COMICS: a community property-based triangle motif clustering scheme

System model and problem formulation

In this section, we present basic theoretical definitions about cutting conditions and the mathematical expressions of motifs. After that, we describe the investigated problems in details.

Comparative conditions for cluster

In this subsection, we introduce four conditions to partition the original large collaboration networks into different clusters.

For a given graph G = (V, E), we define the adjacent matrix H = {h_i,j} as: if there exists an edge between vertices i and j, h_i,j = 1; otherwise, h_i,j = 0.

Network partition is defined as: $\text{𝒫}=\{G_1,G_2,\dots ,G_k\},1\le k\le \left|V\right|$, subject to: (1) ${\cup }_{k=1}^K{G}_k=V$, (2) $G_k{{\displaystyle \cap }}^{\text{}}{G}_t=\varnothing$, ∀k ≠ t, and (3) $G_k\ne \varnothing ,\forall k$. For ∀k, the partition 𝒫 satisfies the x—valid condition, called x—valid cluster partition of G, and x is defined as the following conditions according to Lu et al. (2013): (1) $$\text{Condition\hspace{0.17em}}1:{\displaystyle \sum _{\forall j\in G_k}}{H}_{i\mathrm{,}j}>{\displaystyle \sum _{\forall j\in V\backslash G_k}}{H}_{i\mathrm{,}j}\mathrm{,}\forall i\in G_k\mathrm{,}\forall k\mathrm{.}$$ (2) $$\text{Condition\hspace{0.17em}}2:{\displaystyle \sum _{\forall j\in G_k}}{H}_{i\mathrm{,}j}>{\displaystyle \sum _{\forall j\in G_t}}{H}_{i\mathrm{,}j}\mathrm{,}\forall i\in G_k\mathrm{,}k\ne t\mathrm{.}$$ (3) $$\text{Condition\hspace{0.17em}}3:{\displaystyle \sum _{\forall j\in G_k}}{H}_{i,j}>{\displaystyle \sum _{\forall i\in G_k\mathrm{,}\forall j\in V\backslash G_k}}{H}_{i,j}\mathrm{,}\forall k\mathrm{.}$$ (4) $$\text{Condition\hspace{0.17em}}4:{\displaystyle \sum _{\forall j\in G_k}}{H}_{i,j}>{\displaystyle \sum _{\forall i\in G_k\mathrm{,}\forall j\in G_t}}{H}_{i,j}\mathrm{,}\forall k\ne t\mathrm{.}$$

Conditions 1 and 2 check the validity of clusters at the vertex level to confirm whether the internal degree of each vertex is larger than that of the external degree. Conditions 3 and 4 check the validity of clusters, that is, comparing the total internal degree of each cluster. When large graphs are partitioned under the above-mentioned four conditions, Condition 1 generally results in fewer, but larger subgraphs; Condition 4 will lead to more and smaller communities; Conditions 2 and 3 will cause more and smaller communities than Condition 1, but fewer and larger communities than Condition 4.

Definitions of network triangle motifs

In real networks, the most common high-order structures are small network subgraphs, which are defined as motifs, that is, a set of edges with a small number of nodes. In this paper, we analyze undirected triangular motif-based networks.

Formally, we define a triangle motif by a tuple (B, A), where B is a 3 × 3 binary matrix and $A\subset \{1,2,\cdots \mathrm{,}n\}$ is the set of anchor nodes. The matrix B encodes the edge pattern between the three nodes in triangle motifs, and A represents a relevant subset of nodes to define motif conductance. Then, let χ_A be a selection function, taking the subset of a 3-tuple induced by A. Define set (·) as the operator, which takes a tuple to a set, set (v₁, v₂, v₃) = {v₁, v₂, v₃}. Then, the motif set of an unweighted and undirected graph with adjacency matrix A can be denoted by Eq. (5), where v₁ ≠ v₂ ≠ v₃, (5) $$\text{Tri}(B,A)=\{\text{set}(v),\text{set}({\chi }_A(v))|v\in V^k,A_v=B\}.$$

Here, A_v is the k × k adjacency matrix of the subgraph with k nodes of the order vector v. In this paper, the motifs are undirected and unweighted. The matrix B of motif is symmetrical. Hence, we use (v, χ_A(v)) to denote (set (v), set (χ_A(v))) for convenience. Furthermore, we regard any (v, χ_A(v)) ∈ Tri (B, A) as a motif instance. If A and B are arbitrary or clear from context, we simply denote the motif set by Tri. We define the motifs, that is, χ_A(v) = v, as simple motifs, and others are anchor motifs.

