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Moreno, S., Neville, J., & Kirshner, S. (2018). Tied Kronecker Product Graph Models to Capture Variance in Network Populations. ACM Trans. Knowl. Discov. Data, 12(3), 40 pp.
Abstract: Much of the past work on mining and modeling networks has focused on understanding the observed propel ties of single example graphs. However, in many real-life applications it is important to characterize the structure of populations of graphs. In this work, we analyze the distributional properties of probabilistic generative graph models (PGGMs) for network populations. PGGMs are statistical methods that model the network distribution and match common characteristics of real-world networks. Specifically, we show that most PGGMs cannot relied the natural variability in graph properties observed across multiple networks because their edge generation process assumes independence among edges. Then, we propose the mixed Kronecker Product Graph Model (mKPGM) a scalable generalization of KPGMs that uses tied parameters to increase the variability of the sampled networks, while preserving the edge probabilities in expectation. We compare mKPGM to several other graph models. The results show that learned mKPGMs accurately represent the characteristics of real-world networks, while also effectively capturing the natural variability in network structure.
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Moreno, S., Pfeiffer, J. J., & Neville, J. (2018). Scalable and exact sampling method for probabilistic generative graph models. Data Min. Knowl. Discov., 32(6), 1561–1596.
Abstract: Interest in modeling complex networks has fueled the development of multiple probabilistic generative graph models (PGGMs). PGGMs are statistical methods that model the network distribution and match common characteristics of real world networks. Recently, scalable sampling algorithms for well known PGGMs, made the analysis of large-scale, sparse networks feasible for the first time. However, it has been demonstrated that these scalable sampling algorithms do not sample from the original underlying distribution, and sometimes produce very unlikely graphs. To address this, we extend the algorithm proposed in Moreno et al.(in: IEEE 14th international conference on data mining, pp 440-449, 2014) for a single model and develop a general solution for a broad class of PGGMs. Our approach exploits the fact that PGGMs are typically parameterized by a small set of unique probability valuesthis enables fast generation via independent sampling of groups of edges with the same probability value. By sampling within groups, we remove bias due to conditional sampling and probability reallocation. We show that our grouped sampling methods are both provably correct and efficient. Our new algorithm reduces time complexity by avoiding the expensive rejection sampling step previously necessary, and we demonstrate its generality, by outlining implementations for six different PGGMs. We conduct theoretical analysis and empirical evaluation to demonstrate the strengths of our algorithms. We conclude by sampling a network with over a billion edges in 95s on a single processor.
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Munoz-Herrera, S., & Suchan, K. (2022). Local Optima Network Analysis of Multi-Attribute Vehicle Routing Problems. Mathematics, 10(24), 4644.
Abstract: Multi-Attribute Vehicle Routing Problems (MAVRP) are variants of Vehicle Routing Problems (VRP) in which, besides the original constraint on vehicle capacity present in Capacitated Vehicle Routing Problem (CVRP), other attributes that model diverse real-life system characteristics are present. Among the most common attributes studied in the literature are vehicle capacity and route duration constraints. The influence of these restrictions on the overall structure of the problem and the performance of local search algorithms used to solve it has yet to be well known. This paper aims to explain the impact of constraints present in different variants of VRP through the alterations of the structure of the underlying search space that they cause. We focus on Local Optima Network Analysis (LONA) for multiple Traveling Salesman Problem (m-TSP) and VRP with vehicle capacity (CVRP), route duration (DVRP), and both (DCVRP) constraints. We present results that indicate that measures obtained for a sample of local optima provide valuable information on the behavior of the landscape under modifications in the problem's constraints. Additionally, we use the LONA measures to explain the difficulty of VRP instances for solving by local search algorithms.
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