Fusion and community detection in multi-layer graphs

Abstract

Abstract-Relational data arising in many domains can be represented by networks (or graphs) with nodes capturing entities and edges representing relationships between these entities. Community detection in networks has become one of the most important problems having a broad range of applications. Until recently, the vast majority of papers have focused on discovering community structures in a single network. However, with the emergence of multi-view network data in many realworld applications and consequently with the advent of multilayer graph representation, community detection in multi-layer graphs has become a new challenge. Multi-layer graphs provide complementary views of connectivity patterns of the same set of vertices. Fusion of the network layers is expected to achieve better clustering performance. In this paper, we propose two novel methods, coined as WSSNMTF (Weighted Simultaneous Symmetric Non-Negative Matrix Tri-Factorization) and NG-WSSNMTF (Natural Gradient WSSNMTF), for fusion and clustering of multi-layer graphs. Both methods are robust with respect to missing edges and noise. We compare the performance of the proposed methods with two baseline methods, as well as with three state-of-the-art methods on synthetic and three real-world datasets. The experimental results indicate superior performance of the proposed methods.

A Survey of Community Detection in Complex Networks Using Nonnegative Matrix Factorization

“…Fusion capability Scalability Dynamicity NMTF [21] × × × BNMTF [22] × × × PCSNMF [23] √ × × PSSNMF [24] √ × × HPNMF [25] √ × × HNMF [26] √ × × A 2 NMF [27] √ × × PNMF [28] √ × × JNMF [31] × × × SGNMF [32] × × × MCNMF [33] × × × ReS-NMF [36] × × × BRSNMF [37] × × × SPOCD [38] × × × FSL [40] √ × × JWNMF [41] √ × × NMTFR [42] √ × × CFOND [43] √ × × SCI [44] √ × × ASCD [45] √ × × DII [46] √ × × RSECD [47] √ × × WSSNMTF [50] √ × × NF-CCE [51] √ × × MTRD [53] √ × × LJ-SNMF [54] √ × × S2-jNMF [55] √ × × sE-NMF [ Community detection in large-scale dynamic heterogenous networks is more challenging. This needs to design a method that not only have good information fusion capability and scalability, but also can detect dynamic communities.…”

“…Representative methods Topology networks NMTF [21], BNMTF [22], PCSNMF [23], PSSNMF [24], HPNMF [25], HNMF [26], A 2 NMF [27], PNMF [28] Signed networks JNMF [31], SGNMF [32], MCNMF [33], ReS-NMF [36], BRSNMF [37], SPOCD [38] Attributed networks FSL [40], JWNMF [41], NMTFR [42], CFOND [43], SCI [44], ASCD [45], DII [46], RSECD [47] Multi-layer networks WSSNMTF [50], NF-CCE [51], MTRD [53], LJ-SNMF [54], S2-jNMF [55] Dynamic networks sE-NMF [57], GrENMF [58], Cr-ENMF [59], ECGNMF [60], DGR-SNMF [61], DBNMF [62], C 3 [66], Chimera [70] Large-scale networks BIGCLAM [73], HierSymNMF2 [75], cyclicCDSymNMF [77], OGNMF [79], DRNMFSR [80], TCB [81] are often utilized. One is prior knowledge, also known as semi-supervised information, such as ground-truth community labels, node must-link and cannot-link constraints.…”

“…Unexceptionally, these methods all utilize the joint NMF framework to fuse these so-called multi-view network data. For example, in [50] Gligorijević et al proposed a method named WSSNMTF, which is based on weighted simultaneous symmetric NMTF. This method firstly represents a multi-layer network with N layers by a set of adjacency matrices A (i) ∈ R n×n + (i = 1, 2, ..., N ), and meanwhile introduces a weight matrix Ω (i) ∈ R n×n + defined as follows to accelerate the later decomposition operations.…”

“…S is the number of networks slices GrENMF [50] O(Sn 2 k) Cr-ENMF [51] O(Sn 2 k) ECGNMF [52] O(Sn 2 k) DGR-SNMF [53] O(Sn 2 k) DBNMF [54] O(Sn 2 k) C 3 [58] O(S(n 2 k + nk 2 ) + mk) Chimera [62] O(S(n 2 k + nmk))…”

Mixtures of Block Models for Brain Networks

“…Existing work focuses on learning one block structure from multiple graphs, assuming that (1) the link sets of different single graphs share the same generating mechanism, or (2) incorporating additional information, such as structural guidance on the blocks, from other graphs can enhance block structure detection on a particular network. Seminal multilinear models include RESCAL [18], LMF [28], and later Weighted Simultaneous Symmetric Non-Negative Matrix Tri-Factorization (with Natural Gradient) [12], which are essentially variants of Tucker-decomposition. Seminal work focusing on block structure detection from multigraphs includes GraphFuse based on PARAFAC with sparse latent factors [20] and reformulation of the standard spectral learning model for multiview clustering and semisupervised tasks [19], etc.…”