## 1 Introduction

A multiplex network structure is a comprehensive representation of real world networks considering that it allows for multiple kinds of relations, and encodes them separately. The multilayer nature of these networks substantially changes their structure and dynamics [Bianconi2013, DeDomenico2013, Cardillo2013, Boccaletti2014, DeDomenico2015a, Mahdizadehaghdam2016] in comparison to single layer representations[Karrer2011, Wang2013]. Despite being one of the main topics of network science for over a decade, the community detection problem has only recently been more closely studied in the context of multilayer networks [PeterJ.Mucha2010, DeDomenico2015a, Loe2015, Wilson2016, Valles-Catala2016, Taylor2016, Stanley2016, Afsarmanesh2016, Paul2017, DeBacco2017]. Community detection in multiplex networks has found numerous applications, such as dynamics [Palla2007] and multi-relation [Szell2010, huang2016consensus] in social networks, evolution of granular force networks [Papadopoulos2016], and cognitive states of brain networks [Telesford2016].

It is generally advantageous for community detection to decompose an ordinary network into multiple layers based on additional attributes, and to create a multiplex network for individual layers to potentially unravel entangled structures, such as overlapping communities. It can, however, be difficult to reach this goal with a usually limited knowledge. Carelessly breaking a network into layers can be problematic since it can either retain overlapping communities in a single layer, or decrease detectability of certain communities by breaking them up and distributing them over multiple layers, leading to redundant communities and reduced edge density. This problem arises in many real world multiplex networks. For example, in the networks of protein-genetic interactions, each type of genetic interaction may be used to define a layer, but it is shown to return highly redundant layers, which require recombination [DeDomenico2015].

Some recent studies consider the redundancy phenomenon in multiplex networks [DeDomenico2015, Taylor2016, Stanley2016] and try to resolve it by further aggregating the redundant layers. Domenico et al. [DeDomenico2015] utilize tools from quantum information to identify redundant layers and aggregate them hierarchically, thus simplifying the structure. They discovered that many real world multiplex networks, including protein-genetic interactions, social networks, economical and transportation systems, can be significantly simplified by their proposed technique. Taylor et al. [Taylor2016] showed that the detectability of community structure is significantly improved by aggregating layers generated from the same stochastic block model (SBM), which is a popular probabilistic generative model for describing nodes’ group memberships [Wang1987]. Stanley et al. [Stanley2016] proposed a specific multilayer SBM which partitions layers into sets called strata, each described by a single SBM. Layers in a stratum are treated as multiple realizations of the same community structure, thus improving community detection accuracy. A drawback of layer aggregation is that completely consistent community structure between layers is required and needs to be known a priori, otherwise different communities may overlap when aggregated into a single layer, as shown in Fig. 1. Domenico et al. [DeDomenico2015a] used the concept of modular flow to show that aggregating layers into a single layer may obscure actual organizations, and that highly overlapping communities exist in some real-world networks. While many algorithms are proposed for overlapping community detection in single layer networks [Palla2005, Esquivel2011, Psorakis2011, Yang2013a, Yang2013b, Nguyen2015, Gamble2015], the performances remain mediocre due to the loss of layer information.

Inspired by these works, we consider a general multiplex SBM that allows layers to be ”partially” redundant, in which case layers may share one or more common communities, and have different ones at the same time (lower row in Fig. 1). Our goal is to improve detectability by leveraging the consensus communities without assuming any two layers to belong to the exact same SBM. This not only achieves higher accuracy, but improves detectability of weak consensus communities as well, by combining their information from different layers, which are otherwise too noisy to be detectable individually. Since our model potentially generates a heterogeneous community structure across layers, our method provides a way to detect overlapping communities at theoretically optimal accuracy, when they can be allocated to different layers.

Our method originates from an application of belief propagation algorithm to community detection, as first developed by Decelle et al. [Decelle2011, Decelle2011a].

