Graphs are a pervasive representation that arises naturally in a variety of disciplines; however, their non-Euclidean structure has traditionally proven challenging for machine learning and decision-making approaches. The emergence of the Graph Neural Network learning paradigmScarselli et al. (2009)
and geometric deep learning more broadlyBronstein et al. (2017)
have brought about encouraging breakthroughs in diverse application areas for graph-structured data: relevant examples include combinatorial optimizationVinyals et al. (2015); Bello et al. (2016); Khalil et al. (2017), recommendation systems Monti et al. (2017); Ying et al. (2018) and computational chemistry Gilmer et al. (2017); Jin et al. (2018); You et al. (2018); Bradshaw et al. (2019).
A recent line of work focuses on goal-directed graph construction, in which the aim is to build or modify the topology of a graph (i.e., add a set of edges) so as to maximize the value of a global objective function. Unlike classic graph algorithms, which assume that the graph topology is static, in this setting the graph structure itself is dynamically
changed. As this task involves an element of exploration (optimal solutions are not known a priori), its formulation as a decision-making process is a suitable paradigm. Model-free reinforcement learning (RL) techniques have been applied in the context of the derivation of adversarial examples for graph-based classifiersDai et al. (2018) and generation of molecular graphs You et al. (2018). Darvariu et al. (2020) have formulated the optimization of a global structural graph property as an MDP and approached it using a variant of the RL-S2V Dai et al. (2018) algorithm, showing that generalizable strategies for improving a global network objective can be learned, and can obtain performance superior to prior approaches Beygelzimer et al. (2005); Schneider et al. (2011); Wang and Van Mieghem (2008); Wang et al. (2014) in some cases. However, when applying such approaches to improve the properties of real-world networked systems (such as infrastructure networks), two challenges become apparent:
Inability to account for spatial properties of the graphs: optimizing the topology of the graph alone is only part of the problem in many cases. A variety of real-world networks share the property that nodes are embedded in space, and this geometry has a strong relationship with the types of topologies that can be created Gastner and Newman (2006b); Barthélemy (2011). Since there is a cost associated with edge length, connections tend to be local, and long-range connections must be justified by some gain (e.g., providing connectivity to a hub). Furthermore, objective functions defined over nodes’ positions (such as efficiency) are key for understanding their organization Latora and Marchiori (2001).
Scalability: existing methods based on RL are challenging to scale, due to the sample complexity of current training algorithms, the linear increase of possible actions in the number of nodes, and the complexity of evaluating the global objectives (typically polynomial in the number of nodes). Additionally, training data (i.e., instances of real-world graphs) are scarce and we are typically interested in a specific starting graph (e.g., a particular infrastructure network to be improved).
In this paper, we set out to address these shortcomings. For the first time in this emerging line of work, we consider the construction of spatial graphs as a decision-making process that explicitly captures the influence of space on graph-level objectives, realizable links between nodes, and connection budgets. Furthermore, to address the scalability issue, we propose to use planning methods in order to plan an optimal set of edges to add to the graph, which sidesteps the problem of sample complexity since we do not need to learn a policy. We adopt the Monte Carlo Tree Search framework – specifically, the UCT algorithm Kocsis and Szepesvári (2006) – and show it can be applied successfully in planning graph construction strategies. We illustrate our approach at a high level in Figure 1. Finally, we propose several improvements over the basic UCT method in the context of spatial networks. These relate to important characteristics of this family of problems: namely, their single-agent, deterministic nature; the inherent trade-off between the cost of edges and their contribution to the global objective; and an action space that is linear in the number of nodes in the network. Our proposed approach, Spatial Graph UCT (SG-UCT), is designed with these characteristics in mind.
As objective functions, in this study, we consider the global network properties of efficiency and robustness to targeted attacks. While these represent a variety of practical scenarios, our approach is broadly applicable to any other structural property. We perform an evaluation on synthetic graphs (generated by a spatial growth model) and several real-world graphs (internet backbone networks and metro transportation networks), comparing SG-UCT to UCT as well as a variety of baselines that have been proposed in the past. Our results show that SG-UCT performs best out of all methods in all the settings we tested; moreover, the performance gain over UCT is substantial (24% on average and up to 54% over UCT on the largest networks tested in terms of a robustness metric). In addition, we conduct an ablation study that explores the impact of the various algorithmic components.
