Efficient orbit-aware triad and quad census in directed and undirected graphs

Subgraph counting is a fundamental task in network analysis. Typically, algorithmic work is on total counting, where we wish to count the total frequency of a (small) pattern subgraph in a large input data set. But many applications require local counts (also called vertex orbit counts) wherein, for every vertex v of the input graph, one needs the count of the pattern subgraph involving v. This provides a rich set of vertex features that can be used in machine learning tasks, especially classification and clustering. But getting local counts is extremely challenging. Even the easier problem of getting total counts has received much research attention. Local counts require algorithms that get much finer grained information, and the sheer output size makes it difficult to design scalable algorithms.We present EVOKE, a scalable algorithm that can determine vertex orbits counts for all 5-vertex pattern subgraphs. In other words, EVOKE exactly determines, for every vertex v of the input graph and every 5-vertex subgraph H , the number of copies of H that v participates in. EVOKE can process graphs with tens of millions of edges, within an hour on a commodity machine. EVOKE is typically hundreds of times faster than previous state of the art algorithms, and gets results on datasets beyond the reach of previous methods.Theoretically, we generalize a recent "graph cutting" framework to get vertex orbit counts. This framework generate a collection of polynomial equations relating vertex orbit counts of larger subgraphs to those of smaller subgraphs. EVOKE carefully exploits the structure among these equations to rapidly count. We prove and empirically validate that EVOKE only has a small constant factor overhead over the best (total) 5-vertex subgraph counter.

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“…As an aside, for counting 4-VOCs, EVOKE runs typically in minutes, consistent with previous work [38,41].…”

Section: Resultssupporting

confidence: 86%

Efficiently Counting Vertex Orbits of All 5-vertex Subgraphs, by EVOKE

Pashanasangi

Seshadhri

2020

Proceedings of the 13th International Conference on Web Search and Data Mining

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“…Graph orientation, in particular degeneracy ordering, is a classic idea in counting triangles and motifs in static graphs, pioneered by Chiba-Nizhizeki [8]. Recently, there has been a number of triangle counting and motif counting algorithms inspired by these techniques [10,16,17,31,34,38]. The main benefit of degeneracy ordering is that the out-degree of each vertex becomes small when we orient the static graph based on this ordering.…”

Section: Related Workmentioning

confidence: 99%

“…Vertex ordering is a central idea in triangle counting and motif analysis in general [5,8,16,31,34,38,49]. Let 𝐺 be an undirected static simple graph.…”

Section: Preliminariesmentioning

confidence: 99%

Faster and Generalized Temporal Triangle Counting, via Degeneracy Ordering

Pashanasangi

Seshadhri

2021

Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery &Amp; Data Mining

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Triangle counting is a fundamental technique in network analysis, that has received much attention in various input models. The vast majority of triangle counting algorithms are targeted to static graphs. Yet, many real-world graphs are directed and temporal, where edges come with timestamps. Temporal triangles yield much more information, since they account for both the graph topology and the timestamps.Temporal triangle counting has seen a few recent results, but there are varying definitions of temporal triangles. In all cases, temporal triangle patterns enforce constraints on the time interval between edges (in the triangle). We define a general notion ( 1,3 , 1,2 , 2,3 )-temporal triangles that allows for separate time constraints for all pairs of edges.Our main result is a new algorithm, DOTTT (Degeneracy Oriented Temporal Triangle Totaler), that exactly counts all directed variants of ( 1,3 , 1,2 , 2,3 )-temporal triangles. Using the classic idea of degeneracy ordering with careful combinatorial arguments, we can prove that DOTTT runs in ( log ) time, where is the number of (temporal) edges and is the graph degeneracy (max core number). Up to log factors, this matches the running time of the best static triangle counters. Moreover, this running time is better than existing.DOTTT has excellent practical behavior and runs twice as fast as existing state-of-the-art temporal triangle counters (and is also more general). For example, DOTTT computes all types of temporal queries in Bitcoin temporal network with half a billion edges in less than an hour on a commodity machine. CCS CONCEPTS• Theory of computation → Graph algorithms analysis; • Information systems → Data mining; Social networks; • Mathematics of computing → Graph algorithms.

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“…In practice, the maximum number of nodes within the combinations is usually of 3 for directed networks (Holland & Leinhardt, 1976;Milo, 2002) and 4 or 5 for undirected ones (Ali et al, 2014;Pržulj et al, 2004;Yaveroglu et al, 2015), due to combinatorial limitations. Indeed, both the number of different graphlets and the time of computation increase drastically with the value of k [for a study on the complexity for size 4, see (Ortmann & Brandes, 2017)]. Like (Ali et al, 2014), we choose here not to have a concomitant use of graphlets of different sizes.…”

Section: Graphletsmentioning

confidence: 99%

“…From an algorithmic point of view, graphlet counting is very expensive in terms of time of computation and thus different approaches have been proposed. Some algorithms are designed to exactly compute the number of appearances of each graphlet, with or without the orbits (Wernicke, 2006;Stoica & Prieur, 2009;Hocevar & Demsar, 2014;Pinar et al, 2017;Ortmann & Brandes, 2017) while others propose sampling strategies to have an approximate value for each graphlet (Wernicke & Rasche, 2006;Zhao et al, 2012).…”

Section: Graphletsmentioning

confidence: 99%

Stars, holes, or paths across your Facebook friends: A graphlet-based characterization of many networks

Charbey

Prieur²

2019

Net Sci

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Network science gathers methods coming from various disciplines which sometimes hardly cross the boundaries between these disciplines. Widely used in molecular biology in the study of protein interaction networks, the enumeration, in a network, of all possible subgraphs of a limited size (usually around four or five nodes), often called graphlets, can only be found in a few works dealing with social networks. In the present work, we apply this approach to an original corpus of about 10,000 non-overlapping Facebook ego networks gathered from voluntary participants by a survey application. To deal with so many similar networks, we adapt the relative graphlet frequency to a measure that we call graphlet representativity, which we show to be more effective to classify random networks having slight structural differences. From our data, we produce two clusterings, one of graphlets (paths, star-like, holes, light triangles, and dense), one of networks. The latter is presented with a visualization scheme using our representativity measure. We describe the distinct structural characteristics of the five clusters of Facebook ego networks so obtained and discuss the empirical differences between results obtained with 4-node and 5-node graphlets. We also provide suggestions of follow-ups of this work, both in sociology and in network science.

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Efficient orbit-aware triad and quad census in directed and undirected graphs

Cited by 26 publications

References 33 publications

Efficiently Counting Vertex Orbits of All 5-vertex Subgraphs, by EVOKE

Efficiently Counting Vertex Orbits of All 5-vertex Subgraphs, by EVOKE

Faster and Generalized Temporal Triangle Counting, via Degeneracy Ordering

Stars, holes, or paths across your Facebook friends: A graphlet-based characterization of many networks

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