Effects of preference for attachment to low-degree nodes on the degree distributions of a growing directed network and a simple food-web model

We study the growth of a directed network, in which the growth is constrained by the cost of adding links to the existing nodes. We propose a preferential-attachment scheme, in which a new node attaches to an existing node $i$ with probability $\Pi \left(k_i\right)\propto k_i^{-1}$, where $k_i$ is the number of outgoing links at $i$. We calculate the degree distribution for the outgoing links in the asymptotic regime $(t\to \infty )$, $n_k^{*}$, both analytically and by Monte Carlo simulations. The distribution decays like $k{\mu }^k∕\Gamma \left(k\right)$ for large $k$, where $\mu$ is a constant. We investigate the effect of this preferential-attachment scheme, by comparing the results to an equivalent growth model with a degree-independent probability of attachment, which gives an exponential outdegree distribution. Also, we relate this mechanism to simple food-web models by implementing it in the cascade model. We show that the low-degree preferential-attachment mechanism breaks the symmetry between in- and outdegree distributions in the cascade model. It also causes a faster decay in the tails of the outdegree distributions for both our network growth model and the cascade model.

Figure 1

Illustration of the growth with $N_0=4$ and $m=3$. Nodes 1, 2, 3, and 4 are the initial (source) nodes connected to unspecified nodes outside the network. Nodes 5, 6, and 7 are the nodes added at time steps 1, 2, and 3, respectively. See text for details.

Figure 2

(Color online) Outdegree (predator) distributions, $n_k^{*}$, for the growth model with low-degree preferential attachment, for $m=1$, 3, and 5 with $N_0=10$. The results of the Monte Carlo simulations (the symbols ∗, ×, and +) are in excellent agreement with the theory (the symbols 엯, ◻, and ◇). As $k$ is a discrete variable, the lines connecting the symbols are merely guides to the eye. The simulations were performed up to a system size of $N_{\mathrm{T}}=N_0+{10}^6$ nodes, and averaged over eight independent runs. Inset: The same distributions shown on a log-linear scale. The tails decay faster than an exponential, indicating that the topology of the network is relatively homogeneous.

Figure 3

(Color online) Outdegree (predator) distributions, $n_k^{*}$, for the equivalent growth model with degree-independent probability of attachment, for $m=1$, 3, and 5 with $N_0=10$. The results of the Monte Carlo simulations (the symbols ∗, ×, and +) are in excellent agreement with the theory, which predicts exponential decay (the symbols 엯, ◻, and ◇). Compare to the degree-dependent case in Fig. 2. The simulations were performed up to a system size of $N_{\mathrm{T}}=N_0+{10}^6$ nodes, and averaged over eight independent runs. Inset: The same distributions shown on a log-linear scale along with the theoretical results for the degree-dependent case (gray curves, turquoise online) in Fig. 2 for comparison.

Figure 4

(Color online) Degree distributions for the cascade model with and without low-degree preferential attachment with $d=10$ and $S={10}^4$. The error bars for the Monte Carlo data (averaged over eight runs) are smaller than the symbols. “Total” represents the distribution for the total number of links, $k+k^{\prime }$. (a) The distributions for the original cascade model, in which the probability of attachment is $x=d∕S$ (the symbols, 엯, ◻, and ▿). The in- and outdegree distributions obtained from the Monte Carlo simulations practically overlap in agreement with the theory, Eq. (14) (black line). (b) The distributions for the cascade model with low-degree preferential attachment with $x\left(k_i\right)=d∕\left[S(k_i+1)\right]$. The in-outdegree symmetry of the original cascade model is broken in the new distributions. See text for details.

Figure 5

(Color online) Scaled in- and outdegree distributions for the cascade model with (the symbols, ▵ and 엯) and without (solid circles) low-degree preferential attachment. In the axis titles, $n_{\kappa }$ represents the degree distributions, $n_k^{*}$, $n_{k^{\prime }}^{*}$, and $p^{(\mathit{\text{in}}=\mathit{\text{out}})}\left(k\right)$, and $⟨\kappa ⟩$ represents their respective mean values, which are 5 for the original cascade model, and 2.36 for the modified model. Each distribution is scaled using its respective $⟨\kappa ⟩$. The in- and outdegree distributions have the same mean for each case. Inset: The same distributions shown on a log-linear scale for comparison.

Figure 6

(Color online) The $m$ dependence of $z_m^{*}$ and the convergence of the ratio of the standard deviation of $n_k^{*}$ to its mean, $\sigma ∕⟨k⟩$, vs $m$. The graph shows that the standard deviation is proportional to the mean for large $m$. The ratio $\sigma ∕⟨k⟩$ converges to $\lambda \approx 0.522$ like $1∕m$. The $z_m^{*}$ vs $m$ plot confirms that $z_m^{*}\propto 1∕m$ for large $m$.

Figure 7

(Color online) Scaled degree distributions (thin curves), $m{n}_k^{*}$ vs $k∕m$, for the degree-dependent and degree-independent models (each for $m=10$ and 100), shown along with the asymptotic scaling functions (heavy curves), $p\left(u\right)=\frac{\pi u}{2}\exp (-\pi u^2∕4)$ and $q\left(u\right)=\exp (-u)$ with $u=k∕m$. For large values of $m$, the scaled distributions collapse onto their respective asymptotic scaling functions. Scaled distributions are obtained using the analytical forms, Eqs. (11, C4).