Many real-world networks evolve over time, that is, new contacts appear and old contacts may disappear. They can be modeled as temporal graphs where interactions between vertices (which represent people in the case of social networks) are represented by time-stamped edges. One of the most fundamental problems in (social) network analysis is community detection, and one of the most basic primitives to model a community is a clique. Addressing the problem of finding communities in temporal networks, [TCS 2016] introduced @math -cliques as a natural temporal version of cliques. [SNAM 2017] showed how to adapt the well-known Bron-Kerbosch algorithm to enumerate @math -cliques. We continue this work and improve and extend the algorithm of to enumerate temporal @math -plexes (notably, cliques are the special case @math ). We define a @math - @math -plex as a set of vertices and a time interval, where during this time interval each vertex has in each consecutive @math time steps at least one edge to all but at most @math vertices in the chosen set of vertices. We develop a recursive algorithm for enumerating all maximal @math - @math -plexs and perform experiments on real-world social networks that demonstrate the practical feasibility of our approach. In particular, for the special case of @math -1-plexes (that is, @math -cliques), we observe that our algorithm is on average significantly faster than the previous algorithms by [SNAM 2017] and [IPL 2018] for enumerating @math -cliques.
There has been extensive research on both clique enumeration in temporal graphs @cite_0 @cite_39 @cite_13 and @math -plex enumeration on static graphs @cite_34 @cite_9 @cite_7 @cite_43 @cite_25 @cite_31 @cite_28 . To the best of our knowledge, we are the first to investigate the enumeration of @math -plexes in temporal graphs. We follow up on the work of @cite_0 , where the famous algorithm @cite_24 to enumerate cliques in static graphs was lifted to the temporal setting. The problem of finding @math -cliques in temporal graphs was introduced and motivated by the study of @cite_39 who enumerated contact patterns among high-school students.
@cite_7 @cite_28 @cite_9 @cite_39 @cite_0 @cite_43 @cite_24 @cite_31 @cite_34 @cite_13 @cite_25
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We present simple deterministic algorithms for subgraph finding and enumeration in the broadcast CONGEST model of distributed computation: -- For any constant @math , detecting @math -paths and trees on @math nodes can be done in constantly many rounds, and @math -cycles in @math rounds. -- On @math -degenerate graphs, cliques and @math -cycles can be enumerated in @math rounds, and @math -cycles in @math rounds. In many cases, these bounds are tight up to logarithmic factors. Moreover, we show that the algorithms for @math -degenerate graphs can be improved to optimal complexity @math and @math , respectively, in the supported CONGEST model, which can be seen as an intermediate model between CONGEST and the congested clique. We consider the enumeration of maximal bipartite cliques (bicliques) from a large graph, a task central to many practical data mining problems in social network analysis and bioinformatics. We present novel parallel algorithms for the MapReduce platform, and an experimental evaluation using Hadoop MapReduce. Our algorithm is based on clustering the input graph into smaller sized subgraphs, followed by processing different subgraphs in parallel. Our algorithm uses two ideas that enable it to scale to large graphs: (1) the redundancy in work between different subgraph explorations is minimized through a careful pruning of the search space, and (2) the load on different reducers is balanced through the use of an appropriate total order among the vertices. Our evaluation shows that the algorithm scales to large graphs with millions of edges and tens of millions of maximal bicliques. To our knowledge, this is the first work on maximal biclique enumeration for graphs of this scale. Dynamics of interactions play an increasingly important role in the analysis of complex networks. A modeling framework to capture this is temporal graphs which consist of a set of vertices (entities in the network) and a set of time-stamped binary interactions between the vertices. We focus on enumerating Δ-cliques, an extension of the concept of cliques to temporal graphs: for a given time period Δ, a Δ-clique in a temporal graph is a set of vertices and a time interval such that all vertices interact with each other at least after every Δ time steps within the time interval. Viard, Latapy, and Magnien (ASONAM 2015, TCS 2016) proposed a greedy algorithm for enumerating all maximal Δ-cliques in temporal graphs. In contrast to this approach, we adapt the Bron–Kerbosch algorithm—an efficient, recursive backtracking algorithm which enumerates all maximal cliques in static graphs—to the temporal setting. We obtain encouraging results both in theory (concerning worst-case running time analysis based on the parameter “Δ-slice degeneracy” of the underlying graph) as well as in practice with experiments on real-world data. The latter culminates in an improvement for most interesting Δ-values concerning running time in comparison with the algorithm of Viard, Latapy, and Magnien. Graphs of clique–width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we study the clique–width of perfect graph classes. On one hand, we show that every distance–hereditary graph, has clique–width at most 3, and a 3–expression defining it can be obtained in linear time. On the other hand, we show that the classes of unit interval and permutation graphs are not of bounded clique–width. More precisely, we show that for every @math there is a unit interval graph In and a permutation graph Hn having n2 vertices, each of whose clique–width is at least n. These results allow us to see the border within the hierarchy of perfect graphs between classes whose clique–width is bounded and classes whose clique–width is unbounded. Finally we show that every n×n square grid, @math , n ≥ 3, has clique–width exactly n+1.