Esteban Bautista
email: estbautista@gmail.com

Laurent Brisson
email: laurent.brisson@imt-atlantique.fr

Cécile Bothorel
email: cecile.bothorel@imt-atlantique.fr

Grégory Smits
email: gregory.smits@imt-atlantique.fr

MAD: Multi

Keywords: Scale Anomaly Detection in Link Streams Information systems → Data stream mining, Social networks, Traffic analysis, • Security and privacy → Intrusion/anomaly detection and malware mitigation anomaly detection, temporal networks, model interpretability

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INTRODUCTION

A link stream is a set of triplets (𝑡, 𝑢, 𝑣) modeling that 𝑢 and 𝑣 interacted at time 𝑡. Triplets in a link stream may represent that computer 𝑢 sent a packet to computer 𝑣 at time 𝑡 or that bank account 𝑢 made a transaction to account 𝑣 at time 𝑡. Detection of likely/unlikely interactions that suddenly disappear/appear is an important step towards identifying various events of crucial interest, such as financial frauds, network attacks, or infrastructure failures. For example, consider the case illustrated in Figure 1, depicting interactions between servers and users requesting their services. It is rather normal that the servers frequently exchange traffic between them and also with some regular users. Figure 1 represents such users and servers that frequently interact in blue, while it represents users that connect less frequently in grey. If at a given time the communication pattern depicted in the left panel is observed, the situation can be labeled as normal given that the observed interactions only concern entities that usually interact together. Yet, if the observation corresponds to the one depicted in the right panel, such change in the communication pattern may be an indication of a failure or an attack. Another example can be a bank account that suddenly starts to make transactions to several unexpected accounts. Such behavior may be indicative of a fraud.

To spot the aforementioned events, numerous link stream-based anomaly detection algorithms have been proposed in recent years. In a nutshell, such algorithms can be seen as black boxes that receive two inputs, a query and a context, and answer to the question: how abnormal is the query given the context? The essential differences between the existing algorithms are: (i) the types of queries they accept; (ii) the definition given to the notion of anomaly; and (iii) how the context is exploited. For example, numerous algorithms accept time-stamped edges as queries, yet they differ in the criterion used to label a query as abnormal: some do it when the query implies a sudden change in edge counts [START_REF] Bhatia | Midas: Microcluster-based detector of anomalies in edge streams[END_REF], node embeddings [START_REF] Chang | F-fade: Frequency factorization for anomaly detection in edge streams[END_REF], or walk statistics [START_REF] Eswaran | Sedanspot: Detecting anomalies in edge streams[END_REF], while others do it when the query cannot be well predicted from the past [START_REF] Paudel | Pikachu: Temporal walk based dynamic graph embedding for network anomaly detection[END_REF][START_REF] Fang | Anonymous edge representation for inductive anomaly detection in dynamic bipartite graph[END_REF]. Algorithms addressing coarser resolution A group of servers usually exchange between them and with some external users. Historical data may suggest that interactions between blue-colored nodes are highly likely. The sudden halt of likely traffic and the emergence of unlikely one may be an indication of an attack (hackers have taken control of the machines) or a failure (engineers are troubleshooting).

queries, like time-stamped nodes [START_REF] Guo | Subset node anomaly tracking over large dynamic graphs[END_REF][START_REF] Heard | Bayesian anomaly detection methods for social networks[END_REF] or entire graph snapshots [START_REF] Yoon | Fast and accurate anomaly detection in dynamic graphs with a two-pronged approach[END_REF][START_REF] Eswaran | Spotlight: Detecting anomalies in streaming graphs[END_REF][START_REF] Huang | Laplacian change point detection for dynamic graphs[END_REF][START_REF] Mcneil | Cadence: Communityaware detection of dynamic network states[END_REF], have also been proposed. These algorithms also vary in the way they define an anomaly and use the context. Namely, nodes may be deemed abnormal if they suddenly change their centrality [START_REF] Guo | Subset node anomaly tracking over large dynamic graphs[END_REF] or communication counts [START_REF] Heard | Bayesian anomaly detection methods for social networks[END_REF], while graphs may be considered abnormal if they have sudden densifications [START_REF] Eswaran | Spotlight: Detecting anomalies in streaming graphs[END_REF], spectrum changes [START_REF] Huang | Laplacian change point detection for dynamic graphs[END_REF], or community re-configurations [START_REF] Mcneil | Cadence: Communityaware detection of dynamic network states[END_REF], to list some examples.

In spite of numerous successes, existing anomaly detection algorithms remain not fully satisfactory. In particular, the fact that they only accept time-stamped subgraphs of a specific form as queries (either edges, nodes, or entire graphs) makes them too rigid for several real-world use cases. In many situations it is desirable to question an algorithm if an arbitrary communication topology behaves abnormally: take for instance the example of Figure 1, where one wants to track the communications between a specific group of servers and users; or take the case when transactions between a specific group of bank accounts suspected to belong to a criminal organization have to be monitored. To handle these different cases, an ultimate solution is to dispose of an algorithm capable to respond to queries consisting of arbitrary time-stamped subgraphs. Some algorithms for anomalous subgraph detection have been proposed [START_REF] Fernandes | Wintended: Windowed tensor decomposition for densification event detection in time-evolving networks[END_REF][START_REF] Jiang | Two-stage anomaly detection algorithm via dynamic community evolution in temporal graph[END_REF][START_REF] Tasnádi | A new type of anomaly detection problem in dynamic graphs: An ant colony optimization approach[END_REF], yet such approaches automatically search for subgraphs that meet some criteria, like being a dense community, thus preventing users from querying the algorithms with arbitrary subgraphs. It is also worth noticing that most algorithms targeting coarse-grain queries rely on anomaly definitions that do not satisfactorily account for the highly uncertain and sparse nature of temporal interactions. For example, many works rely on anomaly definitions that are based on changes of node centrality, walk statistics, spectral properties or community structures. These definitions implicitly assume dense link streams that slowly evolve, which may be unrealistic in numerous situations.

The aim of this work is to address the limitations listed above. We introduce MAD, a novel anomaly detection algorithm that (i) accepts arbitrary time-stamped subgraphs as queries; and (ii) labels queries as abnormal if they have/lack many unlikely/likely interactions, thus allowing to handle the data uncertainty and sparsity. MAD is based on a novel multi-scale probabilistic analysis for subgraphs that essentially permits to map a query sub-graph into a set of random variables from which it is possible to identify the scale(s) at which a queried sub-graph cannot be well explained from its past activity. Properties of the proposed contribution are as follows:

• Flexibility: MAD can be used to determine if any arbitrary timestamped subgraph is anomalous, thus making edge, node and graph anomaly detection particular instances of our approach. • Interpretability: the developed multi-scale analysis allows to identify the scale(s) and the nature of the event(s) making a query abnormal. Strictly speaking, the interactions of a slice remain time-stamped, yet in some situations it is useful to strip the time reference from them so that they can be considered as the edges of a graph, that is independent of time. Therefore, given the restriction sets 𝑡 1 = 𝑡 2 = 𝑡 and 𝜙, we let 𝐿(𝑡 : 𝑡, 𝜙) refer to the slice of 𝐿 in which interactions remain time-stamped, while we let 𝐿(𝑡, 𝜙) denote the the case in which the time-stamps are striped-out. Relations are considered directed, hence (𝑢, 𝑣) ≠ (𝑣, 𝑢). Moreover, for the sake of notation lightness, the relations emerging from node 𝑢 are denoted by

E 𝑢 = {(𝑢, 𝑣) : 𝑣 ∈ 𝑉 }.
In this work, we extensively use indicator functions to characterize subsets of a set: a binary function indicating which elements from the set belong or not to the subset. The concept plays an important role in this work given that algorithms working directly with a set 𝐴 ⊆ 𝐵 cannot make decisions based on the elements of 𝐵 not included in 𝐴, as they ignore them; while algorithms working with the indicator function know such information. Formally, the indicator function of 𝐴 ⊆ 𝐵 is denoted by 1 𝐵 𝐴 : 𝐵 → {0, 1}, where 1 𝐵 𝐴 (𝑥) = 1 if 𝑥 ∈ 𝐴 and zero otherwise. Thus, the indicator function of a link stream is given by 1 𝑇 × E 𝐿 (𝑡, 𝑢, 𝑣) = 1 if (𝑡, 𝑢, 𝑣) ∈ 𝐿 and zero otherwise. Also, with the aim of notation lightness, we denote the sum of a function 𝑓 : 𝐵 → R over a sub-domain 𝐶 ⊆ 𝐵 by 𝑓 (𝐶) = 𝑐 ∈𝐶 𝑓 (𝑐). Hence, 1 𝐵 𝐴 (𝐶) = 𝑐 ∈𝐶 1 𝐵 𝐴 (𝑐) for 𝐶 ⊆ 𝐵. Lastly, Q refers to a query and H to a historical context. The nature of Q and H depends on the considered algorithm, as detailed next.

Related Works

The ultimate algorithm for link stream-based anomaly detection is one that receives two arbitrary sub-link streams as inputs, constituting a query Q and a context H , and that responds to the question: can the query be explained from the provided context? See Figure 2 for an illustration. No algorithm is able to explore all the possible ways in which an arbitrary query may be explained from an arbitrary context. Thus, existing algorithms essentially narrow the search by (i) focusing on queries and contexts that adhere to a specific form; and (ii) establishing a specific normality criterion that the query must possess in order to be considered as explained by the context. As a result, there is a rich variety of approaches that focus on different combinations of inputs and normality definitions. We briefly review proposed algorithms, structured according to types of queries they handle. Table 1 presents a summary.

Edge anomaly. Algorithms in this category respond to queries consisting of time-stamped edges. The difference between them lies in the criteria employed to consider a query as abnormal. Namely, MIDAS [START_REF] Bhatia | Midas: Microcluster-based detector of anomalies in edge streams[END_REF] uses the history of the query edge to predict its future activity. It labels the query as abnormal if it appears in a period predicted to be of low activity. F-FADE [START_REF] Chang | F-fade: Frequency factorization for anomaly detection in edge streams[END_REF] uses historical interactions to compute a node embedding that explains edge frequencies. It considers a query to be abnormal if the observed distance between edge ends is implausible. SEDANSPOT [START_REF] Eswaran | Sedanspot: Detecting anomalies in edge streams[END_REF] uses the past to estimate a graph of stable communities. The query is abnormal if its inclusion to such graph breaks random walks statistics. AER-AD [START_REF] Fang | Anonymous edge representation for inductive anomaly detection in dynamic bipartite graph[END_REF] uses the past to train a recurrent neural network for link prediction. It labels a query as abnormal if it is not predicted well. PIKACHU [START_REF] Paudel | Pikachu: Temporal walk based dynamic graph embedding for network anomaly detection[END_REF] extracts temporal random walks from historical interactions to train an encoder for link prediction. It labels a query as abnormal if it is not predicted well.