We give an example of triangle motif definition, as shown in Fig. 1, aiming to cluster the given five-node network by the two triangle motifs. First, we define the motifs by the description in Eq. (5). For motif Tri₁, there are two instances of the motifs in G. Meanwhile, for motif Tri₂, G has three instances, and the anchor sets of each instance is the node whose degree is one.

The definition of the triangle motif conductance replaces an edge with a motif instance of type Tri. We suppose that a given network has been clustered into two subnetworks, that is, g and $\overline{g}$, and the conductance based on motifs can be expressed in Eq. (6), (6) $${\psi }_{\text{Tri}}^{(G)}(S)=\frac{(cu{t}_{\text{Tri}}^{(G)}(g\mathrm{,}\overline{g}))}{min(vo{l}_{\text{Tri}}^{(G)}(g),vo{l}_{\text{Tri}}^{(G)}(\overline{g}))}\mathrm{.}$$

When there is at least one anchor node in S and at least one anchor node exists in $\overline{g}$, a motif instance can be cut. In Eq. (6), $cu{t}_{\text{Tri}}^{(G)}(g\mathrm{,}\overline{g})$ is the number of instance cut. $vo{l}_{\text{Tri}}^{(G)}(g)$ is the number of instances, whose end nodes are in g. To be more specific, following the definition of Tri in Eq. (5), as for the same χ_A(v), there may exist many different values of v, and nodes in χ_A(v) are still counted proportionally into the number of motif instances. This growth tendency of motifs is consistent with the number of nodes in networks. This can prove the availability of motifs in clustering networks.

Definition of motif-base matrices

Given a graph and a set of motif Tri, the motif adjacency matrix W_Tri of graph is shown as: (7) $${(W_{\text{Tri}})}_{ij}={\sum }_{((v\mathrm{,}{\chi }_A(v))\in \text{Tri})}1(\{i,j\}\subset {\chi }_A(v)|i\ne j).$$

Herein, (W_Tri)_ij is the number of motif instances in M, where nodes i and j are both in a triangle motif. In other words, the weight will be added into (W_Tri)_ij if and only if node i and node j both appear in the anchor set. In the collaboration networks, (W_Tri)_ij depends on the number of scholars, who collaborate with both scholar i and scholar j.

Then, the motif diagonal degree matrix D_M is defined as ${(D_{\text{Tri}})}_{ii}={\sum }_{j=1}^n{(W_{\text{Tri}})}_{ij}$. The motif Laplacian can be calculated by L_Tri = D_Tri − W_Tri. Finally, we normalize the motif Laplacian as: (8) $${\Gamma }_{\text{Tri}}=I-D_{\text{Tri}}^{-1/2}{W}_{\text{Tri}}{D}_{\text{Tri}}^{-1/2}.$$

Problem statement

Let G = (V, E) be a connected large network, where V is the node set, and E is the edge set. If G contains several disjoint networks, it can be expressed as G = {G₁, G₂, ⋯, G_n}. The complete triangle is the target motif to analyze the large-scale networks. Our objective is to find the dense and stable disjoined subgraph set $\text{𝒫}=\{G{″}_1,G{″}_2,\dots ,G{″}_m\}$ of the given network by motifs.

