An Introduction to the Clique Percolation Community Detection Algorithm
Interconnected entities can be represented as networks (Barabási, 2011). Each network entails two sets, namely nodes, which are the entities, and edges, which are the connections between entities. Many networks are undirected such that edges simply connect two nodes with each other without any directionality in the edges. For instance, nodes could represent people and edges could represent whether they are friends or not. Or nodes could represent cities and edges could represent whether there is a street connecting the cities. Such networks, in which edges are either present or absent, are called unweighted. In other cases, nodes could represent people and edges could represent the frequency with which these people meet. Or nodes could represent questionnaire items and edges could represent the correlations of these items. In such a case, the edges are not only present or absent, but may count the frequency or strength of interactions. Such networks are called weighted. The edges in the network can further be directed, such that edges can point from one node to another. For instance, the internet is a network of webpages in which a directed edge could represent a hyperlink from one webpage to another. Here, only undirected networks are discussed.
Analyzing the structure of such networks is a key task across sciences. One structural feature that is often investigated is the identification of strongly connected subgraphs in the network, which is called community detection (Fortunato, 2010). Most community detection algorithms thereby put each node in only one community. However, it is likely that nodes are often shared by a number of communities. This could occur, for instance, when a friend is part of multiple groups. One community detection algorithm that is aimed at identifying overlapping communities is the clique percolation algorithm, which has been developed for unweighted (Palla et al., 2005) and weighted networks (Farkas et al., 2007).
The clique percolation algorithm for unweighted networks proceeds as follows:
- First, you identify a network of nodes and edges. I will use an unweighted network with eight nodes.
Unweighted network with eight nodes.
- Second, you identify k-cliques, which are fully connected networks with \(k\) nodes. The smallest possible \(k\) would be \(k = 3\). Otherwise, the cliques would be only edges. Applying \(k = 3\) to the example leads to the identification of six 3-cliques, namely
Six 3-cliques in unweighted network.
- Finally, a community is defined as a set of adjacent \(k\)-cliques, that is, \(k\)-cliques that share exactly \(k-1\) nodes. With \(k = 3\), two 3-cliques are adjacent if they share exactly two nodes (equivalent to an edge). In the example, this implies that there are two communities (see below). The green edges entail the 3-cliques \(a\)–\(b\)–\(c\) and \(a\)–\(b\)–\(d\). The pink edges entail the 3-cliques \(d\)–\(e\)–\(f\), \(d\)–\(e\)–\(g\), \(d\)–\(f\)–\(g\), and \(e\)–\(f\)–\(g\).
Two communities in unweighted network.
This also showcases that the clique percolation algorithm can lead to nodes that are shared by communities. In the current example, node \(d\) is part of both the green and the pink community. Nodes \(a\), \(b\), and \(c\) are part of only the green community and nodes \(e\), \(f\), and \(g\) are part of only the pink community. Importantly, clique percolation can also lead to nodes that are part of no community. These are called isolated nodes, in the current example, node \(h\).
One way to plot the community partition on the original network would be to color the nodes according to the communities they belong to. Shared nodes have multiple colors and isolated nodes remain white.
Community partition by node coloring in unweighted network.
For weighted networks, the algorithm has just one intermediate additional step. Specifically, after identifying the \(k\)-cliques, they are considered further only if their Intensity exceeds a specified threshold \(I\). The Intensity of a \(k\)-clique is defined as the geometric mean of the edge weights, namely
\[ I_C = \Bigg(\prod_{i<j;\,i,j\in C} w_{ij}\Bigg)^{2/k(k-1)} \] Where \(C\) is the clique, \(i\) and \(j\) are the nodes, \(w\) is the edge weight, and \(k\) is the clique size. For instance, a 3-clique with edge weights 0.1, 0.2, and 0.3 would have
\[ I_C = (0.1*0.2*0.3)^{2/3(3-1)} \approx 0.18 \]
To show the influence of \(I\), here is the network from the previous example, but this time I added edge weights.
Weighted network with eight nodes.