Node anomaly. These algorithms address time-stamped nodes as queries, which can be represented by their set of incident edges or by the indicator function of such set. DYNANOM [START_REF] Guo | Subset node anomaly tracking over large dynamic graphs[END_REF] uses a short term context to monitor the evolution of PageRank scores. It labels a query as abnormal if its PageRank score drastically changed. BADSN [START_REF] Heard | Bayesian anomaly detection methods for social networks[END_REF] uses the historical interactions of the query node to model the probability of observing it with a given degree. It labels the query as abnormal if the observed degree is unlikely according to the model. DSEDN [START_REF] Bansal | Density-based structural embedding for anomaly detection in dynamic networks[END_REF] uses past interactions to train an autoencoder that embeds nodes in a way that stable structures over time form clusters. It labels a query as abnormal if it is an outlier in the embedding space. GEABS [START_REF] Jiao | Generative evolutionary anomaly detection in dynamic networks[END_REF] leverages historical interactions to fit a custom-made generative model that jointly accounts for community structure and node popularity. It labels a query as abnormal if its community membership is unstable according to the

Algorithm Query (Q) Context (H ) MIDAS [1] 𝐿(𝑡, (𝑢, 𝑣)) 𝐿(0 : 𝑡 -1, (𝑢, 𝑣)) F-FADE [2] 𝐿(𝑡, (𝑢, 𝑣)) 𝐿(0 : 𝑡 -1, E) SEDANSPOT [3] 𝐿(𝑡, (𝑢, 𝑣)) 𝐿(0 : 𝑡 -1, E) DEGOD [17] 𝐿(𝑡, E 𝑢 ) 𝐿(0 : 𝑡, E) DYNANOM [6] 1 E 𝑢 𝐿 (𝑡,E 𝑢 ) 𝐿(𝑡 -1 : 𝑡, E) DSEDN [15] 1 E 𝑢 𝐿 (𝑡,E 𝑢 ) 𝐿(0 : 𝑡, E) ANOMRANK [8] 1 E 𝐿 (𝑡,E ) 𝐿(𝑡 -2 : 𝑡, E) SPOTLIGHT [9] 𝐿(𝑡, E) 𝐿(0 : 𝑡 -1, E) LAD [10] 1 E 𝐿 (𝑡,E )
𝐿(𝑡 -𝑘 : 𝑡, E) Table 1: Summary of the most representative works proposed in the literature. model. DEGOD [START_REF] Wilmet | Degree-based outliers detection within ip traffic modelled as a link stream[END_REF] uses past interactions to compute the degree distribution of nodes over time. It labels a query as abnormal if it causes the current degree distribution to not match the past.

Graph anomaly. These algorithms accept entire time-stamped slices as queries. In particular, ANOMRANK [START_REF] Yoon | Fast and accurate anomaly detection in dynamic graphs with a two-pronged approach[END_REF] uses a local context to track the evolution of PageRank scores of vertices. It labels a query as abnormal if its first derivatives indicate a drastic change. SPOTLIGHT [START_REF] Eswaran | Spotlight: Detecting anomalies in streaming graphs[END_REF] measures the density of random partitions of historical data to compute a set of reference vectors. It labels a query as abnormal if its corresponding vector is an outlier with respect to the computed references. LAD [START_REF] Huang | Laplacian change point detection for dynamic graphs[END_REF] uses long-term and short-term contexts to predict the spectral shape of the query slice. It labels the query as abnormal if its spectrum is far from the expected values. CADENCE [START_REF] Mcneil | Cadence: Communityaware detection of dynamic network states[END_REF] uses past interactions to identify community structures that are stable over time. It labels a query as abnormal if it implies a community reconfiguration. ODGS [START_REF] Aggarwal | Outlier detection in graph streams[END_REF] uses historical data to fit a community-oriented generative model. A query is abnormal if it contains many inter-community edges.

Problem statement. As it can be seen, proposed algorithms essentially fix a time 𝑡 and a group of relations 𝜙 of a specific form: 𝜙 = (𝑢, 𝑣), 𝜙 = E 𝑢 , or 𝜙 = E and address 𝐿(𝑡, 𝜙) as queries. While this allows to tackle many abnormal events, the fact that 𝜙 must possess a specific form (either edges, nodes, or graphs) drastically limits the flexibility and effectiveness of algorithms in many application scenarios, like the one illustrated in Figure 1 where one may be interested in monitoring an arbitrary group of communications. This calls for an algorithm that allows to set 𝜙 as an arbitrary subgraph. Some algorithms have been proposed to address the case in which 𝜙 is a community clique [START_REF] Fernandes | Wintended: Windowed tensor decomposition for densification event detection in time-evolving networks[END_REF][START_REF] Jiang | Two-stage anomaly detection algorithm via dynamic community evolution in temporal graph[END_REF][START_REF] Tasnádi | A new type of anomaly detection problem in dynamic graphs: An ant colony optimization approach[END_REF]. Yet, such algorithms automatically search for a group of candidate communities, thus preventing users to query the algorithms. Moreover, we stress that the community criterion assumes slices of dense link streams, which is unrealistic in most real-world interaction data. Thus, the aim of this work is to address these two situations: to propose an algorithm that allows to set 𝜙 as an arbitrary subgraph and that does it by taking into account the highly uncertain and sparse nature of interactions.

PROPOSED ALGORITHM

In this section, we introduce MAD, a novel solution to anomaly detection. In addition to addressing arbitrary time-stamped subgraphs as queries, MAD also takes into account that (i) real-world interactions are highly dynamic and uncertain; and (ii) anomalous events can be of different scales. Indeed, it is normal that persons or computers communicating at a time 𝑡 do not communicate at 𝑡 + 1, even though it may not be surprising if it occurs, thus making the data highly dynamic and uncertain. Moreover, a hacker intrusion may reflect at the scale of a few communications, while an infrastructure failure may involve most of them. MAD is a solution to these situations. It is developed as follows. Firstly, Section 3.1 develops a scoring function based on a novel probabilistic and multi-scale analysis of subgraphs. It assumes that interactions are generated by a known random process and defines a set of multi-scale random variables that allow to spot queries deviating from the past in a way that cannot be explained from the usual uncertainty. Section 3.2 addresses the problem of estimating the above random process from the historical context. A stationary test is developed to automatically identify the window length in which the process can be best estimated. In sum, MAD is a multi-scale anomaly detection algorithm that accepts Q = 1 𝜙 𝐿 (𝑡,𝜙 ) and H = 𝐿(𝑡 -𝑁 : 𝑡 -1, 𝜙) as inputs, where 𝜙 is an arbitrary subgraph and 𝑁 is the context duration, and returns an anomaly score denoted by 𝑠𝑐𝑜𝑟𝑒 (Q). H is introduced here as a time window in the interactions that precede the query, but MAD can indifferently consider any time window taken in the history, which is especially useful to capture periodically stationary interaction patterns.

A multi-scale analysis of sub-graphs

Our goal is to assess if the binary state (active or inactive) of a group of relations 𝜙 ⊆ E at time 𝑡 is abnormal. For simplicity, we denote the set of active relations by φ := 𝐿(𝑡, 𝜙) and its binary state function by 1 𝜙 φ := 1 𝜙 𝐿 (𝑡,𝜙 ) . As mentioned above, an important property to take into account concerns the uncertainty relative to each interaction involved in 𝜙. Thus, 𝜙 is interpreted as the result of a random process. In particular, we assume the existence of a function 𝑃 : 𝜙 → [0, 1] so that the state of each 𝑒 𝑖 ∈ 𝜙 is seen as the result of running a Bernoulli trial with probability 𝑃 (𝑒 𝑖 ):

if the trial is successful then 1 𝜙 φ (𝑒 𝑖 ) = 1 and zero otherwise. In the complex networks terminology, this is equivalent to interpret that 1 𝜙 φ is generated by an extended Erdös-Rényi model where the probabilities of edges are individually tuned. Then, we consider an observation 1 𝜙 φ as abnormal if it is unlikely to have been generated by such process. Throughout the rest of this subsection, we assume that 𝑃 is known and that it accurately models normality. In practice, we must estimate 𝑃 from H . Section 3.2 addresses such problem.

Given our assumed probabilistic model, a simple and straightforward way to spot unlikely realizations at the subgraph level consists in computing the exact probability of observing 𝜙, which is given by:

𝑃𝑟 (1 𝜙 φ ) = Π 𝑒 𝑖 ∈ φ 𝑃 (𝑒 𝑖 ) × Π 𝑒 𝑗 ∈𝜙\ φ (1 -𝑃 (𝑒 𝑗 )). (1)
However, while simple, this approach is unsatisfactory for anomaly detection for the following two reasons: (i) small-scale anomalies have little impact in [START_REF] Bhatia | Midas: Microcluster-based detector of anomalies in edge streams[END_REF]; and (ii) expected observations are not ranked as the most probable (hence normal) by [START_REF] Bhatia | Midas: Microcluster-based detector of anomalies in edge streams[END_REF]. Notice that a few abnormal edges may not drive the value of (1) sufficiently low to be considered as a clear anomaly. Moreover, consider a case where 𝜙 = {𝑒 1 , 𝑒 2 , 𝑒 3 } and 𝑃 (𝑒 𝑖 ) = 1/3 for all 𝑒 𝑖 . According to (1), the most probable observation for this setting is the empty subgraph, i.e. when φ = ∅. This is undesirable as the empty subgraph is not the one expected to appear from such process: one success is expected. This raises the question of how to find meaningful anomaly scores that allow to spot either small or large scale anomalies.

Interestingly, an alternative is to use random variables measuring properties of the analyzed subgraphs, like their number of active relations | φ |. The advantage of using random variables is that we can characterize the values they take when they are computed on subgraphs generated by the underlying process. Thus, when rare values are observed, the underlying subgraph can be considered anomalous. The challenge with this approach lies in how to define meaningful random variables that measure properties that can spot all targeted anomalies. For instance, | φ | is a useful random variable that allows to readily spot densification or sparsification events by using its expected value as a reference. However, | φ | alone is not enough to detect all anomalies: an event where 𝑘 likely relations are inactive and 𝑘 unlikely ones are active would be normal under the criterion of | φ |. In the following, we address this challenge by introducing a multi-scale analysis of subgraphs. It defines a group of 𝑀 = |𝜙 | random variables that quantify and compare the activity of the query subgraph at multiple scales. We then characterize their distribution for normal queries in order to spot the scale and group of relations that make an observation anomalous.

