Extremal optimization for graph partitioning

A Graph-Cut-Based Approach to Community Detection in Networks

Networks can be used to model various aspects of our lives as well as relations among many real-world entities and objects. To detect a community structure in a network can enhance our understanding of the characteristics, properties, and inner workings of the network. Therefore, there has been significant research on detecting and evaluating community structures in networks. Many fields, including social sciences, biology, engineering, computer science, and applied mathematics, have developed various methods for analyzing and detecting community structures in networks. In this paper, a new community detection algorithm, which repeats the process of dividing a community into two smaller communities by finding a minimum cut, is proposed. The proposed algorithm is applied to some example network data and shows fairly good community detection results with comparable modularity Q values.

Graph Neural Networks for Maximum Constraint Satisfaction

Many combinatorial optimization problems can be phrased in the language of constraint satisfaction problems. We introduce a graph neural network architecture for solving such optimization problems. The architecture is generic; it works for all binary constraint satisfaction problems. Training is unsupervised, and it is sufficient to train on relatively small instances; the resulting networks perform well on much larger instances (at least 10-times larger). We experimentally evaluate our approach for a variety of problems, including Maximum Cut and Maximum Independent Set. Despite being generic, we show that our approach matches or surpasses most greedy and semi-definite programming based algorithms and sometimes even outperforms state-of-the-art heuristics for the specific problems.

Cluster Structure of Optimal Solutions in Bipartitioning of Small Worlds

Using simulated annealing, we examine a bipartitioning of small worlds obtained by adding a fraction of randomly chosen links to a one-dimensional chain or a square lattice. Models defined on small worlds typically exhibit a mean-field behavior, regardless of the underlying lattice. Our work demonstrates that the bipartitioning of small worlds does depend on the underlying lattice. Simulations show that for one-dimensional small worlds, optimal partitions are finite size clusters for any fraction of additional links. In the two-dimensional case, we observe two regimes: when the fraction of additional links is sufficiently small, the optimal partitions have a stripe-like shape, which is lost for a larger number of additional links as optimal partitions become disordered. Some arguments, which interpret additional links as thermal excitations and refer to the thermodynamics of Ising models, suggest a qualitative explanation of such a behavior. The histogram of overlaps suggests that a replica symmetry is broken in a one-dimensional small world. In the two-dimensional case, the replica symmetry seems to hold, but with some additional degeneracy of stripe-like partitions.

Topological and functional comparison of community detection algorithms in biological networks

Background

Community detection algorithms are fundamental tools to uncover important features in networks. There are several studies focused on social networks but only a few deal with biological networks. Directly or indirectly, most of the methods maximize modularity, a measure of the density of links within communities as compared to links between communities.

Results

Here we analyze six different community detection algorithms, namely, Combo, Conclude, Fast Greedy, Leading Eigen, Louvain and Spinglass, on two important biological networks to find their communities and evaluate the results in terms of topological and functional features through Kyoto Encyclopedia of Genes and Genomes pathway and Gene Ontology term enrichment analysis. At a high level, the main assessment criteria are 1) appropriate community size (neither too small nor too large), 2) representation within the community of only one or two broad biological functions, 3) most genes from the network belonging to a pathway should also belong to only one or two communities, and 4) performance speed. The first network in this study is a network of Protein-Protein Interactions (PPI) in Saccharomyces cerevisiae (Yeast) with 6532 nodes and 229,696 edges and the second is a network of PPI in Homo sapiens (Human) with 20,644 nodes and 241,008 edges. All six methods perform well, i.e., find reasonably sized and biologically interpretable communities, for the Yeast PPI network but the Conclude method does not find reasonably sized communities for the Human PPI network. Louvain method maximizes modularity by using an agglomerative approach, and is the fastest method for community detection. For the Yeast PPI network, the results of Spinglass method are most similar to the results of Louvain method with regard to the size of communities and core pathways they identify, whereas for the Human PPI network, Combo and Spinglass methods yield the most similar results, with Louvain being the next closest.

Conclusions

For Yeast and Human PPI networks, Louvain method is likely the best method to find communities in terms of detecting known core pathways in a reasonable time.

Electronic supplementary material

The online version of this article (10.1186/s12859-019-2746-0) contains supplementary material, which is available to authorized users.

Modular organization of brain resting state networks in chronic back pain patients

Recent work on functional magnetic resonance imaging large-scale brain networks under resting conditions demonstrated its potential to evaluate the integrity of brain function under normal and pathological conditions. A similar approach is used in this work to study a group of chronic back pain patients and healthy controls to determine the impact of long enduring pain over brain dynamics. Correlation networks were constructed from the mutual partial correlations of brain activity's time series selected from ninety regions using a well validated brain parcellation atlas. The study of the resulting networks revealed an organization of up to six communities with similar modularity in both groups, but with important differences in the membership of key communities of frontal and temporal regions. The bulk of these findings were confirmed by a surprisingly naive analysis based on the pairwise correlations of the strongest and weakest correlated healthy regions. Beside confirming the brain effects of long enduring pain, these results provide a framework to study the effect of other chronic conditions over cortical function.

