page_title: Signed graph
section_title: Complex systems
hierarchical_section_title: Signed graph / Applications / Complex systems
caption_attribution_description: English: A three-variable signed digraph representing causal relationships forming a simple trophic system comprised of plants, herbivores and carnivores.
context_page_description: In the area of graph theory in mathematics, a signed graph is a graph in which each edge has a positive or negative sign.
  A signed graph is balanced if the product of edge signs around every cycle is positive. Three fundamental questions about a signed graph are: Is it balanced? What is the largest size of a balanced edge set in it? What is the smallest number of vertices that must be deleted to make it balanced? The first question is easy to solve quickly; the second and third are computationally intractable.
  The name "signed graph" and the notion of balance appeared first in a mathematical paper of Frank Harary in 1953. Dénes Kőnig had already studied equivalent notions in 1936 under a different terminology but without recognizing the relevance of the sign group.
  At the Center for Group Dynamics at the University of Michigan, Dorwin Cartwright and Harary generalized Fritz Heider's psychological theory of balance in triangles of sentiments to a psychological theory of balance in signed graphs.
  Signed graphs have been rediscovered many times because they come up naturally in many unrelated areas.
context_section_description: Using an analytic method initially developed in population biology and ecology, but now used a similar notion in many scientific disciplines, analysis of the properties of signed digraphs has found application in applied formal causal reasoning about the behavior of complex causal systems. Such analyses answer questions about feedback at given levels of the system, and about the direction of variable response given a perturbation to a system at one or more points, variable correlations given such perturbations, the distribution of variance across the system, and the sensitivity or insensitivity of particular variables to system perturbations.
caption: A three-variable signed digraph representing a simple trophic system
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