Let us begin the development of our multi-scale analysis of 1 𝜙 φ by making two assumptions about its domain. Firstly, we assume 𝑀 = |𝜙 | to be a power of two. If 𝜙 lacks relations for this to hold, then we assume that virtual elements 𝑒 𝑖 of probability 𝑃 (𝑒 𝑖 ) = 0 are added into 𝜙 until the assumption holds. We stress that the inclusion of these virtual relations is of pure mathematical convenience and they do not hinder our analysis as those elements are always switched-off in 1 𝜙 φ , which is in agreement with their null probability. Secondly, we assume that that the elements of 𝜙 are indexed in decreasing order of their probability. It is assumed that

𝑃 (𝑒 1 ) ≥ • • • ≥ 𝑃 (𝑒 𝑖 ) ≥ 𝑃 (𝑒 𝑖+1 ) ≥ • • • ≥ 𝑃 (𝑒 𝑀 ) for all 𝑒 𝑖 ∈ 𝜙.
Based on these assumptions, the first step of our multi-scale analysis consists in recursively partitioning 𝜙 at different resolution scales. To do it, we set an initial set E (0) 0 = 𝜙 that we split in halves according to the probability of its elements: the top-half likely relations are assigned to a set E = 𝑒 𝑖 . For a visual reference, see the binary tree structure displayed in Figure 3, where the root node is E (0) 0 and the children nodes represent the partitioned sets. Algebraically, the partitioning rule is:

E (ℓ ) 𝑘 = E (ℓ+1) 2𝑘 ∪ E (ℓ+1) 2𝑘+1 , (2)
where

E (ℓ ) 𝑘 = 𝑒 𝑖 ∈ 𝜙 : 𝑘𝑀 2 ℓ + 1 ≤ 𝑖 ≤ (𝑘 + 1)𝑀 2 ℓ . (3)
Thus, the procedure above partitions 𝜙 into disjoint subgraphs at different resolutions, as indicated by the super-script ℓ. Particularly, 2 ℓ partitions arise at level ℓ and they satisfy the following crucial property: no relation contained in E (ℓ ) 𝑘+1 is more probable than the relations contained in E (ℓ ) 𝑘 . In the second step of our analysis, we leverage this property by defining random variables that compare the activity of

E (ℓ ) 𝑘 with that of E (ℓ )
𝑘+1 . This is a natural approach to spot anomalies at multiple scales, as we know that, by construction,

E (ℓ )
𝑘 should be more active than E (ℓ ) 𝑘+1 . In precise terms, we define the following set of random variables:

𝑠 = 1 √ 𝑀 1 𝜙 φ (𝜙) , (4)
and

𝑤 (ℓ ) 𝑘 = √ 2 ℓ √ 𝑀 1 𝜙 φ E (ℓ+1) 2𝑘 -1 𝜙 φ E (ℓ+1) 2𝑘+1 . ( 5
)
for all 𝑘 and ℓ. In total, (4) and ( 5) define 𝑀 random variables as there are 2 ℓ sets associated to ℓ and this one runs from 0 to log 2 (𝑀) -1. Therefore, by doing this analysis we do not change the size of the problem: we transition from analyzing the state of 𝑀 relations in

𝜙

φ to 𝑀 random variables. Moreover, it is worth noticing that the random variables can be computed in O (𝑀) using the binary tree shown in Figure 3: by setting 1 𝜙 φ in the leaves, successive parents compare the activity of relations appearing in their left and right branches, producing the desired random variables.

Concerning the analysis of the random variables, notice that, on the one hand 𝑠 corresponds to a normalized version of | φ |, which, as mentioned previously, is relevant to detect densification or sparsification events. And on the other hand, the variables 𝑤 (ℓ ) 𝑘 spot the anomalies not captured by 𝑠. They do it by comparing the activity between E (ℓ )

𝑘 and E (ℓ ) 𝑘+1 , where the former has relations that are more probable to appear than the latter. This way, a group of likely relations in E (ℓ ) 𝑘 suddenly disappearing and a group of less likely ones in E (ℓ ) 𝑘+1 suddenly appearing have a strong impact in 𝑤 (ℓ ) 𝑘 . A natural question that may arise is why [START_REF] Fang | Anonymous edge representation for inductive anomaly detection in dynamic bipartite graph[END_REF] activities between such specific choices of subsets of relations are compared only, given that there are many more ways in which two groups, one with elements more probable than the other, may be envisaged to define similar random variables. Our next result demonstrates that the family defined by ( 4) and ( 5) already contains all the necessary details to discern anomalies, as it does not involve any information loss about 1 𝜙 φ .

Proposition 1. Let 1 𝜙 φ and {𝑠, 𝑤 (ℓ ) 𝑘 } denote a binary state function and its associated set of random variables as defined in (4) and (5), respectively. It holds that:

1 𝜙 φ = 1 √ 𝑀 𝑠1 𝜙 𝜙 + ∑ ℓ,𝑘 √ 2 ℓ √ 𝑀 𝑤 (ℓ ) 𝑘 1 𝜙 E (ℓ +1) 2𝑘 -1 𝜙 E (ℓ +1) 2𝑘+1 . (6)
The interesting connection between this multi-scale analysis and the assumed random process is that the first and second theoretical moments of 𝑠 and 𝑤 (ℓ ) 𝑘 can be expressed in terms of 𝑃. This is a crucial property for anomaly detection as it allows to characterize the ranges of values that 𝑠 and 𝑤

(a) E[𝑠] = 1 √ 𝑀 𝑃 (𝜙), (b) E[𝑤 (ℓ ) 𝑘 ] = √ 2 ℓ √ 𝑀 𝑃 E (ℓ+1) 2𝑘 -𝑃 E (ℓ+1) 2𝑘+1 , (c) 𝜎 2 [𝑠] = 1 𝑀 𝑒 𝑖 ∈𝜙 𝑃 (𝑒 𝑖 ) [1 -𝑃 (𝑒 𝑖 )], (d) 𝜎 2 [𝑤 (ℓ ) 𝑘 ] = 2 ℓ 𝑀 𝑒 𝑖 ∈ E (ℓ ) 𝑘 𝑃 (𝑒 𝑖 ) [1 -𝑃 (𝑒 𝑖 )].
From the Chebyshev inequality, we know that the probability that a random process produces observations of a random variable that are 𝜆 standard deviations away from its expectation cannot be larger than 1/𝜆 2 . Hence, our suspicion about an observation should increase quadratically in the number of standard deviations that its random variables values are away from the mean. Based on this property, we can define an anomaly score for each random variable given as the inverse of its Chebyshev bound. If 𝑥 𝑖 denotes the 𝑖-th random variable, then its anomaly score is given as follows:

score(𝑥 𝑖 ) = (𝑥 𝑖 -E[𝑥 𝑖 ]) 2 /𝜎 2 [𝑥 𝑖 ]. (7)
If we aim to favor interpretability, we can return the 𝑀 anomaly scores above as the output of the algorithm, allowing a user to identify which parts of the query subgraph are at the origin of an anomaly. For simplicity, we return a single anomaly score summarizing the total anomaly level of the query. Yet, we stress that, for a more refined study, it is possible to recompute our multi-scale analysis on the queries that our approach identifies as abnormal. We produce anomaly score for the entire query as follows:

score(1 𝜙 φ ) = (𝑠 -E[𝑠]) 2 𝜎 2 [𝑠] + ∑ ℓ,𝑘 𝑤 (ℓ ) 𝑘 -E[𝑤 (ℓ ) 𝑘 ] 2 𝜎 2 [𝑤 (ℓ ) 𝑘 ] . ( 8
)
Figure 3 provides a comprehensive illustration of our multi-scale approach to anomaly detection. In short, MAD takes as input a history of past interactions H and query Q. It constructs a model 𝑃 from H (see Section 3.2) and uses it to set the ordering of the tree. Then, the tree is used to decompose the query into a set of random variables. MAD also leverages the model to estimate the theoretical moments of the multi-scale random variables. Then, it measures how many standard deviations away from the mean the query is in order to set an anomaly score. While MAD sets equal importance to the different random variables involved in the computation of the anomaly score, making both large-scale or small-scale anomalies equally relevant, we stress that it is possible to favor anomalies at any desired scale by giving more weight to the random variables associated to such level.

Estimation of the Model Probabilities

This section addresses the estimation of the model 𝑃 assumed above from the history H . 𝑃 must represent what we intend by normality. In this work, we assume that normal interactions should be locally stationary. Stationarity means that the underlying random process generating the data remains stable over time. Thus, we assume that there is a single model that produced interactions in the recent past and that, in order to consider the interactions at time 𝑡 as normal, they should also be generated by such model. Our challenge is thus to spot the model that produced interactions in the recent past and use it to define normality at time 𝑡. Notice that if we identify a window in which the interactions are stationary, then we can straightforwardly estimate 𝑃 through a simple time averaging. This is because stationarity means that all the observed states of relation 𝑒 𝑖 over time are samples of the same Bernoulli experiment of probability 𝑃 (𝑒 𝑖 ). Hence, 𝑃 (𝑒 𝑖 ) can be estimated from its time samples as:

𝑃 = 1 𝐾 𝐾 ∑ 𝑘=1 1 𝜙 𝐿 (𝑡 -𝑘,𝜙 ) (9)
where 𝐾 is the length of the stationary window. The challenge of estimating 𝑃 therefore lies in identifying a sub-window of length 𝐾 from the context of length 𝑁 in which all the slices are stationary. Notably, we can leverage our multi-scale analysis to design a simple stationarity test that addresses this challenge.

The idea of our stationarity test is an hypothesis testing one: we hypothesize that the window is stationary and then we try to reject the hypothesis using our multi-scale analysis. Assuming stationary means that all the 𝐾 slices within the window are realizations of a same 𝑃, which can be estimated as in [START_REF] Eswaran | Spotlight: Detecting anomalies in streaming graphs[END_REF]. Then, if we compute one of our random variables across all the slices within the window, we must obtain 𝐾 values that are distributed as predicted by Proposition 2, given that they are realizations of the same 𝑃. One can assess if these 𝐾 values are indeed distributed in such a way by comparing their sample moments with the theoretical ones predicted by Proposition 2. The stationarity hypothesis is therefore rejected if these two distributions differ. This test allows us to automatically explore the 𝑁 -length context to identify the sub-window of size 𝐾 in which the stationarity assumption best holds: we simply run a window backwardly, starting from 𝑡 -1, for all possible values of 𝐾 and run the test for each window, retaining the one in which the distributions best match. The match of distributions is quantified by setting a fitness score given by the sum of squared differences between the sample and theoretical variance for each random variable. 1

NUMERICAL EXPERIMENTS

This section evaluates the performance of MAD through experimentations that aim to address the following questions. Q1. Accuracy: how accurately can MAD detect anomalous events consisting compared to baselines? Q2. Flexibility: can MAD handle equally well queries of varying form? Q3. Interpretability: does MAD allow to characterize the signature of abnormal events? The implementation of MAD and code to reproduce the experiments is available in https://gitlab.imt-atlantique.fr/publications1/mad.

