GREAT: GRaphlet Edge-based network AlignmenT
aa r X i v : . [ q - b i o . M N ] O c t GREAT: GRaphlet Edge-based network AlignmenT
Joseph Crawford and Tijana Milenkovi´c ∗ Department of Computer Science and Engineering, Interdisciplinary Center forNetwork Science and Applications, and ECK Institute for Global HealthUniversity of Notre Dame ∗ Corresponding Author (E-mail: [email protected])
Abstract.
Network alignment aims to find regions of topological or functional similarities betweennetworks. In computational biology, it can be used to transfer biological knowledge from a well-studied species to a poorly-studied species between aligned network regions. Typically, existingnetwork aligners first compute similarities between nodes in different networks (via a node costfunction) and then aim to find a high-scoring alignment (node mapping between the networks)with respect to “node conservation”, typically the total node cost function over all aligned nodes.Only after an alignment is constructed, the existing methods evaluate its quality with respect toan alternative measure, such as “edge conservation”. Thus, we recently aimed to directly optimizeedge conservation while constructing an alignment, which improved alignment quality. Here, weapproach a novel idea of maximizing both node and edge conservation, and we also approach thisidea from a novel perspective, by aligning optimally edges between networks first in order to improvenode cost function needed to then align well nodes between the networks. In the process, unlike theexisting measures of edge conservation that treat each conserved edge the same, we favor conservededges that are topologically similar over conserved edges that are topologically dissimilar. Weshow that our novel method, which we call GR aphlet E dge A lignmen T (GREAT), improves uponstate-of-the-art methods that aim to optimize node conservation only or edge conservation only. The goal of network (or graph) alignment in computational biology is to find regions of topologicalor functional similarities between networks of different species. (We note, however, that networkalignment has applications in many domains [1, 2].) The more biological network data is becomingavailable, the more importance the problem of network alignment gains. This is because networkalignment can be used, for example, to transfer functional (e.g., aging-related [3, 4, 5]) knowledgefrom well annotated species to poorly studied ones between aligned network regions.There are two categories of network alignment methods: local network alignment (LNA) andglobal network alignment (GNA). LNA focuses on optimizing similarity between local (small) re-gions of different networks, plus, it allows for a region in one network to be mapped to multi-ple regions in another network. This way, LNA is generally unable to find large conserved sub-graphs between networks, and also, it can lead to many-to-many node mappings between thenetworks [6, 7, 8, 9, 10, 11, 12, 13, 14], which might be motivated biologically, but such align-ments are hard to characterize in terms of topological alignment quality [15, 16]. On the otherhand, GNA aims to optimize global (overall) similarity between different networks, and in gen-eral (although some exceptions exist [17]), it results in one-to-one (i.e., injective) node mapping[2, 3, 4, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. As such, GNA is able to find largeconserved subgraphs, and it also allows for quantifying both topological and biological quality of1he resulting alignments. In this study, we focus on one-to-one GNA due to its recent popularity(and henceforth, we refer to GNA simply as network alignment), but all concepts introduced herecan be applied to LNA as well.We more formally define network alignment as an injective mapping between the nodes of twonetworks that aligns the networks well with respect to a desired criterion. Network alignment isa computationally hard problem to solve due to the underlying subgraph isomorphism problem,which is NP-complete [30]. The subgraph isomorphism problem aims to find out whether somegraph G contains another graph G as its exact subgraph. With this in mind, the networkalignment problem aims to “fit well” G into G when G is not necessarily an exact subgraph of G . Thus, since network alignment is computationally intractable, all existing algorithms aimingto solve this problem are heuristics.In general (although there are some exceptions [28]), existing network alignment methods typ-ically contain two algorithmic components: 1) a node cost function and 2) an alignment strategy[3, 4, 15, 16, 17, 18, 22, 23, 24, 31, 32]. A node cost function finds pairwise topological (potentiallyalso biological, e.g., protein sequence) similarities (or equivalently, costs) between two nodes fromdifferent networks, while the alignment strategy uses these costs to select a high-scoring