Given a node v ∈ V in the network G_i ∈ G, the degree of v is denoted by deg(v) and the neighbor node set of the subgraph G_i in the original networks is denoted by $N_{G_i}$. In the partition phase, G_i can be cutted into a set of subgraphs, $\left\{{G^{\prime }}_{i1},{G^{\prime }}_{i2},\dots ,{G^{\prime }}_{ik}\right\}$. For a node v in a partition subgraph ${G^{\prime }}_{ij}$ of G_i, we use deg_inter(v) and deg_extra(v) to represent the degree within ${G^{\prime }}_{ij}$ and the number of edges between ${G^{\prime }}_{ij}$ and $G_i/{G^{\prime }}_{ij}$, respectively. Con(${G^{\prime }}_{ij}$) is the set of subgraphs that are connected with ${G^{\prime }}_{ij}$ in the partition 𝒫. Variable Q is the modularity score when the original graph G gets the partition 𝒫. In the process of graph partition, we cut the original networks into initial subgraphs under the four conditions. In that way, the network can be cutted into subgraphs with strong internal connectivity and weak external connectivity in both local and global aspects. The modularity score refining subprocedure can optimize the partition. Then, we cluster and analyze the initial dense subgraphs by the complete triangle motif.

COMICS algorithm

In this section, we describe the whole process of our COMICS in details. As shown in Fig. 2, COMICS consists of a series of partition refine strategies and a motif-based clustering procedure, that is, graph partition, modularity refine procedure and motif-based clustering procedure. We first illustrate the graph partition techniques under four conditions and the modularity refine procedure in Algorithm 1. After that, the motif-based clustering procedure is constructed on each subgraph in cutting set. We specify the whole clustering layer algorithm in Algorithm 4, by which we are able to get the close and stable subgraph structures from the original input networks.

Graph partition and modularity refine procedures

To obtain the total information of large networks effectively, we first perform cutting operations in large networks. In this subsection, we explain the graph partition and modularity refine procedures in details. We use the total large graph as the input of the partition procedure, and the procedure returns a set of partition subgraphs of the original graph. The subgraph set is refined in the modularity refine procedure by modularity score. In the graph partition procedure, we take the differences between the internal and external degrees as the degree difference value of a node v, denoted by D(v), that is, (9) $$D(v)=de{g}_{inter}(v)-de{g}_{extra}(v).$$

For all pairs of nodes v and u in networks, if nodes v and u fall in the same subgraph, the quantity of s_vs_u is 1, otherwise, it equals to −1. |E| is the total number of edges in the original network. The value of e_{v, u} is 1, if there exists one edge between nodes v and u, otherwise it is 0. Therefore, the modularity score of a network is defined as Eq. (10): (10) $$Q=\frac{1}{4\left|E\right|}{\displaystyle \sum _{v,u}}(e_{v,u}-\frac{deg\left(v\right)deg\left(u\right)}{2\left|E\right|})(s_v{s}_u+1).$$

As described in Algorithm 1, we take the large networks and the cutting conditions as the input of the graph partition procedure. At the beginning of this procedure, there is only one original graph added in the result partition set R. The number of subgraphs in R is represented as |R|. In each outer loop, each subgraph G_i is chosen to generate new subgraphs, ${G^{\prime }}_{ij}$. Hence, the root node of the new subgraph ${G^{\prime }}_{ij}$ is selected randomly among the nodes that are with the minimum degree of the new subgraph in R. As described in lines 6–12, the loop aims to generate the new graph from the root node from G_i. If at least one neighbor node of the total new subgraph satisfies the validity of the input conditions, the neighbor node is added to the new subgraph ${G^{\prime }}_{ij}$ with its connectivity; otherwise, it means that there is no neighbor with this root node, and we select a new node in G_i as the root to check the connectivity with the original network. The node with the minimum difference between the internal and external degrees will be the root node of ${G^{\prime }}_{ij}$. Then, graph partition operations will be carried until no other nodes from the original networks satisfy the cutting condition.

In line 3 of Algorithm 1, there are two cases when selecting a new root node of the new subgraph: one is that the root node selected in line 4 or line 9 has no neighbor, that is, the degree of root node is 0. It indicates the root node v_root is invalid, and no new subgraph can be added into R. The other situation is that there is no node in G connected with the partition subgraph. In other words, the iteration of adding nodes to the new subgraph stops. Then a new subgraph generated by the root node v_root is added to the subgraph set R. The iteration of the partition procedure stops when the number of subgraphs in R does not increase any more. However, if there exists one graph in R all the time, it illustrates that the root node with the minimum degree is invalid, and we have to choose another root node and restart the iteration.

