Found Graph Data and Planted Vertex Covers
A typical way in which network data is recorded is to measure all interactions involving a specified set of core nodes, which produces a graph containing this core together with a potentially larger set of fringe nodes that link to the core. Interactions between nodes in the fringe, however, are not present in the resulting graph data. For example, a phone service provider may only record calls in which at least one of the participants is a customer; this can include calls between a customer and a non-customer, but not between pairs of non-customers. Knowledge of which nodes belong to the core is crucial for interpreting the dataset, but this metadata is unavailable in many cases, either because it has been lost due to difficulties in data provenance, or because the network consists of "found data" obtained in settings such as counter-surveillance. This leads to an algorithmic problem of recovering the core set. Since the core is a vertex cover, we essentially have a planted vertex cover problem, but with an arbitrary underlying graph. We develop a framework for analyzing this planted vertex cover problem, based on the theory of fixed-parameter tractability, together with algorithms for recovering the core. Our algorithms are fast, simple to implement, and out-perform several baselines based on core-periphery structure on various real-world datasets.
1 Partially measured graphs, data provenance, and planted structure Datasets that take the form of graphs are ubiquitous throughout the sciences [4,23,49], but the graph data that we work with is generally incomplete in certain systematic ways [28,33,34,36,39]. A common type of incompleteness comes from the way in which graph data is generally measured: we observe a set of nodes C and record all the interactions involving this set of nodes. The result is a measured graph G consisting of this core set C together with a a potentially larger set of additional fringe nodes-the nodes outside of C that interact with some node in C. For example, in constructing a social network dataset, we might study the employees of a company and record all of their friendships [54]. From this information, we now have a graph that contains all the employees together with all of their friends, including friends who do not work for the company. This latter group constitutes the set of fringe nodes in the graph. The edge set of the graph G reflects the construction: each edge involves at least one core node, but if two nodes in the fringe have interacted, it is invisible to us and hence not recorded in the data.
E-mail and other communication datasets typically have this core-fringe structure. For example, the widely-studied Enron email graph [24,37,43,55] contains tens of thousands of nodes; however, this graph was constructed from the email inboxes of fewer than 150 employees [35]. The vast majority of the nodes in the graph, therefore, belong to the fringe, and their direct communications are not part of the data. The issue comes up in much larger network datasets as well. For example, a phone service provider has data on calls and messages that its customers make both to each other and to non-customers, but it does not have data on communication between pairs of non-customers. Another example would be a massive online social network may get some information about the contacts of its users with a fringe set comprised of of people not on the system-often including entire countries that do not participate in the platform-but generally not about the interactions taking place in this fringe set. Similarly, Internet measurements at the IP-layer from individual service providers give 32nd Conference on Neural Information Processing Systems (NeurIPS 2018), Montréal, Canada.
1 Partially measured graphs, data provenance, and planted structure Datasets that take the form of graphs are ubiquitous throughout the sciences [4,23,49], but the graph data that we work with is generally incomplete in certain systematic ways [28,33,34,36,39]. A common type of incompleteness comes from the way in which graph data is generally measured: we observe a set of nodes C and record all the interactions involving this set of nodes. The result is a measured graph G consisting of this core set C together with a a potentially larger set of additional fringe nodes-the nodes outside of C that interact with some node in C. For example, in constructing a social network dataset, we might study the employees of a company and record all of their friendships [54]. From this information, we now have a graph that contains all the employees together with all of their friends, including friends who do not work for the company. This latter group constitutes the set of fringe nodes in the graph. The edge set of the graph G reflects the construction: each edge involves at least one core node, but if two nodes in the fringe have interacted, it is invisible to us and hence not recorded in the data.
