We show that the recently introduced label propagation method for detecting communities in complex networks is equivalent to find the local minima of a simple Potts model. Applying to empirical data, the number of such local minima was found to be very high, much larger than the number of nodes in the graph. The aggregation method for combining information from more local minima shows a tendency to fragment the communities into very small pieces.
We report on an exceptionally accurate spin-glass-type Potts model for community detection. With a simple algorithm, we find that our approach is at least as accurate as the best currently available algorithms and robust to the effects of noise. It is also competitive with the best currently available algorithms in terms of speed and size of solvable systems. We find that the computational demand often exhibits superlinear scaling L^1.3 where L is the number of edges in the system, and we have applied the algorithm to synthetic systems as large as 40x10^6 nodes and over 1x10^9 edges. A previou…
Potts model is a powerful tool to uncover community structure in complex networks. Here, we propose a new framework to reveal the optimal number of communities and stability of network structure by quantitatively analyzing the dynamics of Potts model. Specifically we model the community structure detection Potts procedure by a Markov process, which has a clear mathematical explanation. Then we show that the local uniform behavior of spin values across multiple timescales in the representation of the Markov variables could naturally reveal the network's hierarchical community structure. In addi…
Based on brief review of approaches for community identification and measurement for sensitivity characterization, the accuracy and precision of several approaches for detecting communities in weighted networks are investigated. In weighted networks, the community structure should take both links and link weights into account and the partition of networks should be evaluated by weighted modularity $Q^w$. The results reveal that link weight has important effects on communities especially in dense networks. Potts model and Weighted Extremal Optimization (WEO) algorithm work well on weighted netw…
We examine a global disorder transition when identifying community structure in an arbitrary complex network. Earlier, we illustrated [Phil. Mag. 92, 406 (2012)] that "community detection" (CD) generally exhibits disordered (or unsolvable) and ordered (solvable) phases of both high and low computational complexity along with corresponding transitions from regular to chaotic dynamics in derived systems. Using an exact generalized dimensional reduction inequality, multivariate Tutte polynomials, and other considerations, we illustrate how increasing the number of communities q emulates increasin…
Phase transitions in spin glass type systems and, more recently, in related computational problems have gained broad interest in disparate arenas. In the current work, we focus on the "community detection" problem when cast in terms of a general Potts spin glass type problem. As such, our results apply to rather broad Potts spin glass type systems. Community detection describes the general problem of partitioning a complex system involving many elements into optimally decoupled "communities" of such elements. We report on phase transitions between solvable and unsolvable regimes. Solvable regi…
The analysis of stability of community structure is an important problem for scientists from many fields. Here, we propose a new framework to reveal hidden properties of community structure by quantitatively analyzing the dynamics of Potts model. Specifically we model the Potts procedure of community structure detection by a Markov process, which has a clear mathematical explanation. Critical topological information regarding to multivariate spin configuration could also be inferred from the spectral significance of the Markov process. We test our framework on some example networks and find it…
Detecting communities in complex networks accurately is a prime challenge, preceding further analyses of network characteristics and dynamics. Until now, community detection took into account only positively valued links, while many actual networks also feature negative links. We extend an existing Potts model to incorporate negative links as well, resulting in a method similar to the clustering of signed graphs, as dealt with in social balance theory, but more general. To illustrate our method, we applied it to a network of international alliances and disputes. Using data from 1993--2001, it …
Detecting community structure in real-world networks is a challenging problem. Recently, it has been shown that the resolution of methods based on optimizing a modularity measure or a corresponding energy is limited; communities with sizes below some threshold remain unresolved. One possibility to go around this problem is to vary the threshold by using a tuning parameter, and investigate the community structure at variable resolutions. Here, we analyze the resolution limit and multiresolution behavior for two different methods: a q-state Potts method proposed by Reichard and Bornholdt, and a …
We use a Potts model community detection algorithm to accurately and quantitatively evaluate the hierarchical or multiresolution structure of a graph. Our multiresolution algorithm calculates correlations among multiple copies ("replicas") of the same graph over a range of resolutions. Significant multiresolution structures are identified by strongly correlated replicas. The average normalized mutual information, the variation of information, and other measures in principle give a quantitative estimate of the "best" resolutions and indicate the relative strength of the structures in the graph.…
Community structures detection is one of the fundamental problems in complex network analysis towards understanding the topology structures of the network and the functions of it. Nonnegative matrix factorization (NMF) is a widely used method for community detection, and modularity Q and modularity density D are criteria to evaluate the quality of community structures. In this paper, we establish the connections between Q, D and NMF for the first time. Q maximization can be approximately reformulated under the framework of NMF with Frobenius norm, especially when $n$ is large, and D maximizati…