Algorithm 1 cuts large graph into small dense subgraphs. Each loop generates a new subgraph from set R, and the loop stops when no more dense subgraph can be found. To avoid damaging the connectivity of the rest nodes, we check the connectivity of both cutting and remaining parts, if there exists more than one component, we put all subgraphs in set R to the modularity refine procedure in Algorithm 2.

The modularity refine procedure described in Algorithm 2 takes the results of the Algorithm 1 as input to refine the partition results of the original networks. As shown in Algorithm 2, lines 1 and 2 enumerate two connected subgraphs: ${G^{\prime }}_{ij}$ and ${G^{\prime }}_{ik}$ in R. In the following line 3 to line 6, if the two subgraphs are combined to one, it results in much higher modularity score than the original network partition. Then we replace ${G^{\prime }}_{ij}$ and ${G^{\prime }}_{ik}$ by ${G^{\prime }}_{ij}\cup {G^{\prime }}_{ik}$, and the iteration stops until the modularity scores do not increase any more. Variable R₀ is the result set of Algorithm 2, containing a series of refined subgraphs. This procedure of graph partition aims to maintain more structure information of the original large-scare networks, so that the output partition by Algorithm 2 can achieve higher modularity score than the input subgraph set. The operation of merging these two cutting subgraphs increases the internal degrees. Merging two subgraphs into one can also decrease the external degree for the subgraphs in partition 𝒫. If two subgraphs can be merged, the number of edges between them is larger than that cannot be merged. Then as long as the two subgraphs are merged, the external degree of the input network can be reduced.

Triangle motif-based clustering procedure

We take the subgraph set of the original network as the input of the motif-based clustering procedure. As shown in Algorithm 3, its main idea is to find a single cluster in a graph by leveraging target motifs. In this procedure, we cluster the subgraphs in R by the minimum conductance, aiming to find the most stable part with the highest conductance of the given subgraph. The algorithm outputs a partition of nodes in g and $\overline{g}$. The motif conductance is symmetric in the sense that ${\psi }_{\text{Tri}}^{(G)}(G_{\text{Node}})={\psi }_{\text{Tri}}^{(G)}({\overline{G}}_{\text{Node}})$, so that any set of nodes (g and $\overline{g}$) can be interpreted as a cluster. However, it is common that one set is substantially smaller than the other in practice. We take the larger set as a module in the network. Some networks are clustered for specific motivations, such as mining the relationships of a person in the social networks. In that case, Algorithm 3 takes the larger part in the clustering results g and $\overline{g}$ as the cluster results as shown from line 9 to line 12.

In the process of the motif-based clustering, we take the target motif and a subgraph partitioned in R (the output of Algorithm 2) as the input. As shown in Fig. 3, for a given graph and a target motif, we calculate a series of matrices, that is, W_Tri, D_Tri, and ${\Gamma }_{\text{Tri}}$, before weighting the input graph of matrix W_Tri. Therefore, the graph is cut by the minimum conductance ${\psi }_{\text{Tri}}^{(G)}$ expressed in Eq. (6). In line 8 of Algorithm 3, the value of ${\psi }_{\text{Tri}}^{(G)}$ is determined by a series of sorted eigenvalues of the subgraph’s motif Laplacian matrix ${\Gamma }_{\text{Tri}}$. Between the two cut subgraphs of the input graph, the larger one will be chosen as the output result.

Combined COMICS algorithm

In this subsection, we describe the overall algorithm of COMICS in Algorithm 4. It combines all three subprocedures in this subsection.

We take the large-scale networks, target motifs and the given validity conditions as the algorithm input, and obtain the clustering set of the original input network. At beginning, a series of partition and refining operations are carried on input networks under the valid conditions. Then we get a partition with high modularity scores of the original input large networks. Each subgraph in the partition set has a strong internal connectivity and a weak external connectivity, maintaining the stable structure information of the original network. Furthermore, we carry the motif-based clustering operations on the subgraph in the partition set. Finally, we can get the non-overlapping optimal partition of the original graph.

