a part of graph theory which actually deals with graphical drawing and presentation of graphs, briefly touched in Chapter 6, where also simple algorithms are. This book is intended as an introduction to graph theory. Our aim 'applications' that employ just the language of graphs and no theory. The. Basics of Graph Theory. 1 Basic notions. A simple graph G = (V,E) consists of V, a nonempty set of vertices, and E, a set of unordered pairs of distinct elements.
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In recent years, graph theory has established itself as an important suitable both for mathematicians taking courses in graph theory and also for non-. PDF | Introduction to Graph Theory | ResearchGate, the professional network for scientists. Graph Theory. Electronic Edition cс Springer-Verlag New York , This is an electronic version of the second () edition of the above. Springer.
A labeled tree is a tree in which each vertex is given a unique label. The vertices of a labeled tree on n vertices are typically given the labels 1, 2, …, n. A recursive tree is a labeled rooted tree where the vertex labels respect the tree order i.
In a rooted tree, the parent of a vertex v is the vertex connected to v on the path to the root; every vertex has a unique parent except the root which has no parent. A descendant of a vertex v is any vertex which is either the child of v or is recursively the descendant of any of the children of v. A sibling to a vertex v is any other vertex on the tree which has the same parent as v. The height of the tree is the height of the root.
The depth of a vertex is the length of the path to its root root path. This is commonly needed in the manipulation of the various self-balancing trees, AVL trees in particular. The root has depth zero, leaves have height zero, and a tree with only a single vertex hence both a root and leaf has depth and height zero. A k-ary tree is a rooted tree in which each vertex has at most k children. Ordered tree[ edit ] An ordered tree or plane tree is a rooted tree in which an ordering is specified for the children of each vertex.
Given an embedding of a rooted tree in the plane, if one fixes a direction of children, say left to right, then an embedding gives an ordering of the children. Conversely, given an ordered tree, and conventionally drawing the root at the top, then the child vertices in an ordered tree can be drawn left-to-right, yielding an essentially unique planar embedding.
Every tree is a bipartite graph and a median graph. Every tree with only countably many vertices is a planar graph. Every connected graph G admits a spanning tree , which is a tree that contains every vertex of G and whose edges are edges of G.
A graph is bipartite if and only if it contains no cycles of odd length. Since, a tree contains no cycles at all, it is bipartite.
Every connected graph with only countably many vertices admits a normal spanning tree Diestel , Prop. There exist connected graphs with uncountably many vertices which do not admit a normal spanning tree Diestel , Prop.
The number of leaves is at least the maximal vertex degree.
For any three vertices in a tree, the three paths between them have exactly one vertex in common. Every tree has a center consisting of one vertex or two adjacent vertices. The center is the middle vertex or middle two vertices in every longest path.
Similarly, every n-vertex tree has a centroid consisting of one vertex or two adjacent vertices. We demonstrate that compact graph representations require significantly fewer degrees of freedom dof to capture micro-fracture information and further accelerate these models with Machine Learning. Our method has been shown to improve accuracy of predictions with up to four orders of magnitude speedup. Introduction Fractures are a foundational structure in numerous natural and engineered applications that influence our daily lives.
Examples that motivated this study include 1 hydraulic fracturing, which has had a profound impact on US energy independence through the increased availability of unconventional fossil fuels 1 , 2 ; 2 chemical signature from clandestine nuclear weapon testing, where gas migration through fractured rock provides the definitive smoking gun when used in conjunction with conventional seismic methods 3 and remains critical to global security as countries like North Korea continue to conduct low-yield nuclear tests; and 3 predicting the brittle failure of materials such as ceramics and some metals, e.
For all of the examples of fractured systems mentioned here, individual fracture information geometry, orientation etc. In fractured systems, the connections between fractures often dominate system behavior, we refer to this connectivity as the topology of the fracture network. Because the fracture networks are statistically modeled, the topology is inherently uncertain and requires an ensemble of realizations of these fracture networks.
Since the topology of the graph matches the topology of the fracture network, we can interrogate the topological uncertainty with the graph. Moreover, the uncertainty surrounding topological properties network connectivity dominate system behavior. For example, in a system with two large fractures, system behavior is very much dependent on whether these fractures intersect or are connected by smaller fractures. However, such structural information cannot be fully characterized at the macroscale due to the high computational cost incurred in representing the discontinuities formed by the presence of cracks using highly resolved meshes.
While there are high fidelity mesh-based fracture models capable of representing millions of micro-fractures, such as dfnWorks 5 and HOSS 6 both developed by this team , the computational cost for s of model runs to bound the topological uncertainty quickly adds up to petabytes of information and is not feasible 7 , 8.