alignment(out of all possible alignments) typically with respect to the total node cost function over all alignednodes [28]. Then, the quality of the resulting alignment is evaluated with respect to some othertopological measure, which is different than the node cost function that is used to produce thealignment in the first place. (Alignment quality is also measured via a biological measure, such asfunctional enrichment of aligned node pairs [28].) Typically, one measures the amount of conservededges, and multiple measures have been proposed for this purpose, with our recent symmetricsubstructure score (S ) being a superior measure [28]. That is, the goal of existing methods is toalign nodes well in hope that they will align edges well, but only after the alignment is produced.Hence, recently, we introduced a novel algorithm, called MAGNA, which is capable of optimizingedge conservation directly while an alignment is being constructed [28].Here, we approach a novel idea of maximizing both node and edge conservation, and we alsoapproach this idea from a novel perspective, by aligning edges between networks in order to improvenode cost function. These are the two major novelties of our study that distinguish us from theexisting work. In the process, unlike the existing measures of edge conservation that treat eachconserved edge the same, we propose a new measure of edge conservation to favor conserved edgesthat are topologically similar over conserved edges that are topologically dissimilar. This is anotherof our novelties. We note that a method exists that infers plausibly alignable interactions acrossprotein-protein interaction (PPI) networks of different species [25]. However, this method is guided biologically rather than topologically: it aligns PPIs relying on conservation of their constituentdomain interactions, and thus, it aims to address not the problem of subgraph isomorphism noredge conservation but rather that of biological correctness of the aligned edges [25].To simultaneously optimize both node and edge conservation, our new method, which we callGRaphlet Edge AlignmenT (GREAT), first aims to optimally align edges between two networks,and based on the resulting edge alignment, it constructs (as we will show) a more efficient nodecost function compared to state-of-the-art node similarity measures that are typically used for thispurpose. That is, when we use within a given existing alignment strategy our new edge alignment-based node cost function, we get alignments of higher quality with respect to both node and edgeconservation than when we use within the same alignment strategy an existing node cost function.Thus, we improve upon methods that aim to optimize node conservation only. At the same time,GREAT is comparable or superior to MAGNA that aims to optimize edge conservation only.2 Methods
GREAT consists of four algorithmic components: 1) edge cost function, 2) edge alignment strategy,3) node cost function, and 4) node alignment strategy. Edge cost function and edge alignmentstrategy are used to align well edges between two networks, similar to how existing methods alignnodes between two networks based on node cost function and node alignment strategy. Then, theresulting edge alignment is used to compute a novel node cost function, according to which twonodes from different networks are similar if the nodes’ adjacent edges have been aligned and withhigh similarity with respect to the edge cost function. Then, the resulting edge alignment-basednode cost function is used within an existing node alignment strategy to produce an injective nodemapping between the networks. In this way, the output of GREAT can be directly comparedagainst alignments of the existing methods. This section details the four steps of GREAT.
To create pairwise edge scores, GREAT uses the notion of graphlets, as follows.Graphlets are small induced non-isomorphic subgraphs (e.g., a triangle or a square; Figure 1)[33, 34, 35, 36, 37, 38, 39, 40]. A graphlet-based node cost function was already used for networkalignment by three state-of-the-art methods: GRAAL [15], H-GRAAL [16], and MI-GRAAL [22](also, see [3, 4, 31]). This node cost function relies on node graphlet degree vector (node-GDV) [34],which counts the number of graphlets (i.e., their topologically unique node “symmetry groups”,called automorphism orbits; Figure 1) that a node touches. Then, the graphlet-based node costfunction computes topological similarity between extended neighborhoods of two nodes from differ-ent networks by comparing the nodes’ GDVs. Hence, this function is called node-GDV-similarity [35]. Recently, we extended the notion of node-GDV into edge-GDV (Figure 1) and of node-GDV-similarity into edge-GDV-similarity , to allow for quantifying topological similarity betweenextended neighborhoods of two edges rather than nodes [39]. Then, we used edge-GDV-similarityas