Multi-agent networks are often modeled as interaction graphs, where the nodes represent the agents and the edges denote some direct interactions. The robustness of a multi-agent network to perturbations such as failures, noise, or malicious attacks largely depends on the corresponding graph. In many applications, networks are desired to have well-connected interaction graphs with relatively small number of links. One family of such graphs is the random regular graphs. In this paper, we present a decentralized scheme for transforming any connected interaction graph with a possibly non-integer average degree of @math into a connected random @math -regular graph for some @math . Accordingly, the agents improve the robustness of the network while maintaining a similar number of links as the initial configuration by locally adding or removing some edges.
Another group of studies consider the explicit construction of expanders. Expanders can be constructed via graph operations such as zig-zag product (e.g., @cite_42 @cite_20 ), or derandomized graph squaring @cite_27 . Furthermore, for any @math such that @math is a prime power, an explicit algebraic construction method for a family of @math -regular expanders, i.e. Ramanujan graphs, was presented in @cite_1 . In @cite_4 , Watts-Strogatz small-word networks are transformed into quasi Ramanujan graphs by rewiring some of the edges.
@cite_4 @cite_42 @cite_1 @cite_27 @cite_20
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In this paper, we explore spectral properties of a class of regular Cayley graphs known as Ramanujan graphs and prove that the ratio of their algebraic connectivity to that of regular lattices grows exponentially as O(ngamma) with gamma = 1.84plusmn0.05 for networks with average degree of O(log(n)). Explicit construction algorithms exist for Ramanujan graphs that create regular graphs with especial degree and scale that depend on a pair of prime numbers. We introduce a randomized algorithm for construction of a class of fast regular graphs called quasi Ramanujan graphs. These graphs are obtained from finite number of degree balancing operations on Watts-Strogatz small-word networks that are irregular graphs. We show that quasi Ramanujan graphs share similar combinatorial optimality spectral properties as Ramanujan graphs and are not restricted to especial choices of degree and scale. A byproduct of this fact is that the algebraic connectivity ratio of quasi Ramanujan graphs grows exponentially in n as well. Numerical experiments are performed to verify our analytical predictions. Consensus algorithms converge extremely fast on networks with exponentially growing algebraic connectivity ratios. The main contribution of this work is a new type of graph product, which we call the zig-zag product. Taking a product of a large graph with a small graph, the resulting graph inherits (roughly) its size from the large one, its degree from the small one, and its expansion properties from both! Iteration yields simple explicit constructions of constant-degree expanders of arbitrary size, starting from one constant-size expander. Crucial to our intuition (and simple analysis) of the properties of this graph product is the view of expanders as functions which act as "entropy wave" propagators -they transform probability distributions in which entropy is concentrated in one area to distributions where that concentration is dissipated. In these terms, the graph product affords the constructive interference of two such waves. We construct an explicit generating sets @math and @math of the alternating and the symmetric groups, which make the Cayley graphs @math and @math a family of bounded degree expanders for all sufficiently large @math . These expanders have many applications in the theory of random walks on groups and other areas of mathematics. The main concrete result of this paper is the first explicit construction of constant degree lossless expanders. In these graphs, the expansion factor is almost as large as possible: (1—e)D, where D is the degree and e is an arbitrarily small constant. The best previous explicit constructions gave expansion factor D 2, which is too weak for many applications. The D 2 bound was obtained via the eigenvalue method, and is known that that method cannot give better bounds.The main abstract contribution of this paper is the introduction and initial study of randomness conductors, a notion which generalizes extractors, expanders, condensers and other similar objects. In all these functions, certain guarantee on the input "entropy" is converted to a guarantee on the output "entropy". For historical reasons, specific objects used specific guarantees of different flavors. We show that the flexibility afforded by the conductor definition leads to interesting combinations of these objects, and to better constructions such as those above.The main technical tool in these constructions is a natural generalization to conductors of the zig-zag graph product, previously defined for expanders and extractors.