On spectral partitioning of signed graphs

Knyazev, Andrew

doi:10.1137/1.9781611975215.2

Cited by 9 publications

(5 citation statements)

References 23 publications

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“…Most similar to our work here is the work of [22] in which Knyazev uses the repelling Laplacian L r = D − A where D is the diagonal matrix of row sums of the adjacency matrix A, which can take positive and negative values. Knyazev argues that signed graphs can be clustered effectively using the repelling Laplacian from a mechanical perspective, modeling clusters as eigenmodes of a mass-spring system with negative springs.…”

Section: A Spectral Methodsmentioning

confidence: 80%

“…Note that even if λ k+1 < 0, the addition of the k + 1-th dimension may still increase the energy H(D(k +1)) > H(D(k) due to the normalization factor. This is related to the idea proposed in [22] of using the k smallest eigenvectors for the embedding, where k is chosen by looking for the largest eigen-gap, but formalizes this argument in term of a physical energy function.…”

Section: Generalizations To Non-complete Graphsmentioning

confidence: 99%

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SHEEP: Signed Hamiltonian Eigenvector Embedding for Proximity

Babul¹,

Lambiotte²

2023

Preprint

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We introduce a spectral embedding algorithm for finding proximal relationships between nodes in signed graphs, where edges can take either positive or negative weights. Adopting a physical perspective, we construct a Hamiltonian which is dependent on the distance between nodes, such that relative embedding distance results in a similarity metric between nodes. The Hamiltonian admits a global minimum energy configuration, which can be re-configured as an eigenvector problem, and therefore is computationally efficient to compute. We use matrix perturbation theory to show that the embedding generates a ground state energy, which can be used as a statistical test for the presence of strong balance, and to develop an energy-based approach for locating the optimal embedding dimension. Finally, we show through a series of experiments on synthetic and empirical networks, that the resulting position in the embedding can be used to recover certain continuous node attributes, and that the distance to the origin in the optimal embedding gives a measure of node "extremism".

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Section: A Spectral Methodsmentioning

confidence: 80%

Section: Generalizations To Non-complete Graphsmentioning

confidence: 99%

SHEEP: Signed Hamiltonian Eigenvector Embedding for Proximity

Babul¹,

Lambiotte²

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…It should be further noted that the programming language used for implementation and obtaining these ndings is Python 3. * Obtained by applying the Doriean and Mrvar [25] approach to the dataset using Pajek software [47].…”

Section: Resultsmentioning

confidence: 99%

BridgeCut: A New Algorithm for Balanced Partitioning of Signed Networks

Ehsani

Mansouri

2023

Preprint

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Structural analysis of signed networks is an important issue with many applications in the field of complex networks. In this context, balance theory is the main framework that leads naturally to partitioning balanced graphs into two parts with entirely positive edges in each part and only negative edges between them. However, real-world signed networks rarely have a perfectly balanced structure, and detecting near-balanced partitions can be formulated as an optimization problem with different objective functions. However, as an NP-Hard problem with many applications, different approaches have been proposed to it. Nevertheless, many methods ignore negative links and divide the graph into parts considering only positive edges. Emphasizing the importance of negative links and their properties in the partition scheme, we present a new approach, inspired by the Girvan-Newman community detection algorithm, called BridgeCut, in which the negative edges as the bridge set between the two parts are determined in a constructive algorithm. Sorted by edge betweenness, every negative edge is evaluated whether to be selected as a member of the bridge set or not, based on a balance improvement index. Analyzing the time complexity of this algorithm, it is implemented on some benchmark datasets and the results confirm its satisfactory performance. We further extend some measures to evaluate signed links' properties in relevance to their position and role in the whole structure.

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“…We will assume throughout an undirected, unweighted, unsigned, connected and static graph G = (V, E) with vertex and edge sets of size |V| = n and |E| = m, respectively. Spectral clustering methods generalize trivially to weighted graphs and other have extended them to handle signed graphs (Knyazev, 2018), dynamic graphs (Ning et al, 2007), and directed graphs (Van Lierde et al, 2018). The sketching methods we employ generalize to weighted, signed, dynamic, and directed graphs with much less complication, as the key operator is a simple linear projection.…”

Section: Notations and Backgroundmentioning

confidence: 99%

Scaling Graph Clustering with Distributed Sketches

Priest¹,

Dunton²,

Sanders³

2020

Preprint

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The unsupervised learning of community structure, in particular the partitioning vertices into clusters or communities, is a canonical and well-studied problem in exploratory graph analysis. However, like most graph analyses the introduction of immense scale presents challenges to traditional methods. Spectral clustering in distributed memory, for example, requires hundreds of expensive bulksynchronous communication rounds to compute an embedding of vertices to a few eigenvectors of a graph associated matrix. Furthermore, the whole computation may need to be repeated if the underlying graph changes some low percentage of edge updates. We present a method inspired by spectral clustering where we instead use matrix sketches derived from random dimension-reducing projections. We show that our method produces embeddings that yield performant clustering results given a fully-dynamic stochastic block model stream using both the fast Johnson-Lindenstrauss and CountSketch transforms. We also discuss the effects of stochastic block model parameters upon the required dimensionality of the subsequent embeddings, and show how random projections could significantly improve the performance of graph clustering in distributed memory.

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On spectral partitioning of signed graphs

Cited by 9 publications

References 23 publications

SHEEP: Signed Hamiltonian Eigenvector Embedding for Proximity

SHEEP: Signed Hamiltonian Eigenvector Embedding for Proximity

BridgeCut: A New Algorithm for Balanced Partitioning of Signed Networks

Scaling Graph Clustering with Distributed Sketches

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