Random Directed Graphs Research Articles

Giant components in three-parameter random directed graphs

A three-parameter model of a random directed graph (digraph) is specified by the probability of ‘up arrows' from vertex i to vertex j where i < j, the probability of ‘down arrows' from i to j where i ≥ j, and the probability of bidirectional arrows between i and j. In this model, a phase transition—the abrupt appearance of a giant strongly connected component—takes place as the parameters cross a critical surface. The critical surface is determined explicitly. Before the giant component appears, almost surely all non-trivial components are small cycles. The asymptotic probability that the digraph contains no cycles of length 3 or more is computed explicitly. This model and its analysis are motivated by the theory of food webs in ecology.

Chain Lengths in Certain Random Directed Graphs

AbstractWe study the random directed graph with vertex set {1, …, n} in which the directed edges (i, j) occur independently with probability cn/n for i<j and probability zero for i ⩽ j. Let Mn (resp., Ln) denote the length of the longest path (resp., longest path starting from vertex 1). When cn is bounded away from 0 and ∞ as n→∞, the asymptotic behavior of Mn was analyzed in previous work of the author and J. E. Cohen. Here, all restrictions on cn are eliminated and the asymptotic behavior of Ln is also obtained. In particular, if cn/ln(n)→∞ while cn/n→0, then both Mn/cn and Ln/cn are shown to converge in probability to the constant e.

On the diameter of a class of random graphs

The diameter of a class of random directed graphs in which the outdegree of each vertex is constrained to be exactly k is examined. Vertices connect themselves to k other distinct vertices with outwardly directed edges, all possible sets of k vertices being chosen with equal probability. It is shown that the diameter of such a random graph almost surely takes on only one of two values. >

The statistics of random directed graphs

The analysis of random graphs developed by the author, principally as a model for polymerization processes, is extended to the case of directed random graphs, with models of neural nets in mind. The principal novelty of the directed case is the representation of the partition function by a complex rather than a real integral, and the replacement of simple maxima in asymptotic evaluations by an interesting form of saddle point.

An algorithm for finding hamilton cycles in random directed graphs

We describe a polynomial ( O( n 1.5)) time algorithm DHAM for finding hamilton cycles in digraphs. For digraphs chosen uniformly at random from the set of digraphs with vertex set {1, 2, …, n} and m = m( n) edges the limiting probability (as n → ∞) that DHAM finds a hamilton cycle equals the limiting probability that the digraph is hamiltonian. Some applications to random “travelling salesman problems” are discussed.

Almost Sure Fault Tolerance in Random Graphs

We analyze the PMC [10] model for fault tolerant systems by means of random directed graphs. Previous work has shown that the minimum in-degree of the testing digraph had to exceed the expected number of faulty units (vertices). We show, for much sparser digraphs than those described above, that the asymptotic probability of correct diagnosis of a faulty system tends to 1. Specifically we show that if our directed graph has n vertices, arc probability $p = {{(c\log n)} / n}$ and vertex failure probability $q < \frac{1}{2}$ then a simple algorithm will correctly diagnose the system “almost always” when ${{c > 1} / {(1 - q)}}$. However, when ${{c < 1} / {(1 - q)}}$ no algorithm can correctly diagnose the system. We present a similar analysis for a family of directed graphs which are not random. We conclude with analogous results for undirected graphs.

Autocatalytic sets of proteins

This article investigates the possibility that the emergence of reflexively autocatalytic sets of peptides and polypeptides may be an essentially inevitable collective property of any sufficiently complex set of polypeptides. The central idea is based on the connectivity properties of random directed graphs. In the set of amino acid monomer and polymer species up to some maximum length, M, the number of possible polypeptides is large, but, for specifiable "legitimate" end condensation, cleavage and transpeptidation exchange reactions, the number of potential reactions by which the possible polypeptides can interconvert is very much larger. A directed graph in which arrows from smaller fragments to larger condensation products depict potential synthesis reactions, while arrows from the larger peptide to the smaller fragments depict the reverse cleavage reactions, comprises the reaction graph for such a system. Polypeptide protoenzymes are able to catalyze such reactions. The distribution of catalytic capacities in peptide space is a fundamental problem in its own right, and in its bearing on the existence of autocatalytic sets of proteins. Using an initial idealized hypothesis that an arbitrary polypeptide has a fixed a priori probability of catalyzing any arbitrary legitimate reaction to assign to each polypeptide those reactions, if any, which it catalyzes, the probability that the set of polypeptides up to length M contains a reflexively autocatalytic subset can be calculated and is a percolation problem on such reaction graphs. Because, as M increases, the ratio of reactions among the possible polypeptides to polypeptides rises rapidly, the existence of such autocatalytic subsets is assured for any fixed probability of catalysis. The main conclusions of this analysis appear independent of the idealizations of the initial model, introduce a novel kind of parallel selection for peptides catalyzing connected sequences of reactions, depend upon a new kind of minimal critical complexity whose properties are definable, and suggest that the emergence of self replicating systems may be a self organizing collective property of critically complex protein systems in prebiotic evolution. Similar principles may apply to the emergence of a primitive connected metabolism. Recombinant DNA procedures, cloning random DNA coding sequences into expression vectors, afford a direct avenue to test the distribution of catalytic capacities in peptide space, may provide a new means to select or screen for peptides with useful properties, and may ultimately lead toward the actual construction of autocatalytic peptide sets.