a basis for a novel superior network clustering method [39]. Here, we use for the first timeedge-GDV-similarity for network alignment, and we use it as a part of our edge cost function.The other part of our edge cost function is a novel concept of edge graphlet degree centrality (edge-GDC), which we define as a measure the complexity of the extended network neighborhood ofan edge (see below for a formal definition). We introduce edge-GDC to modify the total similarity ofaligning two edges, in order to favor alignment of the densest parts of the networks. Namely, edgeswith a given edge-GDV-similarity and with high edge-GDC (and thus dense network neighborhoods)should be aligned before correspondingly edge-GDV-similar edges with low edge-GDC [16].We define edge-GDC analogously to our existing definition of node-GDC [38]. For a given edge e , we denote the i th coordinate of its edge-GDV (that is, the number of times edge e touches orbit i ) as e i . Then: edge-GDC( e ) = P w i ∗ ln ( e i + 1), where w i ∈ [0 ,
1] is the weight of orbit i thataccounts for dependencies between orbits, and 68 is the total number of edge orbits in 3-5-nodegraphlets (there is an additional orbit for the only 2-node graphlet, i.e., an edge, but we leave outthis orbit, as each edge will participate in exactly one such orbit) [35, 39].With the notions of edge-GDV-similarity and edge-GDC introduced, we define our edge costfunction (ECF), i.e., the total similarity between two edges e and f from different networks, as:ECF = α × edge-GDV-similarity( e, f )+(1 − α ) × edge-GDC( e ) + edge-GDC( f ) max (edge-GDC( G )) + max (edge-GDC( G )) (1)where α is a parameter in [0,1], G and G are the two networks being aligned, and edge-GDC( G i )is the maximum GDC in network G i . For this study, we vary α from 0 to 1 in increments of 0.2.The formula is designed to normalize edge cost function to [0,1] range.3igure 1: All automorphism orbits (i.e., topologically unique “symmetry groups”) of a node (top)and an edge (bottom) in up to 4-node graphlets [34, 39]. We illustrate only up to 4-node graphletsfor esthetics, but all up to 5-node graphlets (with 73 node orbits and 69 edge orbits) are used withinour method. The figure has been adopted and adapted from [39].Given pairwise edge scores computed with respect to the above edge cost function, GREAT feedsthese scores into an existing edge alignment strategy to produce injective edge mapping betweenthe two networks. We use two such strategies: 1) greedy alignment strategy, which maps, oneat a time, the highest-scoring edge pairs in a greedy fashion, and 2) the Hungarian algorithm formaximum weight bipartite matching, which produces optimal edge mapping with respect to ouredge cost function. We use these methods as edge alignment strategies because equivalent (andthus comparable) methods have already been used in the context of network alignment as nodealignment strategies, within e.g., IsoRank [18] and H-GRAAL [16], respectively. Ideally, we wouldhave adjusted more recent and superior node alignment strategies, such as those of MI-GRAAL [22]or GHOST [23], to fit the context of our edge alignment problem. However, generalizing these nodealignment strategies into analogous (and thus comparable) edge alignment strategies is non-trivial,as the current implementations of MI-GRAAL or GHOST either rely on proprietary libraries orare too complex to be extended in any way, respectively. After generating an edge alignment, GREAT continues onto calculating pairwise node scores basedon this alignment, as follows. Let v be a node in graph G and let v be a node in graph G . Let E ′ be the set of aligned edges and let sim ( v , v ) be the similarity between v and v . Then, GREAT’sedge alignment-based node cost function sim ( v , v ) is the sum of similarities (with respect to edgecost function) over all edges in E ′ (Appendix Figure A.1).After GREAT generates node cost function as above, it feeds the resulting node scores intoan existing node alignment strategy to generate an injective node between the two. Here, we usethree such strategies: 1) greedy alignment strategy, 2) Hungarian algorithm for maximum weightbipartite matching, and 3) MI-GRAAL’s alignment strategy. Again, we would have used a morerecent node alignment strategy, such as GHOST’s, but the current implementation of GHOST istoo complex to allow for feeding into its alignment strategy any node cost function but its own. By mixing and matching different options for GREAT’s edge and node alignment strategies, asdiscussed above, we get a total of six different GREAT variations, i.e., aligners (Table 1).4REAT variation Edge alignment strategy Node alignment strategyEGG Greedy GreedyEHG Hungarian GreedyEGH Greedy HungarianEHH Hungarian HungarianEGM Greedy MI-GRAALEHM Hungarian MI-GRAALTable 1: Six variations of GREAT, depending on which edge and node alignment strategy it uses.
