Circuits in random graphs: from local trees to global loops

Abstract

Abstract. We compute the number of circuits and of loops with multiple crossings in random regular graphs. We discuss the importance of this issue for the validity of the cavity approach. On the one side we obtain analytic results for the infinite volume limit in agreement with existing exact results. On the other side we implement a counting algorithm, enumerate circuits at finite N and draw some general conclusions about the finite N behavior of the circuits. IntroductionThe study of random graphs, initiated more than four decades ago, has been since an issue of major interest in probability theory and in statistical mechanics. Examples of random graphs abound [1], from the original Erdös-Renyi model [2], where edges are chosen independently of each other between pairs of a set of N vertices (with a fixed probability of O(1/N)), to the scale-free graphs with power law degree distribution [3], only introduced in recent times. An interesting and useful distribution is the one that generates random c-regular graphs [4]. These are uniformly drawn from the set of all graphs over N vertices, each restricted to have degree c. Random regular graphs can be easily generated when N is large (and c finite) [5], and an instance of 3-regular graph is symbolized in figure 1. Around a randomly picked up vertex called root the graph looks like a regular tree. The probability that there exists a circuit (a self-avoiding closed path) of length L going through the root vanishes when N is sent to infinity and L is kept finite ‡ [4].The purpose of this article is to reach some quantitative understanding of the presence of long circuits in random graphs. Our motivation is at least two-fold.First, improving our knowledge on circuits in random graphs would certainly have positive fall-out on the understanding of equilibrium properties of models of interacting ‡ More precisely, this probability asymptotically departs from zero when log N = O(L). Finite-length loops may be present in the graph, but not in extensive number.

Loops of any size and Hamilton cycles in random scale-free networks

“…Characteristic motifs in a graph and degree correlations are in many real graphs not independent phenomena but they depend on each other as it has been shown for small (up to maximal connectivity) size subgraphs in [11,10,12,13]. Last but not least, it has been observed that the distribution of loop sizes is intimately connected with the thermal properties of lattice models defined on that graph [18,17]. On the other hand, the analytic approach to these models relies on the assumption that locally a random graph can be considered to have a tree like structure [19,20,21], i.e.…”

“…where ℓ = L/N and σ(ℓ) is a function having the maximum at ℓ max = c/(c + 1) whose expression can be found in the literature [16,17]. Regarding Hamilton cycles, i.e.…”

Self-sustaining oscillations in complex networks of excitable elements

“…[13][7] [10] In complex networks, loops are both generic and abundant. [15][14] While short loops of length L ≪ N (where N is the network size) are rare in large random networks, ones with L ln N occur generically in numbers growing exponentially with N . Their number also grows roughly exponentially with L up to a maximum at a length L m ∼ N.…”

Emergence of large cliques in random scale-free networks

“…For example, ER graphs with finite average connectivity have a finite number of finite loops [4,7]. On the contrary scale-free graphs have a number of finite loops which increases with the number N of vertices, provided that γ ≤ 3 [8,9].…”