Enumerating Simple Paths from Connected Induced Subgraphs

Giscard, Pierre-Louis; Rochet, Paul

doi:10.1007/s00373-018-1966-9

Cited by 4 publications

(5 citation statements)

References 28 publications

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“…Our algorithm is based on a recent result from algebraic combinatorics relating the numbers of walks and of simple cycles on any (directed) graph. This result provides an explicit formula for the ordinary generating function of the number γ( ) of simple cycles of length multiplied [17] (…”

Section: Counting Simple Cyclesmentioning

confidence: 97%

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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 Simple Cyclesmentioning

confidence: 97%

“…To find all the simple paths, we rely once more on a recent result from algebraic combinatorics according to which the ordinary generating function of the number π i→j ( ) of simple paths of length from vertex i to j is the ij-entry of [17] (3)…”

Section: Counting Simple Paths and Simple Cycles Visiting A Fixed Vermentioning

confidence: 99%

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

2019

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“…In this expression H is a connected 2 induced subgraph of G, A H its adjacency matrix, |H| the number of vertices in H and |N(H)| the number of neighbours of H in G. A neighbour of H in G is a vertex v of G which is not in H and such that there exists at least one edge, possibly directed, from v to a vertex of H or from a vertex of H to v. The result of Eq. (2.1), as well as further exact formulas for P(z), is presented in [16] and shall not be proven here. .1) is that, if we use a direct algorithm for finding all the connected induced subgraphs of a graph, for example using reverse-search, the time complexity is O N + |E| + N 2 n H (G) , with N and |E| the number of vertices and of edges of the graph, respectively [4,45,11].…”

Section: Core Combinatorial Resultsmentioning

confidence: 94%

“…Both of these observations stem from a matrix extension of Eq. (2.1) which is presented in [16]. This extension provides a matrix P(z) whose entry P(z) i j is the ordinary generating function of the simple paths from i to j (i = j) or of the simple-cycles from i to itself (i = j).…”

Section: Functions On Simple Cycles and Simple Pathsmentioning

confidence: 99%

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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“…In this paper, we present an algorithm for counting the number of paths between two vertices in a graph G, based on the work in [17]. Our algorithm is based on a recent result from algebraic combinatorics relating the numbers of walks and paths on any (directed) graph.…”

Section: Simple Paths and Simple Cycles Counting Algorithmmentioning

confidence: 99%

The All-Paths and Cycles Graph Kernel

Giscard¹,

Wilson²

2017

Preprint

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With the recent rise in the amount of structured data available, there has been considerable interest in methods for machine learning with graphs. Many of these approaches have been kernel methods, which focus on measuring the similarity between graphs. These generally involving measuring the similarity of structural elements such as walks or paths. Borgwardt and Kriegel [1] proposed the all-paths kernel but emphasized that it is NP-hard to compute and infeasible in practice, favouring instead the shortest-path kernel. In this paper, we introduce a new algorithm for computing the all-paths kernel which is very efficient and enrich it further by including the simple cycles as well. We demonstrate how it is feasible even on large datasets to compute all the paths and simple cycles up to a moderate length. We show how to count labelled paths/simple cycles between vertices of a graph and evaluate a labelled path and simple cycles kernel. Extensive evaluations on a variety of graph datasets demonstrate that the all-paths and cycles kernel has superior performance to the shortest-path kernel and state-of-the-art performance overall.

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Enumerating Simple Paths from Connected Induced Subgraphs

Cited by 4 publications

References 28 publications

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

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

Evaluating balance on social networks from their simple cycles

The All-Paths and Cycles Graph Kernel

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