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A mathematical model of dynamic social networks

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Abstract

A mathematical model for dynamic networks is developed that is based on closed, rather than open, sets. For a social network, it seems appropriate to use a neighborhood concept to establish these sets. We then define a rigorous concept of continuous change, and show that it shares some of the properties associated with the continuity of the calculus. We demonstrate that continuity is local in nature, in that if the network change is discontinuous, it will be so at a single point and the discontinuity will be apparent in that point’s immediate neighborhood. Necessary and sufficient criteria for continuity are provided when the change involves only the addition, or deletion, of individual nodes or connections (edges). To illustrate large scale continuous change, we choose a practical process which reduces a complex network to its fundamental cycles, in the course of which most triadically closed subportions are removed. Finally, we explore several variants of the neighborhood concept, and prove that a rigorous notion of fuzzy closure can be defined.

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Notes

  1. In graph theory, Y.η is often called the “open neighborhood of Y” and denoted N(Y), while Y.ρ, denoted N[Y] has been called the “closed neighborhood of Y” (Agnarsson and Greenlaw 2007; Harary 1969). This is a rather different meaning of “closed”.

  2. Because homomorphisms are point functions, not set valued operators, we employ prefix notation.

  3. A graph, or subgraph, is said to be chordal if it contains no cycles of length greater than 3 without a chord (edge) joining two of its points (Jacobson and Peters 1990; McKee 1993).

  4. Figure reprinted with permission from Newman (2006). Copyright (2006) by the American Physical Society.

  5. E.g. the typical \(\epsilon-\delta\) definition of real analysis (Royden 1988).

  6. Many graph theory texts say that Y.η is an “open” neighborhood, c.f (Agnarsson and Greenlaw 2007; Harary 1969).

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  1. Department of Computer Science, University of Virginia, Charlottesville, USA

    John L. Pfaltz

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  1. John L. Pfaltz
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Pfaltz, J.L. A mathematical model of dynamic social networks. Soc. Netw. Anal. Min. 3, 863–872 (2013). https://doi.org/10.1007/s13278-013-0109-9

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