In a consensus protocol an agreement among agents is achieved thanks to the collaborative efforts of all agents, expresses by a communication graph with nonnegative weights. The question we ask in this paper is the following: is it possible to achieve a form of agreement also in presence of antagonistic interactions, modeled as negative weights on the communication graph? The answer to this question is affirmative: on signed networks all agents can converge to a consensus value which is the same for all agents except for the sign. Necessary and sufficient conditions are obtained to describe cases in which this is possible. These conditions have strong analogies with the theory of monotone systems. Linear and nonlinear Laplacian feedback designs are proposed.

Consensus Problems on Networks With Antagonistic Interactions

The energy of a graph

Self-Organized Criticality

We analyse the corpus of user relationships of the Slashdot technology news site. The data was collected from the Slashdot Zoo feature where users of the website can tag other users as friends and foes, providing positive and negative endorsements. We adapt social network analysis techniques to the problem of negative edge weights. In particular, we consider signed variants of global network characteristics such as the clustering coefficient, node-level characteristics such as centrality and popularity measures, and link-level characteristics such as distances and similarity measures. We evaluate these measures on the task of identifying unpopular users, as well as on the task of predicting the sign of links and show that the network exhibits multiplicative transitivity which allows algebraic methods based on matrix multiplication to be used. We compare our methods to traditional methods which are only suitable for positively weighted edges.

The Slashdot Zoo: Mining a Social Network with Negative Edges

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The slashdot zoo: mining a social network with negative edges

A signed graph is a graph with a sign attached to each edge. This article extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the largest Laplacian eigenvalue of a signed graph is investigated, which generalizes the corresponding results on the largest Laplacian eigenvalue of a graph.

On the Laplacian Eigenvalues of Signed Graphs

If G is a graph and λ1,λ2,...,λ
 n
 are its eigenvalues, then the energy of G is defined as E(G)=|λ1|+|λ2|+⋅⋅⋅+|λ
 n
 |. Let S

 n
 
3 be the graph obtained from the star graph with n vertices by adding an edge. In this paper we prove that S

 n
 
3 is the unique minimal energy graph among all unicyclic graphs with n vertices (n≥6).

Unicyclic Graphs with Minimal Energy

Given two graphs G1, with vertices 1, 2, ..., n and edges e1, e2, ..., em, and G2, the edge corona G1 | G2 of G1 and G2 is defined as the graph obtained by taking m copies of G2 and for each edge ek = ij of G, joining edges between the two end-vertices i, j of ek and each vertex of the k-copy of G2. In this paper, the adjacency spectrum and Laplacian spectrum of G1 | G2 are given in terms of the spectrum and Laplacian spectrum of G1 and G2, respectively. As an application of these results, the number of spanning trees of the edge corona is also considered.

/pdf/the-spectrum-of-the-edge-corona-of-two-graphs-1re17zlkos.pdf

The spectrum of the edge corona of two graphs

A signed graph is a graph with a sign attached to each edge. This paper extends some
fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the
relationships between the least Laplacian eigenvalue and the unbalancedness of a signed graph are
investigated.

Yaoping Hou

Papers

On the Laplacian Eigenvalues of Signed Graphs

Unicyclic Graphs with Minimal Energy

The spectrum of the edge corona of two graphs

Bounds for the Least Laplacian Eigenvalue of a Signed Graph

Bounds on the largest eigenvalues of trees with a given size of matching