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Communities are a fundamental aspect of several networks. Through this problem, we will explore some properties of the Louvain algorithm for community detection so named because the authors were all affiliated with the University of Louvain in Belgium at some point.

The original paper from Blondel et al. If you are stuck on this question at any point please refer to the paper; there is a good chance that you will find what you seek there.

We will first explore the idea of modularity. The modularity of a weighted graph is a measure that compares the density of edges within a community to the density of edges between communities. Formally, we define the modularity Q for a given graph as follows:. Note that we treat communities as disjoint. In other words, a given node from a graph can only belong to one community in that graph. Maximizing the modularity of a given graph is a computationally hard problem, so we try different heuristics for this purporse.

One such heuristic is the Louvain algorithm. This algorithm outperforms many similar algorithms in terms of both speed as well as maximum modularity obtained. We will run a few steps of the algorithm on a couple of example networks to gain some insights about its behavior and properties.

Figure 1: Example from Blondel et al. Note how weights for self-edges are assigned in the Community Aggregation phase. An example taken from Blondel et al. Figure 1 illustrates the two phases of the algorithm. Note how weights for self-edges are assigned in the Community Aggregation phase — we will need to use this later.

Consider a node i that is in a community all by itself. Let C represent an existing community in the graph. Node i feels lonely and decides to move into the community C , we will inspect the change in modularity when this happens. This situation can be modeled by a graph Figure 2 with C being represented by a single node. To begin with, C and i are in separate communities colored green and red respectively. Prove that the modularity gain seen when i merges with C i. Hint: Using the community aggregation step of the Louvain method may make computation easier.

In practice, this result is used while running the Louvain algorithm along with a similar related result to make incremental modularity computations much faster. Figure 2: Before merging, i is an isolated node and C represents an existing community. The rest of the graph can be treated as a single node for this problem. Consider the graph G Figure 3 , with 4 cliques of 4 nodes each arranged in a ring.

Assume all the edges have same weight value 1. There exists exactly one edge between any two adjacent cliques. We will manually inspect the results of the Louvain algorithm on this network.

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We also strongly encourage using the online Discussions to discuss exercises with other students. However, do not look at any source code written by others or share your source code with others.

In the first half of this exercise, you will be using support vector machines SVMs with various example 2D datasets. In the next half of the exercise, you will be using support vector machines to build a spam classifier. We will begin by with a 2D example dataset which can be separated by a linear boundary. The script ex6. As part of this exercise, you will also see how this outlier affects the SVM decision boundary.

In this part of the exercise, you will try using different values of the C parameter with SVMs. Informally, the C parameter is a positive value that controls the penalty for misclassified training examples.

A large C parameter tells the SVM to try to classify all the examples correctly. The next part in ex6. Your task is to try different values of C on this dataset. In this part of the exercise, you will be using SVMs to do non-linear classification. In particular, you will be using SVMs with Gaussian kernels on datasets that are not linearly separable. To find non-linear decision boundaries with the SVM, we need to first implement a Gaussian kernel. The Gaussian kernel function is defined as:.

From the figure, you can obserse that there is no linear decision boundary that separates the positive and negative examples for this dataset. However, by using the Gaussian kernel with the SVM, you will be able to learn a non-linear decision boundary that can perform reasonably well for the dataset. If you have correctly implemented the Gaussian kernel function, ex6. Figure 5 shows the decision boundary found by the SVM with a Gaussian kernel.

The decision boundary is able to separate most of the positive and negative examples correctly and follows the contours of the dataset well. In this part of the exercise, you will gain more practical skills on how to use a SVM with a Gaussian kernel.

The next part of ex6. You will be using the SVM with the Gaussian kernel with this dataset. In the provided dataset, ex6data3. The provided code in ex6. Recall that for classification, the error is defined as the fraction of the cross validation examples that were classified incorrectly.

You can use the svmPredict function to generate the predictions for the cross validation set. Many email services today provide spam filters that are able to classify emails into spam and non-spam email with high accuracy.

In this part of the exercise, you will use SVMs to build your own spam filter. The following parts of the exercise will walk you through how such a feature vector can be constructed from an email.