Generating Random Graphs with Tunable Clustering Coefficient
Parikh, Nidhi Kiranbhai
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Most real-world networks exhibit a high clustering coefficientâ the probability that two neighbors of a node are also neighbors of each other. We propose four algorithms CONF-1, CONF-2, THROW-1, and THROW-2 which are based on the configuration model and that take triangle degree sequence (representing the number of triangles/corners at a node) and single-edge degree sequence (representing the number of single-edges/stubs at a node) as input and generate a random graph with a tunable clustering coefficient. We analyze them theoretically and empirically for the case of a regular graph. CONF-1 and CONF-2 generate a random graph with the degree sequence and the clustering coefficient anticipated from the input triangle and single-edge degree sequences. At each time step, CONF-1 chooses each node for creating triangles or single edges with the same probability, while CONF-2 chooses a node for creating triangles or single edge with a probability proportional to their number of unconnected corners or unconnected stubs, respectively. Experimental results match quite well with the anticipated clustering coefficient except for highly dense graphs, in which case the experimental clustering coefficient is higher than the anticipated value. THROW-2 chooses three distinct nodes for creating triangles and two distinct nodes for creating single edges, while they need not be distinct for THROW-1. For THROW-1 and THROW-2, the degree sequence and the clustering coefficient of the generated graph varies from the input. However, the expected degree distribution, and the clustering coefficient of the generated graph can also be predicted using analytical results. Experiments show that, for THROW-1 and THROW-2, the results match quite well with the analytical results. Typically, only information about degree sequence or degree distribution is available. We also propose an algorithm DEG that takes degree sequence and clustering coefficient as input and generates a graph with the same properties. Experiments show results for DEG that are quite similar to those for CONF-1 and CONF-2.
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