Belief propagation is one algorithms in the Bayesian inference framework, which in turn, is known to yield optimal estimates of communities for a network generated by the underlying SBM

[Decelle2011]. Decelle et al. studied detectability transition, and identified a phase transition point in the parameter space, where all community detection algorithms fail. Since then, some extending works using belief propagation have been reported

[Newman2015, Ghasemian2015, Zhang2016, Kawamoto2017]. Ghasemian et al. [Ghasemian2015] extended this method to temporal networks, introducing Dynamic Stochastic Block Model, where nodes gradually change connections and their community memberships over time. While not intended for general temporal networks, our multiplex network model in contrast to [Ghasemian2015], addresses networks that typically encode multiple relations through layers, and the members of a given community remain unchanged irrespectively of the layer the latter occurs in. Our model also does not enforce a temporal order of the layers as in [Ghasemian2015]. Despite aiming for different types of multiplex networks, a simplified version of [Ghasemian2015] is used as a comparison with our model, in presence of homogeneous and heterogeneous community structures. We show that in different situations, both method show their own strength.The outline of the paper is as follows. In Sec. 2.1, we define the problem of community detection and information fusion in multilayer networks. In Sec. 2.2, we present our proposed stochastic model and a simpler model for comparison. In Sec. 2.3, we review the belief propagation algorithm and explain the implementation on the proposed model. In Sec. 3, we show multiple experimental results of the proposed model and discuss its evaluation in detail and its comparison with the simpler model.

## 2 Problem and method

### 2.1 Problem Description

An informed description of our problem of interest is the following: suppose that a multiplex network is given where is the set of nodes and is the set of edges on at the -th layer. We are to identify a collection of node communities, where corresponds to a dense subgraph in at least one layer. Although our problem admits overlapping communities, we assume that the co-occurring communities in each layer are disjoint. Each community may also appear in multiple layers, in which case the resulting data multiplicity can be used to improve community detectability by improving the signal to noise ratio (SNR). However, since the occurrence pattern of the communities is not a priorily known, fusing multiple observations of the same community is not straightforward. For a large part of this paper, we assume that the number of communities is known. However in Section 3.3, we briefly discuss the impact of an incorrect choice of and possible remedies.

### 2.2 Bayesian Solution by Stochastic Modeling

We adopt a Bayesian approach by providing a stochastic generative model for the observed multiplex network, expressed by a likelihood function , as well as a prior distribution on communities. Then, the maximum a-posteriori (MAP) estimate of the communities is obtained by maximizing the a-posteriori distribution, computed according to the Bayes rule:

where is a scaling constant and can be eliminated from optimization. Our generative model utilizes the stochastic block model (SBM), explained in Section 2.2.1, which is widely expressed in terms of node-community labeling. For this reason, we provide an alternative representation of the communities by community labeling of the nodes at different layers . Since, there is a correspondence between possible communities and the labeling , the generative model and the prior can be equivalently expressed in terms of the labeling as and , respectively. We carefully explain this approach, and the resulting stochastic model is given in Section 2.2.2. We can similarly obtain the MAP estimate of and find its corresponding set of communities, which coincides with

, but we resort to a well-known alternative approach, for numerical feasibility. In this approach, we first calculate the marginal probability distribution

of the labels of a single node in a single layer . This is given by(1) |

where we recall that the posterior distribution is calculated by Bayes rule as

(2) |

Next, we obtain the maximum marginal a-posteriori probability (MMAP) label estimates by individually maximizing the resulting posterior marginal distributions for every node:

from which the corresponding community estimates can be easily obtained. It is shown in [Decelle2011] that is an optimal estimate of the original assignment for large networks with the SBM, which is often slightly better than the MAP estimate (ground state) in terms of the number of correct assignments.

Numerical efficiency of the above approach depends on the computation of marginal distributions , which is difficult to perform directly. For example, the denominator in Eq. (2), known as the partition function, cannot be exactly calculated unless the system is extremely small or approximate approaches such as Gibbs sampling are used. In Sec. 2.3, we use a computationally more efficient variational method called belief propagation (BP), which gives the exact marginals as an approximation of the partition function by a product of marginals, and leads to an efficient implementation of the above approach. We next discuss the generative model in detail.

#### 2.2.1 Stochastic Block Model in Single-Layer Network

Stochastic block model (SBM) is commonly used to describe non-overlapping community structures of a single layer network, and plays an important role in our model. Hence, we explain it first. As a generative model, it includes the following parameters: the number of communities , the fraction of the size of each community

, the affinity matrix

showing the probability of an edge between nodes in communities and , and the community assignment for each node .A single-layer network is generated from SBM by first assigning to each node one of the community labels . The probability of a node being assigned to a community label is proportional to the size of the community. Then, a pair are connected ( in the adjacency matrix) with probability independently of other pairs. According to the SBM [Wang1987], if the size of a community is large enough, the community will appear as a block with high probability in the adjacency matrix, under suitable ordering of the nodes.