Datasets. One synthetic and three real world datasets are used (Table 2). The synthetic one is composed of a graph sequence with stable community structure but where some edges appear more frequently than others. It is done by fixing a model 𝑃 and generating a sequence of realizations according to the procedure detailed in Section 3.1. The model 𝑃 consists of a heterogeneous stochastic block model: edges within and between communities have different probabilities. See the supplementary for a detailed description of how the model is set. Real datasets are: Hospital [START_REF] Vanhems | Estimating potential infection transmission routes in hospital wards using wearable proximity sensors[END_REF] containing temporal interactions between patients and health-care workers in a hospital ward (slices of 20-seconds); Emails [START_REF] Kunegis | KONECT -The Koblenz Network Collection[END_REF]: the directed network of emails in the 2016 Democratic National Committee email leak (slices of one minute); Traffic [START_REF] Paxson | Wide area traffic: the failure of poisson modeling[END_REF]: two-hours of TCP traffic between the Lawrence Berkeley Laboratory and the rest of the world (slices of one second). In general, these datasets are very dynamic and sparse.

Anomaly injection. There are no known anomalies within the selected datasets. Therefore, the whole datasets are considerd normal and different abnormal events are added: (i) sudden densifications; (ii) sudden sparsifications; and (iii) sudden rewirings. We inject abnormal events according to the type of queries to be assessed (see the supplementary for a full description):

• Edge anomalies. A relation (𝑢, 𝑣) is randomly selected and attacked at various times. Densification attacks make (𝑢, 𝑣) active at times where it is very infrequent, while sparsification attacks suppress (𝑢, 𝑣) at times where it appears frequently. For each dataset, 50 relations are attacked. • Node anomalies. A randomly selected node is attacked at various times with densifications/sparsifications or rewirings.

The former attack injects/suppresses communications emerging from the attacked node, while the latter redirects its communications towards other nodes. Created edges due to attacks always point towards nodes that the attacked node has already communicated with. For each dataset, 10 nodes are attacked over time. Attacks are bounded to 3 edges. • Graph anomalies. Anomalies here concern densification/sparsification or rewiring events applied to link stream slices chosen at random. For each dataset, 1% of its active slices are attacked. Attacks are bounded to 5 edges. Baselines. Six state-of-the-art algorithms form the baselines. Two for edge anomalies: MIDAS [START_REF] Bhatia | Midas: Microcluster-based detector of anomalies in edge streams[END_REF] and F-FADE [START_REF] Chang | F-fade: Frequency factorization for anomaly detection in edge streams[END_REF]. Two for node anomalies: DynAnom [START_REF] Guo | Subset node anomaly tracking over large dynamic graphs[END_REF] and F-FADE-N [START_REF] Chang | F-fade: Frequency factorization for anomaly detection in edge streams[END_REF], the variant of F-FADE proposed by their authors to address node anomalies. Two for graph anomalies: AnomRank [START_REF] Yoon | Fast and accurate anomaly detection in dynamic graphs with a two-pronged approach[END_REF] and LAD [START_REF] Huang | Laplacian change point detection for dynamic graphs[END_REF].

Accuracy of MAD

This subsection aims to address Q1 and Q2 by assessing the accuracy of MAD and baselines, in AUC score, in the tasks of edge, node and graph anomaly detection. For all experiments we tried numerous hyper-parameters configurations and retained the best ones. See the supplementary for our choices of hyper-parameters.

Edge detection. MAD and edge-anomaly baselines are questioned as follows: for each relation (𝑢, 𝑣) that was attacked, we ask algorithms to produce an anomaly score for (𝑢, 𝑣) at all possible timestamps (attacked and non-attacked times). Algorithms must then return high scores for timestamps at which (𝑢, 𝑣) was attacked. Two versions of each dataset are analyzed, one with injected densifications and one with injected sparsifications.

Results are reported in Additionally, MAD does it by just considering a context based on the past activity of the query edge while F-FADE needs to use all the past link stream interactions. Node detection. MAD and baselines are also evaluated in a node detection setting trough a similar experimental setup: algorithms are asked to determine the abnormality of each node 𝑢 that was attacked for all possible timestamps. F-FADE-N is not able to produce an answer for queries with no communications in them, hence its accuracy is assessed on the subset of scores that it is able to produce. Two versions of each dataset are analyzed, one with injected densifications/sparsifications and one with rewiring events.

Table 4 clearly shows that MAD performs very well in the detection of both sparsification/densification and rewiring events. In particular, MAD is able to detect the rewiring events in the real datasets with almost perfect accuracy. Such a performance of MAD is due to the fact that rewiring events in those very sparse datasets essentially replace their few likely edges with unlikely ones only, making the attacked queries extremely inconsistent with the recent past. It can be observed that F-FADE-N performs well in detecting densifications/sparsifications, even though it only produces an output at times when the queried nodes have communications in them. Thus, a massive event that completely shuts-down a node would be missed by F-FADE-N. DynAnom systematically performs very poorly. This poor performance should not come as a surprise: DynAnom computes anomaly scores based on the stability of the PageRank of nodes, which is clearly not a meaningful feature for such sparse and quickly evolving link streams.

Graph detection. The accuracy of MAD and baselines are evaluated in a graph detection setting by feeding the algorithms with the slices of the attacked datasets. Two versions of each dataset are analyzed, one with injected densifications/sparsifications and one with rewiring events.

Results are reported in Table 5. As it can be seen, MAD performs very well in the detection of both events regardless of the dataset, while LAD performs inconsistently and AnomRank very poorly. As in the node case, attacks make likely edges disappear and unlikely ones appear. Therefore, MAD sees those collective events as hard to explain from the context, explaining its good accuracy. LAD is inconsistent as it depends on the stability of eigenvalue distributions, which is not guaranteed when edges are fully replaced between snapshots. AnomRank relies on PageRank, thus it suffers from the same issues of DynAnom.

Interpretability of MAD

In this subsection, Q3 is addressed by studying how the different attacks influence the individual anomaly scores produced by Equation [START_REF] Heard | Bayesian anomaly detection methods for social networks[END_REF]. This study is conducted by (i) taking our model 𝑃 used to generate synthetic data; (ii) generating a normal graph using the model; (iii) applying different types of attacks on this graph; (iv) performing the multi-scale analysis to each of the resulting graphs; and (v) computing the anomaly scores of each random variable.

Figure 4 displays the distribution of anomaly scores for the different types of attacks. The random variables are ordered so that the left-most ones in the plot are the ones associated with the coarsest scales, i.e. 𝑠 and 𝑤 (0) 0 , and the right-most ones are the ones associated to the finest scales. As it can be seen, the normal graph produces low anomaly scores for most random variables: only few fine-scale ones have large scores, which is due to the inherent uncertainty associated to a graph generated at random. When this graph is subject to a densification attack, it can be seen that a large number of random variables immediately yield large scores. Since the likely edges remain present in the graph and the majority of unlikely ones remains inactive in the attacked graph, the large scores mostly appear at fine scales as most of the activity in the graph remains well explained by the model. Yet, notice that 𝑠 immediately activates, thus pointing the densification. Notice that a sparsification attack suppresses most likely edges and this immediately triggers the scores associated to coarse resolutions, particularly 𝑠 and 𝑤 (0) 0 . The random variables 𝑠 and 𝑤 (0) 0 significantly increase because the activity of this graph does not match the expected one for the former, and because the attack mostly affects the left-side of the tree for the latter. Since a rewiring attack is essentially a combination of a sparsification and a densification, one can remark that the anomaly scores of this event combine the signature of the previous two. Thus, in sum, the different attacks produce different signatures in our anomaly scores, paving the way to study the signature of more complex and real events as further research.

CONCLUSION

In this work we introduced MAD, a Multi-scale Anomaly Detection algorithm for link streams that allows to evaluate if any arbitrary time-stamped subgraph is abnormal. Through a numerical evaluation, we demonstrated that MAD performs significantly better than state-of-the-art alternatives, even when the data at hand is very uncertain and sparse, in the tasks of detecting edges, nodes or graphs that were subject to densification, sparsification and redirection attacks. This flexibility and good accuracy of MAD stems from its scoring mechanism, which builds on a novel probabilistic and multi-scale analysis of sub-graphs that allows to decompose them into a set of random variables that capture anomalies at various resolution scales. This makes MAD not only accurate but also inherently interpretable and theoretically sound. The next step concerns the combination of MAD with an anomaly explanation mechanism to assist final users in the analysis of the found anomalies.

Figure 1 :

 1 Figure 1: Examples of normal and abnormal communication patterns. A group of servers usually exchange between them and with some external users. Historical data may suggest that interactions between blue-colored nodes are highly likely. The sudden halt of likely traffic and the emergence of unlikely one may be an indication of an attack (hackers have taken control of the machines) or a failure (engineers are troubleshooting).

Figure 2 :

 2 Figure 2: Unified view of anomaly detection. Algorithms aim to assess the abnormality of a query in a given context.

( 1 ) 0 = 2 }

 102 {𝑒 1 , . . . , 𝑒 𝑀 and the bottom-half ones to a set E (1) 1 = {𝑒 𝑀 2 +1 , . . . , 𝑒 𝑀 }. This recursive partitioning is applied until singletons are obtained E (log 2 (𝑀 ) ) 𝑖

  𝑘 normally take when they are computed on realizations generated by 𝑃. Our next result states this connection. Proposition 2. Let E[•] and 𝜎 2 [•] denote the expectation and variance operators, respectively. If the functions 1 𝜙 φ are drawn from the generative model defined above, it holds that:

Figure 3 :

 3 Figure 3: Schematic representation of the proposed multi-scale approach for anomaly detection.

Figure 4 :

 4 Figure 4: Distribution of anomaly scores across random variables. Different attacks produce different signatures.

Table 2 :

 2 Datasets statistics.

		Synthetic Hospital Emails Traffic
	Triplets	1.24M	32.4K	30.7K	382K
	Nodes	925	75	1646	1622
	Max. time	5K	17.4K	44.5K	7.2K
	Empty slices	0	7.9K	32K	0
	Activity peak 308	20	76	94

Table 3 .

