SUMMARY To understand relationships between phosphorylation-based signaling pathways, we analyzed 150 deletion mutants of protein kinases and phosphatases in S. cerevisiae using DNA microarrays. Downstream changes in gene expression were treated as a phenotypic readout. Double mutants with synthetic genetic interactions were included to investigate genetic buffering relationships such as redundancy. Three types of genetic buffering relationships are identified: mixed epistasis, complete redundancy, and quantitative redundancy. In mixed epistasis, the most common buffering relationship, different gene sets respond in different epistatic ways. Mixed epistasis arises from pairs of regulators that have only partial overlap in function and that are coupled by additional regulatory links such as repression of one by the other. Such regulatory modules confer the ability to control different combinations of processes depending on condition or context. These properties likely contribute to the evolutionary maintenance of paralogs and indicate a way in which signaling pathways connect for multiprocess control.

The analysis of complex networks permeates all sciences, from biology to sociology. A fundamental, unsolved problem is how to characterize the community structure of a network. Here, using both standard and novel benchmarks, we show that maximization of a simple global parameter, which we call Surprise (S), leads to a very efficient characterization of the community structure of complex synthetic networks. Particularly, S qualitatively outperforms the most commonly used criterion to define communities, Newman and Girvan's modularity (Q). Applying S maximization to real networks often provides natural, well-supported partitions, but also sometimes counterintuitive solutions that expose the limitations of our previous knowledge. These results indicate that it is possible to define an effective global criterion for community structure and open new routes for the understanding of complex networks.

We introduce and explore a new method for inferring hidden geometric coordinates of nodes in complex networks based on the number of common neighbors between the nodes. We compare this approach to the HyperMap method, which is based only on the connections (and disconnections) between the nodes, i.e., on the links that the nodes have (or do not have). We find that for high degree nodes the common-neighbors approach yields a more accurate inference than the link-based method, unless heuristic periodic adjustments (or "correction steps") are used in the latter. The common-neighbors approach is computationally intensive, requiring O(t 4 ) running time to map a network of t nodes, versus O(t 3 ) in the link-based method. But we also develop a hybrid method with O(t 3 ) running time, which combines the common-neighbors and link-based approaches, and explore a heuristic that reduces its running time further to O(t 2 ), without significant reduction in the mapping accuracy. We apply this method to the Autonomous Systems (AS) Internet, and reveal how soft communities of ASes evolve over time in the similarity space. We further demonstrate the method's predictive power by forecasting future links between ASes. Taken altogether, our results advance our understanding of how to efficiently and accurately map real networks to their latent geometric spaces, which is an important necessary step towards understanding the laws that govern the dynamics of nodes in these spaces, and the fine-grained dynamics of network connections.

How to determine the community structure of complex networks is an open question. It is critical to establish the best strategies for community detection in networks of unknown structure. Here, using standard synthetic benchmarks, we show that none of the algorithms hitherto developed for community structure characterization perform optimally. Significantly, evaluating the results according to their modularity, the most popular measure of the quality of a partition, systematically provides mistaken solutions. However, a novel quality function, called Surprise, can be used to elucidate which is the optimal division into communities. Consequently, we show that the best strategy to find the community structure of all the networks examined involves choosing among the solutions provided by multiple algorithms the one with the highest Surprise value. We conclude that Surprise maximization precisely reveals the community structure of complex networks.