Community structure is one of the most important features of real networks and reveals the internal organization of the nodes. Many algorithms have been proposed but the crucial issue of testing, i.e. the question of how good an algorithm is, with respect to others, is still open. Standard tests include the analysis of simple artificial graphs with a built-in community structure, that the algorithm has to recover. However, the special graphs adopted in actual tests have a structure that does not reflect the real properties of nodes and communities found in real networks. Here we introduce a ne…
We demonstrate an exact equivalence between two widely used methods of community detection in networks, the method of modularity maximization in its generalized form which incorporates a resolution parameter controlling the size of the communities discovered, and the method of maximum likelihood applied to the special case of the stochastic block model known as the planted partition model, in which all communities in a network are assumed to have statistically similar properties. Among other things, this equivalence provides a mathematically principled derivation of the modularity function, cl…
Community detection in networks refers to the process of seeking strongly internally connected groups of nodes which are weakly externally connected. In this work, we introduce and study a community definition based on internal edge density. Beginning with the simple concept that edge density equals number of edges divided by maximal number of edges, we apply this definition to a variety of node and community arrangements to show that our definition yields sensible results. Our community definition is equivalent to that of the Absolute Potts Model community detection method (Phys. Rev. E 81, 0…
Multiscale community detection can be viewed from a dynamical perspective within the Markov Stability framework, which uses the diffusion of a Markov process on the graph to uncover intrinsic network substructures across all scales. Here we reformulate multiscale community detection as a max-sum length vector partitioning problem with respect to the set of time-dependent node vectors expressed in terms of eigenvectors of the transition matrix. This formulation provides a geometric interpretation of Markov Stability in terms of a time-dependent spectral embedding, where the Markov time acts as …
We present a critical evaluation of normalized mutual information (NMI) as an evaluation metric for community detection. NMI exaggerates the leximin method's performance on weak communities: Does leximin, in finding the trivial singletons clustering, truly outperform eight other community detection methods? Three NMI improvements from the literature are AMI, rrNMI, and cNMI. We show equivalences under relevant random models, and for evaluating community detection, we advise one-sided AMI under the $\mathbb{M}_{\mathrm{all}}$ model (all partitions of $n$ nodes). This work seeks (1) to start a c…
Grouping objects into clusters based on similarities or weights between them is one of the most important problems in science and engineering. In this work, by extending message passing algorithms and spectral algorithms proposed for unweighted community detection problem, we develop a non-parametric method based on statistical physics, by mapping the problem to Potts model at the critical temperature of spin glass transition and applying belief propagation to solve the marginals corresponding to the Boltzmann distribution. Our algorithm is robust to over-fitting and gives a principled way to …
Community detection is one of the most active fields in complex networks analysis, due to its potential value in practical applications. Many works inspired by different paradigms are devoted to the development of algorithmic solutions allowing to reveal the network structure in such cohesive subgroups. Comparative studies reported in the literature usually rely on a performance measure considering the community structure as a partition (Rand Index, Normalized Mutual information, etc.). However, this type of comparison neglects the topological properties of the communities. In this article, we…
Most methods proposed to uncover communities in complex networks rely on combinatorial graph properties. Usually an edge-counting quality function, such as modularity, is optimized over all partitions of the graph compared against a null random graph model. Here we introduce a systematic dynamical framework to design and analyze a wide variety of quality functions for community detection. The quality of a partition is measured by its Markov Stability, a time-parametrized function defined in terms of the statistical properties of a Markov process taking place on the graph. The Markov process pr…
Among topics of opinion formation it is of interest to observe the characteristics of networks with a priori distinct communities. As an illustration, we report on the citation network(s) unfolded in the recent decades through web available works belonging to selected members of the Neocreationist and Intelligent Design Proponents (IDP) and the Darwinian Evolution Defenders (DED) communities. An adjacency matrix of tagged nodes is first constructed; it is not symmetric. A generalization of considerations pertaining to the case of networks with biased links, directed or undirected, is thus pres…
Label propagation is a heuristic method initially proposed for community detection in networks, while the method can be adopted also for other types of network clustering and partitioning. Among all the approaches and techniques described in this book, label propagation is neither the most accurate nor the most robust method. It is, however, without doubt one of the simplest and fastest clustering methods. Label propagation can be implemented with a few lines of programming code and applied to networks with hundreds of millions of nodes and edges on a standard computer, which is true only for …