 3 It can be seen that MAD systematically performs well in detecting both anomalies, while MIDAS and F-FADE are inconsistent with densifications and they cannot handle sparsification anomalies. Such inconsistent behavior may be due

			MIDAS F-FADE MAD
		Synthetic	0.49	0.53	0.58
	Densification	Hospital	0.50	0.80	0.82
		Emails	0.73	0.98	0.76
		Traffic	0.52	0.56	0.76
		Synthetic	-	-	0.80
	Sparsification	Hospital	-	-	0.85
		Emails	-	-	0.84
		Traffic	-	-	0.89

Table 3 :

 3 Edge anomaly detection performance in AUC.

			F-FADE-N * DynAnom MAD
		Synthetic	0.52	0.56	0.88
	Densification	Hospital	0.82	0.51	0.92
	&	Emails	0.82	0.54	0.84
	Sparsification	Traffic	0.77	0.51	0.74
		Synthetic	0.53	0.52	0.83
	Rewiring	Hospital	0.57	0.54	0.99
		Emails	0.59	0.51	0.99
		Traffic	0.53	0.54	0.99

Table 4 :

 4 Node anomaly detection performance in AUC.

	to the fact that (i) MIDAS considers global aggregates and hence is
	agnostic to short intervals of low activity; and (ii) F-FADE requires
	a stable embedding to produce accurate frequency estimations and
	is only able to attain it for the datasets that have many empty slices.
	Moreover, we stress that MIDAS and F-FADE cannot respond to
	queries consisting of inactive relations, making them unable to spot
	sparsifications. Notice that MAD solves these two issues by being
	able to spot the two anomaly types and in a consistent manner.

* F-FADE-N is evaluated only on the subset of scores that it is able to produce.

Table 5 :

 5 Graph anomaly detection performance in AUC.

			AnomRank LAD MAD
		Synthetic	0.49	0.50	0.76
	Densification	Hospital	0.51	0.80	0.95
	&	Emails	0.54	0.92	0.98
	Sparsification	Traffic	0.43	0.46	0.77
		Synthetic	0.59	0.52	0.63
	Rewiring	Hospital	0.58	0.77	0.94
		Emails	0.53	0.87	0.87
		Traffic	0.44	0.55	0.85

Only the variance is employed as the fact that 𝑃 is estimated using a sample mean implies that the expectations from Proposition

equal the sample ones.

ACKNOWLEDGEMENTS

The authors would like to thank the Carnot Télécom & Société Numérique institute for the financial support.


CONSPECTUS

The alkylation of enolates is one of the backbones of ketone chemistry, yet in practice it suffers from numerous limitations due to problems of regiochemistry, including O-vs C-alkylation, of multiple alkylations, of self-condensation, of competing elimination, and of incompatibility with many polar groups that have to be protected. Over the years, various solutions have been devised to overcome these difficulties, such as the employment of auxiliary ester or sulfone groups to modify the pKa of the enolizable hydrogens, the passage by the corresponding hydrazones, the use of transition-metal catalyzed redox systems to formally alkylate ketones with alcohols, etc.

Most of these hurdles disappear upon switching to a-ketonyl radicals. Radicals are tolerant of most polar functions, and radical additions to flat sp 2 centers is generally easier to accomplish than enolate substitution at tetrahedral sp 3 carbons. The main stumbling block, however, has been a lack of generally applicable methods for the generation and intermolecular capture of aketonyl radicals. We have found over the past years that the degenerative exchange of xanthates represents in many ways an ideal solution to this problem. It overcomes essentially all the difficulties faced by other radical processes, because of its unique ability to reversibly store reactive radicals in a dormant, non-reactive form. The lifetime of the radicals can therefore be significantly enhanced, even in the concentrated medium needed for bimolecular additions, while at the same time regulating their absolute and relative concentrations. The ability to perform intermolecular additions to highly functionalized alkene partners opens up numerous possibilities for the rapid and convergent access to complex structures. Of particular importance is the elaboration of ketones that are prone to self-condensation, such trifluoroacetone, and of base sensitive ketones, such as chloro-and dichloro-acetone, since the products can be used for the synthesis of a myriad fluorinated and heteroaromatic compounds of relevance to medicinal chemistry and agro-chemistry. The formal distal dialkylation of ketones, also of utmost synthetic interest, is readily accomplished, allowing a convenient access to a wide array of useful ketone building blocks. Cascade processes can be implemented and, in alliance with powerful classical reactions (aldol, alkylative Birch reductions, etc.), furnish a quick route to complex polycyclic scaffolds. Furthermore, the presence of the xanthate group in the adducts can be exploited to obtain a variety of arenes and heteroarenes, such as pyrroles, thiophenes, naphthalenes, and pyridines, as well as enones, dienes and cyclopropanes. Last but not least, the reagents and most of the starting materials are exceedingly cheap, and the reactions are safe and easy to scale up. alkylation is indeed a case in point, being complicated by issues of regiochemistry, of multiple alkylations, of self-condensations, etc. (Scheme 1). [START_REF] Caine | Comprehensive Organic Synthesis: Carbon-Carbon σ-Bond Formation[END_REF] There is further little tolerance for unprotected polar functional groups, and various side reactions are caused by the basicity of the enolate.

INTRODUCTION AND CONTEXT

Scheme 1. Side-Products in the Alkylation of Ketones

Some of these difficulties have been alleviated by a better understanding of the role of the counter-cation, aggregation phenomena, and the solvent; 5 by the installation of temporary auxiliary groups (carboxylates, sulfones) and the complementary regioselective monoalkylation of the corresponding dianions (Hauser's rule); 6 by the use of hydrazones proposed by Corey and Enders. [START_REF] Corey | Herstellung und synthetische Verwendung von metallierten Dimethylhydrazonen Regio-und stereoselektive Alkylierung von Carbonylverbindungen[END_REF][START_REF] For | The SAMP-/RAMP-Hydrazone Methodology in Asymmetric Synthesis[END_REF] Condensation of ketones with aldehydes generated in situ by catalytic oxidation of an

1 R O R O 1) Base 2) R'-X R' R O R' R O R' R O R' + + + + + + R O R' R' R O R' R' R O R' R' R' R O R' R' + Aldol reactions etc. + 2 R O X R' R O R' R O R' Y Y
3, X = H, Cl, Br, I, SO 2 Ar 4 6

5

alcohol followed by reduction of the enone thus formed, usually by the same catalyst through the so-called "borrowed hydrogen" mechanism, is limited to simple alcohols and to ketones where there is no issue of regioselectivity (mostly acetophenones). [START_REF] Huang | C-Alkylation of Ketones and Related Compounds by Alcohols: Transition-Metal-Catalyzed Dehydrogenation[END_REF] There are other, less common strategies too numerous to list. Nevertheless, these methods do not overcome the limitations associated with heavily functionalized ketones and/or alkylating agents. There are also specific cases of ketones that are inherently troublesome, such as cyclobutanones and 1,1,1trifluoropropanone that easily self-condense 10 and a-haloketones that are unstable to base.

a-KETONYL RADICALS

Most of these difficulties would disappear if the cumbersome alkylation could be replaced by the addition of an a-ketonyl radical 4 to an alkene 5 to give an "alkylated" product 6. Unfortunately, most of the recorded additions to ordinary alkenes of a-ketonyl radicals, whether generated by oxidation of enols or enolates with metallic salts (especially Mn III salts), by iodine atom transfer starting from an a-iodoketone, or even by the direct oxidation of ketones, are of the intramolecular category, and extension to the vastly more demanding intermolecular arena has met with only limited success. 11 Intermolecular additions to un-activated alkenes are two to three orders of magnitude slower than typical 5-exo cyclizations, [START_REF] Newcomb | Radical Kinetics and Clocks in Encyclopedia of Radicals in Chemistry, Biology and Materials[END_REF] and conditions that favor them (e.g. high concentration) also promote unwanted side reactions. And radicals have manifold possibilities for mischief. From a broader perspective, the problem of intermolecular creation of carbon-carbon bonds starting with un-activated alkenes remains therefore an important synthetic challenge.

THE XANTHATE TRANSFER REACTION

Three decades ago, [START_REF] Delduc | A Convenient Source of Alkyl and Acyl Radicals[END_REF] we discovered a new process based on the radical exchange of dithiocarbonates (xanthates) and related congeners that is especially adapted to the present problem of generating and capturing a-ketonyl radicals. [START_REF] Quiclet-Sire | Fun with Radicals: Some New Perspectives for Organic Synthesis[END_REF] It has a unique ability to store reactive radicals in a dormant form and extends the lifetime of the radicals even in a concentrated medium, while at the same time regulating their absolute and relative concentrations. [START_REF] Zard | Some Intriguing Mechanistic Aspects of the Radical Chemistry of Xanthates[END_REF] To understand these key properties, a brief look at the simplified mechanism in Scheme 2 is necessary.

Scheme (c) By ensuring that radical R• is more stable than radical 9 (neglecting polar effects), it is possible to modulate their relative concentration in the medium so as to favor the fragmentation of intermediate 10 towards radical R• and addition product 11 and thus limit oligomerization of alkene 5. From a polymer point of view, however, the process can be geared to deliberately generate telechelic polymer 14 and then diblock copolymer 16 by further additions to monomer 15. This is the basis of the extremely powerful RAFT/MADIX technology for the controlled synthesis of block copolymers. [START_REF] Barner-Kowollik | Handbook of RAFT Polymerization[END_REF] Over 8000 publications and 500 patents have appeared on the subject and at least two block copolymers are industrially manufactured on a multi-ton scale by this process (Rhodibloc® by Solvay-Rhodia and Asteric® by Lubrizol). [START_REF] Perrier | 50th Anniversary Perspective: RAFT Polymerization-A User Guide[END_REF] (d) If adduct radical 9 is particularly electron rich, it is possible to oxidize it to cation 17 by electron transfer to the peroxide. This cross-over from the radical to the cationic manifold has important synthetic consequences, in particular in cyclizations and intermolecular additions to (hetero)aromatics. The small price to pay is that the peroxide is now acting as both an initiator and an oxidant and must be employed in stoichiometric quantities. Di-lauroyl peroxide (Laurox ® ) or DLP, is an inexpensive and safe initiator.

MONO-ALKYLATION OF KETONES

The application of xanthates for the indirect, regioselective monoalkylation of ketones is summarized in Scheme 3. The reactions can be run in a highly concentrated medium, typically 1-4 M, in a range of solvents, including water, and even neat, and are easy to scale up. In fact, contrary to common experience, the yield often increases upon scale-up.

Scheme 3. Application of the Xanthate Addition to Ketones

Substitution of halides 3 by a xanthate salt is the most common route to a-xanthyl ketones 18. It is also possible to quench lithium enolates with a bis-xanthate (20 ® 21). This seldom applied variant could prove convenient in cases where the a-halo ketone is inaccessible or unreactive.

Xanthate 21 (meso isomer) is readily converted into bicycle 22 by a 6-endo ring closure, [START_REF] Rodriguez | Stereoselective Synthesis of CD-Ring Precursors of Vitamin D Derivatives[END_REF] and seven-and even eight-membered ring ketones are accessible by a similar strategy. [START_REF] Heng | A flexible, Unified Approach to Polycyclic Structures[END_REF] It is, however, the remarkable ability to mediate intermolecular additions to un-activated alkenes that sets this method apart. A few examples from several hundred are displayed in Scheme 4.