To test GREAT’s performance, we use a popular evaluation test [15, 16, 22, 23, 28, 3, 4, 31].Namely, we focus on a high-confidence yeast PPI network with 1,004 proteins and 8,323 PPIs [41],and we produce five additional “synthetic” networks by adding noise to the yeast network. Thenoise is the addition to the original yeast network of x % of low-confidence PPIs from the samedata set [41], where we vary x from 5% to 25% in increments of 5%. We align the original yeastnetwork to each of the synthetic networks with x % noise, resulting in the total of five network pairsto be aligned. Importantly, since all network pairs have the same set of nodes, we know the truenode correspondence (i.e., mapping). Thus, for each considered method, we can measure how wellthe method reconstructs the correspondence, along with evaluating the method’s alignment qualitywith respect to some other measures (Section 2.5).We note the main focus of our paper is a fair evaluation of our new GREAT method againstthe existing methods. If we aimed to predict new biological knowledge, we would have appliedour method to additional networks, such as PPI networks of different species. However, since ourmain focus is method evaluation, we focus on the above network data set because: 1) the originalyeast network is of high confidence and thus trustable; 2) the data encompasses different PPItypes, including PPIs obtained via affinity purification followed by mass spectrometry (AP/MS),and as such is of high coverage, 3) the same data has already been actively used for evaluation ofdifferent network aligners, and 4) we know the true node mapping as well as the actual level ofstructural difference (corresponding to the given percentage of the low-confidence PPIs) betweeneach pair of aligned networks, and hence, we can meaningfully interpret our alignments (wherethis is not the case for networks with unknown node mapping or unknown structural difference)[15, 16, 22, 23, 3, 28, 4]. Ultimately, what matters for a fair evaluation is that all compared methodsare tested on the same data, which is exactly what we do [4]. Let G = ( V , E ) and G ( V , E ) be graphs such that V ≤ V . Let E ′ be the set of edges in G thatexist between the set of nodes in G that are aligned to nodes in G . Then, we measure alignmentquality with respect to the following well-established measures [15, 16, 22, 23, 28, 3, 4, 31]. Node correctness (NC): If f : V → V is the correct node mapping of G to G and h : V → V is an alignment produced by the given method, then N C = |{ u ∈ V : f ( u )= h ( u ) }|| V | × Symmetric substructure score (S ) : Although NC captures the amount of true node mapping, it isstill important to measure the amount of conserved edges. For example, if we map an n -node clique(complete graph) in one network to an n -node clique in another network, there are many possibletopologically correct alignments with respect to S , i.e., alignments that conserve all edges, but5here is a single correct alignment with respect to NC. Plus, true node mapping is not known formost real-world networks; in such cases, NC can not be computed and one needs to rely on measuresof edge conservation. One such measure is S , defined as the percentage of conserved edges out of alledges in E and E ′ combined. More formally, it is defined as follows: S = | E ∩ E ′ || E | + | E ′ |−| E ∩ E ′ | × combines the advantages of both of these measures while addressingtheir drawbacks, and as such, it has been shown to be the superior of the three measures [28].The size of the largest connected common subgraph (LCCS) [15], which we use because of twoalignments with similar S scores, one could expose large, contiguous, and topologically complexregions of network similarity, while the other could fail to do so. Thus, in addition to countingaligned edges, it is important that the aligned edges cluster together to form large connectedsubgraphs rather than being isolated. Hence, a connected common subgraph (CCS) is defined asa connected subgraph (not necessarily induced) that appears in both networks [16]. We measurethe size of the largest CCS (LCCS) in terms of the number of nodes as well as edges. Namely, wecompute the LCCS score as in our recent work [28]. First, we count N , the percentage of nodesfrom G that are in the LCCS. Then, we count E , the percentage of edges that are in the LCCSout of all edges that could have been aligned between the nodes in the LCCS. That is, E is theminimum of the number of edges in the subgraph of G that is induced on the nodes from theLCCS, and the number of edges in the subgraph of G that is induced on the nodes from the LCCS[28]. Finally, we compute their geometric mean as p ( N × E ), in order to penalize alignments thathave small N or small E . Large values of this final LCCS score are desirable. We aim to answer the following: What parameter values to use within GREAT’s edge costfunction to optimally balance between edge-GDV-similarity and edge-GDC (Section 3.1)? Doesedge-based network alignment improve upon comparable traditional node-based network alignment(Section 3.2)? This is the main goal of our study, and achieving it would be sufficient to demonstrateGREAT’s superiority over the traditional methods. Given that we will demonstrate that edgealignment does improve over node alignment, which of the two edge alignment strategies (greedyversus optimal) to use to achieve both high accuracy and low computational complexity (Section3.3)? That is, can we still achieve with fast greedy edge alignment similar accuracy as with sloweroptimal edge alignment? This has important implications for processing large real-world networks. Can GREAT, our edge-based network alignment method, not only beat comparable node-basednetwork alignment methods (question 2 above) but also the most recent and thus advanced existingnetwork aligners (which would only further confirm GREAT’s superiority) (Section 3.4)?
