A graph convolutional fusion model for community detection in multiplex networks

Abstract

Community detection is to partition a network into several components, each of which contains densely connected nodes with some structural similarities. Recently, multiplex networks, each layer consisting of a same node set but with a different topology by a unique edge type, have been proposed to model real-world multi-relational networks. Although some heuristic algorithms have been extended into multiplex networks, little work on neural models have been done so far. In this paper, we propose a graph convolutional fusion model (GCFM) for community detection in multiplex networks, which takes account of both intra-layer structural and inter-layer relational information for learning node representation in an interwoven fashion. In particular, we first develop a graph convolutional auto-encoder for each network layer to encode neighbor-aware intra-layer structural features under different convolution scales. We next design a multiscale fusion network to learn a holistic version of nodes’ representations by fusing nodes’ encodings at different layers and different scales. Finally, a self-training mechanism is used to train our model and output community divisions. Experiment results on both synthetic and real-world datasets indicate that the proposed GCFM outperforms the state-of-the-art techniques in terms of better detection performances.

Appendix A

In this Appendix, we discuss the characteristics of multilayer modularity and the encoding distance with a node corresponding to the same entity in different layers.

Theorem 1

Given the modularity of each single layer $Q_{s}=\sum \limits _{ij} [(\textbf{A}_{ijs} - \gamma _{s} \frac{d_{is} d_{js}}{2m_{s}})] \delta \left( g_{i}, \right. \left. g_{j}\right)$, the multilayer modularity will degenerate into the sum of multiple single layer modularity, i.e. $\sum \limits _{s} Q_{s}$, if a node is assigned with only one community label in its all layers of a multiplex network.

For the multilayer modularity defined by Eq.(21), it allows that a node in each layer can have its own label; While GCFM and some other baseline methods (PMM, MDLPA, CSNMF, DMGI, HDMI) assign a community label to a node in its belonging layers. In order to analyze the GCFM, we can rewrite Eq.(21) as:

$$\begin{aligned} Q_{m} = \frac{1}{2\eta }\sum \limits _{ijsr} [(\textbf{A}_{ijs} - \gamma _{s} \frac{d_{is} d_{jr}}{2m_{s}} ) \delta (s,r) + \delta (i,j) \mathscr {C}_{jsr}] \delta \left( g_{i}, g_{j}\right) . \end{aligned}$$

(A1)

For layer s, r and node i, j, the Eq.(A1) can be divided into four cases:

$\bullet$ Case1: $s=r$ and $i=j$ ($\delta \left( g_{i}, g_{j}\right) =1$ and $C_{jsr}=0$).

$$\begin{aligned} Q^{(1)} = \frac{1}{2\eta }\sum \limits _{is} [(\textbf{A}_{iis} - \gamma _{s} \frac{d_{is}^{2}}{2m_{s}})]. \end{aligned}$$

(A2)

$\bullet$ Case2: $s=r$ and $i \ne j$.

$$\begin{aligned} Q^{(2)} = \frac{1}{2\eta }\sum \limits _{ijs,i \ne j} \left[ \left( \textbf{A}_{ijs} - \gamma _{s} \frac{d_{is} d_{js}}{2m_{s}}\right) \right] \delta \left( g_{i}, g_{j}\right) . \end{aligned}$$

(A3)

$\bullet$ Case3: $s \ne r$ and $i = j$ ($\delta \left( g_{i}, g_{j}\right) =1$).

$$\begin{aligned} Q^{(3)} = \frac{1}{2\eta }\sum \limits _{isr, s \ne r} \mathscr {C}_{jsr}. \end{aligned}$$

(A4)

$\bullet$ Case4: $s \ne r$ and $i \ne j$.

$$\begin{aligned} Q^{(4)} = 0. \end{aligned}$$

(A5)

