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 (technically, they are NP-hard).
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. For instance, they enable one to describe and analyze the geometry of subsets of the classical root systems. They appear in topological graph theory and group theory. They are a natural context for questions about odd and even cycles in graphs. They appear in computing the ground state energy in the non-ferromagnetic Ising model; for this one needs to find a largest balanced edge set in Σ. They have been applied to data classification in correlation clustering.
- The complete signed graph on n vertices with loops, denoted by ±Kno, has every possible positive and negative edge including negative loops, but no positive loops. Its edges correspond to the roots of the root system Cn; the column of an edge in the incidence matrix (see below) is the vector representing the root.
- The complete signed graph with half-edges, ±Kn', is ±Kn with a half-edge at every vertex. Its edges correspond to the roots of the root system Bn, half-edges corresponding to the unit basis vectors.
- The complete signed link graph, ±Kn, is the same but without loops. Its edges correspond to the roots of the root system Dn.
- An all-positive signed graph has only positive edges. If the underlying graph is G, the all-positive signing is written +G.
- An all-negative signed graph has only negative edges. It is balanced if and only if it is bipartite because a circle is positive if and only if it has even length. An all-negative graph with underlying graph G is written −G.
- A signed complete graph has as underlying graph G the ordinary complete graph Kn. It may have any signs. Signed complete graphs are equivalent to two-graphs, which are of value in finite group theory. A two-graph can be defined as the class of vertex sets of negative triangles (having an odd number of negative edges) in a signed complete graph.
The adjacency matrix of a signed graph Σ on n vertices is an n × n matrix A(Σ). It has a row and column for each vertex. The entry avw in row v and column w is the number of positive vw edges minus the number of negative vw edges. On the diagonal, avv = 0 if there are no loops or half-edges; the correct definition when such edges exist depends on the circumstances.
A signed graph is oriented when each end of each edge is given a direction, so that in a positive edge the ends are both directed from one endpoint to the other, and in a negative edge either both ends are directed outward, to their own vertices, or both are directed inward, away from their vertices. Thus, an oriented signed graph is the same as a bidirected graph. (It is very different from a signed digraph.)
The (more correctly, "an") incidence matrix of a signed graph with n vertices and m edges is an n × m matrix, with a row for each vertex and a column for each edge. It is obtained by orienting the signed graph in any way. Then its entry ηij is +1 if edge j is oriented into vertex i, −1 if edge j is oriented out of vertex i, and 0 if vertex i and edge j are not incident. This rule applies to a link, whose column will have two nonzero entries with absolute value 1, a half-edge, whose column has a single nonzero entry +1 or −1, and a loose edge, whose column has only zeroes. The column of a loop, however, is all zero if the loop is positive, and if the loop is negative it has entry ±2 in the row corresponding to its incident vertex.
Any two incidence matrices are related by negating some subset of the columns. Thus, for most purposes it makes no difference which orientation we use to define the incidence matrix, and we may speak of the incidence matrix of Σ without worrying about exactly which one it is.
Negating a row of the incidence matrix corresponds to switching the corresponding vertex.
Switching a vertex in Σ means negating the signs of all the edges incident to that vertex. Switching a set of vertices means negating all the edges that have one end in that set and one end in the complementary set. Switching a series of vertices, once each, is the same as switching the whole set at once.
Switching of signed graphs (signed switching) is generalized from Seidel (1976), where it was applied to graphs (graph switching), in a way that is equivalent to switching of signed complete graphs.
Switching equivalence means that two graphs are related by switching, and an equivalence class of signed graphs under switching is called a switching class. Sometimes these terms are applied to equivalence of signed graphs under the combination of switching and isomorphism, especially when the graphs are unlabeled; but to distinguish the two concepts the combined equivalence may be called switching isomorphism and an equivalence class under switching isomorphism may be called a switching isomorphism class.
Switching a set of vertices affects the adjacency matrix by negating the rows and columns of the switched vertices. It affects the incidence matrix by negating the rows of the switched vertices.
