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Abstract:We propose new measures of shared information, unique information and synergistic information that can be used to decompose the mutual information of a pair of random variables (Y, Z) with a third random variable X. Our measures are motivated by an operational idea of unique information, which suggests that shared information and unique information should depend only on the marginal distributions of the pairs (X, Y ) and (X, Z). Although this invariance property has not been studied before, it is satisfied by other proposed measures of shared information. The invariance property does not uniquely determine our new measures, but it implies that the functions that we define are bounds to any other measures satisfying the same invariance property. We study properties of our measures and compare them to other candidate measures.

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In this paper, we explore the relationship between one of the most elementary and important properties of graphs, the presence and relative frequency of triangles, and a combinatorial notion of Ricci curvature. We employ a definition of generalized Ricci curvature proposed by Ollivier in a general framework of Markov processes and metric spaces and applied in graph theory by Lin-Yau. In analogy with curvature notions in Riemannian geometry, we interpret this Ricci curvature as a control on the amount of overlap between neighborhoods of two neighboring vertices. It is therefore naturally related to the presence of triangles containing those vertices, or more precisely, the local clustering coefficient, that is, the relative proportion of connected neighbors among all the neighbors of a vertex. This suggests to derive lower Ricci curvature bounds on graphs in terms of such local clustering coefficients. We also study curvature dimension inequalities on graphs, building upon previous work of several authors.

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We have performed an empirical comparison of two distinct notions of discrete Ricci curvature for graphs or networks, namely, the Forman-Ricci curvature and Ollivier-Ricci curvature. Importantly, these two discretizations of the Ricci curvature were developed based on different properties of the classical smooth notion, and thus, the two notions shed light on different aspects of network structure and behavior. Nevertheless, our extensive computational analysis in a wide range of both model and real-world networks shows that the two discretizations of Ricci curvature are highly correlated in many networks. Moreover, we show that if one considers the augmented Forman-Ricci curvature which also accounts for the two-dimensional simplicial complexes arising in graphs, the observed correlation between the two discretizations is even higher, especially, in real networks. Besides the potential theoretical implications of these observations, the close relationship between the two discretizations has practical implications whereby Forman-Ricci curvature can be employed in place of Ollivier-Ricci curvature for faster computation in larger real-world networks whenever coarse analysis suffices.

A goal in network science is the geometrical characterization of complex networks. In this direction, we have recently introduced the Forman's discretization of Ricci curvature to the realm of undirected networks. Investigation of this edge-centric network measure, Forman curvature, in diverse model and real-world undirected networks revealed that the curvature measure captures several aspects of the organization of complex undirected networks. However, many important realworld networks are inherently directed in nature, and the definition of the Forman curvature for undirected networks is unsuitable for the analysis of such directed networks. Hence, we here extend the Forman curvature for undirected networks to the case of directed networks. The simple mathematical formula for the Forman curvature of a directed edge elegantly incorporates node weights, edge weights and edge direction. By applying the Forman curvature for directed networks to a variety of model and real-world directed networks, we show that the measure can be used to characterize the structure of complex directed networks. Furthermore, our results also hold in real directed networks which are weighted or spatial in nature. These results in combination with our previous results suggest that the Forman curvature can be readily employed to study the organization of both directed and undirected complex networks. * jost@mis.mpg.de † this also makes the measure suitable for analysis of both unweighted and weighted networks [21]. Since Forman curvature represents a discretization of the classical Ricci curvature which is intrinsically associated with edges of a network, this notion of curvature does not necessitate the technical artifice of extending a measure for the curvature of nodes to the edges [21]. Thus, Forman curvature can be exploited for edge-based analysis of complex networks. Given the definition of the Forman curvature of an edge, one can easily define the Forman curvature of a node in the network by summing or averaging the curvatures of its adjacent edges, somewhat analogous to the concept of scalar curvature in Riemannian geometry [23]. We remark that the Forman curvature for an edge is a local measure dependent on weights of adjacent nodes and edges in the network [21]. Still Forman curvature, a local geometric characteristic, can provide deep insights on the global topology of the network [30]. Moreover, two networks with the same degree distribution can have very different distributions of Forman curvature (Fig. 1).Although, we have successfully introduced Forman curvature to undirected networks [21], several important real networks in nature and society are inherently directed in nature. These include the metabolic networks [7,33,34], gene regulatory networks [35], signaling networks [36], neural networks [37], the world wide web (WWW) [38], online social networks [10] and transportation networks [39,40]. However, the two different discretizations of the Ricci curvature, Ollivier-Ricci curvature and Forman-Ricci curvature, have been...

We first develop a general framework for Laplace operators defined in terms of the combinatorial structure of a simplicial complex. This includes, among others, the graph Laplacian, the combinatorial Laplacian on simplicial complexes, the weighted Laplacian, and the normalized graph Laplacian. This framework then allows us to define the normalized Laplace operator ∆ up i on simplicial complexes which we then systematically investigate. We study the effects of a wedge sum, a join and a duplication of a motif on the spectrum of the normalized Laplace operator, and identify some of the combinatorial features of a simplicial complex that are encoded in its spectrum.