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Null Models and Community Detection in Multi-Layer Networks

Abstract

Multi-layer networks of multiplex type represent relational data on a set of entities (nodes) with multiple types of relations (edges) among them where each type of relation is represented as a network layer. A large group of popular community detection methods in networks are based on optimizing a quality function known as the modularity score, which is a measure of the extent of presence of module or community structure in networks compared to a suitable null model. Here we introduce several multi-layer network modularity and model likelihood quality function measures using different null models of the multi-layer network, motivated by empirical observations in networks from a diverse field of applications. In particular, we define multi-layer variants of the Chung-Lu expected degree model as null models that differ in their modeling of the multi-layer degrees. We propose simple estimators for the models and prove their consistency properties. A hypothesis testing procedure is also proposed for selecting an appropriate null model for data. These null models are used to define modularity measures as well as model likelihood based quality functions. The proposed measures are then optimized to detect the optimal community assignment of nodes (Code available at: https://u.osu.edu/subhadeep/codes/). We compare the effectiveness of the measures in community detection in simulated networks and then apply them to four real multi-layer networks.

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Acknowledgements

This work was supported in part by National Science Foundation grants DMS-1830547, DMS-2015561 and CCF-1934986. We also thank two anonymous reviewers for their extensive comments which have immensely helped us improve the paper.

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Correspondence to Yuguo Chen.

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Appendices

Appendix A

Proof of Theorem 1

Proof.

We start by noting that \(\hat {\theta }_{i} = \frac {k_{i}}{\sqrt {2L}}\) and \(\bar {\theta }_{i} = \frac {\kappa _{i}}{\sqrt {2{\mathscr{L}}}}\). From Chernoff inequality (Theorem A.1.4 of Alon and Spencer (2004)), we have for a given i,

$$ P(|k_{i}-\kappa_{i}| > \epsilon\sqrt{N\bar{\kappa}}) \leq 2\exp\left( -\frac{2\epsilon^{2}N\bar{\kappa}}{N}\right)=2\exp(-2\epsilon^{2}\bar{\kappa}). $$

Taking a union bound over all i,

$$P\left( \sup_{i \in \{1,\ldots,N\}} |k_{i}-\kappa_{i}| > \epsilon\sqrt{N\bar{\kappa}}\right) \leq 2N \exp(-2\epsilon^{2}\bar{\kappa})=\exp(\log (2N)-2\epsilon^{2}\bar{\kappa}) \to 0, \text{ as } N\to \infty, $$

for a sufficiently large C since \(\bar {\kappa } \geq C \log N\) by assumption. Therefore,

$$ P\left( \underset{i \in \{1,\ldots,N\}}{\sup} \frac{|k_{i}-\kappa_{i}|}{\sqrt{2\mathcal{L}}} >\epsilon\right) \to 0 \text{ as } N \to \infty. $$
(A.1)

Now

$$ \begin{array}{@{}rcl@{}} P\left( \left|\frac{k_{i}}{\sqrt{2L}} -\frac{\kappa_{i}}{\sqrt{2\mathcal{L}}}\right| > \epsilon\right) & =& P\left( \left|\frac{k_{i}}{\sqrt{2L}} -\frac{k_{i}}{\sqrt{2\mathcal{L}}} + \frac{k_{i}}{\sqrt{2\mathcal{L}}} - \frac{\kappa_{i}}{\sqrt{2\mathcal{L}}}\right| > \epsilon\right)\\ & \leq& P\left( \left|\frac{k_{i}}{\sqrt{2L}} -\frac{k_{i}}{\sqrt{2\mathcal{L}}}\right| >\epsilon/2\right) + P\left( \left|\frac{k_{i}}{\sqrt{2\mathcal{L}}} - \frac{\kappa_{i}}{\sqrt{2\mathcal{L}}}\right| > \epsilon/2\right)\\ & \leq& P\left( \frac{k_{i}}{\sqrt{2\mathcal{L}}}\left|\frac{\sqrt{2\mathcal{L}}}{\sqrt{2L}}-1\right| >\epsilon/2\right) + P\left( \frac{|k_{i} - \kappa_{i}| }{\sqrt{2\mathcal{L}}}>\epsilon/2\right). \end{array} $$

