Experiments

The proposed method was applied to several biochemistry (transcriptional gene regulation) and ecology (food web) networks to identify bridge and brick network

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motifs. Networks and sources are listed in Table 3. All data and programs (including source code) are available online at ftp://www.csie.cgu.edu.tw/bbm/.

Several engineering (electronic circuit) and social networks (Table 3) were used to

demonstrate that the proposed motif detection method is both general-purpose and robust. It was also compared with Milo et al.‘s [26] original method for complex

network analysis. In electronic circuits consisting of digital fractional multipliers (data from an ISCA89 benchmark) [26], nodes represent logic gates and flip-flops and edges represent directed electronic transmission paths. Experimental results indicate that the s208, s420, and s838 electronic circuit networks contain significant numbers of bridge motifs. Here the low degree of clustering is considered trivial because designers often try to simplify connection structures and numbers of electronic components [77]. The identified feedback bridge motif (consisting of weak-tie links only) fulfills this requirement as described by Kundu et al. [95] (ID = 9) (Figs. 14, Table 3). As its name implies, the feedback bridge motif indicates the existence of a feedback structure without redundancy in the three above-named electronic

circuits—again proven by Kundu et al. [95], who also reported that redundant circuits seldom appear in simple electronic circuits such as s208, s420, and s838. However, they also note that redundant wires and components are frequently added to more

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complex electronic circuits (e.g., s15850, s35932, s38417 and s38584) to prevent accidental system failures. The over-simplification of electronic circuits can result in large numbers of errors [77] or complete system breakdowns when one component fails. Accordingly, it is necessary to add an appropriate level of redundancy as a means of bypassing failed components or substituting for the original path [77, 95].

Strong-tie links represent alternative paths and weak-tie links represent simplified electronic circuits. Combined, simplification and duplication help prevent unexpected system breakdowns.

Fig. 14. Bridge motif ratio profiles for three electrical circuits (s208, s420 and s838).

In the two social networks that were analyzed, nodes represent individuals in a group and edges represent positive sentiments directed from one group member to another based on responses to questionnaire items. Similar characteristics were found between

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two networks, one consisting of prison inmates (N = 67 nodes, E = 110 edges) and the

other of college students in a leadership course (N = 32, E = 96). The inmates responded to the question, ―Who are your closest friends in your cellblock?‖ The

students were asked to name three classmates they would invite to serve on a

committee (correlation coefficient c = 0.92 to 0.96 [96, 97]). According to Milo et al.‘s [16] methods, both social networks belong to the same superfamily. Strong

similarities between the two networks were also identified according to the triad significance profile (TSP) of bridge motifs (c = 0.92) (Fig. 15, Table 3), but not according to the TSP of brick motifs (c = 0.6) (Fig. 16, Table 3). Also found was a significantly higher number of bridge motifs (i.e., more ―nodding acquaintances‖) in the prisoner network. The significantly larger number of brick motifs in the leadership class network indicates that small, strong groups are easily formed. The bridge and brick motifs can be used to further analyze network topological structures, functions, and differences.

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Fig. 15. Bridge motif ratio profiles for two social networks.

Fig. 16. Brick motif ratio profiles for two social networks.

In gene regulation networks for one bacteria (Escherichia coli) and one eukaryote (the yeast Saccharomyces cerevisiae) [26], each node represents a gene or operon that encodes a transcription factor (TF); edges denote the TFs themselves. Many TFs are encoded within operons, therefore directed links represent direct transcriptional modulation from a TF to an operon or from a TF-contained operon to another operon [26]. More bridge than brick subgraphs were found in both networks (they are not called motifs until they reach statistical significance). Furthermore, the two

transcription networks had the same feed-forward bridge motif (ID = 5), indicating that the transcription networks have, at minimum, non-replaceable interactions without intermediate interactions with other genes (Fig. 17 and Table 3). This suggests that the weak-tie link that provides a unique path for controlling the signal

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exerts a significant impact on the signal processing function of transcription networks [26, 80]. When analyzing the relationship between coherent (incoherent) FFLs and brick (bridge) FFLs, I identified E. coli‘s 34 coherent and 8 incoherent FFLs (Table 3) [34, 98, 99]. Accordingly, differences in coherent (incoherent) FFL frequencies cannot be explained simply in terms of the relative abundances of bridge and brick motifs in a network.

Fig. 17. Brick-bridge motif ratio profiles for two regulation networks (one bacteria and one eukaryote).

In the seven analyzed food webs [100], nodes represent groups of species and edges connect predator and prey nodes. Two studies have shown that strong interactions (similar to the definition of weak-tie links used here) between two consecutive levels of a trophic chain have a significant effect on food web stability and dynamics [38, 101]. A strong interaction indicates a strong predator preference for one prey species and a low potential for intermediate species—a phenomenon that supports the

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proposal that weak-tie links exert certain impacts on food webs. Also in the seven food webs, the numbers of bridge motifs were significantly higher than the numbers

of brick motifs, especially feedback (ID = 5) and three-point chains (ID = 2) (Fig. 18, Table 3). This confirms Jordi‘s [38] claim that these two motifs exert significant

impacts on ecosystem food webs. The reason why ecosystems containing these two types of bridge motifs easily become unbalanced is likely because they have many weak links—in other words, it is difficult to find substitute nodes or links for the purpose of preserving ecosystem stability.

Fig. 18. Bridge motif ratio profiles for seven food webs.

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Table 3. Brick and bridge motifs in fourteen real world networks, including edge and node definitions, network sizes, and references.

Category Common Feature

According to the definitions of weighted links and network motifs used in this study and the results of the validation experiments using theoretical complex networks, the presence of bridge and brick motifs in a network is closely associated with network topological structures (especially local connections), but not with network size (i.e., number of nodes). In summary, three experimental predictions were tested to verify