The xanthate and / or the alkene can be complex structures bearing various functional groups, including free hydroxy groups (23), [START_REF] Bacqué | A Flexible Strategy for the Divergent Modification of Pleuromutilin[END_REF] MIDA boronates (25), 21 aminoacids (26), [START_REF] Li | A Convergent Enantioselective Synthesis of Open-Chain, Cyclic, and Fluorinated α-Amino Acids[END_REF] protected hydroxylamines ( 29) and (31) [note that these adducts are derived respectively from O-allyl and N-allyl hydroxylamines)] and hydrazines (33). [START_REF] Quiclet-Sire | A Convergent and Flexible Approach to Hydroxylamines, Hydrazines, and Related Structures[END_REF] Terminal alkenes are the best substrates. With internal alkenes, the addition is slower and complicated by lack of regioselectivity and by abstraction of allylic hydrogens. Strained alkenes, in contrast, such as cyclopropenes,

Scheme 7. Formation of Cyclopropanes

Adducts to vinyl esters and related derivatives are masked 1,4-ketoaldehydes with a vast synthetic potential (Scheme 8). Thus, adduct 60 can be smoothly converted into thiophene 61 [START_REF] Jullien | Convergent Routes to Thiophenes[END_REF] or into pyrrole 62. [START_REF] Quiclet-Sire | A New Practical Variant of the Paal-Knorr Synthesis of Pyrroles[END_REF] Pyrroles 66 and 69 are derived from adducts to Boc-protected azetine 64. [START_REF] Han | A Modular Approach to Substituted Boc-Protected 3-Aminomethylpyrroles[END_REF] Notice the presence of an aminomethyl group in the generally difficult to functionalize position 4 and the possibility of cyclizing this extra-cyclic amine with the aldehyde group in pyrrole 69.

Scheme 8. Formation of Thiophenes and Pyrroles

In the case of (hetero)aromatic a-ketonyl xanthates, the subsequent ring-closure onto the (hetero)aromatic ring can be viewed as the equivalent of the venerable Friedel-Crafts reaction, with the added advantages of mild, neutral experimental conditions and the ability to employ both electron-rich and electron-poor (hetero)arenes. Aza-tetralone 71 [START_REF] Han | A Convergent Route to Substituted Azetidines and to Boc-Protected 4-Aminomethylpyrroles[END_REF] and tricyclic structure

Scheme 9. Addition to (Hetero)Aromatics

Access to tetralones by addition of phenacyl xanthates to alkenes and cyclization proved to be a gold mine. For example, in alliance with the powerful alkylative Birch reduction, complex tricyclic framework 81 can be assembled swiftly from very simple starting materials (Scheme 10). [START_REF] Braun | Applications de la Chimie Radicalaire des Xanthates[END_REF] . The extended lifetime of the intermediate radicals allows exploitation of another mode of attack on aromatic rings, namely the neophyl rearrangement. This is also a reversible process that can be made irreversible by combining the aryl shift with the elimination of a sulfonyl radical (Scheme 13). [START_REF] Georghe | Synthesis of Substituted 3-Arylpiperidines and 3-Arylpyrrolidines by Radical 1,4 and 1,2-Aryl Migrations[END_REF] diketone 116 in 57% yield, after hydrolysis of the regioselectively formed enol silyl ether 115. [START_REF] Ouvry | Convergent Access to Ketones, Vinyl Esters and Vinyl Bromides by a Tin-Free Radical Addition-Intramolecular Hydrogen Atom Tranfer[END_REF] Noteworthy is the ease of generating functional a-aminoketones, known to self-condense to give pyrroles (cf. the biogenesis of porphyrins from aminolevulinic acid).

Scheme 10. An Alliance with the Alkylative Birch Reduction

Scheme 14. A Synthesis of 1,8-Diketone

Protected 1,5-diketone 120 from reaction of xanthate 117 with vinyl dioxolane 118 can be converted into pyridine 122 with the xanthate group remaining intact (Scheme 15). [START_REF] Boivin | An Expedient, Flexible and Convergent Access to Selectively Protected 1,5-Dicarbonyl Compounds. Applications to the Synthesis of 2,6-Disubstituted Pyridines and Thiopyridines[END_REF] The ketal prevents stabilization of the adduct radical by the carbonyl group (as in 119) and the consequent formation of oligomers (cf. 12 in Scheme 2).

Scheme 15. Three Syntheses of Pyridines

Two further totally different routes to 1,5-diketones and to pyridines are depicted in Scheme 15.

In the first, diketones 124 and 125, produced by expulsion of a phosphonyl radical, furnish pyridines 126 and 127. [START_REF] Goh | A Free-Radical Variant for the Synthesis of Functionalized 1,5-Diketones[END_REF] Cyclohexenones 128 should also be accessible through textbook Robinson annulations. The second route takes advantage of the extended lifetime of the radical to force an unusual carbon-carbon bond homolysis to expel a stabilized cumyl radical (see box)

and create a ketone (or an aldehyde) in the process, prior to condensation with ammonium acetate (129®131®132).

ALKYLATION OF TROUBLESOME KETONES

In this section, we shall look more specifically at ketones that are problematic, either because the enolates tend to easily self-condense (cyclobutanones and trifluoroacetone) or because of pronounced sensitivity to base (chloroacetone and 1,1-dichloroacetone).

Only a handful of examples of alkylations of cyclobutanones have been reported, the large majority proceeding through the hydrazone or imine derivatives. Scheme 16). The effect turned out to be spectacular. [START_REF] Heng | Substitution of Chlorocyclobutanones with Xanthates. The Remarkable Effect of Added Base[END_REF] In the presence of just 0.2 equivalent of DABCO, chlorocyclobutanone 142 reacted within 30 minutes to give xanthate 143 in 94% yield.

1,1,1-Trifluoro-2-propanone is another ketone where alkylation is complicated by the low of reactivity of the enolate and the high reactivity of the ketone, a recipe for untoward selfcondensations. An effective remedy through the use of the dimethylhydrazone was published in 2017. 10c Here again, the radical derived from xanthate 144 proved superior to the enolate, as long as precautions are taken to avoid formation of hydrate 145 (Scheme 17). [START_REF] Denieul | Trifluoroacetonyl Radicals: A Versatile Approach to Trifluoromethyl Ketones[END_REF] In contrast to the parent species, the hydrated radical 146 does not enjoy stabilization by conjugation with the carbonyl group and is unable to propagate the chain.

Scheme 17. Synthesis of Trifluoromethyl Ketones

Two examples of complex trifluoromethyl ketones, 148 [START_REF] Denieul | Trifluoroacetonyl Radicals: A Versatile Approach to Trifluoromethyl Ketones[END_REF] and 150, [START_REF] Li | A Convergent Enantioselective Synthesis of Open-Chain, Cyclic, and Fluorinated α-Amino Acids[END_REF] are pictured in Scheme 17.

The latter could be processed into optically pure trifluoromethyl pipecolate 152 with, remarkably, no deleterious interference by the labile hydrogen highlighted in red in protected vinyl glycine 149.

As for chloroacetone and 1,1-dichloroacetone, no normal alkylations appear to have been documented. Only a few aldol reactions and one Michael addition have been recorded with the former. [START_REF] He | L-Proline Amide-Catalyzed Direct Asymmetric Aldol Reaction of Aldehydes with Chloroacetone[END_REF] The radical additions of xanthate 153 proceeded smoothly to give products 155, 51 157, [START_REF] Bergeot | S-(3-Chloro-2-oxo-propyl)-O-ethyl Xanthate: A Linchpin Radical Coupling Agent for the Synthesis of Heterocyclic and Polycyclic compounds[END_REF] and 160, [START_REF] Laot | Fluoroazaindolines by an Uncommon Radical ipso-Substitution of a C-F Bond[END_REF] despite the presence of the ionically sensitive chloromethylketone (Scheme 18).

Adduct 155 was dexanthylated into 156, an apicidin analog, [START_REF] Traore | Flexible Synthesis and Evaluation of Diverse Anti-Apicomplexa Cyclic Peptides[END_REF] whereas condensation of Nacetylthiourea with 157 with furnished aminothiazole 158, [START_REF] Bergeot | S-(3-Chloro-2-oxo-propyl)-O-ethyl Xanthate: A Linchpin Radical Coupling Agent for the Synthesis of Heterocyclic and Polycyclic compounds[END_REF] and trifluoroazaindoline 161 was obtained from adduct 160 by an exceedingly rare homolytic ipso substitution of a fluorine atom. [START_REF] Laot | Fluoroazaindolines by an Uncommon Radical ipso-Substitution of a C-F Bond[END_REF]

Scheme 18. Synthesis of Chloromethyl Ketones

Additions with the even more base sensitive dichloroacetonyl xanthate 163 also proceeded efficiently (examples 164-170 in Scheme 19). [START_REF] Anthore | A Radical Solution to the Alkylation of the Highly Base-Sensitive 1,1-Dichloroacetone. Application to the Synthesis of Z-alkenoates and E, E-dienoates[END_REF] However, the dichloroketone moiety proved too reactive towards tin, silicon, and phosphorus centered radicals, and to ketyl radicals present in the isopropanol/DLP reduction system. Only the triethylborane / water combination discovered by Wood 55 was capable of accomplishing clean dexanthylation (168 ® 171 and 170 ®172).

Scheme 20. Synthesis of (Z)-Enoates and (E),(E)-Dienoates

The formation of (Z) enoates 179 proceeds via an oxyallyl cation 177, according to earlier mechanistic studies by Schamp. oxyallyl cation. Indeed, placing the xanthate group away from the vicinity of the dichloromethyl ketone obviates the need for ozonolysis (169 ® 181 ® 182, Scheme 21). [START_REF] Anthore | A Radical Solution to the Alkylation of the Highly Base-Sensitive 1,1-Dichloroacetone. Application to the Synthesis of Z-alkenoates and E, E-dienoates[END_REF] The phthalimide group stabilizes an adjacent radical and allows in this case a second radical addition to be achieved (180 + 169 ® 181). [START_REF] Quiclet-Sire | The Xanthate Route to Amines, Anilines, and other Nitrogen Compounds[END_REF] Carboxylic acid 185, containing a skipped diene with the sidechain (Z)-alkene, was obtained from the addition-fragmentation product 184. [START_REF] Anthore | A Radical Solution to the Alkylation of the Highly Base-Sensitive 1,1-Dichloroacetone. Application to the Synthesis of Z-alkenoates and E, E-dienoates[END_REF]

Scheme 21. Further Examples of (Z)-Enoates

The present approach expands considerably the pool of available dichloromethyl ketones and opens up numerous possibilities and variations not hitherto easily contemplated. For example, interception of the reactive oxyallyl cation 177 and cyclopropanone 178 by internal nucleophiles could lead to interesting structures, including lactones, lactams, and carbocycles.

