Coin-Flipping, Ball-Dropping, and Grass-Hopping for Generating Random Graphs from Matrices of Edge Probabilities

Abstract

Common models for random graphs, such as Erdős-Rényi and Kronecker graphs, correspond to generating random adjacency matrices where each entry is non-zero based on a large matrix of probabilities. Generating an instance of a random graph based on these models is easy, although inefficient, by flipping biased coins (i.e. sampling binomial random variables) for each possible edge. This process is inefficient because most large graph models correspond to sparse graphs where the vast majority of coin flips will result in no edges. We describe some not-entirely-well-known, but not-entirely-unknown, techniques that will enable us to sample a graph by finding only the coin flips that will produce edges. Our analogies for these procedures are ball-dropping, which is easier to implement, but may need extra work due to duplicate edges, and grass-hopping, which results in no duplicated work or extra edges. Grass-hopping does this using geometric random variables. In order to use this idea on complex probability matrices such as those in Kronecker graphs, we decompose the problem into three steps, each of which are independently useful computational primitives: (i) enumerating non-decreasing sequences, (ii) unranking multiset permutations, and (iii) decoding and encoding z-curve and Morton codes and permutations. The third step is the result of a new connection between repeated Kronecker product operations and Morton codes. Throughout, we draw connections to ideas underlying applied math and computer science including coupon collector problems.

Network structure from rich but noisy data

“…In the particular case in which the posterior distribution factors into independent distributions over each edge-as in all of the models considered here-Monte Carlo sampling of networks is trivial. One simply generates each edge independently with the appropriate probability Q ij , and there exist straightforward algorithms for doing this efficiently [17]. In cases where the edges are not independent, one can generate networks using Markov chain importance sampling [18], in which one repeatedly makes small changes A → A ′ to the network, such as the addition or removal of a single edge, then accepts those changes with the standard Metropolis-Hastings acceptance probability P a = q(A ′ )/q(A) if q(A ′ ) < q(A), 1 otherwise.…”

Generating large scale‐free networks with the Chung–Lu random graph model

“…The main advantage of our generator is its compact, highly modular implementation, which runs very efficiently on MATLAB and other high‐level scientific programming languages, due to the absence of explicit for‐loops. Our algorithm is based on some ideas laid out in [23,38] and, as for the method proposed by Miller and Hagberg [30], has a running time essentially proportional to the number of edges in the graph, over a very large size range.…”

“…The algorithm implements the principle called “ball dropping” by Ramani et al [38]. According to such principle, the algorithm initially generates two random vectors, I and J, whose entries are node indices.…”

Generating large scale-free networks with the Chung-Lu random graph model

“…Our algorithm is based on some ideas laid out in [4,19] and, as for the method proposed in [23], has a running time essentially proportional to the number of edges in the graph. The algorithm implements the principle called "ball dropping" in [29]. Initially, the algorithm generates two random vectors, I and J, whose entries are node indices.…”

“…Even though a wealth of generative models for scale-free random networks is now available, the investigation of methods for modeling networks either analytically or numerically is still an active research direction in network science [19,23,29,31]. In fact, no generative model fits the ever-changing needs of complex network analysis.…”