High-Performance Graph Algorithms

Ever since Euler took a stroll across the bridges of a city then called Königsberg, relationships of entities have been modeled as graphs. Being a useful abstraction when the structure of relationships is the significant aspect of a problem, popular uses of graphs include the modeling of social networks, supply chain dependencies, the internet, customer preferences, street networks or the who-eats-whom (aka food network) in an ecosystem. In parallel computing, the assignment of sub-tasks to processes can massively influence the performance, since data dependencies between processes are significantly more expensive than within them. This scenario has been profitably modeled as a graph problem, with sub-tasks as vertices and their communication dependencies as edges. Many graphs are governed by an underlying geometry. Some are derived directly from a geometric object, such as street networks or meshes from spatial simulations. Others have a hidden geometry, in which no explicit geometry information is known, but the graph structure follows an underlying geometry. A suitable embedding into this geometry would then show a close relationship between graph distances and geometric distances. A subclass of graphs enjoying significant attention are complex networks. Though hard to define exactly, they are commonly characterized by a low diameter, heterogeneous structure, and a skewed degree distribution, often following a power law. The most famous examples include social networks, the hyperlink network and communication networks. Development of new analysis algorithms for complex networks is ongoing. Especially since the instances of interest are often in the size of billions of vertices, fast analysis algorithms and approximations are a natural focus of development. To accurately test and benchmark new developments, as well as to gain theoretical insight about network formation, generative graph models are required: A mathematical model describing a family of random graphs, from which instances can be sampled efficiently. Even if real test data is available, interesting instances are often in the size of terabytes, making storage and transmission inconvenient. While the underlying geometry of street networks is the spherical geometry they were built in, there is some evidence that the geometry best fitting to complex networks is not Euclidean or spherical, but hyperbolic. Based on this notion, Krioukov et al. proposed a generative graph model for complex networks, called Random Hyperbolic Graphs. They are created by setting vertices randomly within a disk in the hyperbolic plane and connecting pairs of vertices with a probability depending on their distance. An important subclass of this model, called Threshold Random Hyperbolic Graphs connects vertices exactly if the distances between vertices is below a threshold. This model has pleasant properties and has received considerable attention from theoreticians. Unfortunately, a straightforward generation algorithm has a complexity quadratic in the number of nodes, which renders it infeasible for instances of more than a few million vertices. We developed four faster generation algorithms for random hyperbolic graphs: By projecting hyperbolic geometry in the Euclidean geometry of the Poincaré disk model, we are able to use adapted versions of existing geometric data structures. Our first algorithm uses a polar quadtree to avoid distance calculations and achieves a time complexity of $\mathcal{O}((n^{3/2} + m)\log n)$ whp. -- the first subquadratic generation algorithm for threshold random hyperbolic graphs. Empirically, our implementation achieves an improvement of three orders of magnitude over a reference implementation of the straightforward algorithm. We extend this quadtree data structure further for the generation of general random hyperbolic graphs, in which all edges are probabilistic. Since each edge has a non-zero probability of existing, sampling them by throwing a biased coin for each would again cost quadratic time complexity. We address this issue by sampling jumping widths within leaf cells and aggregating subtrees to virtual leaf cells when the expected number of neighbors in them is less than a threshold. With this tradeoff, we bound both the number of distance calculations and the number of examined quadtree cells per edge, resulting in the same time complexity of $\mathcal{O}((n^{3/2} + m)\log n)$ also for general random hyperbolic graphs. We generalize this sampling scenario and define Probabilistic Neighborhood Queries, in which a random sample of a geometric point set is desired, with the probability of inclusion depending on the distance to a query point. Usable to simulate probabilistic spatial spreading, we show a significant speedup on a proof of concept disease simulation. Our second algorithm for threshold random hyperbolic graphs uses a data structure of concentric annuli in the hyperbolic plane. For each given vertex, the positions of possible neighbors in each band can be restricted with the hyperbolic law of cosines, leading to a much reduced number of candidates that need to be checked. This yields a reduced time complexity of $\mathcal{O}(n \log^2 n + m)$ and a further order of magnitude in practice for graphs of a few million vertices. Finally, we extend also this data structure to general random hyperbolic graphs, with the same time complexity for constant parameters. The second part of my thesis is in many aspects the opposite of the first. Instead of a hidden geometry, I consider graphs whose geometric information is explicit. Instead of using it to generate graphs, I use their geometric information to decide how to cut them into pieces. Given a graph, the Graph Partitioning Problem asks for a disjoint partition of the vertex set so that each subset has a similar number of vertices and some objective function is optimized. Its many applications include parallelizing a computing task while balancing load and minimizing communication, dividing a graph into blocks as preparation for graph analysis tasks or finding natural cuts in street networks for efficient route planning. Since the graph partitioning problem is NP-complete and hard to approximate, heuristics are used in practice. Our first graph partitioner is designed for an application in quantum chemistry. Computing electron density fields is necessary to accurately predict protein interactions, but the required time scales quadratically with the protein\u27s size, with punitive constant factors. This effectively restricts these density-based methods to proteins of at most a few hundred amino acids. It is possible to circumvent this limitation by computing only parts of the target protein at a time, an approach known as subsystem quantum chemistry. However, the interactions between amino acids in different parts are then neglected; this neglect causes errors in the solution. We model this problem as partitioning a protein graph: Each vertex represents one amino acid, each edge an interaction between them and each edge weight the expected error caused by neglecting this interaction. Finding a set of subsets with minimum error is then equivalent to finding a partition of the protein graph with minimum edge cut. The requirements of the chemical simulations cause additional constraints on this partition, as well as an implied geometry by the protein structure. We provide an implementation of the well-known multilevel heuristic together with the local search algorithm by Fiduccia and Mattheyses, both adapted to respect these new constraints. We also provide an optimal dynamic programming algorithm for a restricted scenario with a smaller solution space. In terms of edge cut, we achieve an average improvement of 13.5% against the naive solution, which was previously used by domain scientists. Our second graph partitioner targets geometric meshes from numerical simulations in the scale of billions of grid points, parallelized to tens of thousands of processes. In general purpose graph partitioning, the multilevel heuristic computes high-quality partitions, but its scalability is limited. Due to the large halos in our targeted application, the shape of the partitioned blocks is also important, with convex shapes leading to less communication. We adapt the well-known $k$-means algorithm, which yields convex shapes, to the partitioning of geometric meshes by including a balancing scheme. We further extend several existing geometric optimizations to the balanced version to achieve fast running times and parallel scalability. The classic $k$-means algorithm is highly dependent on the choice of initial centers. We select initial centers among the input points along a space-filling curve, thus guaranteeing a similar distribution as the input points. The resulting implementation scales to tens of thousands of processes and billions of vertices, partitioning them in seconds. Compared to previous fast geometric partitioners, our method provides partitions with a 10-15% lower communication volume and also a corresponding smaller communication time in a distributed SpMV benchmark