Recall from Section 2.1 that the α parameter controls the contribution of edge-GDV-similarity andedge-GDC in GREAT’s edge cost function. When we test its effect on a comprehensive network set(see below), we find that overall, α of 1 is superior (although α of 0.8 performs relatively well too)(Figure 2). That is, edge-GDV-similarity is overall favored over edge-GDC. We note that this wasnot the case with a comparable node cost function [16], where some contribution of node centralitywas desired, which is why we tested the effect of edge-GDC in GREAT’s edge cost function in thefirst place. Henceforth, in subsequent analyses, we consider only the dominant α of 1.We have performed this analysis on synthetic (geometric and scale-free [36]) networks of differentsizes (we varied the number of nodes from 500 to 1,000, and for a given node size, we varied the6verage degree from four to 12). We have aligned each such network to its noisy counterpart.Here, by noisy counterpart, we mean that in a given synthetic network, we have rewired x % of thenetwork’s edges, where x ∈ { , , , , } . We have done this on the synthetic (geometric andscale-free) network data rather than on our yeast network data from Section 2.4, since we needed totest many different values of α , and doing so on relatively large (dense) yeast networks is more timeconsuming. Also, we wanted to test the effect of α on alignment quality as a function of networksize, and the yeast data does not allow for this. But henceforth, when we test actual alignmentaccuracy, we focus only on the yeast network data.(a) (b) (c)Figure 2: The ranking of the six α values used in GREAT’s edge cost function across all variationsof GREAT and all synthetic network alignments with respect to: (a) NC, (b) S , and (c) LCCS.Recall that α = 1 corresponds to using only edge-GDV-similarity, while α = 0 corresponds to usingonly edge-GDC (Equation 1). Recall that there are different GREAT variations depending on the choices of edge and node align-ment strategies (Table 1). To fairly evaluate whether edge-based network alignment improves uponnode-based network alignment, we benchmark a given variation of GREAT against the compara-ble node-based network alignment method. That is, recall that a given version of GREAT usesedge-GDV-similarity-based edge cost function (corresponding to α of 1), a given (greedy or opti-mal) edge alignment strategy, the edge-alignment-based node cost function, and a given (greedy,optimal (Hungarian), or MI-GRAAL’s) node alignment strategy. Given this, we produce the cor-responding node-based network alignment method as follows: it uses node-GDV-similarity as itsnode cost function and the same node alignment strategy as the given version of GREAT. This way,because edge-GDV-similarity and node-GDV-similarity are as fairly comparable as possible mea-sures of topological similarity of edges and nodes, respectively, and because we are using the samenode alignment strategy in both GREAT and the corresponding node alignment-based method,any difference that we observe between the two methods will be a direct consequence of edge-basednetwork alignment compared to node-based alignment.Consequently, we denote by NG the node-based alignment method that uses the greedy nodealignment strategy and by NH the node-based alignment method that uses the Hungarian nodealignment strategy. The node-based alignment method that uses MI-GRAAL’s node alignmentstrategy is MI-GRAAL itself. Then, we compare each of EGG and EHG to NG, each of EGH andEHH to NH, and each of EGM and EHM to MI-GRAAL (Table 1).Overall, across all network alignment measures, we find that edge alignment is superior to nodealignment (Figure 3 and Appendix Figure A.2). That is, we demonstrate that edge-based networkalignment outperforms node-based network alignment when using comparable cost functions andalignment strategies, which is the main contribution of our study.7a) (b) (c)Figure 3: The ranking of a given edge-based network aligner and the corresponding node-basedaligner across all alignments with respect to all alignment quality measures for: (a) greedy, (b) Hungarian, and (c)
MI-GRAAL’s node alignment strategy. Note that these results are for greedyedge alignment strategy only, but almost identical results are obtained for optimal edge alignmentstrategy, i.e., when comparing EHG with NG, EHH with NH, and EHM with MI-GRAAL.