MDPs and Planning.Markov Decision Processes (MDPs) are widely adopted for effective formalization of decision making tasks. The decision maker, usually called an agent, interacts with an environment. When in a state , the agent must take an action out of the set of valid actions, receiving a reward governed by the reward function . Finally, the agent finds itself in a new state , depending on a transition model
that governs the joint probability distributionof transitioning to state after taking action in state . This sequence of interactions gives rise to a trajectory , which continues until a terminal state is reached. In deterministic MDPs, there exists a unique state s.t. . The tuple defines this MDP, where is a discount factor. We also define a policy , a distribution of actions over states. There exists a spectrum of algorithms for constructing a policy, ranging from model-based algorithms (which assume knowledge of the MDP) to model-free algorithms (which require only samples of agent-environment interactions). In the cases in which the full MDP specification or a model are available, planning can be used to construct a policy, for example using forward search (Russell and Norvig, 2010, Chapter 10).
Monte Carlo Tree Search. Monte Carlo Tree Search (MCTS) is a model-based planning technique that addresses the inability to explore all paths in large MDPs by constructing a policy from the current state
. Values of states are estimated through the returns obtained by executing simulations from the starting state. Upper Confidence Bounds for Trees (UCT), a variant of MCTS, consists of a tree search in which the decision at each node is framed as an independent multi-armed bandit problem. At decision time, thetree policy of the algorithm selects the child node corresponding to action that maximizes , where is the sum of returns obtained when taking action in state , is the number of parent node visits, the number of child node visits, and is a constant that controls the level of exploration Kocsis and Szepesvári (2006). In the standard version of the algorithm, the returns are estimated using a random default policy when expanding a node. MCTS has been applied to great success in connectionist games such as Morpion Solitaire Rosin (2011), Hex Nash (1952); Anthony et al. (2017), and Go, which was previously thought computationally intractable Silver et al. (2016, 2018).
Spatial Networks. We define a spatial network as the tuple . is the set of vertices, and is the set of edges. is a function that maps nodes in the graph to a set of positions . We require that admits a metric , i.e., there exists a function defining a pairwise distance between elements in . The tuple defines a space, common examples of which include Euclidean space and spherical geometry. associates a weight with each edge: a positive real-valued number that denotes its capacity.
Global Objectives in Spatial Networks. We consider two global objectives for spatial networks that are representative of a wide class of properties relevant in real-world situations. Depending on the domain, there are many other global objectives for spatial networks that can be considered, to which the approach that we present is directly applicable.
Efficiency. Efficiency is a metric quantifying how well a network exchanges information. It is a measure of how fast information can travel between any pair of nodes in the network on average, and is hypothesized to be an underlying principle for the organization of networks Latora and Marchiori (2001). Efficiency does not solely depend on topology but also on the spatial distances between the nodes in the network. We adopt the definition of global efficiency as formalized by Latora and Marchiori, and let , where denotes the cumulative length of the shortest path between vertices and . To normalize this quantity, we divide it by the ideal efficiency , and possible values are thus in .111It is worth noting that efficiency is a more suitable metric for measuring the exchange of information than the inverse average path length between pairs of nodes. In the extreme case where the network is disconnected (and thus some paths lengths are infinite), this metric does not go to infinity. More generally, this metric is better suited for systems in which information is exchanged in a parallel, rather than sequential, way Latora and Marchiori (2001). Efficiency can be computed in by aggregating path lengths obtained using the Floyd-Warshall algorithm.222In practice, this may be made faster by considering dynamic shortest path algorithms, e.g. Demetrescu and Italiano (2004).
Robustness. We consider the property of robustness, i.e., the resilience of the network in the face of removals of nodes. We adopt a robustness measure widely used in the literature Albert et al. (2000); Callaway et al. (2000) and of practical interest and applicability based on the largest connected component (LCC), i.e., the component with most nodes. In particular, we use the definition in Schneider et al. (2011), which considers the size of the LCC as nodes are removed from the network. We consider only the targeted attack case as previous work has found it is more challenging Albert et al. (2000); Darvariu et al. (2020). We define the robustness measure as , where denotes the fraction of nodes in the LCC of after the removal of the first nodes in the permutation (in which nodes appear in descending order of their degrees). Possible values are in . This quantity can be estimated using Monte Carlo simulations and scales as .
