In some graphs, the edges only go one way. These are called directed graphs. Some graphs consist of multiple distinct segments which are not connected by edges. These graphs are disconnected. Other graphs may contain multiple edges between the same pairs of vertices, or vertices which are connected to themselves loops.
For simplicity we will only think about undirected and connected graphs without multiple edges and loops in this course. We can create new graphs from an existing graph by removing some of the vertices and edges. The result is called a subgraph. Here are a few examples of graphs and subgraphs:. The order of a graph is its number of vertices. The degree of a vertex in a graph is the number of edges which meet at that vertex.
MTH750U - Graphs and Networks - 12222/20
Change Language English Vietnamese. Send us Feedback Please let us know if you have any feedback and suggestions, or if you find any errors and bugs in our content. The space is hidden to us, kind of like a black box. We can think of the new points that we add to the graph as being sampled from this hidden space, where the distribution defined by graph KNN lives. So for K-means, we will have to replace the centroids with a different notion.
There are a number of possibilities, but the simplest is the following: Given a set of vertices in a graph, we will say that the radius of a vertex is the largest distance or smallest similarity to any other point in the set. The radius center of a set of vertices in the graph will be the vertex with smallest radius. With this definition, we can translate K-means directly into the graph setting. Second, make a list of the radius centers of each of the resulting sets of vertices.
As with vector K-means, this will give us a small number of vertices that are more or less evenly distributed throughout the graph. However, we can again think of these Voronoi cells as living in some unseen data space that contains the vertices of the graph. Before I end this post, I want to point out one more thing: If we were to form a graph from a vector data set, using the Gaussian function to define the weights, then convert our similarity scores back to distances as described above, we would find that a lot of the distances had gotten bigger, possibly quite a bit bigger.
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If we could see the underlying probability density function that defined the data set, then these paths would mostly stick to the dark parts of the probability cloud. If the data set forms a very curved shape, as in the Figure to the left, then these types of distance could be off by quite a bit.
In the Figure, the intrinsic distance between the two red vertices, indicated by the green path, is much longer than the extrinsic distance, which is indicated by the dotted blue line. Since these two notions of distance can be so different, this begs the question: Which of them is better for understanding the data set? There is no good answer to this question, partially because both distances depend heavily on the way the data is normalized and a number of other factors.
Thus both types of distance could be potentially quite bad. On the other hand, with the correct parameters and normalization, each type of distance will encode very different types of information about the data. In the Figure above, using only the extrinsic structure of the data set would make it hard to distinguish from a Gaussian blob. Pingback: Statistics vs.
Heuristics The Shape of Data. You are commenting using your WordPress. You are commenting using your Google account. You are commenting using your Twitter account. You are commenting using your Facebook account. Notify me of new comments via email. Notify me of new posts via email.
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The Shape of Data. Skip to content. Home About Table of Contents. Graphs and networks Posted on August 13, by Jesse Johnson. Share this: Twitter Facebook.
Graphs and Networks, MATH20150
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