In benchmark tests, it is common to set if , and if . The constants and are selected such that the fraction is between 0 to 1, so as to control the community quality in the generated network. means no connections between two different communities, which represents a high quality community structure. A high value () means that the connection densities inside and outside the blocks are not significantly distinct, usually reflecting a noisy and weak community structure.

#### 2.2.2 Generalization to Multiplex Networks

Now we generalize SBM to multiplex networks. The idea behind our generative model for multiplex networks is that the same community may appear in multiple layers. Each layer takes a subset of a collection of communities , denoted by . If communities and (here and are community labels), it is required that , meaning that overlapping communities are not allowed in any layer. Also, we assume that when a community exists in multiple layers, it refers to the same group of nodes, so that the definition of is independent of the layers. We call these requirements Well Partitioned Property (WPP), and it is an interlayer constraint. WPP has real world relevance a good case being the social network. We can build a multiplex social network using different types of relations, such as contacts, collegial interaction, common interests, etc., in order to disentangle overlapping community structures. However, communities may exist across multiple layers, e.g. a group of close friends may be reflected as the same community in both the rock music network and the soccer fan network. Meanwhile, in these two layers, other people may form inconsistent community structures. In conclusion, we want to build a model, such that only if consistent communities exist between layers, they will be matched and fused.

Under WPP, we may define the community label vector

for all nodes in layer similarly to the single-layer model in Sec. 2.2.1:The community can be easily recovered from the labels by collecting every node labeled by :

The generative model proceeds as follows: the community label vector for nodes in layer is generated from SBM parameters, under the interlayer constraint WPP. The adjacency matrix of layer is then independently generated as an ordinary SBM. We propose to formulate the probability of a multiplex network and community labels , conditioned on a set of SBM parameters as,

Proposed model:

(3) | ||||

In the following, we break down the formulation and explain each component. We start with a factorized form of the likelihood function, assuming the set of parameters given,

(4) | ||||

where is a multiplex layer.

If we look at the product term, is the probability that a layer of the network being generated by a community structure . Same as the single-layer SBM, introduced in [Decelle2011],

(5) | ||||

The other term, , is the probability distribution over all community patterns satisfying the interlayer constraints from WPP. We express these constraints by a product of local indicator functions over the associated community assignment labels . Therefore if at least one of the indicator functions is zero (local WPP condition is not satisfied), will be zero. Specifically, we propose the distribution of community patterns as:

(6) | ||||

where is a suitable normalization constant. The local constraint is an indicator function of the state (community label) of the copies of 2 nodes in 2 different layers, ( means node in layer ). The function checks whether the associated labels satisfy WPP, and equals one if the following occurs, and is zero otherwise:

This set of conditions summarize when the labels of two nodes satisfy WPP, as we will discuss in detail next. In practice, given a certain number of communities , we can build a list of all possible combinations that satisfy the above constraint and set . Therefore, the process of evaluating the function by verifying the above constraint, can be significantly simplified by storing a look-up table. The look-up table is simple to build for moderate with a complexity of , and only needs to be computed once for a certain value.

#### 2.2.3 Characterizing WPP

We are able to proof that a multi-layer community structure satisfies WPP, if and only if, for the labels of every pair of nodes and every two layers, the value of function equals one and hence . The general proof is in the appendix.

Here we show a simple example to demonstrate one of the constraints.

Fig. 2 shows a situation where community structures in two layers are different (each connected component in a layer is a community). According to the connectivity patterns, we observe that and as are in the same community in layer 1, while they are in different communities in the second layer. We conclude that, in the second layer, neither node , nor node can be assigned to the same community as the one in the first layer, and hence at least 3 communities are required for a correct assignment. This simple intuition is reflected in the definition of (the case of ), where .