It is worth noting that the value of the objective functions typically increases the more edges exist in the network (the complete graph has both the highest possible efficiency and robustness). However, constructing a complete graph is wasteful in terms of resources, and so it may be necessary to balance the contribution of an edge to the objective with its cost. The method that we propose explicitly accounts for this trade-off, which is widely observed in infrastructure and brain networks Gastner and Newman (2006a); Bullmore and Sporns (2012).
3 Proposed Method
In this section, we first formulate the construction of spatial networks in terms of a global objective function as an MDP. Subsequently, we propose a variant of the UCT algorithm (SG-UCT) for planning in this MDP, which exploits the characteristics of spatial networks.
3.1 Spatial Graph Construction as an MDP
Spatial Constraints in Network Construction. Spatial networks that can be observed in the real world typically incur a cost to edge creation. Take the example of a power grid: the cost of a link can be expressed as a function of its geographic distance as well as its capacity. It is vital to consider both aspects of link cost in the process of network construction. We let denote the cost of edge and be the cost of a set of edges . We consider to capture the notion that longer, higher capacity connections are more expensive – although different notions of cost may be desirable depending on the domain. To ensure fair comparisons between various networks, we normalize costs to be in .
Problem Statement. Let be the set of labeled, undirected, weighted, spatial networks with nodes. We let be an objective function, and be a modification budget. Given an initial graph , the aim is to add a set of edges to such that the resulting graph satisfies:
MDP Formulation. We next define the key elements of the MDP:
State: The state is a 3-tuple containing the spatial graph , an edge stub , and the remaining budget . can be either the empty set or the singleton , where . If the edge stub is non-empty, it means that the agent has “committed" in the previous step to creating an edge originating at the edge stub node.
Action: For scalability to large graphs, we let actions correspond to the selection of a single node in (thus having at most choices). We enforce spatial constraints in the following way: given a node , we define the set of connectable nodes that represent realizable connections. We let:
which formalizes the idea that a node can only connect as far as a proportion of its longest existing connection, with fixed based on the initial graph . This has the benefit of allowing long-range connections if they already exist in the network. Given an unspent connection budget , we let the set consist of those connectable nodes whose cost is not more
than the unspent budget.333Depending on the type of network being considered, in practice there may be different types of constraints on the connections that can be realized. For example, in transportation networks, there can be obstacles that make link creation impossible, such as prohibitive landforms or populated areas. In circuits and utility lines, planarity is a desirable characteristic as it makes circuit design cheaper. Such constraints can be captured by the definition of and enforced by the environment when providing the agent with available actions . Conversely, defining recovers the simplified case where no constraints are imposed. Letting the degree of node be , available actions are defined as:
Transitions: The deterministic transition model adds an edge to the graph topology every two steps. Concretely, we define it as , where
Reward: The final reward is defined as and all intermediary rewards are .
Episodes in this MDP proceed for an arbitrary number of steps until the budget is exhausted and no valid actions remain (concretely, ). Since we are in the finite horizon case, we let . Given the MDP definition above, the problem specified in Equation 1 can be reinterpreted as finding the trajectory that starts at such that the final reward is maximal – actions along this trajectory will define the set of edges .
The formulation above can, in principle, be used with any planning algorithm for MDPs in order to identify an optimal set of edges to add to the network. The UCT algorithm, discussed in Section 2, is one such algorithm that has proven very effective in a variety of settings. We refer the reader to Browne et al. (2012) for an in-depth description of the algorithm and its various applications. However, the generic UCT algorithm assumes very little about the particulars of the problem under consideration, which, in the context of spatial network construction, may lead to sub-optimal solutions. In this section, we identify and address concerns specific to this family of problems, and formulate the Spatial Graph UCT (SG-UCT) variant of UCT in Algorithm 1. The evaluation presented in Section 4 compares SG-UCT to UCT and other baselines, and contains an ablation study of SG-UCT’s components.
Best Trajectory Memorization (BTM). The standard UCT algorithm is applicable in a variety of settings, including multi-agent, stochastic environments. For example, in two-player games, an agent needs to re-plan from the new state that is arrived at after the opponent executes its move. However, the single-agent (puzzle), deterministic nature of the problem considered means that there is no-need to re-plan trajectories after a stochastic event: the agent can plan all its actions from the very beginning in a single step. We thus propose the following modification over UCT: memorizing the trajectory with the highest reward found during the rollouts, and returning it at the end of the search. We name this Best Trajectory Memorization, shortened BTM. This is similar in spirit (albeit much simpler) to ideas used in Reflexive and Nested MCTS for deterministic puzzles, where the best move found at lower levels of a nested search are used to inform the upper level Cazenave (2007, 2009).
