General: Bridge and Brick Network Motif-Detecting Algorithm

As shown in Figure 7, a link-weighted value that is dependent on the number of all possible paths between two linked nodes equals the summation of the reciprocal values of all possible path lengths except for the link itself. This is expressed as



and length(pathi (a, b)) ≤ average network diameter. The length of one path represents its total number of nodes.

ShortestPath(a, b) = Min(length(path_i (a, b)))

Fig. 7. Link-weighted value calculating example. The link-weighted value weight

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(a, b) of edge (a, b) is 0 while weight (b, c)

This definition implies clustering, with any increase in the number of possible paths resulting in an increase in the clustering degree between two linked nodes.

Furthermore, the concepts and algorithms discussed in this dissertation are

generalizable to non-directed networks. To ensure that the proposed method can be applied to any complex network, the link-weighted values calculated by the network motif detection method are derived from the number of all possible paths between two linked nodes within all network topological and local connection structures (no preset link quantity). This definition is similar to that of betweenness [43, 45]—effects resulting from the removal of network links. Accordingly, the proposed link-weighted value calculation method is assumed to represent the importance of each link in a real network [46, 47].

Also considered were the interactive strengths of individual links in a quantitative real network. To validate the proposal for weighted links, they were compared with

quantitative links. However, interactive quantitative links are defined by

category-specific functions such as proteins, genes, species, and so on. It is difficult to specify the overall impacts of these links on protein-protein interaction networks [48]

and food webs. For example, the number of links between tigers and wild oxen does

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not reflect the significance of their connection within an overall food web.

Furthermore, each complex network type has its own measure for interactive strength.

A switching algorithm (i.e., AB, CD becomes AD, CB if AD and CB

do not exist) was used to create random networks according to any given degree sequence [16, 26]. Results from previous studies indicate that these random networks have the same number of nodes and edges, as well as node in-degrees (incoming edges) and out-degrees (outgoing edges) that are identical to those of real networks.

Furthermore, randomized networks preserve the same number of appearances of all (n-1) node subgraphs as in real (original) networks [16]. The threshold that

determines the strength of an edge (link) is the mean weighted value of all edges in a

random network ensemble. Accordingly, 1,000 random networks were generated to serve as a control. Edges were labeled ―weak‖ when their weighted values in these or

real networks were smaller than the threshold minus a double standard deviation (p = 0.01); all other edges were labeled ―strong.‖ Researchers can define criteria for strong

and weak links according to their own needs. Finally, all possible motifs were located,

and their distributions in real and random networks were compared.

Milo et al.‘s method [16] for identifying bridge and brick motifs in complex networks

was expanded to include the following steps:

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1. Calculate the weighted value of each link in a network of interest and an ensemble of random networks to calculate the significance of n-node subgraphs. The goal is to maintain the same number of appearances for all (n – 1) node subgraphs as in the original network.

2. Label all weighted links in the network of interest and random network ensemble as

―strong‖ or ―weak‖ according to a benchmark of two standard deviations from the

mean weighted value of all links in the ensemble. Links with weighted values below the benchmark are considered weak.

3. Identify all n-node bridge/brick subgraph types in the network of interest and random network ensemble.

4. Mark all n-node bridge/brick subgraph types by calculating their numbers in the network of interest and random network ensemble. Each n-node bridge/brick

subgraph type is selected as a representative motif only if its frequency in the network of interest far exceeds its frequency in the ensemble.

These steps can assist research efforts to understand the functions and roles of identified motifs in a real network and to analyze the dynamic behaviors of complex networks. Regarding method robustness, the proposed approach emphasizes the global and local topological properties of each real network rather than the specific

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functions of different network types.

Motif frequency can be used to measure levels of similarity between two networks of

interest. In addition, it is possible to calculate the Z-scores for all bridge/brick motifs and significance profiles (SPs) in a network by expanding Milo et al.‘s [26, 49, 50]

methods. As shown in the following formula, ZScore(Bridgei) represents the statistical

significance of the i^th kind of bridge motif in a network:

( ) ( )

where Nreal(Bridgei) represents the time of appearance of the i^th type of bridge motif in a network, and <Nrandom(Bridgei)> and STD(Nrandom(Bridgei)) respectively represent the mean and standard deviation of the time of appearance of the i^th type of bridge motif in a randomized network ensemble. In the next equation, SP(Bridge_i) is the vector of Z_Score(Bridge_i) normalized to a length of 1. This normalization emphasizes the relative significance of the i^th type of bridge motif rather than the absolute significance. Z_Score(Brick_i) and SP(Brick_i) can be derived in the same manner:





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2.1.2 Specific: Bridge and Brick Network Motif-Detecting