Time complexity analysis

In this section, we analyze the time complexity of COMICS. The main clustering layer includes the following three phases: graph partition, graph refining and motif-clustering. We assume m and n as the number of edges and nodes in the network. Here, d(v) is the degree of node v and d_max is the maximal degree in the network.

Graph partition procedure: To get and parse the information from large networks effectively, we first apply cutting operations on the large-scale networks. We analyze the worst case of the graph partition subprocedure. In the first cutting iteration, the root node is one of the nodes with the smallest degree in the graph, and its time complexity is O(n). For each new subgraph g generated by root node v, we check the connectivity of the new-added nodes, including the internal and external links with subgraph g and check the corresponding conditions before partitioning. The time complexity of this procedure is $O(n^2{d}_{max}^2)$.

Modularity refining procedure: In the subgraph refining procedure, we use modularity score Q (Newman, 2006) to get a suitable partition. The required time of iterations is up to the number of subgraphs in the result sets R, which are generated by the graph partition procedure. We define the number of subgraphs in R as p, and the runtime of the procedure is determined by the step of calculating Q, whose computation complexity is O[(m + n)n]. Because the first refining subgraph need to check the other p − 1 subgraphs and the second one checks the remaining p − 2. Hence, the total checking times is p²/2. Therefore, the computation complexity of the refining procedure is O[p²(m + n)n/2].

Triangle motif-based clustering procedure: In general, the time complexity of the algorithm is determined by the construction of the adjacency matrix and the solution of the eigenvector. For simplicity, we consider that we can access network edges within O(1), and modify matrix entries within O(1). The complexity of calculating the eigenvectors through Laplacian matrix is O((m + n) (log n)^O(1)), and sorting the eigenvector indexes can be finished in time O(n log n). For a motif with three nodes, we can compute W_M in $\Theta$(n³) in a complete graph with n nodes. Therefore, the computation complexity of the motif-based analysis procedure is O(n³).

According to the description above, the time complexity of COMICS is O(rn³), where r is the number of subgraphs in the partition set R, and n is the number of nodes.

Experiment settings

In this subsection, we describe the settings of our experiments from three aspects, that is, time cost, clustering accuracy and academic teamwork behavior analysis with complete triangle motif in academic areas. In academic collaboration networks, we consider two algorithms. The Facebook social networks do not contain any statistical information. Therefore, we merely compare our method with K-means algorithm in the social network:

K-means clustering algorithm (Ding & He, 2004): This method proves that principal components are the continuous solutions to cluster membership indicators for K-means clustering. It takes principal component analysis into the clustering process, which is suitable for the scholar science and social data sets.

Co-authorship algorithm (Reyes-Gonzalez, Gonzalez-Brambila & Veloso, 2016): This algorithm considers all the principal investigators and collaborators, and defines knowledge footprints¹ of the groups to calculate the distances between scholars and the group. Based on the distance, the academic groups can be detected in an accurate way. This method iterates all the researchers with their collaborator and institution similarities until they are assigned to a academic team can be applied to understand the self-organizing of research teams and obtain the better assessment of their performances.

To demonstrate the runtime efficiency and the accuracy of our clustering results in large-scale networks, we divide the APS and MAG data sets into different parts with various sizes by years, respectively, so that we can get the collaboration networks with distinct number of nodes (from 1,000 to 200,000). Considering the integrality and veracity of the academic research teams in data sets, we take the whole APS and MAG data sets as the collaboration networks to detect the collaborative relationships.

Evaluation metrics

To evaluate and analyze the accuracy of network clustering results of our proposed COMICS, we use two metrics, that is, compactness and separation, to evaluate node closeness in clustering results and the distances among clusters. In academic collaboration networks, we combine the statistical paper publishing data with network structures together, and calculate three metrics to find the characteristics discovered through the target triangle motif to uncover the hidden collaboration patterns and teamwork of scholars in academic networks.