Reduction of dynamical biochemical reaction networks in computational biology

Biochemical networks are used in computational biology, to model the static and dynamical details of systems involved in cell signaling, metabolism, and regulation of gene expression. Parametric and structural uncertainty, as well as combinatorial explosion are strong obstacles against analyzing the dynamics of large models of this type. Multi-scaleness is another property of these networks, that can be used to get past some of these obstacles. Networks with many well separated time scales, can be reduced to simpler networks, in a way that depends only on the orders of magnitude and not on the exact values of the kinetic parameters. The main idea used for such robust simplifications of networks is the concept of dominance among model elements, allowing hierarchical organization of these elements according to their effects on the network dynamics. This concept finds a natural formulation in tropical geometry. We revisit, in the light of these new ideas, the main approaches to model reduction of reaction networks, such as quasi-steady state and quasi-equilibrium approximations, and provide practical recipes for model reduction of linear and nonlinear networks. We also discuss the application of model reduction to backward pruning machine learning techniques

COMPUTER SIMULATION AND COMPUTABILITY OF BIOLOGICAL SYSTEMS

The ability to simulate a biological organism by employing a computer is related to the ability of the computer to calculate the behavior of such a dynamical system, or the "computability" of the system.* However, the two questions of computability and simulation are not equivalent. Since the question of computability can be given a precise answer in terms of recursive functions, automata theory and dynamical systems, it will be appropriate to consider it first. The more elusive question of adequate simulation of biological systems by a computer will be then addressed and a possible connection between the two answers given will be considered. A conjecture is formulated that suggests the possibility of employing an algebraic-topological, "quantum" computer (Baianu, 1971b) for analogous and symbolic simulations of biological systems that may include chaotic processes that are not, in genral, either recursively or digitally computable. Depending on the biological network being modelled, such as the Human Genome/Cell Interactome or a trillion-cell Cognitive Neural Network system, the appropriate logical structure for such simulations might be either the Quantum MV-Logic (QMV) discussed in recent publications (Chiara, 2004, and references cited therein)or Lukasiewicz Logic Algebras that were shown to be isomorphic to MV-logic algebras (Georgescu et al, 2001)

