Spectral Analysis of Signed Graphs for Clustering, Prediction and Visualization

Kunegis, Jérôme; Schmidt, Stephan; Lommatzsch, Andreas; Lerner, Jürgen; Luca, Ernesto William De; Albayrak, Şahin

doi:10.1137/1.9781611972801.49

Cited by 300 publications

(309 citation statements)

References 18 publications

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“…In particular, a signed graph is exactly balanced (i.e., tensions are completely absent) if and only if all its cycles are positive (16). As such, structural balance is intrinsically a property of the network as a whole, not fragmentable into elementary subgraphs.From a computational point of view, verifying if a signed undirected network is exactly balanced is an easy problem, which can be answered in polynomial time (17)(18)(19). When instead a graph is not exactly balanced, one can compute a distance to exact balance (i.e., a measure of the amount of unbalance in the network).…”

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confidence: 99%

“…From a computational point of view, verifying if a signed undirected network is exactly balanced is an easy problem, which can be answered in polynomial time (17)(18)(19). When instead a graph is not exactly balanced, one can compute a distance to exact balance (i.e., a measure of the amount of unbalance in the network).…”

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confidence: 99%

“…An alternative approach is taken in ref. 18, where spectral properties of the Laplacian are investigated. For connected signed graphs, the magnitude of the smallest eigenvalue of the Laplacian is indicative of how unbalanced a network is-i.e., of how much frustration is encoded in the cycles of the networks.…”

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Computing global structural balance in large-scale signed social networks

Facchetti

Iacono

Altafini

2011

Proc. Natl. Acad. Sci. U.S.A.

351

287

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Structural balance theory affirms that signed social networks (i.e., graphs whose signed edges represent friendly/hostile interactions among individuals) tend to be organized so as to avoid conflictual situations, corresponding to cycles of negative parity. Using an algorithm for ground-state calculation in large-scale Ising spin glasses, in this paper we compute the global level of balance of very large online social networks and verify that currently available networks are indeed extremely balanced. This property is explainable in terms of the high degree of skewness of the sign distributions on the nodes of the graph. In particular, individuals linked by a large majority of negative edges create mostly "apparent disorder," rather than true "frustration."combinatorial optimization | social network theory O nline social networks are examples of large-scale communities of interacting individuals in which local ties between users (friend, fan, colleague, but also friend/foe, trust/distrust, etc.) give rise to a complex, multidimensional web of aggregated social behavior (1-4). For such complex networks, the emergence of global properties from local interactions is an intriguing subject, so far investigated mostly at structural and topological level (2,(5)(6)(7)(8). In social network theory (9-11), however, the content of the relationships is often even more important than their topology, and this calls for the development of appropriate analytical and computational tools, able to extrapolate content-related features out of the set of interactions of a social community. Obtaining efficient tools is particularly challenging when, as in social networks retrieved from online media, the size of the community is very big, of the order of 10 5 individuals or higher.A global property that has recently attracted some attention (1,(12)(13)(14) is determining the structural balance of a signed social network. Structural (or social) balance theory was first formulated by Heider (15) in order to understand the structure and origin of tensions and conflicts in a network of individuals whose mutual relationships are characterizable in terms of friendship and hostility. It was modeled in terms of signed graphs by Cartwright and Harary (16); see refs. 10 and 11 for an overview of the theory. The nodes of the graph represent users and the positive/ negative edges their friendly/hostile relationships. It has been known for some time how to interpret structural balance on such networks (16): The potential source of tensions are the cycles of the graph (i.e., the closed paths beginning and ending on the same node), notably those of negative sign (i.e., having an odd number of negative edges). It follows that the concept of balance is not related to the actual number of negative edges on the cycles but only to their parity; see Fig. 1 for an illustration on basic graphs. In particular, a signed graph is exactly balanced (i.e., tensions are completely absent) if and only if all its cycles are positive (16). As such, structural balance ...

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Computing global structural balance in large-scale signed social networks

Facchetti

Iacono

Altafini

2011

Proc. Natl. Acad. Sci. U.S.A.

351

287

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“…In [10], degree based features such as the number of incoming positive and negative links of a node and triad based features that include the structure information of a triad are defined manually and extracted from the network to represent the nodes for sign prediction in signed network. Another work in [21] extends spectral analysis for signed network. In this paper, we study the novel problem of learning embedding for signed social network by utilizing social theory.…”

Section: Related Workmentioning

confidence: 99%

“…• SC [21]: A spectral clustering algorithm is proposed where a signed version of Laplacian matrix is defined. In this experiment, for the link prediction purpose, we choose the top-d eigen-vetors corresponding to the smallest eigenvalues of the signed Laplacian matrix as the low dimensional vector representations of nodes.…”

Section: Signed Link Prediction In Signed Socialmentioning

confidence: 99%

Signed Network Embedding in Social Media

Wang

Tang

Aggarwal³

et al. 2017

Proceedings of the 2017 SIAM International Conference on Data Mining

199

162

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Network embedding is to learn low-dimensional vector representations for nodes of a given social network, facilitating many tasks in social network analysis such as link prediction. The vast majority of existing embedding algorithms are designed for unsigned social networks or social networks with only positive links. However, networks in social media could have both positive and negative links, and little work exists for signed social networks. From recent findings of signed network analysis, it is evident that negative links have distinct properties and added value besides positive links, which brings about both challenges and opportunities for signed network embedding. In this paper, we propose a deep learning framework SiNE for signed network embedding. The framework optimizes an objective function guided by social theories that provide a fundamental understanding of signed social networks. Experimental results on two realworld datasets of social media demonstrate the effectiveness of the proposed framework SiNE.

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Structured Networks and Coarse‐Grained Descriptions

Delvenne

Lambiotte

Barahona

2019

Advances in Network Clustering and Blockmodeling

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This chapter discusses the interplay between structure and dynamics in complex networks. Given a particular network with an endowed dynamics, our goal is to find partitions aligned with the dynamical process acting on top of the network. We thus aim to gain a reduced description of the system that takes into account both its structure and dynamics.In the first part, we introduce the general mathematical setup for the types of dynamics we consider throughout the chapter. We provide two guiding examples, namely consensus dynamics and diffusion processes (random walks), motivating their connection to social network analysis, and provide a brief discussion on the general dynamical framework and its possible extensions.In the second part, we focus on the influence of graph structure on the dynamics taking place on the network, focussing on three concepts that allow us to gain insight into this notion. First, we describe how time scale separation can appear in the dynamics on a network as a consequence of graph structure. Second, we discuss how the presence of particular symmetries in the network give rise to invariant dynamical subspaces that can be precisely described by graph partitions. Third, we show how this dynamical viewpoint can be extended to study dynamics on networks with signed edges, which allow us to discuss connections to concepts in social network analysis, such as structural balance.In the third part, we discuss how to use dynamical processes unfolding on the network to detect meaningful network substructures. We then show how such dynamical measures can be related to seemingly different algorithm for community detection and coarse-graining proposed in the literature. We conclude with a brief summary and highlight interesting open future directions.

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Spectral Analysis of Signed Graphs for Clustering, Prediction and Visualization

Cited by 300 publications

References 18 publications

Computing global structural balance in large-scale signed social networks

Computing global structural balance in large-scale signed social networks

Signed Network Embedding in Social Media

Structured Networks and Coarse‐Grained Descriptions

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