Distribution of shortest path lengths in a class of node duplication network models

We present analytical results for the distribution of shortest path lengths (DSPL) in a network growth model which evolves by node duplication (ND). The model captures essential properties of the structure and growth dynamics of social networks, acquaintance networks, and scientific citation networks, where duplication mechanisms play a major role. Starting from an initial seed network, at each time step a random node, referred to as a mother node, is selected for duplication. Its daughter node is added to the network, forming a link to the mother node, and with probability $p$ to each one of its neighbors. The degree distribution of the resulting network turns out to follow a power-law distribution, thus the ND network is a scale-free network. To calculate the DSPL we derive a master equation for the time evolution of the probability $P_t(L=\ell )$, $\ell =1,2,\cdots$, where $L$ is the distance between a pair of nodes and $t$ is the time. Finding an exact analytical solution of the master equation, we obtain a closed form expression for $P_t(L=\ell )$. The mean distance ${\langle L\rangle }_t$ and the diameter ${\mathrm{\Delta }}_t$ are found to scale like $lnt$, namely, the ND network is a small-world network. The variance of the DSPL is also found to scale like $lnt$. Interestingly, the mean distance and the diameter exhibit properties of a small-world network, rather than the ultrasmall-world network behavior observed in other scale-free networks, in which ${\langle L\rangle }_t\sim lnlnt$.

Figure 1

Illustration of the corded ND model. A random node, referred to as a mother node M (empty circle) is selected for duplication. The newly created daughter node D (empty circle) forms a deterministic edge (solid line) to the mother node, and with probability $p$ it forms a probabilistic edge (dashed line) to each one of the neighbors of M. In this example, D forms links to its two sister nodes, denoted by S, but does not form a link to its grandmother node, denoted by GM. In this illustration, all the other edges (solid lines) are deterministic edges.

Figure 2

Two instances of corded ND networks of size $N=50$, with $p=0.1$ (a) and $p=0.4$ (b). For the sake of comparison, both instances are formed around the same backbone tree (solid lines). The probabilistic edges (dashed lines) essentially decorate the tree. Upon formation, a probabilistic edge shortens the distance between its two ends from $L=2$ to 1, forming a triangle. Increasing $p$ makes the network denser.

Figure 3

Illustrations of two local network structures which give rise to double degeneracy $G=2$ in the first step of the shortest path between the daughter node D on the right and a target node T, which resides further down the branch, on the left. In both illustrations, solid lines correspond to deterministic edges, which belong to the backbone tree, while dashed lines correspond to probabilistic edges. (a) An alternate path of length $L=2$ is formed by probabilistic edges between nodes D and GM, via a sister node S. This path is degenerate with the primary path which resides fully on the backbone tree. Such structure may form in two distinct sequences of events. One possibility is that S is an older sister which was connected to GM upon formation. When D forms it connects probabilistically to S and completes the alternate path. In the other possibility, S is a younger sister of D, conditioned on D not forming a probabilistic edge to GM upon formation. When S is formed, it connects simultaneously to GM and to D, thus forming the alternate path. (b) In this structure, the distance between D and GGM along the backbone tree is $L=3$, while the shortest path in the entire network is of length $L=2$. This is achieved by two consecutive probabilistic shortcuts, one from M to GGM (created upon formation of M) and the other from D to GM (created upon formation of D). Note that, in this case, the existence of sisters of D (younger or older) makes no difference.

Figure 4

The parameter $\eta$ as a function of the probability $p$. This parameter represents the probability that the distance between the daughter node D and a random target node T is equal to the distance between the mother node M and T, namely, $\eta =P(L_{\text{DT}}=L_{\text{MT}})$. Hence, the probability that $L_{\text{DT}}=L_{\text{MT}}+1$ is given by $1-\eta$. The theoretical results (solid line), obtained from Eq. (29), are found to be in good agreement with the simulation results (circles). For small values of $p$, where the shortest paths are most likely to be unique, $\eta$ is equal to $p$. As $p$ is increased, the shortest paths become degenerate. As a result, $\eta$ acquires a nonlinear dependence on $p$, making it larger than $p$.

Figure 5

The DSPL of the corded ND network of $N_t={10}^4$ nodes with (a) $p=0.1$, (b) $p=0.2$, (c) $p=0.3$, and (d) $p=0.4$. The theoretical results (solid lines), obtained from Eqs. (36) and (37), are found to be in good agreement with the results of computer simulations (circles), obtained by averaging over 100 instances. As $p$ is increased, the distances become shorter and the DSPL becomes narrower, consistent with Eqs. (58) and (74). The agreement is better for smaller values of $p$. In fact, Eqs. (36) and (37) are exact, while the deviation from the simulation results are due to the underestimate of $\eta$, as can be seen in Fig. 4.

Figure 6

The mean shortest path length ${\langle L\rangle }_t$ of the corded ND network as a function of network size $N_t$. The theoretical results (solid lines), obtained from Eq. (58), where $\eta$ is taken from Eq. (29), are generally in good agreement with the simulation results (symbols), confirming the logarithmic dependence on the network size. As $p$ is increased, the mean shortest path length decreases. As in Fig. 5, the deviation between the theory and simulation increases as $p$ is increased, due to the fact that the exact value of $\eta$ is not known. For clarity, we focus on network sizes in the range ${10}^2\le N_t\le {10}^4$.

Figure 7

The diameter ${\mathrm{\Delta }}_t$ of the corded ND network as a function of network size $N_t$. The theoretical results (solid lines), obtained from Eq. (68), where $\eta$ is taken from Eq. (29), are found to be in good agreement with the simulation results (symbols). The results confirm the logarithmic dependence of the diameter on the network size. As $p$ is increased, the diameter decreases.

Figure 8

The standard deviation of the DSPL, ${\sigma }_t$, as a function of network size $N_t$. The theoretical results (solid lines), obtained from Eq. (75), where $\eta$ is taken from Eq. (29), are found to be in good agreement with the simulation results (symbols).

Figure 9

The mean distance $\langle L\rangle ={\langle L\rangle }_t$, expressed in units of $lnN=ln{N}_t$, namely, $\langle L\rangle /lnN\simeq 2(1-\eta )$, of the corded ND network as a function of the parameter $p$ (dashed line), and the corresponding ratio $\langle L\rangle /lnN=1/ln[(\langle K^2\rangle -\langle K\rangle )/\langle K\rangle ]$ for a configuration model network with the same degree distribution (solid line), where $\langle K\rangle$ is given by Eq. (5) and $\langle K^2\rangle$ is given by Eq. (6).