Note that \(2{\mathscr{L}} = n \bar {\kappa }\). Then for any i, \(\frac {k_{i}}{\sqrt {2{\mathscr{L}}}} = \frac {\kappa _{i}}{\sqrt {2{\mathscr{L}}}}+o_{p}(1) \leq 1 + o_{p}(1) = O_{p}(1)\), since \(\frac {\kappa _{i}}{\sqrt {2{\mathscr{L}}}} = \theta _{i} \leq 1\) by model assumption. Moreover, since \(L={\sum }_{i,j} L_{ij}\) is the sum of N2 independent random variables,

$$ \begin{array}{@{}rcl@{}} P(|2L-2\mathcal{L}| >\epsilon 2\mathcal{L}) \leq \exp\left( -\frac{2\epsilon^{2} n^{2} \bar{\kappa}^{2}}{n^{2}}\right) \to 0 \text{ as } N \to \infty. \end{array} $$

Therefore, \( \frac {2L}{2{\mathscr{L}}} \overset {p}{\to } 1. \) Since the function \(\frac {1}{\sqrt {x}}\) is continuous at x = 1, by continuous mapping theorem

$$ \frac{\sqrt{2\mathcal{L}}}{\sqrt{2L}} \overset{p}{\to} 1. $$

Therefore, the quantity

$$ \frac{k_{i}}{\sqrt{2\mathcal{L}}}|\frac{\sqrt{2\mathcal{L}}}{\sqrt{2L}}-1| = O_{p}(1) o_{p}(1) = o_{p}(1) \text{ for all } i. $$
(A.2)

Combining (A.2) and (A.1) we have the result

$$ \begin{array}{@{}rcl@{}} P\left( \underset{i \in \{1,\ldots,N\}}{\sup} \left|\frac{k_{i}}{\sqrt{2L}} -\frac{\kappa_{i}}{\sqrt{2\mathcal{L}}}\right| > \epsilon\right) & \leq& P\left( \underset{i \in \{1,\ldots,N\}}{\sup} \frac{k_{i}}{\sqrt{2\mathcal{L}}}\left|\frac{\sqrt{2\mathcal{L}}}{\sqrt{2L}}-1\right| \!>\!\epsilon/2\right) \\&&+ P\left( \underset{i \in \{1,\ldots,N\}}{\sup} \frac{|k_{i} - \kappa_{i}| }{\sqrt{2\mathcal{L}}}>\epsilon/2\right) \\ & \to &0. \end{array} $$

Proof of Theorem 2

Proof.

We follow the same proof technique as in Theorem 1. Recall

$$ \hat{\theta}^{(m)}_{i} = \frac{k_{i}^{(m)}}{\sqrt{2L^{(m)}}}, \quad \text{ and } \bar{\theta}^{(m)}_{i} = \frac{\kappa_{i}^{(m)}}{\sqrt{2\mathcal{L}^{(m)}}}, $$
(A.3)

Since \(A_{ij}^{(m)}\) are independent binary random variables, from Chernoff inequality, we have for a given i and m and 𝜖 > 0,

$$ P\left( |k_{i}^{(m)}- \kappa_{i}^{(m)}| > \epsilon\sqrt{2\mathcal{L}^{(m)}}\right) \leq 2\exp\left( -\frac{2\epsilon^{2} 2\mathcal{L}^{(m)}}{N}\right) \leq 2\exp\left( -\frac{4\epsilon^{2} 2\mathcal{L}^{\prime}}{N}\right), $$

where \({\mathscr{L}}^{\prime } =\min \limits _{m} {\mathscr{L}}^{(m)} \geq C N \log (MN)\) by assumption. Taking a union bound over all i and m,

$$ \begin{array}{@{}rcl@{}} P\left( \underset{\underset{m \in \{1,\ldots,M\}}{i \in \{1,\ldots,N\},}}{\sup} \frac{|k_{i}^{(m)}- \kappa_{i}^{(m)}|}{{\sqrt{2\mathcal{L}^{(m)}}}} > \epsilon\right) \!\!\!&\leq&\!\!\! 2NM \exp\left( -\frac{4\epsilon^{2} \mathcal{L}^{\prime}}{N}\right)\\ \!\!\!&=&\!\!\!\exp(\log (2MN)-4\epsilon^{2}C \log (MN)) \to 0,\\ \end{array} $$
(A.4)

for a sufficiently large C.