Compactness and separation (Halkidi, Batistakis & Vazirgiannis, 2002) are used to evaluate the accuracy of clustering results by different methods. Compactness is a widely used metric to quantify the tightness of clustering subgraphs, representing the distances between nodes in a subgraph. Separation calculates the distances among the cores of different subgraphs. That is, if a clustering subgraph is with lower compactness value and higher separation value, the subgraph can be detected effectively. Compactness is expressed by Eq. (11), (11) $$Compactness=\frac{1}{\left|R\right|}{\displaystyle \sum _{v_i\in \Omega }\left|v_i-w\right|}.$$

Here, R is the clustering result set, v_i is one of the nodes in the subgraph, and w is defined as the core of the subgraph cluster, because w is the node with the maximum degree in a cluster. The value of |v_i − w| means the shortest distance between node v_i and the cluster core node w. SP is defined as in Eq. (12). (12) $$\text{Separation}=\frac{2}{k^2-k}{\displaystyle \sum _{i=1}^k}{\displaystyle \sum _{j=i+1}^k}|w_i-w_j|,$$ wherein, k is the number of subgraphs in the result set and w_i is the core of the given subgraph i, which is the same as w_j. The value of |w_i − w_j| equals to the shortest distance between w_i and w_j.

In collaboration networks, we assume the clusters as academic teams, in which scholars work together. Therefore, three metrics are defined to analyze the collaboration behaviors through triangle motif: TCV, TPV, and MSV.

TCV: This metric reflects the tightness and volatility among members in a team. For one scholar i in a team, we define the TCV as follows, (13) $${\sigma }_{co}=\frac{{\sum }_i^n{(c{o}_i-c{o}_{ave})}^2}{n}\mathrm{.}$$

Herein, n is the number of team members, co_i is the number of scholars that scholar i has collaborated within the same team, and co_ave is the average number of collaborators collaborated with scholars in a team.

TPV: An academic team with high performance refers that the members in team have published a large number of paper. Similarly, in a stable team, the gaps of published paper numbers among team members are small. To evaluate the academic levels and stability of a team, we define TPV as follows: (14) $${\sigma }_{qtt}=\frac{{\sum }_i^n{(q_i-q_{ave})}^2}{n}\mathrm{,}$$ where σ_qtt means scholar i’s variance of publishing papers in the detected team, q_i is the number of papers that scholar i has published, and q_ave is the average number of papers in the team.

MSV: This metric calculates the difference of motif number that the scholar nodes are included in the collaboration networks. We define the MSV as follows, (15) $${\sigma }_{\text{primitive}}=\frac{{\sum }_i^n{(t_i-t_{\text{ave}})}^2}{n}\mathrm{.}$$

Herein, t_i is the number of target motif that scholar i owns, and t_ave is the average motifs of a team.

To uncover the collaboration patterns mined by triangle motifs among scholars in academic teams, we use the above three arguments to analyze relationships between productions and motifs of the clustered academic teams.

Analysis in academic collaboration networks

After analyzing the time complexity and effectiveness of our system above, in this subsection, we analyze the clustering results with the triangle motifs in academic collaboration networks. The results prove the triangle motif structures can reflect the hidden statistical information and connections with network structures. For example, as the analysis results show, collaboration patterns as well as the correlations of network structure and team productions can be summarized in the academic collaboration networks.

We regard the cluster results of each academic collaboration network as an academic team. Then the values of three variances, that is, TPV, TCV, and MSV are calculated, and the results are shown in Figs. 6A and 6B. Hence, we can see that the number of high-order triangle motif can reflect the performance of an academic team to some extent.

According to Figs. 6A and 6B, we conclude that the TPV and TCV are both proportional to the MSV. Meanwhile, the TPV is also approximately positive linear with the MSV. That means, the lower the MSV is in a cluster team, the performance of team members are in smaller gaps. Therefore, it can be concluded that the value of MSV can reflect the gap of collaboration relationships in teams and performance of team members. However, we can infer that the scholars with few number of complete triangle motifs, have collaborated with only few scholars in the team. Those scholars are probably students or new team members, resulting in the high collaboration and paper variances. Hence, in collaboration networks, we can use MSV to evaluate the gaps of team collaboration relationships and the performance of team members. The two teamwork gaps in different periods represent the stability and volatility of academic teams.