We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

This manuscript resulted from discussions at a workshop sponsored by the U.S. Department of Energy (DOE), Office of Science (SC), Office of Basic Energy Sciences (BES), Chemical Sciences, Geosciences, and Biosciences (CSGB) Division that was held in Leesburg, Virginia, in May 2016. We are grateful to James Rustad for his leadership, contributions to discussions at the workshop, and encouragement and support during the preparation of this review. S. E. L. appreciates support in organizing and conducting the workshop and preparing the paper from Grant DE‐FG02‐03ER15430 from the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division. Sandia National Laboratories (SNL) is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. DOE's National Nuclear Security Administration under Contract DENA0003525. Pacific Northwest National Laboratory (PNNL) is a multiprogram national laboratory operated by Battelle Memorial Institute for the U.S. DOE. Contributions from ORNL, SNL and PNNL are based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division. The views expressed in the article do not necessarily represent the views of the U.S. DOE or the U.S. Government. J. L. U. acknowledges funding by the German Science Foundation Project NE 822/34‐1|UR 64/17‐1. We also value discussions with P. Eichhubl, A. Fall, and J. F. W. Gale, contributions to workshop preparation from R. A. Schultz, and discussion and comments from R. Cygan, S.F. Forstner, Q. Wang, and journal reviewers. No data were used in the preparation of this manuscript.

https://agupubs.onlinelibrary.wiley.com/doi/epdf/10.1029/2019RG000671

The Role of Chemistry in Fracture Pattern Development and Opportunities to Advance Interpretations of Geological Materials

Glossary of fault and other fracture networks

Graph theory in the geosciences

https://hal.archives-ouvertes.fr/hal-01765560/document

Spatial arrangement of faults and opening-mode fractures

We propose a mapping from fracture systems consisting of intersecting fracture sheets in three dimensions to an abstract network consisting of nodes and links. This makes it possible to analyze fracture systems with the methods developed within modern network theory. We test the mapping for two-dimensional geological fracture outcrops and find that the equivalent networks are small-world and dissasortative. By anlayzing the Discrete Fracture Network model, which is used to generate artifical fracture networks, we also find small world networks. However, the networks turn out to be assortative.

The Topology of Fracture Networks

We propose a mapping from fracture systems consisting of intersecting fracture sheets in three dimensions to an abstract network consisting of nodes and links. This makes it possible to analyze fracture systems with the methods developed within modern network theory. We test the mapping for two-dimensional geological fracture outcrops and find that the equivalent networks show small-world characteristics and are dissasortative. By analyzing the Discrete Fracture Network model, which is used to generate artificial fracture outcrop networks, we also find small world networks. However, the networks turn out to be assortative.

/pdf/topology-of-fracture-networks-3pah84t7kf.pdf

Topology of fracture networks

Gas composition tracking in transient pipeline flow

The topology of two discrete fracture network models is compared to investigate the impact of constrained fracture growth. In the Poissonian discrete fracture network model the fractures are assigned length, position and orientation independent of all other fractures, while in the mechanical discrete fracture network model the fractures grow and the growth can be limited by the presence of other fractures. The topology is found to be impacted by both the choice of model, as well as the choice of rules for the mechanical model. A significant difference is the degree mixing. In two dimensions the Poissonian model results in assortative networks, while the mechanical model results in disassortative networks. In three dimensions both models produce disassortative networks, but the disassortative mixing is strongest for the mechanical model.

/pdf/topological-impact-of-constrained-fracture-growth-sc3w1neo5h.pdf

Topological impact of constrained fracture growth

Fracturing and refreezing of sea ice in the Kara sea are investigated using complex networkanalysis. By going to the dual network, where the fractures are nodes and their intersectionslinks, we gain access to topological features which are easy to measure and hence comparewith modeled networks. Resulting network reveal statistical properties of the fracturing process.The dual networks have a broad degree distribution, with a scale-free tail, high clusteringand efficiency. The degree-degree correlation profile shows disassortative behavior, indicatingpreferential growth. This implies that long, dominating fractures appear earlier than shorterfractures, and that the short fractures which are created later tend to connect to the longfractures.The knowledge of the fracturing process is used to construct growing fracture network (GFN)model which provides insight into the generation of fracture networks. The GFN model isprimarily based on the observation that fractures in sea ice are likely to end when hitting existingfractures. Based on an investigation of which fractures survive over time, a simple model forrefreezing is also added to the GFN model, and the model is analyzed and compared to the realnetworks.

/pdf/fracture-networks-in-sea-ice-3dgydkaivj.pdf

Sigmund Mongstad Hope

Papers

The Topology of Fracture Networks

Topology of fracture networks

Gas composition tracking in transient pipeline flow

Topological impact of constrained fracture growth

Fracture networks in sea ice