Now similar to the arguments in the proof of Theorem 1, for any 𝜖 > 0 and given i and m,

$$ \begin{array}{@{}rcl@{}} P\left( \left|\frac{k_{i}^{(m)}}{\sqrt{2L^{(m)}}} -\frac{\kappa_{i}^{(m)}}{\sqrt{2\mathcal{L}^{(m)}}}\right| > \epsilon\right) & \leq P\left( \frac{k_{i}^{(m)}}{\sqrt{2\mathcal{L}^{(m)}}}\left|\frac{\sqrt{2\mathcal{L}^{(m)}}}{\sqrt{2L^{(m)}}}-1\right| >\epsilon/2\right) + P\left( \frac{|k_{i}^{(m)} - \kappa_{i}^{(m)}| }{\sqrt{2\mathcal{L}^{(m)}}}>\epsilon/2\right). \end{array} $$

Taking supremum over i and m we have

$$ \begin{array}{@{}rcl@{}} P\left( \underset{\underset{m \in \{1,\ldots,M\}}{i \in \{1,\ldots,N\},}}{\sup}\left|\frac{k_{i}^{(m)}}{\sqrt{2L^{(m)}}} -\frac{\kappa_{i}^{(m)}}{\sqrt{2\mathcal{L}^{(m)}}}\right| > \epsilon\right) &\leq& P\left( \underset{\underset{m \in \{1,\ldots,M\}}{i \in \{1,\ldots,N\},}}{\sup}\frac{k_{i}^{(m)}}{\sqrt{2\mathcal{L}^{(m)}}}\left|\frac{\sqrt{2\mathcal{L}^{(m)}}}{\sqrt{2L^{(m)}}}-1\right| >\epsilon/2\right) \\ && + P\left( \underset{\underset{m \in \{1,\ldots,M\}}{i \in \{1,\ldots,N\},}}{\sup}\frac{|k_{i}^{(m)} - \kappa_{i}^{(m)}| }{\sqrt{2\mathcal{L}^{(m)}}}>\epsilon/2\right). \end{array} $$

From Eq. A.4, we have \(\frac {k_{i}^{(m)}}{\sqrt {2{\mathscr{L}}^{(m)}}} = \frac {\kappa _{i}^{(m)}}{\sqrt {2{\mathscr{L}}^{(m)}}}+o_{p}(1) \leq O_{p}(1) + o_{p}(1) = O_{p}(1)\), since \(\frac {\kappa _{i}}{\sqrt {2{\mathscr{L}}}} = \theta _{i} \leq 1\) by model assumption. Moreover, since the convergence in Eq. A.4 holds for all i and m, this result also holds for all i and m. Finally,

$$ \begin{array}{@{}rcl@{}} P\left( \underset{m \in \{1,\ldots,M\}}{\sup} \{|2L^{(m)}-2\mathcal{L}^{(m)}| >\epsilon 2\mathcal{L}^{(m)}\}\right) \leq \exp\left( \log M-\frac{8\epsilon^{2} (\mathcal{L}^{\prime})^{2}}{N^{2}}\right) \to 0, \end{array} $$

since \({\mathscr{L}}^{\prime } =\min \limits _{m} {\mathscr{L}}^{(m)} \geq C N \log (MN).\) Therefore,

$$ \underset{m \in \{1,\ldots,M\}}{\sup}\sqrt{\frac{2L^{(m)}}{2\mathcal{L}^{(m)}}} \overset{p}{\to} 1. $$

Therefore,

$$ \begin{array}{@{}rcl@{}} \underset{\underset{m \in \{1,\ldots,M\}}{i \in \{1,\ldots,N\},}}{\sup}\frac{k_{i}^{(m)}}{\sqrt{2\mathcal{L}^{(m)}}}\left|\frac{\sqrt{2\mathcal{L}^{(m)}}}{\sqrt{2L}^{(m)}}-1\right| = O_{p}(1)o_{p}(1) = o_{p}(1). \end{array} $$
(A.5)

Hence combining results in Eqs. A.4 and A.5 we have the desired result. □

Proof of Theorem 3

Proof.

