On the Number of Cycles in a Graph

Movarraei, Nazanin; Boxwala, Samina

doi:10.4236/ojdm.2016.62005

Cited by 3 publications

(2 citation statements)

References 14 publications

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“…When only short simple cycles on undirected graphs are of interest, these may be counted via a set of special identities involving the adjacency matrix. This approach was pioneered by Harary and Manvel in the 1970s and has remained popular ever since [21,1,10,34]. In particular, Alon, Yuster and Zwick presented an algorithm for evaluating these identities up to = 7 in O(N ω ) time and O(N 2 ) space [1].…”

Section: Counting Using Immanantsmentioning

confidence: 99%

A General Purpose Algorithm for Counting Simple Cycles and Simple Paths of Any Length

2019

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We describe a general purpose algorithm for counting simple cycles and simple paths of any length on a (weighted di)graph on N vertices and M edges, achieving a time complexity of O N + M + ω + ∆ |S | . In this expression, |S | is the number of (weakly) connected induced subgraphs of G on at most vertices, ∆ is the maximum degree of any vertex and ω is the exponent of matrix multiplication. We compare the algorithm complexity both theoretically and experimentally with most of the existing algorithms for the same task. These comparisons show that the algorithm described here is the best general purpose algorithm for the class of graphs where ( ω−1 ∆ −1 +1)|S | ≤ |Cycle |, with |Cycle | the total number of simple cycles of length at most , including backtracks and self-loops. On Erdős-Rényi random graphs, we find empirically that this happens when the edge probability is larger than circa 4/N . In addition, we show that some real-world networks also belong to this class. Finally, the algorithm permits the enumeration of simple cycles and simple paths on networks where vertices are labeled from an alphabet on n letters with a time complexity of O N + M + n ω + ∆ |S | . A Matlab implementation of the algorithm proposed here is available for download.

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Section: Counting Using Immanantsmentioning

confidence: 99%

A General Purpose Algorithm for Counting Simple Cycles and Simple Paths of Any Length

2019

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“…The properties of Eq. (2.1) contrast it with other analytical formulas for counting simple cycles of small lengths[22,36,1]. First, these formulas work only on undirected graphs.…”

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Evaluating balance on social networks from their simple cycles

Giscard

Rochet

Wilson

2017

Journal of Complex Networks

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Signed networks have long been used to represent social relations of amity (+) and enmity (-) between individuals. Group of individuals who are cyclically connected are said to be balanced if the number of negative edges in the cycle is even and unbalanced otherwise. In its earliest and most natural formulation, the balance of a social network was thus defined from its simple cycles, cycles which do not visit any vertex more than once. Because of the inherent difficulty associated with finding such cycles on very large networks, social balance has since then been studied via other means. In this article we present the balance as measured from the simple cycles and primitive orbits of social networks. We specifically provide two measures of balance: the proportion R of negative simple cycles of length for each 20 which generalises the triangle index, and a ratio K which extends the relative signed clustering coefficient introduced by Kunegis. To do so, we use a Monte Carlo implementation of a novel exact formula for counting the simple cycles on any weighted directed graph. Our method is free from the double-counting problem affecting previous cycle-based approaches, does not require edge-reciprocity of the underlying network, provides a gray-scale measure of balance for each cycle length separately and is sufficiently tractable that it can be implemented on a standard desktop computer. We observe that social networks exhibit strong inter-edge correlations favouring balanced situations and we determine the corresponding correlation length ξ . For longer simple cycles, R undergoes a sharp transition to values expected from an uncorrelated model. This transition is absent from synthetic random networks, strongly suggesting that it carries a sociological meaning warranting further research.

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