Therefore, $Q_{m}$ can be represented as $Q^{(1)}+Q^{(2)}+Q^{(3)}+Q^{(4)}$. $Q^{(3)}$ and $Q^{(4)}$ are constants and $C_{jsr}$ is proven ineffective in our problem setting. It works on the case that each node in each layer is assigned with an independent community label. $Q^{(1)}$ and $Q^((2))$ can be merged as:

$$\begin{aligned} Q^{(1)}+Q^{(2)} = \frac{1}{2\eta }\sum \limits _{s}\sum \limits _{ij} \left[ \left( \textbf{A}_{ijs} - \gamma _{s} \frac{d_{is} d_{js}}{2m_{s}}\right) \right] \delta \left( g_{i}, g_{j}\right) , \end{aligned}$$

(A6)

which means the sum of each single layer modularity $Q_{s}=\sum \limits _{ij} [(\textbf{A}_{ijs} - \gamma _{s} \frac{d_{is} d_{js}}{2m_{s}})] \delta \left( g_{i}, g_{j}\right)$. To sum up, our model is to design an optimization function $g(\cdot )$ to maximize the sum of monoplex modularity:

$$\begin{aligned} \mathop {\arg \max }\limits _{g} \sum \limits _{s} Q_{s}. \end{aligned}$$

(A7)

Theorem 2

The GCA can decrease the encoding distance of same node in different layers, i.e. $\Vert \hat{\textbf{h}}_{i,1}-\hat{\textbf{h}}_{i,2}\Vert _{2}$, if they have similar topological structures in their own layers.

Given a multiplex network, we focus on the k-th GCA scale. Denote $\textbf{h}_{i,l}$ as the encoding of node i of the l-th layer (denoted as (i,l)) in $(k-1)$-th scale and we have $\hat{\textbf{h}}_{i,l}=ReLU(\sum _{j \in N_{i,l}} \frac{\textbf{h}_{j,l}}{\sqrt{d_{i,l}}\sqrt{d_{j,l}}}\textbf{W})$, where $N_{i,l}$ is the neighborhoods of node (i,l). Assume $ReLU(\cdot )$ as $\sigma (x)=x$ and $\textbf{W}=\textbf{I}$. $\hat{\textbf{h}}_{i,l}$ can be regarded as the average of neighbor encodings. Now we focus on node i in a two-layer multiplex network, i.e. $\hat{\textbf{h}}_{i,1}$ and $\hat{\textbf{h}}_{i,2}$. For $\hat{\textbf{h}}_{i,1}$, it can be divided into three parts: (a) Self node encoding $\frac{\textbf{h}_{i,l}}{d_{i,l}}$. (b) The encodings of generalized common neighbors of (i,1) and (i,2), where generalized common neighbors mean that nodes correspond to the same entity. It can be represented as $\frac{\textbf{B}_{1}}{\sqrt{d_{i,1}}}$, where $\textbf{B}_{1}=\sum _{u \in N_{i,1} \cap N_{i,2}} \frac{\textbf{h}_{u,1}}{\sqrt{d_{u,1}}}$ is the sum of common neighbor encodings. (c) The other non-common neighbor encodings, denoted as $\frac{\textbf{D}_{1}}{\sqrt{d_{i,1}}}$, where $\textbf{D}_{1}=\sum _{v \in N_{i,1} - N_{i,1} \cap N_{i,2}} \frac{\textbf{h}_{v,1}}{\sqrt{d_{v,1}}}$. Assume each node in the two layer has the same initial encoding in $(k-1)$-th scale, i.e. $\textbf{h}_{i,1}=\textbf{h}_{i,2}$. Then we measure the distance of $\hat{\textbf{h}}_{i,1}$ and $\hat{\textbf{h}}_{i,2}$:

$$\begin{aligned} \begin{aligned}&\Vert \hat{\textbf{h}}_{i,1}-\hat{\textbf{h}}_{i,2}\Vert _{2}\\&\quad =\Vert \left( \frac{\textbf{h}_{i,1}}{d_{i,1}}-\frac{\textbf{h}_{i,2}}{d_{i,2}}\right) +\left( \frac{\textbf{B}_{1}}{\sqrt{d_{i,1}}}-\frac{\textbf{B}_{2}}{\sqrt{d_{i,2}}}\right) + \left( \frac{\textbf{D}_{1}}{\sqrt{d_{i,1}}}-\frac{\textbf{D}_{2}}{\sqrt{d_{i,2}}}\right) \Vert _{2}\\&\quad \le \left\| \frac{\textbf{h}_{i,1}}{d_{i,1}}-\frac{\textbf{h}_{i,2}}{d_{i,2}} \right\| _{2} + \left\| \frac{\textbf{B}_{1}}{\sqrt{d_{i,1}}}-\frac{\textbf{B}_{2}}{\sqrt{d_{i,2}}} \right\| _{2} + \Vert \frac{\textbf{D}_{1}}{\sqrt{d_{i,1}}}-\frac{\textbf{D}_{2}}{\sqrt{d_{i,2}}} \Vert _{2}\\&\quad \le \left| \frac{d_{i,2}-d_{i,1}}{d_{i,1}d_{i,2}}\vert \Vert \textbf{h}_{i,1} \right\| _{2} \\&\qquad +\sum _{u \in N_{i,1} \cap N_{i,2}} \left| \frac{\sqrt{d_{i,2}d_{u,2}}-\sqrt{d_{i,1}d_{u,1}}}{\sqrt{d_{i,1}d_{u,1}d_{i,2}d_{u,2}}}\right| \left\| \textbf{h}_{u,1} \right\| _{2} \\&\qquad +\left( \left\| \frac{\textbf{D}_{1}}{\sqrt{d_{i,1}}}\right\| _{2} +\left\| \frac{\textbf{D}_{2}}{\sqrt{d_{i,2}}}\right\| _{2}\right) . \end{aligned} \end{aligned}$$

(A8)

From Eq.(A8) we can derive the upper bound of the distance between $\hat{\textbf{h}}_{i,1}$ and $\hat{\textbf{h}}_{i,2}$. The first term measures the degree difference of (i,1) and (i,2), where $\vert \frac{d_{i,2}-d_{i,1}}{d_{i,1}d_{i,2}}\vert \le 1$. If $d_{i,1} \approx d_{i,2}$, the effect of the first term can be ignored; While if the two nodes have no similar degrees such as $d_{i,2} \gg d_{i,1} = 1$, it may achieve the upper bound 1. The second term represents the influence of the common neighbors. For each item $\vert \frac{\sqrt{d_{i,2}d_{u,2}}-\sqrt{d_{i,1}d_{u,1}}}{\sqrt{d_{i,1}d_{u,1}d_{i,2}d_{u,2}}}\vert \le 1$, we assume (i,1) and (i,2) have similar degrees. Specially, if $d_{u,2} \gg d_{u,1}$, this item will become $\vert \frac{1}{\sqrt{d_{i,1}d_{u,1}}} \vert$ which is controlled by the degree of node (i,1) and the degree of its common neighbor (u,1), both normally larger than 1. So usually $\vert \frac{\sqrt{d_{i,2}d_{u,2}}-\sqrt{d_{i,1}d_{u,1}}}{\sqrt{d_{i,1}d_{u,1}d_{i,2}d_{u,2}}}\vert < 1$. The third term shows the influence of the non-common neighbors. For the upper bound of each item $\vert \frac{1}{\sqrt{d_{i,1}d_{v,1}}} \vert$ and $\vert \frac{1}{\sqrt{d_{i,2}d_{v,1}}} \vert$, if $d_{v,1} \gg d_{i,1}$ and $d_{v,2} \gg d_{i,2}$, the effect of the third term can be ignored. Besides, there is certain restrictive correlation between the second term and the third term, which is $d_{i,1}=\vert u \in N_{i,1} \cap N_{i,2} \vert + \vert v \in N_{i,1} - N_{i,1} \cap N_{i,2} \vert$.

In a multiplex network, we hope that the same node in different layers have approximate performance if they have similar topology so that we can fuse their encodings and obtain a consistent community label. Equation(A8) proves that our GCA can decrease the encoding distance of same node in different layers with similar topology, i.e node degree, common neighbor and one-hop neighbor degree, which will be benefit for the task of community detection.