In the notation of the theorem,

$$ \hat{\theta}_{i}=\frac{{\sum}_{m} k_{i}^{(m)}}{\sqrt{2L}}, \quad \hat{\beta}_{m}=\frac{L^{(m)}}{L}, \quad \quad \text{ and } \quad \quad \bar{\theta}_{i}=\frac{{\sum}_{m} \kappa_{i}^{(m)}}{\sqrt{2\mathcal{L}}}, \quad \bar{\beta}_{m}=\frac{\mathcal{L}^{(m)}}{\mathcal{L}}. $$
(A.6)

Note that by assumption, \({\mathscr{L}} = \underset {m}{\sum } {\mathscr{L}}^{(m)} \geq C NM \log N\). Since \(A_{ij}^{(m)}\) are independent binary random variables, and \(\underset {m}{\sum } k_{i}^{(m)} = {\sum }_{m} {\sum }_{j} A_{ij}^{(m)}\), from Chernoff inequality, we have for any i and for any 𝜖 > 0,

$$ P\left( \left|\underset{m}{\sum} k_{i}^{(m)}- \underset{m}{\sum} \kappa_{i}^{(m)}\right| > \epsilon\sqrt{2\mathcal{L}}\right) \leq 2\exp\left( -\frac{2\epsilon^{2} 2\mathcal{L}}{NM}\right). $$

Taking a union bound over all i,

$$ \begin{array}{@{}rcl@{}} &&P\left( \underset{i \in \{1,\ldots,N\}}{\sup} \left|\underset{m}{\sum} k_{i}^{(m)}- \underset{m}{\sum} \kappa_{i}^{(m)}\right| > \epsilon\sqrt{2\mathcal{L}}\right) \leq 2N \exp\left( -\frac{4\epsilon^{2} \mathcal{L}}{NM}\right) \\ &\leq& \exp(\log(2N)-4\epsilon^{2}C \log N) \to 0, \end{array} $$

as \( N\to \infty \), for a sufficiently large C. Therefore,

$$ P\left( \underset{i \in \{1,\ldots,N\}}{\sup} \frac{\left|{\sum}_{m} k_{i}^{(m)}- {\sum}_{m} \kappa_{i}^{(m)}\right|}{\sqrt{2\mathcal{L}}} >\epsilon\right) \to 0 \text{ as } N \to \infty. $$

As a consequence of the above result, we have \(\frac {{\sum }_{m} k_{i}^{(m)}}{\sqrt {2{\mathscr{L}}}} = \frac {{\sum }_{m} \kappa _{i}^{(m)}}{\sqrt {2{\mathscr{L}}}}+o_{p}(1) =O_{p}(1)\) (in particular, bounded by 2 with high probability) for all i, since \(\frac {{\sum }_{m} \kappa _{i}^{(m)}}{\sqrt {2{\mathscr{L}}}} = \bar {\theta }_{i} \leq 1\) by model assumption. Further,

$$ \begin{array}{@{}rcl@{}} P(|2L-2\mathcal{L}| >\epsilon 2\mathcal{L}) \leq 2 \exp\left( -\frac{2\epsilon^{2} N^{2}M^{2} (\log N)^{2}}{N^{2}M}\right) \to 0. \end{array} $$
(A.7)

Then similar arguments as the proof of Theorem 1 lead to the result

$$ \begin{array}{@{}rcl@{}} P\left( \underset{i \in \{1,\ldots,N\}}{\sup} |\frac{\ {\sum}_{m} k_{i}^{(m)}}{\sqrt{2L}} -\frac{{\sum}_{m} \kappa_{i}^{(m)}}{\sqrt{2\mathcal{L}}}| > \epsilon\right) \to 0. \end{array} $$

Next we prove the result for the estimators of the βm parameters. Clearly, since L(m) is the sum of N2 independent random variables,

$$ \begin{array}{@{}rcl@{}} P\left( \underset{m \in \{1,\ldots,M\}}{\sup} \{|2L^{(m)} - 2\mathcal{L}^{(m)}| >\epsilon 2\mathcal{L}\}\right) \!\leq\! \exp\left( \log M - \frac{8\epsilon^{2} \mathcal{L}^{2}}{N^{2}}\right) \to 0, \end{array} $$
(A.8)

since \({\mathscr{L}} \geq C NM \log N.\)