Cost-Sensitive Default Policy.
The standard default policy used to perform out-of-tree actions in the UCT framework is based on random rollouts. While evaluating nodes using this approach is free from bias, rollouts can lead to high-variance estimates, which can hurt the performance of the search. Previous work has considered hand-crafted heuristics and learned policies as alternatives, although, perhaps counter-intuitively, learned policies may lead to worse resultsGelly and Silver (2007). As initially discussed in Section 2
, the value of the objective functions we consider grows with the number of edges of the graph. We thus propose the following default policy for spatial networks: sampling each edge with probability inversely proportional to its cost. Formally, we let the probability of edgebeing selected during rollouts be proportional to , where denotes the level of bias. reduces to random choices, while selects the minimum cost edge. This is very inexpensive from a computational point of view, as the edge costs only need to be computed once, at the start of the search.
Action Space Reduction. In certain domains, the number of actions available to an agent is large, which can greatly affect scalability. Previous work in RL has considered decomposing actions into independent sub-actions He et al. (2016), generalizing across similar actions by embedding them in a continuous space Dulac-Arnold et al. (2015), or learning which actions to eliminate via a supervision signal provided by the environment Zahavy et al. (2018). Previous work on planning considers progressively widening the search based on a heuristic Chaslot et al. (2008) or learning a partial policy for eliminating actions in the search tree Pinto and Fern (2017).
Concretely, with the MDP definition used, the action space grows linearly in the number of nodes. This is partly addressed by the imposed connectivity constraints: once an edge stub is selected (equivalently, at odd values of), the branching factor of the search is small since only connectable nodes need to be considered. However, the number of actions when selecting the origin node of the edge (even values of ) remains large, which might become detrimental to performance as the size of the network grows (we illustrate this in Figure 2). Can this be mitigated?
We consider limiting the nodes that can initiate connections to a specific subset – which effectively prunes away all branches in the search tree that are not part of this set. Concretely, let a reduction policy be a function that, given the initial graph , outputs a strict subset of its nodes.444Learning a reduction policy in a data-driven way is also possible; however, the supervision signal needed (i.e., results of node rankings over multiple MCTS runs) is very expensive to obtain. Furthermore, since we prioritize performance on specific graph instances over generalizable policies, simple statistics may be sufficient. Still, a learned reduction policy that predicts an entire set at once may be able to identify better subsets than individual statistics alone. We consider this a worthwhile direction for future work. Then, we modify our definition of allowed actions as follows: under a reduction policy , we define
We investigate the following class of reduction policies: a node is included in if and only if it is among the top nodes ranked by a local node statistic . More specifically, we consider the listed below, where
. Since the performance of reduction strategies may depend heavily on the objective function, we treat it as a hyperparameter to be optimized.
Degree (DEG): ; Inverse Degree (ID): ; Number of Connections (NC):
Best Edge (BE): ; BE Cost Sensitive (BECS):
Average Edge (AE): ; AECS: .
4.1 Experimental Protocol
Definitions of Space and Distance. For all experiments in this paper, we consider the unit 2D square as our space, i.e. we let and the distance be Euclidean distance. In case the graph is defined on a spherical coordinate system (as is the case with physical networks positioned on Earth), we use the WGS84 variant of the Mercator projection to project nodes to the plane; then normalize to the unit plane. For simplicity, we consider uniform weights, i.e., .
Synthetic and Real-World Graphs. As a means of generating synthetic graph data, we use the popular model proposed by Kaiser and Hilgetag (2004), which simulates a process of growth for spatial networks. Related to the Waxman model Waxman (1988), in this model the probability that a connection is created is proportional to its distance from existing nodes. The distinguishing feature of this model is that, unlike e.g. the Random Geometric Graph Dall and Christensen (2002), this model produces connected networks: a crucial characteristic for the types of objectives we consider. We henceforth refer to this model as Kaiser-Hilgetag (shortened KH). We use and , which yields sparse graphs with scale-free degree distributions – a structure similar to road infrastructure networks. We also evaluate performance on networks belonging to the following real-world datasets, detailed in Table 1: Internet (a dataset of internet backbone infrastructure from a variety of ISPs Knight et al. (2011)) and Metro (a dataset of metro networks in major cities around the world Roth et al. (2012)). Due to computational budget constraints, we limit the sizes of networks considered to .
