community's design reach - a decentralized system design, which is not solely driven by static physical models and sound engineering principals, but more dynamically, perhaps sub-optimally at initial deployment and socially-influenced in its evolution. To complement today's Internet system, this thesis proposes a Decentralized Communication System (DCS) architecture with the following characteristics: flat physical topologies with numerous compute oriented and communication intensive nodes in the network with many of these nodes operating in multiple functional roles; self-organizing virtual structures formed through alternative mobility scenarios and capable of serving ad hoc networking formations; emergent operations and control with limited dependency on centralized control and management administration. Today, decentralized systems are not commercially scalable or viable for broad adoption in the same way we have to come to rely on the Internet or telephony systems. The premise in this thesis is that DCS can reach high levels of resilience, usefulness, scale that the industry has come to experience with traditional centralized systems by exploiting the following properties: (i.) network density and topological diversity; (ii.) self-organization and emergent attributes; (iii.) cooperative and dynamic infrastructure; and (iv.) node role diversity. This thesis delivers key contributions towards advancing the current state of the art in decentralized systems. First, we present the vision and a conceptual framework for DCS. Second, the thesis demonstrates that such a framework and concept architecture is feasible by prototyping a DCS platform that exhibits the above properties or minimally, demonstrates that these properties are feasible through prototyped network services. Third, this work expands on an alternative approach to network clustering using hierarchical virtual clusters (HVC) to facilitate self-organizing network structures. With increasing network complexity, decentralized systems can generally lead to unreliable and irregular service quality, especially given unpredictable node mobility and traffic dynamics. The HVC framework is an architectural strategy to address organizational disorder associated with traditional decentralized systems. The proposed HVC architecture along with the associated promotional methodology organizes distributed control and management services by leveraging alternative organizational models (e.g., peer-to-peer (P2P), centralized or tiered) in hierarchical and virtual fashion. Through simulation and analytical modeling, we demonstrate HVC efficiencies in DCS structural scalability and resilience by comparing static and dynamic HVC node configurations against traditional physical configurations based on P2P, centralized or tiered structures. Next, an emergent management architecture for DCS exploiting HVC for self-organization, introduces emergence as an operational approach to scaling DCS services for state management and policy control. In this thesis, emergence scales in hierarchical fashion using virtual clustering to create multiple tiers of local and global separation for aggregation, distribution and network control. Emergence is an architectural objective, which HVC introduces into the proposed self-management design for scaling and stability purposes. Since HVC expands the clustering model hierarchically and virtually, a clusterhead (CH) node, positioned as a proxy for a specific cluster or grouped DCS nodes, can also operate in a micro-capacity as a peer member of an organized cluster in a higher tier. As the HVC promotional process continues through the hierarchy, each tier of the hierarchy exhibits emergent behavior. With HVC as the self-organizing structural framework, a multi-tiered, emergent architecture enables the decentralized management strategy to improve scaling objectives that traditionally challenge decentralized systems. The HVC organizational concept and the emergence properties align with and the view of the human brain's neocortex layering structure of sensory storage, prediction and intelligence. It is the position in this thesis, that for DCS to scale and maintain broad stability, network control and management must strive towards an emergent or natural approach. While today's models for network control and management have proven to lack scalability and responsiveness based on pure centralized models, it is unlikely that singular organizational models can withstand the operational complexities associated with DCS. In this work, we integrate emergence and learning-based methods in a cooperative computing manner towards realizing DCS self-management. However, unlike many existing work in these areas which break down with increased network complexity and dynamics, the proposed HVC framework is utilized to offset these issues through effective separation, aggregation and asynchronous processing of both distributed state and policy. Using modeling techniques, we demonstrate that such architecture is feasible and can improve the operational robustness of DCS. The modeling emphasis focuses on demonstrating the operational advantages of an HVC-based organizational strategy for emergent management services (i.e., reachability, availability or performance). By integrating the two approaches, the DCS architecture forms a scalable system to address the challenges associated with traditional decentralized systems. The hypothesis is that the emergent management system architecture will improve the operational scaling properties of DCS-based applications and services. Additionally, we demonstrate structural flexibility of HVC as an underlying service infrastructure to build and deploy DCS applications and layered services. The modeling results demonstrate that an HVC-based emergent management and control system operationally outperforms traditional structural organizational models. In summary, this thesis brings together the above contributions towards delivering a scalable, decentralized system for Internet mobile computing and communications

