The configuration model for Barabasi-Albert networks

We develop and test a rewiring method (originally proposed by Newman) which allows to build random networks having pre-assigned degree distribution and two-point correlations. For the case of scale-free degree distributions, we discretize the tail of the distribution according to the general prescription by Dorogovtsev and Mendes. The application of this method to Barabasi-Albert (BA) networks is possible thanks to recent analytical results on their correlations, and allows to compare the ensemble of random networks generated in the configuration model with that of “real” networks obtained from preferential attachment. For $\beta\ge 2$ ($\beta$ is the number of parent nodes in the preferential attachment scheme) the networks obtained with the configuration model are completely connected (giant component equal to 100%). In both generation schemes a clear disassortativity of the small degree nodes is demonstrated from the computation of the function $k_{nn}$. We also develop an efficient rewiring method which produces tunable variations of the assortativity coefficient $r$, and we use it to obtain maximally disassortative networks having the same degree distribution of BA networks with given $\beta$. Possible applications of this method concern assortative social networks.

💡 Research Summary

The paper presents a comprehensive methodology for generating random networks that simultaneously match a prescribed degree distribution and a target two‑point correlation structure. Building on Newman’s degree‑preserving rewiring algorithm, the authors adapt the classic configuration model to incorporate analytical correlation data derived for Barabási‑Albert (BA) networks.

First, the authors address the discretization of a scale‑free degree distribution (P(k)∝k⁻³) with a power‑law tail. They discuss three practical schemes—Cumulation, Random hubs, and a probability‑transformation method—to convert the continuous distribution into an exact set of integer node degrees for a finite network of size N. The Cumulation approach accumulates fractional expected counts until a whole node is created, thereby preserving the rare but crucial high‑degree hubs that appear in real BA graphs. The Random hubs method introduces stochasticity in the tail, while the probability‑transformation method guarantees exactly N nodes by sampling from the cumulative distribution function.

With the degree sequence fixed, the wiring phase deviates from the fully random stub‑matching of the traditional configuration model. Instead, nodes are processed in descending order of degree; each node’s stubs are connected to randomly chosen available stubs of other nodes. This partially deterministic step yields a preliminary link list L that already respects the degree sequence but does not yet reproduce the desired correlations.

The core of the technique is the subsequent rewiring stage. Two links (a,b) and (c,d) are selected uniformly at random. Let A, B, C, D denote the excess degrees (degree minus one) of the four involved nodes. Using the target correlation matrix e₀_{jk} (computed from the exact conditional probabilities P(h|k) for BA networks, as derived by Fotouhi and Rabbat), the algorithm evaluates
E₁ = e₀_{AB}·e₀_{CD} and E₂ = e₀_{AC}·e₀_{BD}.
If E₁ = 0 the swap (a,b),(c,d) → (a,c),(b,d) is performed unconditionally. Otherwise a probability P = E₂/E₁ is computed; the swap is accepted if P ≥ 1 or if a uniform random number ξ < P. This acceptance rule biases the rewiring toward configurations that increase the likelihood under the target joint degree distribution, while preserving each node’s degree. The process is repeated roughly 10³ times per node (≈10⁷ attempts for N=2500), which empirically yields convergence: over half of the proposed swaps succeed, and the network’s empirical joint degree matrix closely matches e₀_{jk}.

Applying this framework to BA networks with different attachment parameters β (the number of parent nodes each new node connects to) reveals striking structural differences. For β ≥ 2 the average degree ⟨k⟩ = 2β is sufficiently large that the rewiring phase eliminates isolated components; the resulting graphs are fully connected (giant component = 100 %). For β = 1 (⟨k⟩ = 2) the final graphs retain a giant component of about 0.69 N, with the remainder consisting mainly of isolated triples. The authors explain this by noting that the conditional probability P(1|1)=0 in BA1 prevents degree‑1 nodes from linking directly, yet the rewiring can still produce small disconnected motifs because the knowledge of P(h|k) alone does not fully constrain higher‑order structure.

The paper validates the success of the correlation reconstruction by examining the average nearest‑neighbor degree function kₙₙ(k)=∑ₕ h P(h|k). In the generated networks kₙₙ(k) decreases sharply for low k, reaches a minimum near k≈0.2 n (where n is the maximum degree), and then rises slowly for large k. This mirrors the mixed assortative/disassortative pattern previously reported for BA graphs: low‑degree nodes are strongly disassortative, while high‑degree hubs exhibit a mild assortative tendency.

Beyond reproducing BA correlations, the authors extend the rewiring scheme to tune the assortativity coefficient r. By adjusting the acceptance rule they can drive r to more negative values while keeping the degree distribution unchanged. They demonstrate the construction of maximally disassortative networks (r≈−0.09) that share the same power‑law exponent (γ=3) as the original BA graphs.

To illustrate the practical impact of these structural differences, the authors simulate the Bass innovation diffusion model on several network ensembles: BA (β=1–5), an uncorrelated configuration model, a deliberately disassortative network (built via Newman’s original recipe), and an assortative network (constructed with their own method). Diffusion peak times are reported for three imitation coefficients q (0.30, 0.38, 0.48). The results show that BA1 networks achieve the fastest diffusion, underscoring how the specific mixture of degree heterogeneity and correlation pattern can accelerate spreading processes.

In conclusion, the paper makes three substantive contributions: (1) a robust, computationally efficient pipeline for generating random graphs with both a prescribed degree sequence and a target joint-degree matrix; (2) a detailed empirical comparison between such synthetic graphs and genuine BA networks, highlighting the role of β in determining global connectivity and component structure; and (3) a flexible rewiring mechanism that can systematically vary the assortativity coefficient, opening avenues for generating realistic assortative social networks or maximally disassortative technological networks. The methodology bridges a gap between theoretical random‑graph models and the nuanced correlation structures observed in real‑world complex systems.