Setup. For all experiments, we allow agents a modification budget equal to a proportion of the total cost of the edges of the original graph, i.e., . We use . We let for synthetic graphs and
for real-world graphs, respectively. Confidence intervals are computed using results ofruns, each initialized using a different random seed. Rollouts are not truncated. We allow a number of node expansions per move equal to (a larger number of expansions can improve performance, but leads to diminishing returns), and select as the move at each step the node with the maximum average value (commonly referred to as MaxChild). Full details of the hyperparameter selection methodology and the values used are provided in the Appendix.
Baselines. The baselines we compare against are detailed below. These methods represent the major approaches that have been considered in the past for the problem of goal-directed graph modifications: namely, considering a local node statistic Beygelzimer et al. (2005), a shallow greedy criterion Schneider et al. (2011), or the spectral properties of the graph Wang and Van Mieghem (2008); Wang et al. (2014). We do not consider previous RL-based methods since they are unsuitable for the scale of the largest graphs taken into consideration, for which the time needed to evaluate the objective functions makes training prohibitively expensive.
Random (): Randomly selects an available action.
Greedy (): Selects the edge that gives the biggest improvement in : formally, edge that satisfies . We also consider the cost-sensitive variant GreedyCS, for which the gain is offset by the cost: .
MinCost (): Selects edge that satisfies .
LBHB (): Adds an edge between the node with Lowest Betweenness and the node with Highest Betweeness; formally, letting the betweeness centrality of node be , this strategy adds an edge between nodes and .
LDP (): Adds an edge between the vertices with the Lowest Degree Product, i.e., vertices that satisfy .
ERes (): Adds an edge between vertices with the highest pairwise Effective Resistance, i.e., nodes that satisfy . is defined as , where is the pseudoinverse of the graph Laplacian Wang et al. (2014).
4.2 Evaluation Results
Synthetic Graph Results. In this experiment, we consider KH graphs each of sizes . The obtained results are shown in the top half of Table 2. We summarize our findings as follows: SG-UCT outperforms UCT and all other methods in all the settings tested, obtaining 13% and 32% better performance than UCT on the largest synthetic graphs for the efficiency and robustness measures respectively. For , UCT outperforms all baselines, while for the performance of the Greedy baselines is superior to UCT. Interestingly, MinCost yields solutions that are superior to all other heuristics and comparable to search-based methods while being very cheap to evaluate. Furthermore, UCT performance decays in comparison to the baselines as the size of the graph increases.
Real-world Graph Results. The results obtained for real-world graphs are shown in the bottom half of Table 2 (an extended version, split by individual graph instance, is shown in Table 5 in the Appendix). As with synthetic graphs, we find that SG-UCT performs better than UCT and all other methods in all settings tested. The differences in performance between SG-UCT and UCT are 10% and 39% for and respectively. We note that the Greedy approaches do not scale to the larger real-world graphs due to their complexity: edges need to be considered at each step, in comparison to the required by UCT and SG-UCT.
Ablation Study. We also conduct an ablation study in order to assess the impact of the individual components of SG-UCT, using the same KH synthetic graphs. The obtained results are shown in Table 3, where SG-UCTBTM denotes UCT with best trajectory memorization, SG-UCTMINCOST denotes UCT with the cost-based default policy, SG-UCT for in denotes UCT with a particular reduction policy , and represents the percentage of original nodes that are selected by . We find that BTM indeed brings a net improvement in performance: on average, 5% for and 11% for . The benefit of the cost-based default policy is substantial (especially for ), ranging from 4% on small graphs to 27% on the largest graphs considered, and grows the higher the level of bias. This is further evidenced in Figure 3, which shows the average reward obtained as a function of . In terms of reduction policies, even for a random selection of nodes, we find that the performance penalty paid is comparatively small: a 60% reduction in actions translates to at most 15% reduction in performance, and as little as 5%; the impact of random action reduction becomes smaller as the size of the network grows. The best-performing reduction policies are those based on a node’s gains, with BECS and AECS outperforming UCT with no action reduction. For the objective, a poor choice of bias can be harmful: prioritizing nodes with high degrees leads to a 32% reduction in performance compared to UCT, while a bias towards lower-degree nodes is beneficial.
