Network theory

About: Network theory is a research topic. Over the lifetime, 2257 publications have been published within this topic receiving 109864 citations.

A new approach to betweenness centrality based on the Shapley Value

Abstract: In many real-life networks, such as urban structures, protein interactions and social networks, one of the key issues is to measure the centrality of nodes, i.e. to determine which nodes and edges are more central to the functioning of the entire network than others. In this paper we focus on betweenness centrality --- a metric based on which the centrality of a node is related to the number of shortest paths that pass through that node. This metric has been shown to be well suited for many, often complex, networks. In its standard form, the betweenness centrality, just like other centrality metrics, evaluates nodes based on their individual contributions to the functioning of the network. For instance, the importance of an intersection in a road network can be computed as the difference between the full capacity of this network and its capacity when the intersection is completely shut down. However, as recently argued in the literature, such an approach is inadequate for many real-life applications, as, for example, multiple nodes can fail simultaneously. Thus, what would be desirable is to refine the existing centrality metrics such that they take into account not only the functioning of nodes as individual entities but also as members of groups of nodes. One recently-proposed way of doing this is based on the Shapley Value --- a solution concept in cooperative game theory that measures in a fair way the contributions of players to all the coalitions that they could possibly participate in. Although this approach has been used to extend various centrality metrics, such an extension to betweenness centrality is yet to be developed. The main challenge when developing such a refinement is to tackle the computational complexity; the Shapley Value generally requires an exponential number of operations, making its use limited to a small number of player (or nodes in our context). Against this background, our main contribution in this paper is to refine the betweenness centrality metric based on the Shapley Value: we develop an algorithm for computing this new metric, and show that it has the same complexity as the best known algorithm due to Brandes [7] to compute the standard betweenness centrality (i.e., polynomial in the size of the network). Finally, we show that our results can be extended to another important centrality metric called stress centrality.

Interpreting Embodied Mathematics Using Network Theory: Implications for Mathematics Education

Abstract: Working from the premise that mathematics knowledge can be described as a complex unity, we develop the suggestion that network theory provides a useful frame for informing understandings of disciplinary knowledge and content learning for schooling. Specifically, we use network theory to analyze associations among mathematical concepts, focusing on their embodied nature and their reliance on metaphor. After describing some of the basic suppositions, we examine the structure of the network of metaphors that underlies embodied mathematics, the dynamics of this network, and the effect of these dynamics on mathematical understanding. Finally, implications for classroom teaching and curriculum are discussed. We conjecture that it is both instructive and important to use the network structure of mathematical knowledge to shed light on both cognition in mathematics and on mathematics education.

Randomizing bipartite networks: the case of the World Trade Web

Abstract: Within the last fifteen years, network theory has been successfully applied both to natural sciences and to socioeconomic disciplines. In particular, bipartite networks have been recognized to provide a particularly insightful representation of many systems, ranging from mutualistic networks in ecology to trade networks in economy, whence the need of a pattern detection-oriented analysis in order to identify statistically-significant structural properties. Such an analysis rests upon the definition of suitable null models, i.e. upon the choice of the portion of network structure to be preserved while randomizing everything else. However, quite surprisingly, little work has been done so far to define null models for real bipartite networks. The aim of the present work is to fill this gap, extending a recently-proposed method to randomize monopartite networks to bipartite networks. While the proposed formalism is perfectly general, we apply our method to the binary, undirected, bipartite representation of the World Trade Web, comparing the observed values of a number of structural quantities of interest with the expected ones, calculated via our randomization procedure. Interestingly, the behavior of the World Trade Web in this new representation is strongly different from the monopartite analogue, showing highly non-trivial patterns of self-organization.

Linear Graph Theory-A Fundamental Engineering Discipline

Abstract: Current techniques for formulating the mathematical characteristics of physical systems vary greatly from one type to another (mechanical, electrical, thermal, etc.). Of these techniques, those used in electrical network analysis have proven to be the more orderly and generally applicable as evidenced by repeated efforts on the part of the system analyst to establish first an electrical analog of the system in question. This paper presents the basis of an operational concept of system analysis embracing all types of systems, and presents an orderly, sure, and relatively simple basis for extending the discipline of linear graph theory (abstracted form of network theory) to the analysis and synthesis of all types of lumped-parameter systems without the artifice of analogies. It is indicated that these procedures and concepts also provide a means for extending electrical network theory beyond current applications to include systems of multiterminal components.

In Search of a Network Theory of Innovations: Relations, Positions, and Perspectives

Abstract: As a complement to Nelson & Winter’s (1977) article entitled “In Search of a Useful Theory of Innovation,” a sociological perspective on innovation networks can be elaborated using Luhmann’s social-systems theory, on the one hand, and Latour’s “sociology of translations,” on the other. Because of a common focus on communication, these perspectives can be recombined as a set of methodologies. Latour’s sociology of translations specifies a mechanism for generating variation in relations (“associations”), whereas Luhmann’s systems perspective enables the specification of (functionally different) selection environments such as markets, professional organizations, and political control. Selection environments can be considered as mechanisms of social coordination that can “self-organize” — beyond the control of human agency — into regimes in terms of interacting codes of communication. Unlike relatively globalized regimes, technological trajectories are organized locally in “landscapes.” A resulting “duality of structure” (Giddens, 1979) between the historical organization of trajectories and evolutionary self-organization at the regime level can be expected to drive innovation cycles. Reflexive translations add a third layer of perspectives to (i) the relational analysis of observable links that shape trajectories, and (ii) the positional analysis of networks in terms of latent dimensions. These three operations can be studied in a single framework, but using different methodologies. Latour’s first-order “associations” can then be analytically distinguished from second-order “translations” in terms of requiring other communicative competencies. The resulting operations remain “infra-reflexively” nested, and can therefore be used for innovative reconstructions of previously constructed boundaries.