We observe that the constraints in , when utilized in a Bayesian learning algorithm, ensure that distinct communities in different layers will not be assigned the same label and not be confounded as one community, so that the structural information will not be mixed up and obscured. This is, according to our example, due to the fact that assigning the same labels to unequal communities will lead to violation of constraints, and make corresponding functions zero and consequently a zero-value posterior distribution . Another role of the constraints is to equally assign consistent communities in different layers, and fuse the structural information to improve detectability. This is illustrated in our example, depicted in Fig. 3, where only the community for node is consistent between two layers, and our goal is to assign to the copies of node the same community label. Notice that in total, 4 communities are involved in this example. If we set , any community assignment with will violate the constraints, which in turn will force in the Bayesian learning algorithm. For example, let and . Since , according to the constraint where , we know that and , and we let and , and therefore . Similarly using the same constraint, we know . We derive that , and due to the community structure in layer 2, . Then again using the constraint of on , we derive that . We find out that is not able to choose from any of the four community labels without violating the constraints. However if we set , we can find community assignments with while satisfying the constraints (for example ), in which case, the communities for node in the two layers will be independently treated and detectability cannot be improved. This also demonstrates the important role of the number of communities as a design parameter. Although we may not know a priori the actual number of communities, this number can be estimated [Decelle2011]. We will discuss later (in Sec. 3.3) how the number of communities affects detection results.

#### 2.2.4 A Prototypical Multiplex Model

To discuss the performance of our proposed approach in Sec. 2.3.2, we present a simpler ”correlated model” without overlapping communities, but with variable and correlated ones in different layers. The correlated model is similar to the DSBM (Dynamic Stochastic Black Model) introduced in [Ghasemian2015]. This model achieves the best performance when the communities in different layers are the same, since layer consistency is used as prior knowledge, much like the layer aggregation method in [Taylor2016]. However, the presence of such a strong prior information is not always realistic, and this model only serves as an oracle bound for our proposed model as in Eq. (3).

We now modify the above SBM model to a correlated multilayer structure, following the same Bayesian description as in Eqs. (4) and (5), nevertheless different from our model in Eq. (3), in that the community assignment prior is instead given by:

(7) |

where is a factor function for the correlation of community assignment of the same node in two layers, indicating the probability of different combinations:

where is the probability of consistent community labels between the same node in two layers. In a special case, if we constrain the number of both layers and communities to 2, when , node labels between two layers are correlated, when , anti-correlated, and when , uncorrelated. Note that when , it allows the same community label to correspond to different sets of nodes in different layers. For , is still the threshold above which the communities become correlated, but then needs to be normalized to be the real probability. Similarly to Eq. (3), we propose the following Bayesian model:

(8) | ||||

Unlike WPP, this model assumes variable communities and correlation between community assignments of a single node between layers. This may be too ideal relative to Eq. (3), since it adds to the model some privileged prior knowledge which is uncommon in real scenarios. We will later compare the model in Eq. (8) with the constrained multiplex model proposed in Eq. (3).

### 2.3 BP algorithm for multilayer community detection

Belief Propagation is an efficient message-passing method for inference problems. Message-passing appears in various contexts, and with various references, such as sum-product algorithm, belief propagation, Kalman filter and cavity method which is used to compute phase diagrams of spin glass systems. Yedidia et al.

[Yedidia2002, Yedidia2005] gave a detailed introduction to Belief Propagation and its connection to free energy.We use the BP algorithm for calculating the marginal posterior distributions as explained in Section 2.2. To that end, we will represent our model in Eq. (3) as a factor graph. A factor graph is composed of factor nodes and variable nodes. Each variable node corresponds to an actual node in our multiplex network. A factor node corresponds to a factor in Eq. (3

). In a tree-like Bayesian network, each factor can also be interpreted as a conditional probability distribution

. Here corresponds to a variable node and denotes its parent nodes [Yedidia2002]. A factor node is connected to its contributing variables, therefore connecting a variable node and all its parent variable nodes. As seen in Eq. (3), two types of factor nodes arise in our case: constraint () nodes, connected to four variable nodes, and the remaining SBM nodes, connected to two variables (See Fig. 4).In BP, ”messages” are reciprocally sent between variable nodes and factor nodes. These messages are a set of equations about the estimates of the conditional marginals. These equations are self-consistent in the sense that they will converge to a consistent solution upon repeatedly iterating. On factor graphs, messages from variable nodes to factor nodes are different from the reversed ones and are given by:

(9) | ||||

where denotes the neighbors of the variable node except , and denotes the neighbors of the factor node except node . Basically, a variable-to-factor message is proportional to the product of all other incoming messages to the variable node, while a factor-to-variable message is the posterior marginal distribution of the variable based on the individual factor, and assuming other incoming messages to the factor as independent priors.