Overall, aligning edges optimally with the Hungarian strategy is expected to outperform (in termsof accuracy) aligning edges with the greedy strategy. However, Hungarian method is much slowerthan the greedy method, with complexity of O ( x ) for the former and O ( x ) for the latter, where x isthe number of elements (in this case, edges) to be aligned. So, the question is to what extent optimaledge alignment improves compared to greedy alignment, and whether this increase in accuracy isworth the drastic increase in running time.To fairly evaluate this, we compare EGG to EHG, EGH to EHH, to EGM and EHM. In Table 2,we show representative running times and alignment accuracy scores (in terms of S measure) ofEGH and EHH, and in Figure 4 we show systematic results for all GREAT versions while takinginto account all measures of alignment quality. As illustrated, aligning edges with the Hungarianmethod yields to only 0%-14% increase (depending on the network data) in accuracy comparedaligning edges greedily, but it leads to extremely large 5,271%-13,407% increase in running time(Table 2). Further, in the systematic analysis, we find that in 47%-73% of all cases (depending onnode alignment strategy) accuracy of greedy edge alignment is within 5% of accuracy of optimaledge alignment (Figure 4), and the percentages are even higher for being within 10% accuracy(Appendix Figure A.3). Thus, we believe that the huge increase in computational complexity ofoptimal edge alignment does not justify incremental increase in its accuracy. Henceforth, especiallyfor large networks, we suggest aligning edges greedily. (Given the resulting edge alignment-basednode cost function, one still might want to align nodes optimally.) Here, we compare GREAT (the best of all variations) against the following recent powerful align-ers: MI-GRAAL [22], GHOST [23], NETAL [24], and MAGNA [28]. We use default parameters(suggested in the original publications) for all methods. We also tried SPINAL [32], but it did notreturn injective node mappings on our network data as the other methods and was thus excluded.We find that GREAT is overall the best aligner across all alignment quality measures (Figure5 (a) and Appendix Figure A.4). This is especially true in terms of NC, which is the ultimatemeasure of alignment accuracy – GREAT is always the best of all methods (Figure 5 (b)). Interms of S , only MAGNA is the best ranked in more cases than GREAT (Figure 5 (c)). However,8a) (b) (c)Figure 4: The ranking of a given greedy edge aligner and the corresponding optimal edge aligneracross all alignments with respect to all alignment quality measures for: (a) greedy, (b) Hungarian,and (c)
MI-GRAAL’s node alignment strategy, for “within 5% accuracy”. By this, we mean thatthe greedy edge aligner’s score is within 5% of the optimal edge aligner’s score.Alignment EGH EHH Percentage IncreaseTime S Time S Time S Yeast-Yeast5% 3h 49m 24s 91.97% 220h 20m 03s 92.33% 5,271% 0.34%Yeast-Yeast10% 4h 16m 08s 71.91% 432h 29m 14s 74.05% 10,031% 2.98%Yeast-Yeast15% 5h 00m 38s 54.70% 570h 48m 22s 60.13% 11,292% 9.93%Yeast-Yeast20% 5h 07m 06s 45.04% 691h 18m 40s 46.77% 13,407% 3.84%Yeast-Yeast25% 5h 37m 59s 34.45% 755h 36m 06s 39.25% 13,314% 13.93%Table 2: Computational complexity versus accuracy of greedy versus optimal edge alignment. Weshow the amount of CPU time it took GREAT variations EGH and EHH to generate an alignmentand S score of the resulting alignment, for each pair of yeast networks. We measure the increase ineither running time or accuracy of EHH (i.e., optimal edge alignment) over EGH (i.e., greedy edgealignment) as the difference of the result of EHH and the result of EGH, divided by the result ofEGH. All alignments were ran on the same server with 16 2.3GHz processors and 24GB of RAM.this is not surprising, as MAGNA directly optimizes S during alignment construction and is thusexpected to dominate the other methods with respect to this measure. Nonetheless, GREAT stilloutperforms MAGNA is 40% of the cases with respect to S . Finally, in terms of LCCS, GREAT isagain superior to almost all methods, including MAGNA (Figure 5 (d)). Exceptions are GHOSTand NETAL, but GREAT still performs comparably to these methods, in the sense that all threemethods rank as the best or the second best in equal number (60%) of all cases.Thus, we do not only demonstrate that GREAT is superior to