Limitations. While we view our work as an important step towards improving evidence-based decision making in infrastructure systems, we were not able to consider realistic domain-specific concerns due to our abstraction level. Another limitation is the fact that, due to our limited computational budget, we opted for hand-crafted reduction policies. Particularly the impact of the number of nodes selected by the reduction policy was not explored. Furthermore, as already discussed in Section 3.2, a learning-based reduction policy able to predict an entire set at once may lead to better performance. We believe that it is possible to scale our approach to substantially larger networks by considering a hierarchical representation (i.e., viewing the network as a graph of graphs) and by further speeding up the calculation of the objective functions (currently the main computational bottleneck).
Societal Impact and Implications. The proposed method is suitable as a framework for the optimization of a variety of infrastructure networks, a problem that has potential for societal benefit. Beyond the communication and metro networks used in our evaluation, we list road networks, water distribution networks, and power grids as relevant application areas. We cannot foresee any immediate negative societal impacts of this work. However, as with any optimization problem, the objective needs to be carefully analyzed and its implications considered before real-world deployment: for example, it may be the case that optimizing for a specific objective might lead to undesirable results such as the introduction of a “rich get richer” effect Merton (1968), resulting in well-connected nodes gaining more and more edges than less well-connected ones.
In this work, we have addressed the problem of spatial graph construction: namely, given an initial spatial graph, a budget defined in terms of edge lengths, and a global objective, finding a set of edges to be added to the graph such that the value of the objective is maximized. For the first time among related works, we have formulated this task as a deterministic MDP that accounts for how the spatial geometry influences the connections and organizational principles of real-world networks. Building on the UCT framework, we have considered several concerns in the context of this problem space and proposed the Spatial Graph UCT (SG-UCT) algorithm to address them. Our evaluation results show performance substantially better than UCT (24% on average and up to 54% in terms of a robustness measure) and all existing baselines taken into consideration.
This work was supported by The Alan Turing Institute under the UK EPSRC grant EP/N510129/1. The authors declare no competing financial interests with respect to this work.
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Appendix A Additional Evaluation Details
Impact of Action Subsets on UCT Results. We include an additional experiment related to the Action Space Reduction problem discussed in Section 3.2. We consider, starting from the same initial graph, a selection of subsets of size 40% of all nodes, obtained with a uniform random . We show the empirical distribution of the reward obtained by UCT with different sampled subsets in Figure 4. Since the subset that is selected has an important impact on performance, a reduction policy yielding high-reward subsets is highly desirable (effectively, we want to bias subset selection towards the upper tail of the distribution of obtained rewards).
Hyperparameters. Hyperparameter optimization for UCT and SG-UCT is performed separately for each objective function and synthetic graph model / real-world network dataset. For synthetic graphs, hyperparameters are tuned over a disjoint set of graphs. For real-world graphs, hyperparameters are optimized separately for each graph. We consider an exploration constant . Since the ranges of the rewards may vary in different settings, we further employ two means of standardization: during the tree search we instead use as the final reward , and further standardize by multiplying with the average reward observed at the root in the previous timestep – ensuring consistent levels of exploration. The hyperparameters for the ablation study are bootstrapped from those of standard UCT, while for the SG-UCTMINCOST variant is optimized separately. These results are used to reduce the hyperparameter search space for SG-UCT for both synthetic and real-world graphs. Values of hyperparameters used are shown in Table 4. For estimating we use Monte Carlo simulations.
Extended Results. Extended results for real-world graphs, split by graph instance, are shown in Table 5.
Appendix B Reproducibility
Implementation. We implement all approaches and baselines in Python using a variety of numerical and scientific computing packages Hunter , Hagberg et al. , McKinney et al. , Paszke et al. , Waskom , while the calculations of the objective functions (efficiency and robustness) are performed in a custom C++ module as they are the main speed bottleneck. In a future version, we will release the implementation as Docker containers together with instructions that enable reproducing (up to hardware differences) all the results reported in the paper, including tables and figures.
Data Availability. In terms of the real-world datasets, the Internet dataset is publicly available without any restrictions and can be downloaded via the Internet Topology Zoo website, http://www.topology-zoo.org/dataset.html. The Metro dataset was originally collected by Roth et al. , and was licensed to us by the authors for the purposes of this work. A copy of the Metro dataset can be obtained by others by contacting its original authors for licensing (see https://www.quanturb.com/data).
Infrastructure and Runtimes. Experiments were carried out on an internal cluster of 8 machines, each equipped with 2 Intel Xeon E5-2630 v3 processors and 128GB RAM. On this infrastructure, all experiments reported in this paper took approximately 21 days to complete.