E-mail and other communication datasets typically have this core-fringe structure. For example, the widely-studied Enron email graph [24,37,43,55] contains tens of thousands of nodes; however, this graph was constructed from the email inboxes of fewer than 150 employees [35]. The vast majority of the nodes in the graph, therefore, belong to the fringe, and their direct communications are not part of the data. The issue comes up in much larger network datasets as well. For example, a phone service provider has data on calls and messages that its customers make both to each other and to non-customers, but it does not have data on communication between pairs of non-customers. Another example would be a massive online social network may get some information about the contacts of its users with a fringe set comprised of of people not on the system-often including entire countries that do not participate in the platform-but generally not about the interactions taking place in this fringe set. Similarly, Internet measurements at the IP-layer from individual service providers give only a partial view of the global Internet network [58,60]. Therefore, graph data often takes the form illustrated in Figure 1a: the nodes are divided into a core and a fringe, and we only see edges that involve a member of the core. Thus, the core set is a vertex cover of the underlying graph-since a vertex cover, by definition, is a vertex set incident to all edges.
Often, a graph dataset is annotated with metadata about which nodes belong to the core, and this is crucial for correctly interpreting the data. But there are a number of important contexts where this metadata is unavailable, and we do not know which nodes constitute the core. In other words, at some point, we have "found data" that has core-fringe structure, but the labels identifying the core nodes are missing. One reason for this scenario is that metadata is lost over time for a wide range of reasons: data is repeatedly shared and manipulated, managers of datasets change jobs, and hard drives are decommissioned. This is a central theme in data provenance, lineage, and preservation [11,44,56,59], an especially challenging issue in modern digitization efforts [38] and large-scale data management [32]. A concrete example is the following. Suppose a telecommunications company shares an anonymized dataset of telephone call records with a university but does not include on which nodes were customers and which were the fringe set of non-customers. By the time a student begins to analyze the data and realizes the metadata is missing and important for analysis, the researchers at the company who assembled the dataset have left. At this point, there is no easy way to reconstruct the metadata.
These issues also arise in current research on counter-surveillance [29]. Intelligence agencies may intercept data from adversaries conducting surveillance and build a graph to determine which communications the adversaries were recording. In different settings, activist groups may petition for the release of surveillance data by governments or infer it from other sources [47]. In these cases, the "found data" consists of a communication graph in which an unknown core subset of the nodes was observed, and the remainder of the nodes in the graph (the fringe) are there simply because they communicated with someone in the core. In such situations, there generally is not any annotation to distinguish the core from the fringe. In this case, the core nodes are the compromised ones, and identifying the core from the data can help to warn the vulnerable parties, hide future communications, or disseminate misinformation.
Planted Vertex Covers. Here we study the problem of recovering the set of core nodes in found graph data, motivated by the range of settings above. We can view this as a planted vertex cover problem: we are given a graph G in which an adversary knows a specific vertex cover C. We do not know C, but we want to output a set that is as close to C as possible. The property of being "close" to C corresponds to a performance guarantee that we will formulate in different ways. We may want to output a set not much larger than C that is guaranteed to completely contain it, or we may want a small set that is guaranteed to have substantial overlap with C. A simple instance of the task is depicted in Figure 1b, after the explicit labeling of the core nodes has been removed from Figure 1a.
Generically, planted problems arise when some hidden structure (like the vertex cover in our case) has been "planted" in a larger input, and the goal is to find the structure. Planted problems are typically core fringe (a) Graph dataset built from a small core.
(b) The dataset without the core labeled. The resulting data contains these core nodes together with a potentially much larger fringe, consisting of all other nodes that had an interaction with some member of the core. (b) Knowing which nodes are in the core is important for interpreting the dataset. However, in many cases, the graph is "found data" and this metadata is not available. This can arise from challenges in data provenance that lead to the loss of the metadata or when only partial information is available in contexts such as counter-surveillance. We study how accurately we can recover the core, despite limited information about how the dataset was constructed. Algorithmically, this leads to a planted vertex cover problem.
based on formal frameworks in which the input is generated by a highly structured probabilistic model. This is true in some of the most studied planted problems, such as the planted clique problem [5,20,25,46] and the recovery problem for stochastic block models [1,2,3,7,19,48].