The computational complexity of BP is low. To obtain a marginal probability distribution of an objective node in graphs with no loops, one starts from all the leaves and uses all messages only once, toward the objective node. In practice, one starts with random initial messages, and let them update iteratively, until they converge to a fixed point, or until they meet a stopping criterion. Hence, for a fixed number of iterations, the computation time is . In a generated sparse graph where we fix the average degree, the computation time is . After convergence, the marginal distribution (also called belief) of a node can be calculated using all incoming messages:

(10) |

While, in the presence of cycles, messages may theoretically require infinite iterations to converge, BP has been observed to perform well in graphs that are locally tree-like even if they have many loops[Decelle2011]

. Notice that in loopy graphs, the order of message passing is arbitrary and often heuristic.

#### 2.3.1 Message Passing for Single-Layer SBM

For single layer networks, ordinary SBM is used to describe community structures. In [Decelle2011] it is shown that since each factor is exactly connected to two variables, the two steps in Eq. (9) can be combined to yield a single node-to-node message passing step as follows:

where is a normalization constant. is the fraction of the size of the community (assigned to node ), which represents local evidence for node . is the rescaled connection probability between nodes in communities and respectively i.e., ( is the number of nodes). Finally, is an element of the adjacency matrix of the network.

Decelle et al. [Decelle2011] further use the following mean field approximation to simplify the influence from unconnected nodes,

(11) |

where is an external field, and expressed as,

Here is the belief at node for community label , corresponding to our objective in Eq. (1). The belief at node is written as,

Clearly, Eq. (11) bears a similar structure to a combination of the two steps in Eq. (9). Note that the inner summation part in parentheses in Eq. (11) is in the form of a message from a factor node to a variable node i.e., the second line in Eq. (9), while the outside product manifests message passing in the first line of Eq. (9) from a variable node to a factor node. We observe that the message (11), being from variable to variable , essentially bypasses the factor node lying between these two variable nodes, hence further reducing complexity.

#### 2.3.2 Multiplex Network as a Message Passing Model

Since the interlayer constraint function in Eq. (6) is defined by 4 variable nodes rather than pairwise interaction, we can no longer combine the two messages in Eq. (9) and directly write inter-layer messages between variable nodes. We instead opt to explicitly write inter-layer messages from variable nodes to factor nodes. For the sake of consistency, we do the same for intralayer messages. The factor graph is illustrated in Fig. 4. The message update equations are shown below. Proposed update equations:

Intra-layer message:

(12) | ||||

Inter-layer message:

(13) | ||||

where represents intra-layer neighbors of node , and the inter-layer ones (the constraint-checking factors). is the external field in layer , referring to the single layer version in Eq. (11).

To calculate the node belief:

(14) | ||||

Note that the marginal posteriors are given by the beliefs as . For experimental purposes, and clarity, we write down the belief propagation equations for a two layer network, similarly for the following model. The associated resulting message passing algorithm is shown below as a pseudocode. The ”for” loops, which update the messages, can be easily executed in a parallel or distributed fashion for large networks. In our experiments, a serial version of the algorithm is implemented. In each step, one edge is randomly selected without replacement and the corresponding message is updated, which influences the following updates of other edges.

#### 2.3.3 An Oracle Limit: Correlated Variable Communities

We now show the message passing expression for the correlated-community model in Eq. (8). The message paths are illustrated in Fig. 5, highlighting inter-layer messages and intra-layer ones. Since every factor node connects only two variable nodes, we can bypass the factor nodes and write messages between variable nodes as in the figure.