fairly comparable node-basednetwork alignment methods, which differ from GREAT in a single aspect (edge versus node align-ment), but also, it is superior to the recent state-of-the-art network alignments, which differ fromGREAT in more than one aspect. As such, incorporating into the design of GREAT the recentmethods’ algorithmic ideas could potentially improve GREAT’s performance even further. We have presented GREAT, our novel alignment method that aims to maximize both node andedge conservation, that does so by first aligning edges well in order to then align nodes well based ontheir adjacent edges being aligned well, and that in the process favors similar conserved edges overdissimilar conserved edges. We have demonstrated that GREAT, the edge-based network align-9 a) (b)(c) (d)
Figure 5: The ranking of GREAT (the best of all variations) and very recent and thus advancedexisting network aligners over all alignments with respect to: (a) all alignment quality measurescombined, (b)
NC, (c) S , and (d) LCCS.ment method, improves upon comparable node-based network alignment methods, confirming ourhypothesis that aligning edges prior to aligning nodes would improve alignment quality comparedto aligning nodes only. In other words, we have demonstrated superiority of GREAT over methodsthat aim to maximize node conservation only, such as MI-GRAAL and GHOST. At the same time,we have demonstrated superiority of GREAT over a recent approach that aims to optimize edgeconservation only and that treats each edge the same, namely MAGNA. Finally, we have shownthat GREAT overall outperforms an additional recent state-of-the-art approach, namely NETAL.Thus, GREAT (and its modified version that would also account for functional, e.g., proteinsequence, similarities between nodes in addition to their topological similarities) has importantimplications for real-world applications of network alignment to biological networks of differentspecies, as well as to networks in other domains, such as social networks or natural languageprocessing. For example, in computationally biology, GREAT can be used to transfer aging-relatedknowledge from well-annotated model species to poorly-annotated human, thus deepening ourcurrent knowledge about human aging [3, 4, 5]. Or, it could have implications on user privacy inonline social networks, as network alignment can be used to de-anonymize such network data [2].
Acknowledgements
We thank Dr. Horst Bunke for useful discussions regarding GREAT, and Dr. Seyed ShahriarArab for his assistance with running NETAL. This work was supported by the National ScienceFoundation CCF-1319469 and EAGER CCF-1243295 grants.10 eferences [1] J. Li, J. Tang, Y. Li, and Q. Luo. RiMOM: A dynamic multistrategy ontology alignment framework.
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GREAT: GRaphlet Edge-based network AlignmenT
Joseph Crawford and Tijana Milenkovi´c ∗ Department of Computer Science and Engineering, Interdisciplinary Center for Network Science and Appli-cations, and ECK Institute for Global Health, University of Notre Dame ∗ Corresponding Author (E-mail: [email protected])
Figure A.1: Illustration of GREAT’s edge alignment-based node cost function. To measure sim-ilarity between two red nodes in networks A and B, GREAT first identifies all edges adjacent tothe red nodes that are aligned to each other, along with their similarity scores with respect to edgecost function. In this case, let us assume that red edge in A is aligned to red edge in B with scoreof 0.9, blue edge in A is aligned to blue edge in B with score of 0.8, and green edge in A is alignedto green edge in B with score of 0.7, while all black edges in the larger of the two networks, i.e.,network B, are unaligned. Then, GREAT’s edge alignment-based node cost function is the sum ofsimilarities (with respect to edge cost function) over all aligned edges.Appendix Page 1igure A.2: Alignment quality results of all six variations of GREAT edge-based network alignmentmethod and the corresponding three node-based network alignment methods, for each of the fivenetwork pairs and with respect to each of the three alignment quality measures.(a) (b) (c)Figure A.3: The ranking of a given greedy edge aligner and the corresponding optimal edge aligneracross all alignments with respect to all alignment quality measures for: (a) greedy, (b)
Hungarian,and (c)(c)