Not All Scale-Free Networks Are Born Equal: The Role of the Seed Graph in PPI Network Evolution

The (asymptotic) degree distributions of the best-known “scale-free” network models are all similar and are independent of the seed graph used; hence, it has been tempting to assume that networks generated by these models are generally similar. In this paper, we observe that several key topological features of such networks depend heavily on the specific model and the seed graph used. Furthermore, we show that starting with the “right” seed graph (typically a dense subgraph of the protein–protein interaction network analyzed), the duplication model captures many topological features of publicly available protein–protein interaction networks very well.
Author Summary The interactions among proteins in an organism can be represented as a protein–protein interaction (PPI) network, where each protein is represented with a node, and each interaction is represented with an edge between two nodes. As PPI networks of several model organisms become available, their topological features attract considerable attention. It is believed that the available PPI networks are (1) “small-world” networks, and (2) their degree distribution is in the form of a “power law.” In other words, (1) it is possible to reach from a protein to any other protein in only a small (approximately six) number of hops, and (2) although most proteins have only a few interactions (one or two), there are a few proteins with many more interactions (200 or more) and that act as “hubs.” It has thus been tempting to develop simple mathematical network generators with topological features similar to those of the available PPI networks. One such model, the “duplication model,” is based on Ohno's model of genome growth. It starts with a small “seed network” and grows by “duplicating” one of the existing nodes at a time, with an identical set of interactions; a randomly selected subset of these interactions is then deleted, and a few new interactions are added at random. It has been mathematically proven that the duplication model provides a small-world network and also has a power-law degree distribution. What we show in this paper is that by choosing the “right” seed network, many other topological features of the available PPI networks can be captured by the duplication model. The right seed network in this case turns out to include two sizable “cliques” (subnetworks where all node pairs are connected) with many interactions in between. In this paper, we also consider the preferential attachment model, which again grows by adding to a seed network one node at a time and connecting the new node to every other node with probability proportional to the existing degree of the second node. Because the preferential attachment model also provides a small-world network and has a power-law degree distribution, it has been considered equivalent to the duplication model. We show that the two models are vastly different in terms of other topological features we consider, and the preferential attachment model cannot capture some key features of the available PPI networks.