## 3 Detectability transition of constrained multiplex networks

### 3.1 Homogeneous multiplex network

In this section, we report the results of the Bayesian method in Section 2.2 with the message-passing algorithms developed in Sec. 2.3.2 and 2.3.3. We set the experimental scenario to consist of a 2-layer network with 200 nodes, where each layer is randomly generated according to a SBM. The nodes are partitioned into two communities of equal size, which are present in both layers. This is a result of the probability having the same labels between two layers, i.e. or 1 in the correlated model, all the while simultaneously satisfying the WPP. For each algorithm, we observe community detectability transition by varying in the SBM. The transition is quantitatively characterized by a normalized agreement score (referring to ”agreement” in [Decelle2011]),

where is the ground truth community labels, is one of the permutations of estimated community labels , and is the size of the largest community. is called agreement score and represents the overlap between estimated community labels and ground truth.

For the correlated-community model in Sec. 2.2.4, we observe transitions curves under various value of in the algorithm in Sec. 2.3.3. In Fig. 6, for , detectability transition is similar to that in a single layer (red dash line) [Decelle2011], because we are practically treating them as independent layers. Except for or 1, high correlation (such as ) or anti-correlation (such as ) between labels increases detectability significantly. We conjecture that the poor performance for or 1 is due to its low tolerance of wrong intermediate label, leading to a lower chance of convergence to the correct fixed point. The fluctuation of the or 1 curves also indicates that the convergence is not stable in these cases, especially considering the loopy factor graph.

This naive assumption that all nodes in different layers have correlated community labels is, however, the same as directly connecting corresponding nodes in two layers without any further weight adjustment over messages. In this case, all nodes in each layer are assumed as uniformly correlated. This assumption from the correlated model is reasonable for certain types of multiplex networks such as temporal networks. However, to account for heterogeneous structure, and a more realistic case of unknown prior knowledge of consistent communities, it will be more suitable to use our generative model with label constraint.

Fig. 6 shows that a two-layer network is enough to exhibit the strength of the correlated model. To directly compare the constrained multiplex model in Sec. 2.2.2 with the correlated one in Sec. 2.2.4, we follow the same experimental setting as in Fig. 6, and test both methods on the homogeneous double layer network. We make sure that each layer has the same community structure and is independently generated by the same SBM parameters: 200 nodes which are divided into two equal communities. Note that we do not generate the network from the correlated model, although the correlated model fits it. We vary to observe the detectability transitions. The result is shown in Fig. 7, where we include the transition curve for a single layer (red line) as a reference. Similarly to the correlated model (blue line), the constrained model (black line with circle marks) fails around similar values. They both perform much better (fail for larger ) than a single layer.

Note that in the correlated model, we know a priori that the community labels are correlated between two layers. In the constrained model, we, however, do not specifically have that prior knowledge. Just by enforcing WPP constraints and limiting the number of communities to 2, we can still achieve a similar performance improvement. This is beneficial for real world networks, since in practice we often have limited prior information about consistent communities. Indeed, in this experiment, this prior knowledge may also be inferred in the correlated model, setting interlayer correlation as a parameter and using the EM algorithm [Dempster1977]. However, in more complex cases where, for example, community structure in two layers can not be simply described by a single correlation parameter, the correlated model will face difficulty, as we will show in the next section.

One may suspect that as long as the blocks are consistent, the detectability can be automatically improved regardless of such correlation being available to the model. This is clearly not the case for the correlated model as in Fig. 6, since setting , does not include correlation in the model, and the performance is poorer and similar to a single layer setting.

### 3.2 Heterogeneous multiplex network

The constrained model being the only model that naturally generates heterogeneous networks, shows the advantage over the correlated model or single layer networks. In the following we compare the community detection performance between the constrained model and the correlated model on heterogeneous networks. We construct a double layer network of 200 nodes, with An example of the synthetic network is shown in Fig. 8. In the first layer, the first 100 nodes form a community and the remaining 100 nodes are assigned to another community. In the second layer the first 100 nodes still form a community but the remaining 100 nodes are divided into two equal communities. By limiting the total number of communities to , we expect the belief of the first 100 nodes in both layers to converge to the same label, and the remaining 100 nodes in two layers to converge to three different labels (refer to Fig. 8). We refer to this expected result as WPP-satisfying labels and other results as error.

