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

Giscard, Pierre-Louis; Kriege, Nils M.; Wilson, Richard C.

doi:10.1007/s00453-019-00552-1

Cited by 20 publications

(23 citation statements)

References 39 publications

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“…5 For a more detailed analysis of the algorithmic implementation of Eq. (2.1) together with comparison with other algorithms for the same task, see [38]. 2.5 × 10 11 for the total number of simple cycles, which we verify analytically to be exact.…”

Section: Core Combinatorial Resultssupporting

confidence: 52%

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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Section: Core Combinatorial Resultssupporting

confidence: 52%

Evaluating balance on social networks from their simple cycles

Giscard

Rochet

Wilson

2017

Journal of Complex Networks

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“…While this study provided an overview of the state-of-the-art of the numerical computations on signed graphs and the vast range of applications to which it can be applied, it is by no means an exhaustive survey on the applications of the frustration index. From a computational perspective, this and some other recent studies [40,39] call for more advanced computational models that put larger networks within the reach of exact analysis. As another future research direction, one may consider formulating edge-based measures of stability for directed signed networks based on theories involving directionality and signed ties [61,87].…”

Section: Resultsmentioning

confidence: 98%

Balance and frustration in signed networks

Aref

2018

Journal of Complex Networks

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The frustration index is a key measure for analysing signed networks, which has been underused due to its computational complexity. We use an exact optimisation-based method to analyse frustration as a global structural property of signed networks coming from diverse application areas. In the classic friendenemy interpretation of balance theory, a by-product of computing the frustration index is the partitioning of nodes into two internally solidary but mutually hostile groups. The main purpose of this paper is to present general methodology for answering questions related to partial balance in signed networks, and apply it to a range of representative examples that are now analysable because of advances in computational methods. We provide exact numerical results on social and biological signed networks, networks of formal alliances and antagonisms between countries, and financial portfolio networks. Molecular graphs of carbon and Ising models are also considered. The purpose served by exploring several problems in this paper is to propose a single general methodology for studying signed networks and to demonstrate its relevance to applications. We point out several mistakes in the signed networks literature caused by inaccurate computation, implementation errors or inappropriate measures. Cornyn (R-TX) Corzine (D-NJ) Craig (R-ID) DeWine (R-OH) Dodd (D-CT) Domenici (R-NM) Durbin (D-IL) Edwards (D-NC) Ensign (R-NV)

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“…if the graph is sparse. More precisely, it has been shown in [15], that an algorithm based on the formulas of Theorem 1 for counting simple cycles and paths of length up to ℓ achieves an asymptotic running time of O N + M + ℓ ω + ℓ∆ |S ℓ | and uses O(N + M ) space. In this expression, N is the number of vertices of the graph, M is the number of edges, |S ℓ | is the number of (weakly) connected induced subgraphs of G on at most ℓ vertices, ∆ is the maximum degree of any vertex and ω is the exponent of matrix multiplication.…”

Section: Theorem 1 the Matrix Generating Series Of Open And Closed Simentioning

confidence: 99%

“…In this expression, N is the number of vertices of the graph, M is the number of edges, |S ℓ | is the number of (weakly) connected induced subgraphs of G on at most ℓ vertices, ∆ is the maximum degree of any vertex and ω is the exponent of matrix multiplication. Extensive comparisons with all existing techniques for counting simple cycles and paths [15], show that the formulas of Theorem 1 yield the best general purpose algorithm for this task whenever (ℓ ω−1 ∆ −1 + 1)|S ℓ | ≤ |Cycle ℓ |, with |Cycle ℓ | the total number of simple cycles of length at most ℓ, including backtracks and self-loops [15]. When this condition is not met the best general purpose algorithm is brute force search.…”

Section: Theorem 1 the Matrix Generating Series Of Open And Closed Simentioning

confidence: 99%

“…A remarkable consequence is an expression that links the Hamiltonian paths of a graph to its dominating connected sets. While the problem of counting simple cycles and paths parametrised by length remains #W[1]-complete, the formulas we obtain form the base of a novel algorithm [15] that has proven to be efficient enough to effectively tackle the problem, up to length 20, on real-world networks [13,14].…”

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Enumerating Simple Paths from Connected Induced Subgraphs

Giscard

Rochet

2018

Graphs and Combinatorics

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We present an exact formula for the ordinary generating series of the simple paths between any two vertices of a graph. Our formula involves the adjacency matrix of the connected induced subgraphs and remains valid on weighted and directed graphs. As a particular case, we obtain a relation linking the Hamiltonian paths and cycles of a graph to its dominating connected sets.Keywords Directed graph · self-avoiding walks · simple cycles · Hamiltonian paths · dominating sets · labeled adjacency matrix 1 IntroductionCounting simple paths, that is trajectories on a graph that do not visit any vertex more than once, is a problem of fundamental importance in enumerative combinatorics [20] with numerous applications, e.g. in sociology [23,9]. Several general purpose methods for counting simple paths have been discovered over the last 60 years, which make use of the inclusion-exclusion principle [3,4,6,18,19] or variants such as finite-difference sieves [2] and recursive expressions involving the adjacency matrix [1,16,21,23]. More rare but also worth mentionning are approaches using different tools such as zeon algebras [24] or immanantal equations [10].While some of these theoretical results have been used to propose algorithms for counting simple cycles or paths of fixed length, the problem remains #W[1]-complete and is generally beyond reach of existing techniques

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

Cited by 20 publications

References 39 publications

Evaluating balance on social networks from their simple cycles

Evaluating balance on social networks from their simple cycles

Balance and frustration in signed networks

Enumerating Simple Paths from Connected Induced Subgraphs

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