[Paper Review] Directed random graphs with given degree distributions

Ningyuan Chen, Mariana Olvera‐Cravioto|arXiv (Cornell University)|Jul 10, 2012

Gene expression and cancer classification4 citations

TL;DR

This paper proposes a method to generate simple directed random graphs with prescribed in- and out-degree distributions by constructing nearly i.i.d. degree sequences with equal in- and out-degree sums, then applying a directed configuration model. It proves that, under finite variance conditions, the probability of obtaining a simple graph remains bounded away from zero, and conditional on simplicity, the empirical degree distributions converge to the target distributions as the graph size grows.

ABSTRACT

Given two distributions F and G on the nonnegative integers we propose an algorithm to construct in- and out-degree sequences from samples of i.i.d. observations from F and G, respectively, that with high probability will be graphical, that is, from which a simple directed graph can be drawn. We then analyze a directed version of the configuration model and show that, provided that F and G have finite variance, the probability of obtaining a simple graph is bounded away from zero as the number of nodes grows. We show that conditional on the resulting graph being simple, the in- and out-degree distributions are (approximately) F and G for large size graphs. Moreover, when the degree distributions have only finite mean we show that the elimination of self-loops and multiple edges does not significantly change the degree distributions in the resulting simple graph.

Motivation & Objective

• To develop a method for generating simple directed random graphs with pre-specified in- and out-degree distributions.
• To ensure that the in- and out-degree sequences are graphical with high probability by balancing their sums while preserving i.i.d. sampling properties.
• To analyze the directed configuration model and show that the resulting simple graphs retain the target degree distributions asymptotically.
• To establish conditions under which the degree distributions remain approximately unchanged after removing self-loops and multiple edges.

Proposed method

• Construct in- and out-degree sequences as i.i.d. samples from target distributions F and G, then adjust them to ensure equal sums using a residual-balancing technique.
• Apply a directed configuration model by randomly pairing in-stubs and out-stubs to form directed edges.
• Use a rejection sampling approach or edge-erasing procedure to eliminate self-loops and multiple edges, ensuring the final graph is simple.
• Prove that the probability of obtaining a simple graph remains bounded away from zero when F and G have finite variance.
• Analyze the effect of edge-removal on degree distributions using concentration inequalities and conditional probability bounds.
• Employ tools from probability theory, including the strong law of large numbers, bounded convergence, and Fatou’s lemma, to establish asymptotic convergence of empirical degree distributions.

Experimental results

Research questions

• RQ1 Can we generate a simple directed random graph with prescribed in- and out-degree distributions using a tractable and scalable algorithm?
• RQ2 Under what conditions is the probability of obtaining a simple graph through random pairing bounded away from zero as the number of nodes increases?
• RQ3 Does the removal of self-loops and multiple edges significantly alter the in- and out-degree distributions in the final graph?
• RQ4 How close are the empirical in- and out-degree distributions in the resulting simple graph to the target distributions F and G?

Key findings

• When the in- and out-degree distributions F and G have finite variance, the probability of generating a simple graph via the directed configuration model remains bounded away from zero as n → ∞.
• Conditional on the graph being simple, the empirical in- and out-degree distributions converge almost surely to the target distributions F and G as the number of nodes grows.
• Even when only finite mean is assumed, the degree distributions in the final simple graph remain approximately unchanged after edge-removal, with the probability of altering a node's degree by more than one vanishing as n increases.
• The method ensures that the constructed degree sequences are graphical with high probability by balancing the sums of in- and out-degrees while preserving near-i.i.d. sampling characteristics.
• The convergence of empirical degree distributions to the target distributions is established via probabilistic bounds, including the use of Fatou’s lemma and concentration inequalities.