Synthetic Graphs

Several methods designed to generate synthetic graphs have been proposed in literature. They produce graphs with different properties, depending on its intended use. Some of the most important methods are described next. Gilbert [196] propose a method to create a Random plane network by employing a Poisson process to place vertices with a density 𝐷 in a given area. Then those vertices are connected together in pairs when their distance does not exceed a given range 𝑅. Barabási and Albert [197], [198]

propose a method to build random networks. It is based on the observation that diverse networks expand continuously by adding new vertices and those new vertices are most likely attached to well connected vertices. The Yule process, or Preferential Attachment, [199], [200] can also be used to generate random graphs. The idea behind this method is that new edges will likely be placed with vertices with high degrees. Newman et al. [201] propose several models to create social network graphs such as unipartite networks and bipartite networks with very good approximations. The Erdős-Rényi model has two variants 𝐺(𝑛,𝑝) [202] and 𝐺(𝑛,𝑀) [203], The first variant creates a graph connecting the 𝑛 vertices randomly with a probability 𝑝. The second variant chooses a random graph from the set of graphs with 𝑛 nodes and 𝑀 edges. The Delaunay, or Delone, triangulation [76], [77] creates a triangular graph by

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connecting a set 𝑃 of vertices such as no other vertex is located inside the circumcircle generated by any triangle in the graph.

The experimental evaluation of DSHEM does not include any synthetic random graph generated by the methods presented above. They are not suited for the evaluation of DSHEM because it is essential to have full control over the graph geometry to investigate how it influences the quality of the partition and performance of the algorithm. The previous methods generate synthetic random graphs which emulate their real word counterparts. They are generally used when no real data, or a limited amount of it, is available. In this case, a big collection of real life graphs is publicly available making the synthetic random graphs redundant. The decision to not use this type of graphs was driven by these factors.

Thus, to have a controlled environment to assess the performance of DSHEM, several synthetic graphs with specific characteristics have been used. The synthetic graphs used for this work are based on the most common geometries found in FEA. Triangular and quadrangular geometry graphs are created in several ways to evaluate the performance of DSHEM. They are divided into regular and irregular, as well as 2D and 3D graphs. It is possible to control the degree of randomness with these synthetic graphs and evaluate how it affects the partitions generated by DSHEM. The range of sizes varies from small to large, as with the real life graphs. A description is presented next.

Regular

The regular synthetic graphs are used to perform a controlled evaluation of the comportment of DSHEM under several conditions. It is more evident how the geometry and size of the graph may affect its behavior, as well as the execution time. With that in mind, three different types of graphs have been selected to evaluate DSHEM; one quadrangular and two triangular geometries. The first type is a square graph, composed of regular squares as depicted in Figure 7.4. It size is measured by the number of vertices per side and it is given by 𝑛×𝑛, in this example 3 × 3.

Figure 7.4. Square graph with 3 vertices per side.

Figure 7.5 depicts the second type of synthetic graph with a triangular geometry. Its size is also measured by the number of vertices per side, i.e., 𝑛×𝑛.

A second triangular graph with a modified geometry is also employed for the experimental analysis.

Figure 7.6 presents a 3 × 3 dense triangular square graphs; (𝑛 −1) × (𝑛 −1) extra vertices have been added, and connected with edges, to the triangular square mesh to create the dense graph. Its size is given by the number of external vertices per side: 𝑛×𝑛.

3D versions of all three regular graphs are also created to evaluate DSHEM. Figure 7.7 shows the process to create a 3D square graph with a size of 2 × 2. It starts with the original graph as shown in a).

Next, in b), the graph is replicated to have 𝑛 copies. Finally, edges are generated between the vertices of the different copies of the graph as depicted in c).

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Figure 7.5. Triangular square graph with 3 vertices per side.

Figure 7.6. Dense triangular square graph with 3 vertices per side.

Figure 7.7. Creation of a 3D square graph with 2 vertices per side.

The creation of the 3D versions of the triangular square and dense triangular square graphs is described in detail in the appendix. It is important to note that the coordinate information is not used during the partitioning process; it is useful only for visualization purposes.

Irregular

Introducing irregularity to the graphs helps understanding the robustness of the algorithm during the partitioning process. The irregular synthetic graphs are based on the regular synthetic versions. All the edges in the graph are visited and removed with a given uniform probability. Several probabilities have been used to build different irregular graphs, giving them a degree or randomness. The resulting graph is then used for the experimental analysis to assess the performance of DSHEM and compare it with SHEM. It is important to understand how the edges are removed from the graph. Once a regular graph is created, all edges are duplicated; it is the normal way the graph is stored. All edges are visited in a sequential order and removed with a given probability. This method, visits edges (𝑢,𝑣) and (𝑣,𝑢) independently; one edge could be removed while the other remains. After visiting all edges, the graph may be corrupted due to the fact that some of the edges are no longer duplicated; the graph is then corrected by duplicating those edges. The final result is a correct graph with some of the edges being removed. If the initial probability to remove an edge was 10%, the resulting graph will have about 5%

(final probability) of the edges removed.

Throughout this work, the probability values for irregular graphs refer to the initial probability unless it is otherwise stated. Figure 7.8 shows an irregular dense triangular square graphs. It is built after the regular version by removing edges with a final probability of 25%. The same procedure is applied to the

a) b) c)

Chapter 7. Evaluation Methodology