Network Partitioning in Distributed Agent-Based Models

Agent-Based Models (ABMs) are an emerging simulation paradigm for modeling complex systems, comprised of autonomous, possibly heterogeneous, interacting agents. The utility of ABMs lies in their ability to represent such complex systems as self-organizing networks of agents. Modeling and understanding the behavior of complex systems usually occurs at large and representative scales, and often obtaining and visualizing of simulation results in real-time is critical. The real-time requirement necessitates the use of in-memory computing, as it is difficult and challenging to handle the latency and unpredictability of disk accesses. Combining this observation with the scale requirement emphasizes the need to use parallel and distributed computing platforms, such as MPI-enabled CPU clusters. Consequently, the agent population must be partitioned across different CPUs in a cluster. Further, the typically high volume of interactions among agents can quickly become a significant bottleneck for real-time or large-scale simulations. The problem is exacerbated if the underlying ABM network is dynamic and the inter-process communication evolves over the course of the simulation. Therefore, it is critical to develop topology-aware partitioning mechanisms to support such large simulations. In this dissertation, we demonstrate that distributed agent-based model simulations benefit from the use of graph partitioning algorithms that involve a local, neighborhood-based perspective. Such methods do not rely on global accesses to the network and thus are more scalable. In addition, we propose two partitioning schemes that consider the bottom-up individual-centric nature of agent-based modeling. The First technique utilizes label-propagation community detection to partition the dynamic agent network of an ABM. We propose a latency-hiding, seamless integration of community detection in the dynamics of a distributed ABM. To achieve this integration, we exploit the similarity in the process flow patterns of a label-propagation community-detection algorithm and self-organizing ABMs. In the second partitioning scheme, we apply a combination of the Guided Local Search (GLS) and Fast Local Search (FLS) metaheuristics in the context of graph partitioning. The main driving principle of GLS is the dynamic modi?cation of the objective function to escape local optima. The algorithm augments the objective of a local search, thereby transforming the landscape structure and escaping a local optimum. FLS is a local search heuristic algorithm that is aimed at reducing the search space of the main search algorithm. It breaks down the space into sub-neighborhoods such that inactive sub-neighborhoods are removed from the search process. The combination of GLS and FLS allowed us to design a graph partitioning algorithm that is both scalable and sensitive to the inherent modularity of real-world networks

Belief Propagation and Bethe approximation for Traffic Prediction

We define and study an inference algorithm based on "belief propagation" (BP) and the Bethe approximation. The idea is to encode into a graph an a priori information composed of correlations or marginal probabilities of variables, and to use a message passing procedure to estimate the actual state from some extra real-time information. This method is originally designed for traffic prediction and is particularly suitable in settings where the only information available is floating car data. We propose a discretized traffic description, based on the Ising model of statistical physics, in order to both reconstruct and predict the traffic in real time. General properties of BP are addressed in this context. In particular, a detailed study of stability is proposed with respect to the a priori data and the graph topology. The behavior of the algorithm is illustrated by numerical studies on a simple traffic toy model. How this approach can be generalized to encode superposition of many traffic patterns is discussed.Comment: Inria Report, 29 pages, 7 figure

Robust Temporally Coherent Laplacian Protrusion Segmentation of 3D Articulated Bodies

In motion analysis and understanding it is important to be able to fit a suitable model or structure to the temporal series of observed data, in order to describe motion patterns in a compact way, and to discriminate between them. In an unsupervised context, i.e., no prior model of the moving object(s) is available, such a structure has to be learned from the data in a bottom-up fashion. In recent times, volumetric approaches in which the motion is captured from a number of cameras and a voxel-set representation of the body is built from the camera views, have gained ground due to attractive features such as inherent view-invariance and robustness to occlusions. Automatic, unsupervised segmentation of moving bodies along entire sequences, in a temporally-coherent and robust way, has the potential to provide a means of constructing a bottom-up model of the moving body, and track motion cues that may be later exploited for motion classification. Spectral methods such as locally linear embedding (LLE) can be useful in this context, as they preserve "protrusions", i.e., high-curvature regions of the 3D volume, of articulated shapes, while improving their separation in a lower dimensional space, making them in this way easier to cluster. In this paper we therefore propose a spectral approach to unsupervised and temporally-coherent body-protrusion segmentation along time sequences. Volumetric shapes are clustered in an embedding space, clusters are propagated in time to ensure coherence, and merged or split to accommodate changes in the body's topology. Experiments on both synthetic and real sequences of dense voxel-set data are shown. This supports the ability of the proposed method to cluster body-parts consistently over time in a totally unsupervised fashion, its robustness to sampling density and shape quality, and its potential for bottom-up model constructionComment: 31 pages, 26 figure