DISTAL DIALKYLATION OF KETONES

Xanthate addition also provides a practical solution to the synthesis of a,a'-dialkyl ketones, which are oftentimes troublesome to attain by enolate chemistry. Symmetrical bis-trialkoxysilane 188, a potentially useful crosslinking agent for sol-gel formation, is obtained after reduction of bis-adduct 187 starting with di-xanthate 186 (Scheme 22). [START_REF] Quiclet-Sire | Versatile Route to Functionalized Trialkoxysilanes[END_REF] Aminolysis of bis-adduct 189 and acidification leads to spiro-dithioketal 190. 59b In contrast to their ubiquitous oxygen analogs, spiro-dithioketals are extremely rare (less than half a dozen examples). Cyanohydrin benzoate 270 and dithiane oxide 271 lead to aldehydes upon unmasking, even though they do not correspond to formal "alkylations" in the a-position (Scheme 29). 70

Xanthates 277 and 278 give rise to 1,2-ketoaldehydes and 2,3-diketones, respectively, and the adducts are useful building blocks for numerous heteroaromatics. 71

Scheme 29. Formation of Aldehydes

The slightly provocative title, given tongue in cheek to attract the eye of the reader, not imply that a-ketonyl radicals will eventually displace enolate chemistry. Introducing an amethyl group or accessing aldol type products is relatively easy via enolates but would be tedious to accomplish with the radicals. Nevertheless, the ability to generate and to capture aketonyl radicals intermolecularly enables retrosynthetic disconnections not available with ionic and organometallic chemistry. Besides ketones, xanthates can be used in the same manner to modify numerous other important functional groups, such as esters, lactones, nitriles, amides,

  the equilibrium will usually favor stabilized radicals 8 and 10, active radicals R• and 9 are most of the time reversibly stored in an inactive dormant state. Their absolute steady state concentration remains constantly extremely low, thus curtailing unwanted radical-radical interactions.

Scheme 4 . 29 ,

 429 Scheme 4. Examples of a-Ketonyl Radical Additions

  74[START_REF] Kaoudi | An Expedient Construction of 7-Membered Rings Adjoining Aromatic Systems[END_REF] (NPhth = phthalimide) are two such examples (Scheme 9). The peroxide acts as initiator and as oxidant for the intermediate cyclized radical (e. g., 70), thus restoring the temporarily lost aromaticity. A few instances of Minisci-type intermolecular additions of a-ketonyl xanthates to heteroarenes have been reported (75 ® 76).[START_REF] Osornio | Intermolecular, Oxidative Radical Alkylation of Heteroaromatic Systems under "Tin-Free[END_REF]

Scheme 13 .

 13 Scheme 13. Synthesis of Arylpiperidines

56

 56

Scheme 19 .

 19 Scheme 19. Synthesis of Dichloromethyl Ketones

Scheme 27 .264

 27 Scheme 27. Addition-Fragmentation Route to Ketones

  The Xanthate Route to Ketones. When the Radical is Better than the Enolate. Laboratoire de Synthèse Organique, CNRS UMR 7652 Ecole Polytechnique, 91128 Palaiseau Cedex, France

	Samir Z. Zard*

"

  In the beginning God created the heaven and the earth. And the carbonyl group. No other functional group can rival its versatility and utility" Few organic chemists would dispute this statement by Prof. Philip Kocieński. 1 Within the family of carbonyl derivatives, ketones and aldehydes play a central role in organic synthesis. 2 Not only are they ubiquitous in natural products, they are also frequently key strategic elements in synthetic planning. Yet unsolved problems remain, despite the vast amount of work devoted to their chemistry. "Alkylation of enolates can be difficult to control" is the head of section in the classical textbook by Vollhardt and Schore.

[START_REF] Vollhardt | [END_REF]

Chemo and regioselective (and stereoselective)

2. Mechanism of the Xanthate Addition to Alkenes

	(a) The reaction of R• with its precursor, xanthate 7, is fast, reversible, and degenerate. It leads
	to stabilized radical 8, which is too bulky to react rapidly with other radicals and is unable to
	disproportionate (no b-hydrogens). It can therefore only fragment back to R•, hence its
	degeneracy. The continuous regeneration of R• increases considerably its lifetime in the
	medium, allowing it to undergo reactions that are kinetically slow, in particular bimolecular
	additions to un-activated alkene 5. Adduct radical 9 is in turn reversibly captured by the xanthate
	to give intermediate 10, a dormant species for the same reasons as for radical 8.

Scheme 11. Syntheses of Naphthalenes

  Addition-cyclisation of phenacyl xanthates to vinyl esters shortens considerably the synthesis of hydroxytetralone natural products (e.g., shinanolone, O-methyl asparvenone and norparvulenone).[START_REF] Vargas | Total Synthesis of 10-Norparvulenone and of O-Methylasparvenone Using a Xanthate-Mediated Free Radical Addition-Cyclisation Sequence[END_REF] Naphthalenes with almost any desired substitution pattern can be readily obtained. Some examples are deployed in Scheme 11.[START_REF] Cordero-Vargas | Synthesis of Substituted Naphthalenes from a-Tetralones Generated by a Xanthate Radical Addition-Cyclisation Sequence[END_REF] The pivalate group is sufficiently robust to withstand various modifications prior to the aromatization step While attempting the conversion of a-tetralone oxime 98 into a naphthylacetamide 100 by the Schroeter reaction [path (a), Scheme 12],[START_REF] Schroeter | Oxim-Umlagerungen in der Tetralon-Reihe[END_REF] we were surprised to find that the major product was tricyclic compound 102. In this novel transformation, the oxime behaves as an a-electrophile via common intermediate 99 [path(b)].[START_REF] Quiclet-Sire | Oxime Derivatives as a-Electrophiles. From a-Tetralone Oximes to Tetracyclic Frameworks[END_REF]

						Boc	N	Boc	N
	N 71, 57% DLP PhthN 83 (not isolated) O H OEt N t-Bu OPiv N S SC(=S)OEt O O 70 S HN O (stoichiometric) DLP (stoichiometric) Boc 2,6-lutidine OEt S N OPiv S EtOAc, reflux OEt 65 Me O PhthN (DLP) X DCE, reflux DLP N H Me O NPhth (DLP) DCE-PhCl (1:1) DCE, reflux DLP (stoichiometric) PhCl, reflux O Cl 94, 56% N H t-Bu O O N H Me PhNHNH 2 polyphosphoric acid p-TSA 100 °C reflux 73, 88% 74, 45% DLP (stoichiometric) DCE, reflux N H O H 76, 44% (71% brsm) S S OEt X OPiv toluene reflux N Ac Ph O O 36 84, X = F, 49% 85, X = Cl, 81% Cl 1) BnNH 2 PO(OEt) 2 2) AlCl 3 3) Ac 2 O 93, 75% CO 2 Me S S t-Bu NH O O N H S S O 54, X = F N 82, X = Cl EtO X OH 72 N H O H 75 F 86, 84% MeO 2 C AcOH PyHBr 3 NaH, toluene
							O		OH
								Br	p-TSA	Br
	F 87, 61%	OPiv p-TSA	F	89	OPiv	toluene reflux	F	90, 69% (from 84)
	toluene reflux CO 2 Me	NCCH 2 CO 2 Et K 2 CO 3 acetone then citric acid	O	Li 2 CO 3 , LiBr DMF, 140 °C NH 2	F	OH
	F								CO 2 Et	(from 84) 91, 57% OPiv
	88, 67%				F
						92, 51% (from 84)
	O	S	S	OEt	OAc		O	(stoichiometric) DLP	O
	47			(DLP) DCE, reflux	77, 94%		S	OEt DCE, reflux	78, 45%	OAc
						OAc	S	1) NH 4 OH 2) TsCl / Py
	O				1) K, NH 3 Et 2 O / t-BuOH		O	KOH	O
					2) LiBr				MeOH
	81, 65 % (81% brsm)	3) MeI		80, 95%	79, 93%	OTs

Scheme 12. Oximes as a-Electrophiles

  [START_REF] Debien | From a Remarkable Manifestation of Polar Effects in a Radical Fragmentation to the Convergent Synthesis of Highly Functionalized Ketones[END_REF]

	O 117 S	S	OEt	Ph O DCE, reflux 118 O (DLP)	O	O Ph SC(=S)OEt O 120, 82%
				O	Ph				S	OEt	H 2 O-DCM CF 3 CO 2 H
	O			O				S	AcONH 4	O	O
	t-Bu	119		Ph		N 122, 82% Ph	AcOH, reflux	121, 78%	Ph SC(=S)OEt
			O			O	S		OEt	O
				PO(OEt) 2			S	36	PO(OEt) 2
	EtO 2 C EtO 2 C		n 123, n = 1,2 O	(DLP) DCE, reflux	EtO 2 C EtO 2 C	n	O (EtO) 2 P=O
									N	O	O
										AcONH 4
	EtO 2 C EtO 2 C	128, n = 1,2 n	EtO 2 C EtO 2 C	AcOH, reflux 127, n = 2, 76% 126, n = 1, 81% n	EtO 2 C EtO 2 C 125, n = 2, 80% n 124, n = 1, 80%
				AcO			O	Me	AcO	O
						S	S	HO 130	CF 3	Me
	AcO			H	H OAc	EtO 129			DLP EtOAc, reflux	AcO	H	H OAc	131, 77% O CF 3
	Via								AcONH 4, AcOH reflux
				Me						AcO	Me
	Rsteroid	HO	CF 3					N	CF 3
										H	H
						AcO			OAc	132, 53%

  10a,b In contrast to the enolates, the radicals are well-behaved, as indicated by the formation of adducts 135 and 139, two convenient precursors to cyclohexene 136 and cycloheptene 140, respectively (Scheme 16).[START_REF] Binot | Intermolecular additions of cyclobutanone derived radicals. A convergent, highly efficient access to polycyclic cyclobutane containing structures[END_REF]

	S O	S		O	134	PO(OEt) 2 CO 2 Me	MeO 2 C S t-Bu S O	O PO(OEt) 2 NaH	S O t-Bu	S	CO 2 Me
	t-Bu	O	O	133	(DLP) DCE, reflux		O	O	135, 69% THF, rt	O	O	136, 85% (dr = 1:1)
										PO(OEt) 2
		S		OEt				EtO 2 C			S	OEt	S	OEt
	O	S					PO(OEt) 2	O		S	EtO 2 C	S
						138	CO 2 Et				NaH
						(DLP) DCE, reflux				THF, 0 °C	140, 55%
												(dr = 1:1)
		137								139, 68%
												S
	Scheme 16. Reactions of Cyclobutanes Unlike open chain or unstrained cyclic a-chloro-ketones, a-chloro-cyclobutanones proved unreactive towards substitution by xanthate salts. More success was found starting with the Cl O Cl R Nu Et Et O Et Et Additive DMF, rt O SNa t-Bu Additive: none, yield : 0% Additive: DABCO (0.2 equiv.), yield 94% (0.5 hour) Me Me S O S bromo analogs, but these substitutions were often capricious. O R t-Bu 142 143 141