The sign of a path is the product of the signs of its edges. Thus a path is positive only if there are an even number of negative edges in it (where zero is even). In the mathematical balance theory of Frank Harary, a signed graph is balanced when every cycle is positive. He proves that a signed graph is balanced when (1) for every pair of nodes, all paths between them have the same sign, or (2) the graph partitions into a pair of subgraphs, each consisting of positive edges, but connected by negative edges. The theorem was published by Harary in 1953. It generalizes the theorem that an ordinary (unsigned) graph is bipartite if and only if every cycle has even length.
A simple proof uses the method of switching. To prove Harary's theorem, one shows by induction that Σ can be switched to be all positive if and only if it is balanced.
A weaker theorem, but with a simpler proof, is that if every 3-cycle in a signed complete graph is positive, then the graph is balanced. For the proof, pick an arbitrary node n and place it and all those nodes that are linked to n by a positive edge in one group, called A, and all those linked to n by a negative edge in the other, called B. Since this is a complete graph, every two nodes in A must be friends and every two nodes in B must be friends, otherwise there would be a 3-cycle which was unbalanced. (Since this is a complete graph, any one negative edge would cause an unbalanced 3-cycle.) Likewise, all negative edges must go between the two groups.
Give each vertex a value of +1 or −1; we call this a state of Σ. An edge is called satisfied if it is positive and both endpoints have the same value, or it is negative and the endpoints have opposite values. An edge that is not satisfied is called frustrated. The smallest number of frustrated edges over all states is called the frustration index (or line index of balance) of Σ. Finding the frustration index is hard, in fact, it is NP-hard. Aref et al. suggest binary programming models that are capable of computing the frustration index of graphs with up to 105 edges in a reasonable time. One can see the NP-hard complexity by observing that the frustration index of an all-negative signed graph is equivalent to the maximum cut problem in graph theory, which is NP-hard. The reason for the equivalence is that the frustration index equals the smallest number of edges whose negation (or, equivalently, deletion; a theorem of Harary) makes Σ balanced. (This can be proved easily by switching.)
The frustration index is important in a model of spin glasses, the mixed Ising model. In this model, the signed graph is fixed. A state consists of giving a "spin", either "up" or "down", to each vertex. We think of spin up as +1 and spin down as −1. Thus, each state has a number of frustrated edges. The energy of a state is larger when it has more frustrated edges, so a ground state is a state with the fewest frustrated energy. Thus, to find the ground state energy of Σ one has to find the frustration index.
There are two matroids associated with a signed graph, called the signed-graphic matroid (also called the frame matroid or sometimes bias matroid) and the lift matroid, both of which generalize the cycle matroid of a graph. They are special cases of the same matroids of a biased graph.
The frame matroid (or signed-graphic matroid) M(G) has for its ground set the edge set E. An edge set is independent if each component contains either no circles or just one circle, which is negative. (In matroid theory a half-edge acts exactly like a negative loop.) A circuit of the matroid is either a positive circle, or a pair of negative circles together with a connecting simple path, such that the two circles are either disjoint (then the connecting path has one end in common with each circle and is otherwise disjoint from both) or share just a single common vertex (in this case the connecting path is that single vertex). The rank of an edge set S is n − b, where n is the number of vertices of G and b is the number of balanced components of S, counting isolated vertices as balanced components. This matroid is the column matroid of the incidence matrix of the signed graph. That is why it describes the linear dependencies of the roots of a classical root system.
The extended lift matroid L0(G) has for its ground set the set E0 the union of edge set E with an extra point, which we denote e0. The lift matroid L(G) is the extended lift matroid restricted to E. The extra point acts exactly like a negative loop, so we describe only the lift matroid. An edge set is independent if it contains either no circles or just one circle, which is negative. (This is the same rule that is applied separately to each component in the signed-graphic matroid.) A matroid circuit is either a positive circle or a pair of negative circles that are either disjoint or have just a common vertex. The rank of an edge set S is n − c + ε, where c is the number of components of S, counting isolated vertices, and ε is 0 if S is balanced and 1 if it is not.