Graph partitioning induced phase transitions

We study the percolation properties of graph partitioning on random regular graphs with N vertices of degree k. Optimal graph partitioning is directly related to optimal attack and immunization of complex networks. We find that for any partitioning process (even if nonoptimal) that partitions the graph into essentially equal sized connected components (clusters), the system undergoes a percolation phase transition at f = fc = 1-2/k where f is the fraction of edges removed to partition the graph. For optimal partitioning, at the percolation threshold, we find S approximately N 0.4 where S is the size of the clusters and l approximately N 0.25 where l is their diameter. Also, we find that S undergoes multiple nonpercolation transitions for f<fc.

Phase transitions in the coloring of random graphs

We consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color. Using the cavity method, we present a detailed and systematic analytical study of the space of proper colorings (solutions). We show that for a fixed number of colors and as the average vertex degree (number of constraints) increases, the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses. First, at the clustering transition, the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard. Afterward, the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one. Another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state. Eventually, above the coloring threshold, no more solutions are available. We compute all the critical connectivities for Erdos-Rényi and regular random graphs and determine their asymptotic values for a large number of colors. Finally, we discuss the algorithmic consequences of our findings. We argue that the onset of computational hardness is not associated with the clustering transition and we suggest instead that the freezing transition might be the relevant phenomenon. We also discuss the performance of a simple local Walk-COL algorithm and of the belief propagation algorithm in the light of our results.

Network community structure and loop coefficient method

A modular structure, in which groups of tightly connected nodes could be resolved as separate entities, is a property that can be found in many complex networks. In this paper, we propose a algorithm for identifying communities in networks. It is based on a local measure, so-called loop coefficient that is a generalization of the clustering coefficient. Nodes with a large loop coefficient tend to be core inner community nodes, while other vertices are usually peripheral sites at the borders of communities. Our method gives satisfactory results for both artificial and real-world graphs, if they have a relatively pronounced modular structure. This type of algorithm could open a way of interpreting the role of nodes in communities in terms of the local loop coefficient, and could be used as a complement to other methods.

Community detection in complex networks using extremal optimization

We propose a method to find the community structure in complex networks based on an extremal optimization of the value of modularity. The method outperforms the optimal modularity found by the existing algorithms in the literature giving a better understanding of the community structure. We present the results of the algorithm for computer-simulated and real networks and compare them with other approaches. The efficiency and accuracy of the method make it feasible to be used for the accurate identification of community structure in large complex networks.

Continuous extremal optimization for Lennard-Jones clusters

We explore a general-purpose heuristic algorithm for finding high-quality solutions to continuous optimization problems. The method, called continuous extremal optimization (CEO), can be considered as an extension of extremal optimization and consists of two components, one which is responsible for global searching and the other which is responsible for local searching. The CEO's performance proves competitive with some more elaborate stochastic optimization procedures such as simulated annealing, genetic algorithms, and so on. We demonstrate it on a well-known continuous optimization problem: the Lennard-Jones cluster optimization problem.

Extremal optimization at the phase transition of the three-coloring problem

We investigate the phase transition in vertex coloring on random graphs, using the extremal optimization heuristic. Three-coloring is among the hardest combinatorial optimization problems and is equivalent to a 3-state anti-ferromagnetic Potts model. Like many other such optimization problems, it has been shown to exhibit a phase transition in its ground state behavior under variation of a system parameter: the graph's mean vertex degree. This phase transition is often associated with the instances of highest complexity. We use extremal optimization to measure the ground state cost and the "backbone," an order parameter related to ground state overlap, averaged over a large number of instances near the transition for random graphs of size n up to 512. For these graphs, benchmarks show that extremal optimization reaches ground states and explores a sufficient number of them to give the correct backbone value after about O (n(3.5)) update steps. Finite size scaling yields a critical mean degree value alpha(c) =4.703 (28). Furthermore, the exploration of the degenerate ground states indicates that the backbone order parameter, measuring the constrainedness of the problem, exhibits a first-order phase transition.

Random geometric graphs

We analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present. The critical connectivity is found numerically by examining the size of the largest cluster. We derive an analytical expression for the cluster coefficient, which shows that the graphs are distinctly different from standard random graphs, even for infinite dimensionality. Insights relevant for graph bipartitioning are included.