If \(I = 0.17\), only two cliques (\(a\)–\(b\)–\(c\), \(a\)–\(b\)–\(d\)) would survive the threshold. If only these are further considered, the clique percolation method would find only one community (the formerly green community) and four nodes would be isolated (\(e\), \(f\), \(g\), \(h\)). If, however, \(I = 0.09\), all cliques would survive the threshold, leading to the same community partition as for the unweighted network.
Surviving cliques in weighted network depending on I.
The program that the developers of the clique percolation algorithm designed, CFinder, adds yet another intermediate step to this procedure, even though this intermediate step is not described in Farkas et al. (2007). Specifically, the Intensity threshold \(I\) is applied not only to the \(k\)-cliques but additionally to the overlap of the \(k\)-cliques. For instance, in the case of the 3-cliques in our example, the overlap of the 3-cliques \(a\)–\(b\)–\(c\) and \(a\)–\(b\)–\(d\) is the edge \(a\)–\(b\). If \(I = 0.17\), the two 3-cliques would survive the threshold. Subsequently, it is checked whether the \(a\)–\(b\) edge also exceeds the threshold. In the current case, it would not, because the edge weights is only 0.1. Given that the edge does not exceed the threshold, the two 3-cliques are not considered adjacent. Therefore, instead of putting them in the same community, the two 3-cliques are considered to be two separate communities with two shared nodes in CFinder.
Optimizing k and I
It is challenging to optimize \(k\) and \(I\) for the clique percolation algorithm. For unweighted networks, only an optimal \(k\) needs to be found, as \(I\) is not used. Hence, when describing ways to optimize \(k\) and \(I\) for weighted networks, the same guidelines apply to unweighted networks, yet, only for \(k\). Palla et al. (2005) and Farkas et al. (2007) provide guidelines based on ideas in percolation theory, which are better illustrated for weighted networks. Specifically, for each \(k\), a large \(I\) will lead to the exclusion of most if not all \(k\)-cliques for further consideration. The other extreme, a very low \(I\), will lead to the inclusion of most if not all \(k\)-cliques for further consideration and these \(k\)-cliques then often combine to a gigantic component to which all nodes belong. The optimal \(I\) for each \(k\) is just above the point of the emergence of the gigantic component. Percolation theory supports that at this point, the size distribution of the communities follows a power-law. An indicator of this point with the broadest possible distribution is the ratio of the largest to second largest community sizes. \(I\) and \(k\) are optimal, if the largest community is twice as large as the second largest.
Notably, applying this threshold requires a large number of communities, otherwise this point might not be very stable. For somewhat smaller, yet still rather large networks, Farkas et al. (2007) propose to instead check the variance of the community sizes after excluding the largest community. The formula they provide is
\[ \chi = \sum^{n_\alpha \ne n_{max}} \frac{n_\alpha^2}{\big(\sum^\beta n_\beta\big)^2} \]
where \(\chi\) is the variance, \(n_\alpha\) is a set of communities excluding the largest one, and \(n_\beta\) is a set of communities that does neither include \(n_\alpha\) nor the largest community. For instance, if there are four communities with 9, 7, 6, and 3 nodes, respectively, then one would exclude the largest community, and determine the variance of the last three by
\[ \chi = \frac{7^2}{(6+3)^2} + \frac{6^2}{(7+3)^2} + \frac{3^2}{7+6^2} \approx 1.02 \]
The logic behind \(\chi\) is as follows. When \(I\) is higher, there will be many equally small communities of size \(k\). When \(I\) is very low, there will be a gigantic community and many equally small ones. Thus, after excluding the largest community, for higher and smaller \(I\), \(\chi\) will be small. In contrast, when the community size distribution is broad (i.e., close to a power-law), then the variance of the communities (after excluding the largest community) will be higher. Thus, the point of the maximal variance is another way to optimize \(I\) for different \(k\).
