Alessio Conte

Sublinear-space bounded-delay enumeration for massive network analytics: Maximal cliques

Due to the sheer size of real-world networks, delay and space become quite relevant measures for the cost of enumeration in network analytics. This paper presents efficient algorithms for listing maximum cliques in networks, providing the first sublinear-space bounds with guaranteed delay per enumerated clique, thus comparing favorably with the known literature.

Finding All Maximal Cliques in Very Large Social Networks

The detection of communities in social networks is a challenging task. A rigorous way to model communities considers maximal cliques, that is, maximal subgraphs in which each pair of nodes is connected by an edge. State-of-the-art strategies for nding maximal cliques in very large networks decompose the network in blocks and then perform a distributed computation. These approaches exhibit a trade-o between eciency and completeness: decreasing the size of the blocks has been shown to improve eciency but some cliques may remain undetected since high-degree nodes, also called hubs, may not t with all their neighborhood into a small block. In this paper, we present a distributed approach that, by suitably handling hub nodes, is able to detect maximal cliques in large networks meeting both completeness and eciency. The approach relies on a two-level decomposition process. The rst level aims at recursively identifying and isolating tractable portions of the network. The second level further decomposes the tractable portions into small blocks. We demonstrate that this process is able to correctly detect all maximal cliques, provided that the sparsity of the network is bounded, as it is the case of real-world social networks. An extensive campaign of experiments conrms the

Fast Enumeration of Large k-Plexes

K-plexes are a formal yet flexible way of defining communities in networks. They generalize the notion of cliques and are more appropriate in most real cases: while a node of a clique C is connected to all other nodes of C, a node of a k-plex may miss up to k connections. Unfortunately, computing all maximal k-plexes is a gruesome task and state-of-the-art algorithms can only process small-size networks. In this paper we propose a new approach for enumerating large k-plexes in networks that speeds up the search by several orders of magnitude, leveraging on (i) methods for strongly reducing the search space and (ii) efficient techniques for the computation of maximal cliques. Several experiments show that our strategy is effective and is able to increase the size of the networks for which the computation of large k-plexes is feasible from a few hundred to several hundred thousand nodes.

Listing Acyclic Orientations of Graphs with Single and Multiple Sources

We study enumeration problems for the acyclic orientations of an undirected graph with n nodes and m edges, where each edge must be assigned a direction so that the resulting directed graph is acyclic. When the acyclic orientations have single or multiple sources specified as input along with the graph, our algorithm is the first one to provide guaranteed bounds, giving new bounds with a delay of \(O(m\cdot n)\) time per solution and \(O(n^2)\) working space. When no sources are specified, our algorithm improves over previous work by reducing the delay to O(m), and is the first one with linear delay.

Clique covering of large real-world networks

The edge clique covering (ecc) problem deals with discovering a set of (possibly overlapping) cliques in a given network, such that each edge is part of at least one of these cliques. We address the ecc problem from an alternative perspective reconsidering the quality of the cliques found, and proposing more structured criteria with respect to the traditional measures such as minimum number of cliques. In the case of real-world networks, having millions of nodes, such as social networks, the possibility of getting a result is constrained to the running time, which should be linear or almost linear in the size of the network. Our algorithm for finding eccs of large networks has linear-time performance in practice, as our experiments show on real-world networks whose number of nodes ranges from thousands to several millions.

Listing Acyclic Subgraphs and Subgraphs of Bounded Girth in Directed Graphs

The girth of a directed graph is the length of its shortest directed cycle. We consider the problem of generating all subgraphs of girth at least g in a directed graph G with n vertices and m edges. This generalizes the problem of generating acyclic subgraphs (i.e., with no directed cycle), that correspond to the subgraphs of girth at least \(n+1\). The problem of finding the acyclic subgraph with maximum size or weight has been thoroughly studied, however to the best of our knowledge there is no known efficient enumeration algorithm. We propose polynomial delay algorithms for listing both induced and edge subgraphs with girth g in time O(n) per solution; both improve upon a naive solution, respectively by a factor O(nm) and \(O(m^2)\). Furthermore, this work is on the line of existing research for extracting acyclic structures from graphs.

Efficient Enumeration of Maximal k-Degenerate Subgraphs in a Chordal Graph

In this paper, we consider the problem of listing the maximal k-degenerate induced subgraphs of a chordal graph, and propose an output-sensitive algorithm using delay \(O(m\cdot \omega (G))\) for any n-vertex chordal graph with m edges, where \(\omega (G) \le n\) is the maximum size of a clique in G. The problem generalizes that of enumerating maximal independent sets and maximal induced forests, which correspond to respectively 0-degenerate and 1-degenerate subgraphs.

Enumerating Cyclic Orientations of a Graph

Acyclic and cyclic orientations of an undirected graph have been widely studied for their importance: an orientation is acyclic if it assigns a direction to each edge so as to obtain a directed acyclic graph (DAG) with the same vertex set; it is cyclic otherwise. As far as we know, only the enumeration of acyclic orientations has been addressed in the literature. In this paper, we pose the problem of efficiently enumerating all the cyclic orientations of an undirected connected graph with n ver-tices and m edges, observing that it cannot be solved using algorithmic techniques previously employed for enumerating acyclic orientations. We show that the problem is of independent interest from both combinato-rial and algorithmic points of view, and that each cyclic orientation can be listed with˜Owith˜ with˜O(m) delay time. Space usage is O(m) with an additional setup cost of O(n 2) time before the enumeration begins, or O(mn) with a setup cost of˜Oof˜ of˜O(m) time.

A Fast Algorithm for Large Common Connected Induced Subgraphs

We present a fast algorithm for finding large common subgraphs, which can be exploited for detecting structural and functional relationships between biological macromolecules. Many fast algorithms exist for finding a single maximum common subgraph. We show with an example that this gives limited information, motivating the less studied problem of finding many large common subgraphs covering different areas. As the latter is also hard, we give heuristics that improve performance by several orders of magnitude. As a case study, we validate our findings experimentally on protein graphs with thousands of atoms.

Directing Road Networks by Listing Strong Orientations

A connected road network with N nodes and L edges has \(K \le L\) edges identified as one-way roads. In a feasible direction, these one-way roads are assigned a direction each, so that every node can reach any other [Robbins ’39]. Using O(L) preprocessing time and space usage, it is shown that all feasible directions can be found in O(K) amortized time each. To do so, we give a new algorithm that lists all the strong orientations of an undirected connected graph with m edges in O(m) amortized time each, using O(m) space. The cost can be deamortized to obtain O(m) delay with \(O(m^2)\) preprocessing time and space.