Determining the distance to monotonicity of a biological network: a graph-theoretical approach

Iacono, Giovanni; Ramezani, Fahimeh; Soranzo, Nicola; Altafini, Claudio

doi:10.1049/iet-syb.2009.0040

Cited by 54 publications

(135 citation statements)

References 20 publications

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“…The heuristic we have introduced in ref. 33 is, however, able to produce fairly tight upper and lower bounds for δ (henceforth δ up and δ low ), also for very large signed graphs. This local search algorithm is described in some detail in the SI Text and in ref.…”

Section: Computation Of Global Balancementioning

confidence: 99%

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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.

360

298

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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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Section: Computation Of Global Balancementioning

confidence: 99%

“…This local search algorithm is described in some detail in the SI Text and in ref. 33. The outcome of the algorithm is a gauge transformation of the adjacency matrix J into the equivalent J σ :…”

Section: Computation Of Global Balancementioning

confidence: 99%

Computing global structural balance in large-scale signed social networks

Facchetti

Iacono

Altafini

2011

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

360

298

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“…(2) is often considered as frustration index [26], true frustration, or merely frustration [20,27]. However, in this context, the frustration and its minimum are considered separately.…”

Section: Preliminaries a Notationsmentioning

confidence: 99%

“…If we set k = 2, the optimal solution is the best two-clustering of a graph where the number of inconsistent edges is equal to the distance of a graph from SB [= F 2 (G)]. Iacono et al proposed a graph-theoretic approach to approximate F 2 (G), which has been originally stated as "distance from monotonicity" for biological networks [27]; note that monotonicity has the same mathematical implication as SB. The algorithm has been further applied to social networks validating that their distance from SB is significantly lower than those of sign-shuffled counterparts [20,34].…”

Section: A Structural Balance and Clusteringmentioning

confidence: 99%

“…Another achievement of Ref. [27] is a scalable algorithm that calculates a lower bound for F 2 (G), which determines, at most, how far is the proposed solution from the optimal value. For k > 2, Chiang et al [35] proposed a scalable k-clustering algorithm by transforming an objective function similar to F k (G, C) (along with some other objectives) into weighted kernel k-means.…”

Section: A Structural Balance and Clusteringmentioning

confidence: 99%

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Mesoscopic analysis of online social networks: The role of negative ties

2014

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A class of networks are those with both positive and negative links. In this manuscript, we studied the interplay between positive and negative ties on mesoscopic level of these networks, i.e., their community structure. A community is considered as a tightly interconnected group of actors; therefore, it does not borrow any assumption from balance theory and merely uses the well-known assumption in the community detection literature. We found that if one detects the communities based on only positive relations (by ignoring the negative ones), the majority of negative relations are already placed between the communities. In other words, negative ties do not have a major role in community formation of signed networks. Moreover, regarding the internal negative ties, we proved that most unbalanced communities are maximally balanced, and hence they cannot be partitioned into k nonempty sub-clusters with higher balancedness (k ≥ 2). Furthermore, we showed that although the mediator triad + + − (hostile-mediator-hostile) is underrepresented, it constitutes a considerable portion of triadic relations among communities. Hence, mediator triads should not be ignored by community detection and clustering algorithms. As a result, if one uses a clustering algorithm that operates merely based on social balance, mesoscopic structure of signed networks significantly remains hidden.

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Partitioning Signed Networks

Traag

Doreian

Mrvar

2019

Advances in Network Clustering and Blockmodeling

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Signed networks appear naturally in contexts where conflict or animosity is apparent. In this book chapter we review some of the literature on signed networks, especially in the context of partitioning. Most of the work is founded in what is known as structural balance theory. We cover the basic mathematical principles of structural balance theory. The theory yields a natural formulation for partitioning. We briefly compare this to other partitioning approaches based on community detection. Finally, we analyse an international network of alliances and conflicts and discuss the implications of our findings for structural balance theory.We are concerned with signed networks, where each link is associated with either a positive (+) or negative sign (−). More generally, weights w ij could be used. Although weights are often assumed to be positive, we explicitly allow them also to be negative. For simplicity, we deal primarily with non-weighted networks, but most concepts used here can be adapted easily to the weighted case. I. NOTATIONWhile we try to be as consistent as possible with the general notation used throughout this book, we require some additional notation because signed networks have signs for arcs and edges. We denote a directed signed network by G = (V, A − , A + ) where A − ⊆ V × V are the negative links and A + ⊆ V × V the positive links. We assume that A − ∩A + = ∅, so that no link is both positive and negative. We exclude loops on nodes. Many studied signed networks are directed. Some are not, including the network we study here. Similarly, an undirected signed network is denoted by G = (V, E − , E + ) where E − ⊆ V ×V are the negative links and E + ⊆ V ×V the positive links. As for the directed case, E − ∩ E + = ∅.We present our initial discussion in terms of directed signed networks. However, if we restrict ourselves to undirected graphs, then (i, j) ∈ E ± is identical to (j, i) ∈ E ± . Also, we assume that there are no self-loops, i.e. no (i, i) exists. For edges, the signs on them are symmetrical by definition.We define the adjacency matrices A + and A − . We set A + ij = 1 whenever (i, j) ∈ A + and A + ij = 0 otherwise. Similarly, A − ij = 1 whenever (i, j) ∈ A − and A − ij = 0 otherwise. We denote the signed adjacency matrix A = arXiv:1803.02082v1 [physics.soc-ph]

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Determining the distance to monotonicity of a biological network: a graph-theoretical approach

Cited by 54 publications

References 20 publications

Computing global structural balance in large-scale signed social networks

Computing global structural balance in large-scale signed social networks

Mesoscopic analysis of online social networks: The role of negative ties

Partitioning Signed Networks

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