Figure 1

(Color online) Pictorial description of our model for a competitive cluster growth process. To make a clear picture of our model, we used as a substrate network for the growth a third generation Apollonian network [19]. The matching graph form of this network makes it particularly suitable for a plane representation. In the first step of the growth model (a), a few nodes (the large circle, square, triangle, and star) are chosen to be the seeds of the growing clusters. Each of the seeds is the first node of a different cluster. All the remaining nodes (small dots) do not get assigned to any of the clusters in the beginning; these nodes will be the ones accessible to the growth. The dashed lines link either a pair of nodes that already belong to one of the clusters or a pair of nodes that are not in any cluster yet. At this time step these lines do not participate in the growth process. The thick lines link one of the seeds to an accessible node, and any of the thick lines can be chosen with the same probability to channel the growth of a cluster. In this way, the growth rate of a particular cluster is proportional to its perimeter, that is, the number of connections from one of the nodes already incorporated into the cluster to an accessible node. In the second step (b) a new node is incorporated into the cluster of squares and new thick lines are added to the perimeter of this cluster. The process continues until every one of the nodes has been incorporated into one of the clusters (c). At this point the size of all the clusters is computed.Reuse & Permissions

Figure 2

(Color online) Normalized cluster size distributions obtained for the competitive growth model under a random ER network topology. These distributions were obtained for networks of ${10}^5$ nodes. We performed a larger number of runs for the cases with less competing clusters in order to have ${10}^5$ cluster samples for each case. The distributions obtained under this topology follow approximately an exponential distribution with a characteristic cluster size depending on the number $n_s$ of competing clusters. The dashed lines are the functions $n_s{e}^{-n_{s}n}$. The continuous lines are a fit for the function $P\left(n\right)=C{n}_s^{1-\gamma }{n}^{-\gamma }\exp (-n_{s}n)$, with the parameters $C=0.85\pm 0.03$ and $\gamma =0.2\pm 0.05$ obtained for the collapsed data shown in the inset. This figure shows that for the random networks the form of the distribution is completely determined by the density of nodes chosen to be seeds for the growth process.Reuse & Permissions

Figure 3

(Color online) Normalized cluster size distributions obtained when the growth process is performed in networks built with the preferential attachment schema. We used networks with ${10}^7$ nodes and obtained ${10}^5$ cluster samples. Different from the case of random networks, here we cannot produce a data collapse for the distributions. We note, however, that for any value of $n_s$ we have a broad distribution with clusters of all sizes ranging from just a few nodes to most of the network. Due to network size constraints, the distributions display a truncation in the limit of large cluster sizes. As observed in the ER networks, the exponential truncation moves to smaller cluster sizes as the number of seeds increases. The continuous line is a power-law fit, $P\left(n\right)\sim n^{-\alpha }$, with the exponent $\alpha =1.2$ obtained for the scaling region of the distribution for $n_s={10}^2$. The slope of the distribution becomes more steep as the number of competing clusters in the network grows.Reuse & Permissions

Figure 4

(Color online) Normalized cluster size distribution when our model is performed on Apollonian networks. The distributions were obtained with the 15th generation of the Apollonian network that corresponds to $N=7174456$. Each curve was sampled from an ensemble of ${10}^5$ clusters. We observe a power-law decay with an exponent $\alpha \approx 1$. The continuous line is a fit for the scaling region of data with $n_s=100$ initial seeds. As in the case of the preferential attachment networks, here we observe a truncation in the large cluster size region of the distributions, with an onset controlled by the number of seeds $n_s$. The periodical steps observed in the shape of the distributions are reminiscent of the self-similar structure resulting from the hierarchical construction of the Apollonian network.Reuse & Permissions