Introduction to the Modeling and Analysis of Complex Systems

15.6. GENERATING RANDOM GRAPHS 323Random graphs can also be generated by randomizing existing graph topologies.Such network randomization is particularly useful when you want to compare a certainproperty of an observed network with that of a randomized “null” model. For example,imagine you found that information spreading was very slow in the network you werestudying. Then you could test whether or not the network’s specific topology caused thisslowness by simulating information spreading on both your network and on its randomizedcounterparts, and then comparing the speed of information spreading between them.There are several different ways to randomize network topologies. They differ regardingwhich network properties are to be conserved during randomization. Some examples areshown below (here we assume that g is the original network to be randomized):• gnm_random_graph(g.number_of_nodes(), g.number_of_edges()) conserves thenumbers of nodes and edges, nothing else.• random_degree_sequence_graph(g.degree().values()) conserves the sequenceof node degrees, in addition to the numbers of nodes and edges. As noted above,this function may not generate a graph within a given number of trials.• expected_degree_graph(g.degree().values()) conserves the number of nodesand the sequence of “expected” node degrees. This function can always generate agraph very quickly using the efficient Miller-Hagberg algorithm [64]. While the generatednetwork’s degree sequence doesn’t exactly match that of the input (especiallyif the network is dense), this method is often a practical, reasonable approach forthe randomization of large real-world networks.• double_edge_swap(g) conserves the numbers of nodes and edges and the sequenceof node degrees. Unlike the previous three functions, this is not a graphgeneratingfunction, but it directly modifies the topology of graph g. It conducts whatis called a double edge swap, i.e., randomly selecting two edges a − b and c − d,removing those edges, and then creating new edges a − c and b − d (if either pairis already connected, another swap will be attempted). This operation realizes aslight randomization of the topology of g, and by repeating it many times, you cangradually randomize the topology from the original g to a completely randomizednetwork. double_edge_swap(g, k) conducts k double edge swaps. There is alsoconnected_double_edge_swap available, which guarantees that the resulting graphremains connected.