Therefore, many researchers have turned to reduced order models ROM to represent these systems, but a general framework linking attributes in the high-fidelity models to the induced ROM is still lacking. The novel research contribution highlighted here is the general framework we have developed, demonstrated on two separate applications in brittle geomaterials, where the common theme is the importance of the underlying fracture network structure in governing the dominant physics.
Although the applications mentioned here appear to be very different problems occurring at different scales, the technical challenge is similar-representing the relevant physics on the graph through ML algorithms and quantifying the dominant topological uncertainty. Graph theory is a powerful tool for interrogating structured systems.
Across many disciplines, ML has proven to simplify and expedite previously computationally intensive processes by learning from available data and knowledge. Combined with graph theory, ML approaches can effectively tackle a broad set of problems in fractured systems where quantifying uncertainties due to the topology is critical. In this study we bridge the knowledge gap between the discrete micron-mm and continuum cm-m scales efficiently by exploiting the underlying structure of fracture networks.
We formulate compact graph representations of fracture networks that avoid detailed meshing and require 2—3 orders of magnitude fewer degrees of freedom dof to capture micro-fracture information.
Recent work in network theory has shown its utility for problems such as diffusion and percolation 9 as well as failure problems 10 , which are similar to topics we explore. By combing ML with graph theory, we develop an approach that efficiently tackles a broad set of problems in fractured systems where structure and topology are critical.
Our method seamlessly lends itself to an uncertainty quantification UQ framework that requires a fraction of the computational resources. In order to demonstrate the robustness and utility of the method, we apply it to two important geophysical problems, flow through fractured media and fracture propagation.
Our critical advance is to integrate computational physics, machine learning and graph theory to make a paradigm shift from computationally intensive grid-based models to efficient graphs. Our graph-based algorithms have made it possible to directly extract geophysical and topological features used in the ML algorithms to predict key phenomena that drive the underlying physics.
We investigate several topological metrics using graph representations and identify those that are appropriate for the different applications we consider. The graph-based algorithms make it possible to extract geophysical and topological features for use in the ML algorithms to predict key phenomena that drive the underlying physics. Our key finding is that appropriately configured graph-based reduced order models can maintain the accuracy of the high-fidelity models with up to 4 orders of magnitude speedup in computational cost.
We also harness the power of ML algorithms to reveal previously neglected, but key microstructural effects and derive accurate upscaled parameters for use in continuum models. For example, continuum-scale material models often only consider one dominant crack orientation, or just one crack and no interactions. We demonstrate that combining ML and graph-based approaches makes such a framework possible. Results Our approach is based on verifying the following hypotheses: 1 Primary flow paths can be identified a priori with graph-based methods, confining computational power to critical regions of interest; 2 Predictive uncertainty is dominated by the topology as a result of structural effects; and 3 Dominant emergent phenomena related to fracture interaction and coalescence can be predicted using ML methods that use feature importance identification mechanisms since the geometry and topology of the fracture networks are directly represented in the graphs as features.
We demonstrate our advances in proving these hypotheses in the next three sub-sections.
Ascertaining the Topological Characteristics of a Fracture Network using graph-based physics solutions and ML-based pruning We first address the hypothesis regarding the pruning of a fracture network to only include the regions that participate significantly in the governing physics. We take on the challenging task of identifying primary flow paths through a fracture network a priori, without conducting computationally intensive mesh-based computations. High-fidelity simulations can then be used efficiently to focus on the primary flow path without including the extraneous parts of the domain where little or no flow occurs.
Here our Quantity of Interest QOI is the first passage time of a solute being transported along with the flow field.
For the exposition of our methods, we adopt a Lagrangian setting where the solute plume is represented by a cloud of tracer particles and the breakthrough curve BTC is the cumulative density function of the time it takes for a particle to travel from the inlet boundary to the outlet boundary. Field and laboratory experiments of flow through fracture networks indicate that flow channeling is a common feature through fractured subsurface systems 11 strongly suggesting the existence of primary flow pathways.
Casting the discrete fracture network DFN as a graph representation allows us to identify relevant sub-networks of the entire network based solely on topology, and here we present three ways to prune the domain — specifically 2-core, shortest paths, and an ML classification approach. Representing the fracture network as a graph allows us to use existing graph theoretic algorithms while introducing a rich feature set that can be leveraged by ML algorithms.
In this graph-representation, fractures in the DFN are represented as nodes in the graph and if two fractures intersect then there is an edge in the graph connecting the corresponding nodes 7. Fracture apertures vary between fractures and are positively correlated to the fracture radius a common assumption in DFN modeling supported by field observations 13 , 14 , The inset in Fig. The first pruning algorithm isolates the 2-core of the graph, which is the maximal subgraph such that every node has degree 2 or more 16 , as a relevant part of the domain that participates in the flow, shown in Fig.