Nevertheless, also a stable estimate of \(\chi\) requires a moderate number of communities. If the network is very small, such that only a few communities can be expected, also the \(\chi\) threshold is unreliable. For instance, networks that represent psychological constructs such as attitudes (Dalege et al., 2016) or emotions (Lange, Dalege, Borsboom, Kleef, & Fischer, 2020), entail typically only up to 25 nodes. Moreover, it would be undesirable to have, for instance, two communities of which one is twice as large as the other. Instead, equally sized communities are preferable. An alternative way to optimize \(I\) for different \(k\) for very small networks would be to rely on the entropy of the community partition, for instance, an entropy measure based on Shannon Information (Shannon, 1948). The idea is to find the most surprising community partition, defined as a low probability of knowing to which community a randomly picked node belongs. If there is only one community, surprisingness would be zero, because it would be certain that a randomly picked node belongs to this community. If there are, for instance, two communities of sizes 15 and 3, it would still not be very surprising, because most likely a randomly picked node will belong to the larger community. Surprisingness would be higher, however, when the communities are equal in size. Entropy based on Shannon Information captures this idea in the equation
\[ Entropy = -\sum_{i=1}^N p_i * \log_2 p_i \]
where \(N\) is the number of communities and \(p_i\) is the probability of being in community \(i\). For the four community sizes above (9, 7, 6, 3), the result would be
\[ \begin{aligned} Entropy &= -\Bigg(\Big(\frac{9}{9+7+6+3} * log_2 \frac{9}{9+7+6+3}\Big) \\ & + \Big(\frac{7}{9+7+6+3} * log_2 \frac{7}{9+7+6+3}\Big) \\ & + \Big(\frac{6}{9+7+6+3} * log_2 \frac{6}{9+7+6+3}\Big) \\ & + \Big(\frac{3}{9+7+6+3} * log_2 \frac{3}{9+7+6+3}\Big)\Bigg) \\ &\approx 1.91 \end{aligned} \]
As pointed out, with only one community, this equation equals zero. When calculating entropy, isolated nodes are treated as another pseudo-community and shared nodes are split equally among the communities they belong to (e.g., a node shared by two communities is added as 0.5 nodes to each of them). As such, entropy favors equally sized communities with few isolated nodes in very small networks. This is exactly what would be desired. The \(I\) that has the maximum entropy for the respective \(k\) optimizes \(k\) and \(I\).
Because for very small networks, a specific amount of communities can occur just by chance alone, permutation tests may complement the calculation of entropy. Specifically, the edges of the network are randomly shuffled a number of times and for each of these random permutations, the highest entropy of a range of \(I\) values is determined separately for each \(k\). Over all permutations, this analysis leads to a distribution of entropy values for each \(k\) separately. Only entropy values that exceed the confidence interval of the distribution of each \(k\) can be considered more surprising than would already be expected by chance alone. For large and very large networks, the entropy threshold should, however, not be used. As it prefers equally sized community sizes, it would penalize the broad power-law distribution of community sizes that is preferred for larger networks.
More recently, Santiago et al. (Santiago, Soares, Quintero, & Jamieson, 2022) used simulations to compare the performance of different thresholds for optimizing \(k\) and \(I\). They argued convincingly that even entropy may often fail to detect the best community partition. Their simulations instead supported the conclusion that thresholds based on fuzzy modularity outperform entropy. Details are available in their manuscript.
Finally, for unweighted networks, the optimization of \(k\) and \(I\) is similar. Yet, given the limited range of possible \(k\) (e.g., 3 to 6) many thresholds will not be very sensitive. Nevertheless, a small \(k\) (e.g., \(k = 3\)) will often result in the emergence of a giant component to which most of the nodes belong. Increasing \(k\) (e.g., \(k = 6\)) will often lead to a large number of smaller communities. \(k\) is optimal when there is no giant component for very large networks, when \(\chi\) is maximal for large networks, or when entropy is maximal for very small networks.
Summary
In sum, the clique percolation algorithm proceeds by identifying \(k\)-cliques and by putting adjacent \(k\)-cliques (i.e., \(k\)-cliques that share \(k - 1\) nodes) into a community. For weighted networks, the Intensity of the \(k\)-cliques must further exceed \(I\). If not, they are removed from consideration. Finally, in the CFinder program, the \(k - 1\) overlap of \(k\)-cliques must further exceed \(I\). If not, the \(k\)-cliques are considered separately and not as a single community. Optimal \(k\) and \(I\) can be determined with the ratio test for large networks, \(\chi\) for small networks, or entropy for very small networks.