We performed 100 independent trials of tests using both models, and count the fraction of the tests that result in WPP-satisfying labels. As in Fig. 9, when , our constrained model yields WPP-satisfying labels in some of the trials, while the correlated model is able to achieve that only for . Also, the constrained model has significantly higher likelihood to yield correct labels, i.e., has the messages converge to the correct point, when . Note that in this experiment, for each layer, we do not limit the number of communities to the correct value (i.e. two communities for layer 1 and three for layer 2), which means each node in a layer will freely choose from 4 different labels. If we detect communities independently in two layers, which corresponds to setting no constraint, the chance of WPP-satisfying labels is no more than , where is -permutation of . Our result does show an advantage in identifying consistent communities in heterogeneous networks, while the correlated model is unsuitable for this task. The detection error may be attributed to local minima which violate the constraint (WPP) to some degree, with, however, sufficient resilience for the messages to converge. In practice, we can run the algorithm multiple times and choose the results that more likely converged to a correct point.

In Fig. 10, for both constrained model and correlated model, we examine the agreement score between prediction and ground truth. That is because in this more complex experiment, it is not as straightforward to define a normalized agreement score as in Fig. 6 and Fig. 7. As stated above, not every trial will converge to the correct point, we therefore select for both models the top 20 trials that satisfy WPP better (without using ground truth information). Specifically, for each trial we count how many pairs of nodes satisfy WPP locally, by calculating function over the inferred labels of pairs of nodes. We observe in Fig. 10 that for , the agreement score of the constrained model is remarkably higher than the correlated model. The performance advantage benefits from a high fraction of WPP-satisfying results using the constrained model for , as shown in Fig. 9. When this benefit vanishes, for , the constrained model gets similar or worse agreement score than the correlated model. Note that again, the proposed constrained model does not utilize the knowledge that the first 100 nodes have correlated community labels, while the correlated model is supplied with this prior information. The reason of the better performance for is that, the constrained model manages to fuse information for the first 100 nodes in two layers, meanwhile leaving the remaining 100 nodes intact, while the correlated model tends to unify the entire community structure in the two layers, hence corrupting the remaining 100 nodes. The poorer performance of the constrained model in the noisier range, we suspect, is due to an optimization in stability caused by many more constraints and factor nodes in the graphical model. On the other hand, the correlated model has a simpler form and is less susceptible to the stability issue.

In this section, we have compared our constrained model with a basic correlated model, and results show a higher modeling capability of the constrained model, in presence of heterogeneous community structures. Although the correlated model is simpler, the assumption of a uniform label correlation between two layers does not naturally generate multiplex networks with diverse relations, where only a portion of communities are correlated or consistent. Hence, the correlated model (similar to [Ghasemian2015]) is more appropriate for smoothly evolving temporal networks, and the constrained model we proposed is typically suitable for multiplex networks with different types of relations, where the layers are not necessarily uniformly correlated. In principle, the basic correlated model can be extended so that different nodes can have their own interlayer correlation, and the flexibility of the correlated model can be much greater. However, we expect inference difficulty for such model, given the significantly larger number of free parameters, unless the parameters are properly constrained. Such model design will require nontrivial work and is interesting for future works.

Since the goal of the proposed algorithm is that of fusing consistent communities across layers in general networks and of improving detectability, we are not aware of a directly comparable algorithm that is designed for the exact same goal. Nevertheless, we provide in passing a comparison with a popular multilayer community detection algorithm, Genlouvain [jutla2011generalized], which maximizes a multilayer modularity function. For the same experiments in this section, when , Genlouvain converges correctly only 3 out of 100 trials, while our proposed algorithm has over 40% success rate. Genlouvain performs similarly to the correlated model in this particular test. The reason is that Genlouvain requires interlayer coupling parameters, which, when not given, and can only be assumed to be uniform. In contrast, our proposed constrained model implicitly infers interlayer coupling through factor nodes.

### 3.3 Impact of a known number of communities

For a single layer network, any that is larger than or equal to the actual value will fit the model well. For example, by setting while performing the BP algorithm in a network generated from SBM with 2 communities, we are allowing each node to choose from 3 distinct community labels. However when the messages have converged, generally most nodes will tend to choose from only 2 of the labels, leaving one barely used. Therefore, the general practice is to opt for a larger , until the free energy of the model stops decreasing [Decelle2011].

This is in contrast to the constrained multiplex networks. In the experiment of a homogeneous multiplex network, only gives the best performance according to the detectability transition curve. To show this effect, we generate such a 2-layer, 2-community network, with high noise . (The noise is so high that when we perform BP algorithm on one of the layers with , the community detection is affected and all 3 labels may have a significant presence among nodes, making the decision of difficult.) Then we run the algorithm with being 2,3, and 4.