Graph Pattern Matching on Symmetric Multiprocessor Systems

Graph-structured data can be found in nearly every aspect of today's world, be it road networks, social networks or the internet itself. From a processing perspective, finding comprehensive patterns in graph-structured data is a core processing primitive in a variety of applications, such as fraud detection, biological engineering or social graph analytics. On the hardware side, multiprocessor systems, that consist of multiple processors in a single scale-up server, are the next important wave on top of multi-core systems. In particular, symmetric multiprocessor systems (SMP) are characterized by the fact, that each processor has the same architecture, e.g. every processor is a multi-core and all multiprocessors share a common and huge main memory space. Moreover, large SMPs will feature a non-uniform memory access (NUMA), whose impact on the design of efficient data processing concepts should not be neglected. The efficient usage of SMP systems, that still increase in size, is an interesting and ongoing research topic. Current state-of-the-art architectural design principles provide different and in parts disjunct suggestions on which data should be partitioned and or how intra-process communication should be realized. In this thesis, we propose a new synthesis of four of the most well-known principles Shared Everything, Partition Serial Execution, Data Oriented Architecture and Delegation, to create the NORAD architecture, which stands for NUMA-aware DORA with Delegation. We built our research prototype called NeMeSys on top of the NORAD architecture to fully exploit the provided hardware capacities of SMPs for graph pattern matching. Being an in-memory engine, NeMeSys allows for online data ingestion as well as online query generation and processing through a terminal based user interface. Storing a graph on a NUMA system inherently requires data partitioning to cope with the mentioned NUMA effect. Hence, we need to dissect the graph into a disjunct set of partitions, which can then be stored on the individual memory domains. This thesis analyzes the capabilites of the NORAD architecture, to perform scalable graph pattern matching on SMP systems. To increase the systems performance, we further develop, integrate and evaluate suitable optimization techniques. That is, we investigate the influence of the inherent data partitioning, the interplay of messaging with and without sufficient locality information and the actual partition placement on any NUMA socket in the system. To underline the applicability of our approach, we evaluate NeMeSys against synthetic datasets and perform an end-to-end evaluation of the whole system stack on the real world knowledge graph of Wikidata

Monotone and near-monotone biochemical networks

Monotone subsystems have appealing properties as components of larger networks, since they exhibit robust dynamical stability and predictability of responses to perturbations. This suggests that natural biological systems may have evolved to be, if not monotone, at least close to monotone in the sense of being decomposable into a “small” number of monotone components, In addition, recent research has shown that much insight can be attained from decomposing networks into monotone subsystems and the analysis of the resulting interconnections using tools from control theory. This paper provides an expository introduction to monotone systems and their interconnections, describing the basic concepts and some of the main mathematical results in a largely informal fashion

Developments in multiscale ONIOM and fragment methods for complex chemical systems

Multiskalenprobleme werden in der Computerchemie immer allgegenwärtiger und bestimmte Klassen solcher Probleme entziehen sich einer effizienten Beschreibung mit den verfügbaren Berechnungsansätzen. In dieser Arbeit wurden effiziente Erweiterungen der Multilayer-Methode ONIOM und von Fragmentmethoden als Lösungsansätze für derartige Probleme entwickelt. Dabei wurde die Kombination von ONIOM und Fragmentmethoden im Rahmen der Multi-Centre Generalised ONIOM entwickelt sowie die eine Multilayer-Variante der Fragment Combinatio Ranges. Außerdem wurden Schemata für elektronische Einbettung derartiger Multilayer-Systeme entwickelt. Der zweite Teil der Arbeit beschreibt die Implementierung im Haskell-Programm "Spicy" und demonstriert Anwendungen derartiger Multiskalen-Methoden

Biological relevance of charge transfer branching pathways in photolyases

In PhrA, a class III CPD photolyase, two branching tryptophan charge transfer pathways have been characterized in the mechanism of FAD photoreduction. To provide a molecular explanation of the charge transfer abilities of both pathways, we performed simulations where the protein motion and the positive charge are simultaneously propagated. Our computational approach reveals that one pathway drives a very fast charge transfer whereas the other pathway provides a very good thermodynamic stabilization of the positive charge. During the simulations, the positive charge firstly moves on the fast triad, allowing the stabilization of reduced FAD. After one nanosecond, we observe a nearly equal probability to find the charge at ending tryptophan of either pathway. Our results highlight the role of the protein environment, which manages the association of a kinetic and a thermodynamic pathways to trigger a fast and efficient FAD photoreduction