Source and target nodes that represent the inflow and outflow boundaries are shown in red and blue respectively and connect to nodes that represent fractures which intersect those boundaries. The graph full network is shown semi-transparent for reference. An alternate way to prune the domain is by retaining only the shortest path in the network from the source to the target, which is shown along with the equivalent graph in Fig.
In this case, the resulting shortest path network is made up of only 7 fractures. Figure 1 A modest sized fracture network with fractures. Insets show the DFN models corresponding to the reduced graph representations. Full size image In order to test how well the sub-networks represent the original DFN, we perform a comparison of upscaled properties. The BTC computed under the same boundary conditions for flow through the full blue and the 2-core red , and shortest path black fracture networks are plotted together in Fig.
This similarity, which can be observed by plotting the complement of the BTC Fig. This is consistent with discarding trees in the graph that cause dispersion into and out of dead ends leading to late arrivals.
Full size image Transport through the shortest path black line Fig. These results show accurate graph-based models are capable of identifying primary flow paths and hence an appropriate reduced domain based on the application of interest. We explored how increasing the number of shortest paths retained influenced the accuracy of predicting the first breakthrough times 7. We demonstrated how to incorporate network properties into this selection for more robust predictions.
We also performed ML on our graph-based models to better identify the sub-network that corresponds to fractures along the primary flowing paths We used supervised classification methods, specifically support vector machine and random forest algorithms, to identify the flowing backbones from our DFN models.
In contrast to DFN models that can take 10 s of hours per realization, these ML methods require only minutes to train and, after training, require merely seconds to identify the flowing backbone. The ML approach provides more pruning than the 2-core method while retaining accuracy. These results make major strides towards proving our first hypothesis: primary flow paths can be identified a priori with graph-based methods, confining computational power to critical regions of interest.
Quantification of Topological Uncertainty In an effort to further reduce computational burden, we exploit graph-based reduced order models as an appealing mesh-free alternative, where flow and transport calculations are performed on the equivalent graph representation. Our recently developed graph Laplacian solver can simulate transport of conservative solutes through a fracture network 18 by mapping intersections to nodes and fracture segments to edges, and up to 4 orders of magnitude computational speedup is achieved with accuracy tradeoffs.
These graph representations include in-fracture attributes, e. Deviations in transport properties on the graph from the high-fidelity model are systematic. We take advantage of the systematic nature of the deviations by using a Bayesian UQ methodology 19 that quantifies system uncertainties represented by the deviations in the BTCs even when our computationally efficient graph-based reduced order models are not an exact representation of the high-fidelity model.
Furthermore, and nontrivially, our Bayesian calibration approach accurately quantifies the uncertainty in the predictions of calibrated QOIs.
We demonstrate our approach on an ensemble of high-fidelity DFN simulations, generated in the same manner as the network in Fig. We use a subset of the networks to learn the discrepancy and calibration terms and the rest for testing the quality of predictions. Our Bayesian methodology corrects the deviation using a single calibration parameter, learned with uncertainty, to shift the BTC in time, and adds a discrepancy function to minimize any deviations thereafter.
The resulting mean BTC is shown in Fig. Finally Fig. The corresponding ensemble uncertainty predicted using the corrected graph-based BTCs is shown in black. These results demonstrate our second hypothesis that predictive uncertainty is dominated by structural effects but spans topological uncertainty space. Full size image Dynamic Fracture Propagation Next, we exploit the nascent field of dynamic graphs combined with ML to develop reduced order models for the more complex case where fractures evolve with time 8.
Currently, reduced order formulations, which include semi-analytical models and continuum approximations, do not account for crack interactions leading to significant errors in failure predictions, particularly resulting in non-conservative predictions. Times to failure are typically over-predicted resulting in failure before it is expected.
Here, we define time to failure to be the amount of time that elapses between when the loading process begins and when a connected fracture spans the entire sample, e. The eventual goal of these simulations is to predict the evolution of the effective moduli of the material as cracks grow and coalesce leading to failure of the material.
Figure 4 Comparison of HOSS simulation to ML predictions and the graphical representation are shown here at top an early time and bottom failure. The solid line lines in the bottom right panel represents the path to failure which is accurately predicted by the Random Forest model. The dashed indicates crack growth and coalescence in the HOSS simulations which are not captured in the Random Forest model. Full size image The first step in formulating a more accurate material model is learning how crack interactions influence the time to failure, and determining characteristics of preferential paths to failure.
We generate crack growth and interaction data from running several simulations of HOSS, a computationally expensive, high-fidelity crack evolution model that can resolve individual micro-cracks unlike the macro-scale continuum models.
HOSS accounts for interactions between micro-cracks in addition to coalescence and growth damage evolution mechanisms. We identify key features in the crack growth data orientation, geometry, etc. The data is partitioned for training and validation purposes and tested on simple fracture systems.