Other kinds of "signed graph"
Sometimes the signs are taken to be +1 and −1. This is only a difference of notation, if the signs are still multiplied around a circle and the sign of the product is the important thing. However, there are two other ways of treating the edge labels that do not fit into signed graph theory.
The term signed graph is applied occasionally to graphs in which each edge has a weight, w(e) = +1 or −1. These are not the same kind of signed graph; they are weighted graphs with a restricted weight set. The difference is that weights are added, not multiplied. The problems and methods are completely different.
The name is also applied to graphs in which the signs function as colors on the edges. The significance of the color is that it determines various weights applied to the edge, and not that its sign is intrinsically significant. This is the case in knot theory, where the only significance of the signs is that they can be interchanged by the two-element group, but there is no intrinsic difference between positive and negative. The matroid of a sign-colored graph is the cycle matroid of the underlying graph; it is not the frame or lift matroid of the signed graph. The sign labels, instead of changing the matroid, become signs on the elements of the matroid.
In this article we discuss only signed graph theory in the strict sense. For sign-colored graphs see colored matroids.
A signed digraph is a directed graph with signed arcs. Signed digraphs are far more complicated than signed graphs, because only the signs of directed cycles are significant. For instance, there are several definitions of balance, each of which is hard to characterize, in strong contrast with the situation for signed undirected graphs.
Signed digraphs should not be confused with oriented signed graphs. The latter are bidirected graphs, not directed graphs (except in the trivial case of all positive signs).
As with unsigned graphs, there is a notion of signed graph coloring. Where a coloring of a graph is a mapping from the vertex set to the natural numbers, a coloring of a signed graph is a mapping from the vertex set to the integers. The constraints on proper colorings come from the edges of the signed graph. The integers assigned to two vertices must be distinct if they are connected by a positive edge. The labels on adjacent vertices must not be additive inverses if the vertices are connected by a negative edge. There can be no proper coloring of a signed graph with a positive loop.
When restricting the vertex labels to the set of integers with magnitude at most a natural number k, the set of proper colorings of a signed graph is finite. The relation between the number of such proper colorings and k is a polynomial in k. This is analogous to the chromatic polynomial of unsigned graphs.
In social psychology, signed graphs have been used to model social situations, with positive edges representing friendships and negative edges enmities between nodes, which represent people. Then, for example, a positive 3-cycle is either three mutual friends, or two friends with a common enemy; while a negative 3-cycle is either three mutual enemies, or two enemies who share a mutual friend. According to balance theory, positive cycles are balanced and supposed to be stable social situations, whereas negative cycles are unbalanced and supposed to be unstable. According to the theory, in the case of three mutual enemies, this is because sharing a common enemy is likely to cause two of the enemies to become friends. In the case of two enemies sharing a friend, the shared friend is likely to choose one over the other and turn one of his or her friendships into an enemy.
Antal, Krapivsky and Reder consider social dynamics as the change in sign on an edge of a signed graph. The social relations with previous friends of a divorcing couple are used to illustrate the evolution of a signed graph in society. Another illustration describes the changing international alliances between European powers in the decades before the First World War. They consider local triad dynamics and constrained triad dynamics, where in the latter case a relationship change is made only when the total number of unbalanced triads is reduced. The simulation presumed a complete graph with random relations having a random unbalanced triad selected for transformation. The evolution of the signed graph with N nodes under this process is studied and simulated to describe the stationary density of friendly links.
Balance theory has been severely challenged, especially in its application to large systems, on the theoretical ground that friendly relations tie a society together, while a society divided into two camps of enemies would be highly unstable. Experimental studies have also provided only weak confirmation of the predictions of structural balance theory.
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.
Correlation clustering looks for natural clustering of data by similarity. The data points are represented as the vertices of a graph, with a positive edge joining similar items and a negative edge joining dissimilar items.
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