[START_REF] Binot | Intermolecular additions of cyclobutanone derived radicals. A convergent, highly efficient access to polycyclic cyclobutane containing structures[END_REF]

From scattered observations in the literature, especially by Hassner,

[START_REF] Hassner | Cine Substitution in 2-Oxabicyclo[4.2.0octanones and Subsequent Unusual Rearrangements[END_REF]

it appeared that addition of an organic base into the medium could facilitate the substitution through the formation of an oxyallyl anion (e. g., 141,

Spiroketals by Distal Dialkylation of Acetone For

  the distal dialkylation of higher ketones, it is not necessary to start with a haloacetonyl xanthate for the simple reason that the sequential addition directly from a di-xanthate can be controlled.[START_REF] Anthore-Dalion | A Radical Bidirectional Fragment Coupling Route to Unsymmetrical Ketones[END_REF] To take the case of butanone di-xanthate 203, the secondary radical 204 is 3-4 kcal/mol more stable than primary radical 206 and the exchange of xanthate is much faster than addition of either radical to the alkene, which must now be used as the limiting reagent (Scheme 24). A difference of 3-4 kcal/mol in stability translates into a factor of the order of 200 in the steady-state concentration of the two radicals, in favor of the secondary radical 204. Therefore, just on statistical grounds, the addition will take place preferentially from the more substituted end. The feasibility is demonstrated by the transformation of 203 into 207 and then into 208 and finally reduction into dialkylated ketone 209. Ketones 210-212 are examples that further underscore the functional group tolerance.

	used in the manufacture of novel polyamides (e. g., 191) by polycondensation with typical AcO AcO S OEt S OEt
	O (dr = 1:1) S Me 184, 89% NPhth Et 3 N H 2 O / THF (1:1) rt, 2 hrs Me Scheme 22. Symmetrical Distal Dialkylation of Acetone (DLP) EtOAc, reflux (dr = 1:1) 185, 73% Me Me S EtO S OEt (Z) 183 EtOAc, reflux S Me Me OH (DLP) O skipped diene 163 O Cl Cl S H N O 191 H n N H N 7 7 4 N S TBD = N O Cl O NH NEt 3 MeOH rt, 22 h 169 S OEt S HN O Cl NPhth S OEt S HN O Cl OMe O 181, 52% 182,79% Cl Cl (Z) 180 diamines. 59b O S S OEt Si(OtBu) 3 X Si (DLP) EtOAc, reflux tBuO OtBu OtBu S S OEt O Si tBuO X tBuO tBuO 187, X = -SC(S)OEt, 82% 188, X = H, 70% (Me 3 Si) 3 SiH (AIBN) PhMe: c-C 6 H 12 (1:1) reflux CO 2 Me 8 (DLP) EtOAc, reflux MeO 2 C CO 2 Me O SCSOEt SCSOEt 7 7 186 189, 78% 1) H 2 NCH 2 CH 2 NH 2 2) H 3 O + S S CO 2 Me MeO 2 C 100 °C to 150 °C 190, 52% H 2 N NH 2 TBD (5 mol%) 7 7 5 193, 79% (over 2 steps) Cl EtOCSSK acetone (DLP) DCE, reflux DLP Isopropanol reflux Cl O OAc S O (DLP) DCE, reflux O X OAc X AcO AcO O O HO 1) KOH / MeOH 2) H 2 SO 4 S S S OEt S S EtO 195, X = -SC(S)OEt, 78% 196, X = H, 84% 192 194 197, 72% O O O Me(CH 2 ) 4 O O t-Bu O O O Me(CH 2 ) 4 O O Me(CH 2 ) 4 OH OEt Br Cl AcO AcO 153 Scheme 23. O Cl 198 199 200 (dr = 1:1.2) 201 (dr = 4:1) 202
	Replacing the chlorine by a xanthate in adducts derived from chloroacetonyl xanthate 153
	enables a second radical addition and provides access to unsymmetrically substituted ketones, as
	shown by the sequence 153 ® 192 ® 193 ® 195 (Scheme 23). 61 Reductive removal of both
	xanthates, saponification of the acetates and acidification affords 6,6-spiroacetal 197 in good
	yield. Spiroketals 198-202 of different ring sizes can be made by merely using the appropriate,
	eventually non-racemic, unsaturated (protected) alcohols as the alkene partners. Instead of
	reductive elimination, the xanthate group can be modified to provide more richly decorated
	spiroketals as exemplified by the bromine and the dichlorovinyl appendages in spiroketals 200
	and 201 (note, moreover, the ethoxy substituent in spiroketal 202 hinting to the possibility of
	constructing trispiroketals).

60

Compound 190 can be

Scheme 24. Distal Dialkylation of Butanone In

  this manner, ketones of generic formula 213 can converted into vastly more complex ketones 215, with the examples in Scheme 25 being illustrative.[START_REF] Anthore-Dalion | A Radical Bidirectional Fragment Coupling Route to Unsymmetrical Ketones[END_REF] Interestingly, the ketoester products can be converted into alkynes by nitrosation of the corresponding isoxazolinones (221 ® 224 ®225).Emmons condensations on the other, providing a general access to enones.[START_REF] Corbet | A Highly Conjunctive a-Keto Phosphonate: Application to the Synthesis of Pyridine Alkaloids Xestamines C[END_REF] The short synthesis of xestamine E 232 is one application (Scheme 26). Reagent 227 is in fact the synthetic equivalent to unmanageable enone radical 233. Xanthates 234 and 235 from the laboratory of the late Jean Boivin are two other equivalents to radical 233, with R = R' = H, but only one simple example (238) of enone formation is described in a Master Thesis.[START_REF] Nguyen | Nouvelles Réactions Radicalaires Appliquées à la Formation d'Hétérocycles Azotés[END_REF] Terminal enones of much greater complexity should be accessible by this route. Scheme 27).[START_REF] Debien | A Modular Approach to Saturated and α, β-Unsaturated Ketones[END_REF] The two examples, 252 and 256, involve a-ketonyl xanthates, in keeping with the theme of this Account, even though a ketone is ultimately produced whatever functional groups are borne by the attacking radical.

	O EtOAc, reflux S (DLP) Me Me 207, 82% S NPhth O S S OEt S EtO S PhthN O Me PhthN S EtO OEt S 203 Me 210 O Me O O Cl NPhth Scheme 25. Distal Dialkylation of Ketones H 3 PO 2, Et 3 N (AIBN) dioxane, reflux OAc (DLP) EtOAc, reflux O Me S EtO S PhthN OAc 208, 77% S OAc OEt S 209, 88% O S Me S OEt O Me S S OEt O Me S S OEt 204 205 206 O Me O CO 2 Et EtO 2 C O Me B O O O O N Me SiMe 3 211 212 UNSATURATED KETONES Conjunctive reagent 227 undergoes radical addition on one side and Horner-Wadsworth-O Z 213 O S R R Y Z O R Y Z S S S EtO S OEt 1) (DLP) EtOAc, reflux 2) (DLP) EtOAc, reflux Y reductive dexanthylation S OEt S EtO S 214 215 O PhthN O PhthN O EtO 2 C CO 2 Me CO 2 Me O O AcNH AcO AcO AcO O Me Me NPhth OMe OMe O O p-ClC 6 H 4 NPhth Me 3 Si O EtO 2 C O Me O EtO 2 C EtO 2 C EtO 2 C O EtO 2 C N Boc O p-ClC 6 H 4 O PhthN O O O O EtO 2 C O p-ClC 6 H 4 O N O O 224, 75% O O p-ClC 6 H 4 225, 60% S O OEt S S EtO S CO 2 Et synthetic equivalent to: NH 2 OH. HCl AcONa, EtOH NaNO 2 , FeSO 4 AcOH, H 2 O OH F 3 C F 3 C Ph Ph Ph 216 217 218 219 220 222 223 221 223 (R = CO 2 Et) 226 MISCELLANEOUS ROUTES TO KETONES Two other routes to ketones will be discussed briefly. The first is based on an addition-fragmentation process, whereby an otherwise strong carbon-oxygen bond is broken Me OMe ( ) 8 O P S OEt S O EtO OEt O P O EtO OEt (DLP) DCE, reflux X N ( ) 7 OMe Me Pyridine-3-carboxaldehyde OMe N N ( ) 7 OMe Me N 231, 91% 232, 70% xestamine E homolytically (247 ® 249, N NaH, THF, rt 1) NaBH 4, CeCl 3 .7H 2 O, MeOH, rt O N ( ) 7 Me 2) H 2 , PtO 2, EtOAc, rt

[START_REF] Anthore-Dalion | A Radical Bidirectional Fragment Coupling Route to Unsymmetrical Ketones[END_REF]

In this transformation we discovered some time ago,

[START_REF] Zard | New Syntheses of Alkynes: A Tale of Serendipity and Design[END_REF]

the ferrous sulfate suppresses a competing radical side reaction (incongruous with the present Account!). Overall, di-xanthate 223 with R = CO2Et can be viewed as the synthetic equivalent of diradical 226, an unknown species that would be exceedingly difficult to generate and almost impossible to handle.

Scheme 26. Synthesis of Enones

Lee described the synthesis and radical additions of xanthates 239 and 240 but did not extend the sequence to enones 243 and 244.

66

Interestingly, phosphoranes 242 can be transformed into aketoesters and a-ketoamides 246 by a protocol devised by Wasserman.

[START_REF] Wasserman | Cyanomethylene)phosphoranes as Novel Carbonyl 1,1-Dipole Synthons: An Efficient Synthesis of a-Keto Acids, Esters, and Amides[END_REF]

anilides, lactams, Weinreb amides, thiolactones, imides, (hetero)arenes, as well as combinations thereof, and others that have yet to be scrutinized.

AUTHOR INFORMATION

ACKNOWLEDGMENT

The author expresses his heartfelt thanks to all the co-workers who participated enthusiastically in this work and whose names appear in the references. A particular debt is owed to Dr Béatrice Sire (Ecole Polytechnique), who made important contributions to most of our past and present projects. Financial support by the C. N. R. S., Ecole Polytechnique, ANR, MRT, China Scholarship Council, CONACyt, Sanofi-Aventis, Rhodia, Laboratoires Servier is gratefully acknowledged. This article is dedicated with respect and admiration to Professor Elias J. Corey (Harvard) on the occasion of his 90th birthday.