On the other hand, Eq. A.7 shows that

$$ \left|\frac{L}{\mathcal{L}}-1\right| =o_{p}(1). $$

Now

$$ P\left( \left|\frac{L^{(m)}}{L} -\frac{\mathcal{L}^{(m)}}{\mathcal{L}}\right| > \epsilon\right) \leq P\left( \left|\frac{L^{(m)}- \mathcal{L}^{(m)}}{\mathcal{L}}\right| >\epsilon/2\right) + P\left( \frac{L^{(m)}}{\mathcal{L}}\left|\frac{\mathcal{L}}{L}-1\right| >\epsilon/2\right). $$
(A.9)

Since for any m, \( L^{(m)} ={\mathscr{L}}^{(m)} + o_{p}(1)\), and \({\mathscr{L}}^{(m)} \leq {\mathscr{L}}\), we have \(\frac {L^{(m)}}{{\mathscr{L}}} =O_{p}(1)\), i.e., bounded (by 2) in high probability. Therefore, in the last term of Eq. A.9,

$$ \frac{L^{(m)}}{\mathcal{L}}\left|\frac{\mathcal{L}}{L}-1\right| =O_{p}(1)o_{p}(1) = o_{p}(1), $$

for any m, while in first term on the right hand side of Eq. A.9, \(\left |\frac {L^{(m)}- {\mathscr{L}}^{(m)}}{{\mathscr{L}}}\right |\) is also op(1) for any m by Eq. A.8. Therefore, combining the two results leads to the result. □

Approximations Without Assuming Self-Loops

While the model with self-loops is commonly used in the literature due to simplified computations (Arcolano et al. 2012; Karrer and Newman, 2011; Newman, 2016), we do note that such a model may not be appropriate for graphs that do not contain self-loops. Here we estimate the expected error in the estimators if the model does not allow for self-loops. For the ID model, plugging in the proposed estimator into the likelihood equations leads to

$$ \frac{{\sum}_{j} A_{ij}^{(m)}}{\hat{\theta}^{(m)}_{i}} - \underset{j}{\sum} \hat{\theta}_{j}^{(m)} + \hat{\theta}_{i}^{(m)}= \frac{k_{i}^{(m)}}{\sqrt{2L^{(m)}}}. $$

The expected error can be approximated with standard assumptions on growth rates of degrees widely employed in the literature. First we note that a first order Taylor series approximation gives

$$ E[\hat{\theta}^{(m)}_{i}] = E\left[\frac{k_{i}^{(m)}}{\sqrt{2L^{(m)}}}\right] \approx \frac{E[k_{i}^{(m)}]}{\sqrt{2E[L^{(m)}}]}. $$

It is common in the literature to assume that expected degrees in sparse networks scale with \(O(\log N)\). Therefore \(E[k_{i}^{(m)}] = O(\log N)\) and \(E[L^{(m)}] = O(N\log N)\). Therefore the extent of error in each of the likelihood equation is \(O(\sqrt {\frac {\log N}{N}})\).

Plugging in the estimators for the SD model in the likelihood equation for the SD model leads to the following estimate of errors:

$$ \begin{array}{@{}rcl@{}} \frac{\partial l }{\partial \theta_{i}} & :\ \ \frac{{\sum}_{m} {\sum}_{j} A_{ij}^{(m)}}{\hat{\theta}_{i}} - \underset{m}{\sum}{\sum}_{j} \hat{\theta}_{j} + \underset{m}{\sum} \hat{\theta}_{i} = \underset{m}{\sum} \hat{\theta}_{i}, \quad \quad i = \{1,{\ldots} N\}, \\ \frac{\partial l }{\partial \beta_{m}} & :\ \ \frac{{\sum}_{i<j} A_{ij}^{(m)}}{\hat{\beta}_{m}} - \sum\limits_{i<j} \hat{\theta}_{i}\hat{\theta}_{j} = L-L = 0, \quad \quad m=\{1,\ldots, M\}. \end{array} $$

Therefore there is no error in the second set of likelihood equations and the error in the first set can be quantified with the above growth rate assumptions. In particular the extent of error in each of the likelihood equations in the first set is O(M).

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Paul, S., Chen, Y. Null Models and Community Detection in Multi-Layer Networks. Sankhya A (2021). https://doi.org/10.1007/s13171-021-00257-0

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Keywords

  • Configuration model
  • degree corrected multi-layer stochastic block model
  • expected degree model
  • multi-layer network
  • multiplex network
  • multi-layer null models.

AMS (2000) subject classification

  • Primary 62F10
  • 62F40
  • 62R07
  • Secondary 62H30
  • 90B15