Motivation: Network alignment (NA) aims to find a node mapping that conserves similar regions between compared networks. NA is applicable to many fields, including computational biology, where NA can guide the transfer of biological knowledge from well-to poorly-studied species across aligned network regions. Existing NA methods can only align static networks. However, most complex real-world systems evolve over time and should thus be modeled as dynamic networks. We hypothesize that aligning dynamic network representations of evolving systems will produce superior alignments compared to aligning the systems' static network representations, as is currently done. Results: For this purpose, we introduce the first ever dynamic NA method, DynaMAGNA þþ. This proof-of-concept dynamic NA method is an extension of a state-of-the-art static NA method, MAGNAþþ. Even though both MAGNAþþ and DynaMAGNAþþ optimize edge as well as node conservation across the aligned networks, MAGNAþþ conserves static edges and similarity between static node neighborhoods, while DynaMAGNAþþ conserves dynamic edges (events) and similarity between evolving node neighborhoods. For this purpose, we introduce the first ever measure of dynamic edge conservation and rely on our recent measure of dynamic node conservation. Importantly, the two dynamic conservation measures can be optimized with any state-of-theart NA method and not just MAGNAþþ. We confirm our hypothesis that dynamic NA is superior to static NA, on synthetic and real-world networks, in computational biology and social domains. DynaMAGNAþþ is parallelized and has a user-friendly graphical interface. categorized as pairwise or multiple (Faisal et al., 2015b; Guzzi and Milenkovi c, 2017; Vijayan and Milenkovi c, 2016) . Pairwise NA aligns two networks while multiple NA can align more than two networks at once. While multiple NA can capture conserved network regions between more networks than pairwise NA, which may lead to deeper biological insights compared to pairwise NA, multiple NA is computationally much harder than pairwise NA since the complexity of the NA problem typically increases exponentially with the number of networks. This is why in this paper we focus on pairwise NA, but our ideas can be extended to multiple NA as well. Henceforth, we refer to global and pairwise NA simply as NA. Existing NA methods can only align static networks. This is because in many domains and applications, static network representations are often used to model complex real-world systems, independent of whether the systems are static or dynamic. However, most real-world systems are dynamic, as they evolve over time. Static networks cannot fully capture the temporal aspect of evolving systems. Instead, such systems can be better modeled as dynamic networks (Holme, 2015) . For example, a complex system such as a social network evolves over time as friendships are made and lost. Static networks cannot model the changes in interactions between nodes over time, while dynamic networks can capture the times during which the friendships begin and end. Other examples of systems that can be more accurately represented as dynamic networks include communication systems, human or animal proximity interactions, ecological systems, and many systems in biology that evolve over time, including brain or cellular functioning. In particular, regarding the latter, while cellular functioning is dynamic, current computational methods (including all existing NA methods) for analyzing systems-level molecular networks, such as PINs, deal with the networks' static representations. This is in part due to unavailability of experimental dynamic molecular network data, owing to limitations of biotechnologies for data collection. Yet, as more dynamic molecular (and other real-world) network data are becoming available, there is a growing need for computational methods that are capable of analyzing dynamic networks (Przytycka and Kim, 2010; Przytycka et al., 2010) , including methods that can align such networks. The question is: how to align dynamic networks, when the existing NA methods can only deal with static networks? To allow for this, we generalize the notion of static NA to its dynamic counterpart. Namely, we define dynamic NA as a process of comparing dynamic networks and finding similar regions between such networks, while exploiting the temporal information explicitly (unlike static NA, which ignores this information). We hypothesize that aligning dynamic network representations of evolving real-world systems will produce superior alignments compared to aligning the systems' static network representations, as is currently done. To test this hypothesis, we introduce the first ever method for dynamic NA. Our proposed dynamic NA method, DynaMAGNA þþ, is a proof-of-concept extension of a state-of-the-art static NA method, MAGNAþþ (Vijayan et al., 2015). Saraph and Milenkovi c (2014) and Vijayan et al. (2015) compared MAGNAþþ to state-of-the-art static NA methods at the time: IsoRank (Singh et al., 2007) , MI-GRAAL (Kuchaiev and Pr zulj, 2011), and GHOST (Patro and Kingsford, 2012) . The comparisons were made on synthetic as well as real-world PINs, in terms of both topological and functional i182 V.Vijayan et al.
DynaMAGNA++ REF aligns two dynamic networks.
Alignment of dynamic networks
Bioinformatics
Computer Science Physics Biology Medicine