It might seem inevitable that planted problems should require such probabilistic assumptions-how else could an algorithm guess which part of the graph corresponds to the planted structure, if there are no assumptions on what the "non-planted" part of the graph looks like? But the vertex cover problem turns out to be different, and it is possible to solve what, surprisingly, can be described as a "worstcase" planted problem. With extremely limited assumptions, we can provide provable guarantees for approximately recovering unknown vertex covers. Specifically, we make only two assumptions (both necessary in some form, though relaxable): (i) the planted vertex cover is inclusion-wise minimal and (ii) its size is upper-bounded by a known quantity k. We think of k as small compared to the total number of nodes, which is consistent with many datasets. Among other results, we use fixed-parameter tractability to show that there is an algorithm operating on arbitrary graphs that can output a set of f (k) nodes (independent of the size of the graph) guaranteed to contain the planted vertex cover. We obtain further bounds with additional structural assumptions.
We pair our theoretical guarantees with experiments showing the efficacy of these methods on realworld networks with core-fringe structure. Using our theory, we develop a natural heuristic based on constructing a union of minimal vertex covers from random initializations. Our simple algorithm provides superior recovery performance and running time compared to a number of competitive baselines. Among these, we show improvements over a line of well-developed heuristics for detecting core-periphery structure [15,31,53,63]-a sociological notion in which a network has a dense core and a sparser periphery, generally for reasons of differential status rather than measurement effects.
This paper considers graphs that are partitioned into two separate components called a "core" (C) and a "fringe" (F), where all edges have at least one endpoint in C. The partitioning is supposed to be unknown, and the goal of the paper is to come up with an algorithm that recovers it.  Strong aspects of the paper are:  - The paper first performs a theoretical analysis, deriving various conditions for the set C and upper and lower bounds for its size under various assumptions. Based on the theory, it then proposes an algorithm for recovering C, and shows that this algorithm outperforms various baselines.  - The theoretical analysis is sound and intuitive. The theory nicely and naturally leads to an algorithm that is indeed surprisingly simple and performs well empirically.   - The paper is very well written, the motivation is clearly explained, the theoretical results are clearly structured and nicely followable. Most of the theoretical results are relatively straightforward with a few exceptions (Lemma 3 and Theorem 2).   - The considered problem is a special case of "core-periphery" structure identification in networks (e.g. Zhang et al, 2015, Phys Review E; their ref 61). The contribution rests on the assumption that no links at all exist between nodes in the periphery. This assumption leads to an elegant algorithm that can convincingly outperform others when the assumption holds. The techniques are potentially of interest for other researchers in the networks field.  I have three main criticisms:  - The restriction to no edges in F seems very restrictive and I am not convinced by the given examples that this is a problem that would be of interest in the real world. In the used datasets, the authors dropped the available labels of C and F. In the motivation given in the beginning, they say that in some applications these labels are lost, but can they point to any real datasets that have this property or even better, analyze one and present the results?   - It is perhaps not too surprising that the proposed algorithm outperforms baseline given that the baseline methods were developed for a more general scenario and thus do not take advantage of the absence of edges in F. It would have been interesting to see how the proposed algorithm would fare on more general community structure detection problems such as the blog link data shown in Zhang et al, 2015, Phys Review E. Would the algorithm still provide a useful approximation in this case, or would it break down completely?  - Finally, I wonder if NIPS is the appropriate avenue for this research area. Networks has traditionally not been a topic with a very strong presence at NIPS as far as I am aware, and more appropriate venues that would might to mind would be KDD, ICDM. A possible sign of mismatch is that the paper does not cite any paper from NIPS or any other ML/AI venue. Perhaps the authors can explain in their rebuttal why they chose this specific conference?   Minor points: - I would have liked to see more justification for the choice of the number of clusters to be generated (fixed here at 300). Is this important for performance? Under which circumstances would I need to set this higher or lower?  - It was hard to compare the numbers in Table 2 due to different units, can this be standardized to one unit or alternatively, can at least the fastest algorithm be highlighted in each row?   = After author response =  The authors have presented some new data that addresses my second criticism and since no other reviewer had any doubts about suitability for NIPS, I retract that criticism as well. The authors cite personal communications as evidence for the importance of the considered setting, which still does not convince me entirely. However, I am satisfied with the response overall and will revise my score up by 1.