As shown in Fig. 11, the performance is getting poorer as increases. Specifically, at , most nodes have close-to-one probability of some label, and the selected labels match well among two layers. For , the labels still tend to match across layers, but for nodes from 101 to 200, two labels are competing with each other (blue circles and yellow asterisks). For , even the labels are not correctly matched. This is because the constraint factors, more specifically WPP, allows the same communities in two layers to be assigned different labels when . We therefore cannot combine their information to increase the signal-to-noise ratio. The fact that using the correct will give a distinctive performance, also enables us to more reliably select .

### 3.4 Practical considerations and more layers

A common challenge in belief propagation algorithm for general graphical model is the presence of a fair number of short loops. Specifically, in our model, the interlayer factor nodes introduce many short loops in our factor graph, both within layer and between layers. These short loops result in a quick convergence to undesirable points, and message update equations become more approximate, due to the influence of being overly amplified. To cope with this, we slightly modify message update equations. Specifically, instead of making the product over all incoming messages from neighboring interlayer factor nodes , we sample and multiply a fraction of incoming interlayer messages, which also conveniently reduces the computational load. Meanwhile, we can also change the values of function from to, for example, , to relax the constraint. By applying these modifications, we observe a more reliable and stable convergence to the correct point in our experiments. In Figure. 12, we find multiple combinations of learning parameters value and , where the constrained model has over chance to converge to the correct point. These points form a continuous band in the parameter space.

Generally for networks with node, layers and

edges per layer, the number of message passing per epoch is

, which is dense. For multiplex networks with layers, our original idea needs in total different interlayer factor nodes between pairs of layers, since we do not assume sequential layers. Viewing from the scale of layers, messages between all pairs of layers form high level loops, making it even more difficult to converge correctly. We address this difficulty by adopting the idea of alternating projection. Specifically, in each iteration, we optimize messages in every two layers at a time, while freezing other layers, until all pairs are updated. In this way, we break the high level loops among the layers, and decompose the problem into several subproblems , which are more studied and have better convergence behavior. Another possibility to reduce the complexity is to incorporate this complex structure into a single factor node, an extended constraint function that covers all the layers at once, instead of just two layers, so the number of interlayer messages will scale linearly to the number of layers. We show an experiment of a 3-layer heterogeneous network to compare these two strategies. The 3-layer network has 90 nodes in each layer, including 5 different communities in total, while a common one exists between layer 1 and 2, and between layer 2 and 3 respectively. We find that only optimizing two layers at a time has a significant advantage in improving the speed and chance of convergence to a correct point (48 correct convergence out of 100 trials). Similar to the experiment in Section 3.2, the correlated model will fail to deal with such heterogeneous structures.## 4 Conclusions

We developed a belief propagation algorithm for community detection in general multiplex networks. We considered a case where natural label constraints exist. This case corresponds to a potentially heterogeneous community structure for different layers, a likely scenario for real-world networks.

As a comparison, we also considered a correlated model where community labels are uniformly correlated across the layers, for homogeneous multiplex networks. Relying on Bayesian inference, our method is theoretically optimal for networks described by our proposed probability model. For the correlated model, combining information from two layers significantly improves detectability due to the additional prior information. More importantly, for the label constrained model, we showed that using just label WPP constraints and limiting the number of communities, we can achieve a similar performance improvement as that of the correlated model, without rather restrictive prior assumptions. Furthermore, the constrained model is able to assign correct labels to heterogeneous commnuity structures, and achieve a much better detection accuracy than the correlated model over some parameter space. This is especially beneficial for detecting sparse and noisy communities in multiplex networks, such as social networks and biological neural networks. Our current constrained model assumes a homogeneous structure within each community. For networks with specific topologies, we can apply modified SBM in our model, such as degree-corrected SBM for social networks

[Karrer2011, Newman2015]. Future directions also include improving factor graph design and interlayer message passing efficiency, and applications to real world networks, with the proper numerical efficiencies.## Acknowledgments

We thank Han Wang for helpful discussions. We would like to acknowledge the support of U.S. Army Research Office: Grant # W911NF-16-2-0005.

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