1 Real networks
One of the first practical studies on graphs can be dated back to the original work of Moreno [51] in the 30s. Since then, there has been a growing interest in graph analysis associated with strong developments in the modelling and the processing of these data. Graphs are now used in many scientific fields. In Biology [54, 2, 7], for instance, metabolic networks can describe pathways of biochemical reactions [41], while in social sciences networks are used to represent relation ties between actors [66, 56, 36, 34]. Other examples include powergrids [71] and the web [75]. Recently, networks have also been considered in other areas such as geography [22] and history [59, 39]. In machine learning, networks are seen as powerful tools to model problems in order to extract information from data and for prediction purposes. This is the object of this paper. For more complete surveys, we refer to [28, 62, 49, 45].
In this section, we introduce notations and highlight properties shared by most real networks. In Section 2, we then consider methods aiming at extracting information from a unique network. We will particularly focus on clustering methods where the goal is to find clusters of vertices. Finally, in Section 3, techniques that take a series of networks into account, where each network is seen as an object, are investigated. In particular, distances and kernels for graphs are discussed.
1.1 Notations
A graph is first characterised by a set of vertices and a set of edges between pairs of vertices. The graph is said to be directed if the pairs in are ordered, undirected otherwise. A graph with self loops is made of vertices which can be connected to themselves. The degree of a vertex is the total number of edges connected to , with self loops counted twice. In most applications, only the presence or absence of an edge is characterised. However, edges can also be weighted by a function for any set . More generally arbitrary labelling functions can be defined on both the vertices and the edges, leading to labelled graphs.
A graph is usually described by an adjacency matrix where is the value associated to the edge between the pair. It is equal to zero in the absence of relationship between the nodes. In the case of binary graphs, the matrix is binary and indicates that the two vertices are connected. If the graph is directed then is symmetric that is for all .
We use interchangeably the vocabulary from graph theory introduced above and a less formal vocabulary in with a graph is called a network and a vertex a node. In general, the network is the real world object while the graph is its mathematical representation, but we have a more relaxed use of the terms.
1.2 Properties
A remarkable characteristic of most real networks is that they share common properties [20, 3, 67, 54]. First, most of them are sparse i.e. the number of edges present in not quadratic in the number of vertices, but linear. Thus, the mean degree remains bounded when increases and the network density, defined as the ratio between the number of existing edges over the number of potential edges, tends to zero. Second, while some vertices of a real network can have few connections or no connection at all with the other vertices, most vertices belong to a single component, so called giant component, where it is always possible to find a path, i.e. a set of adjacent connected edges, connecting any pair of nodes. Nodes can be disconnected from this component, forming significantly smaller components. Finally, we would like to highlight the degree heterogeneity and small world properties. The first property states that few vertices have a lot of links, while most of the vertices have few connections. Therefore, scale free distributions are often considered to model the degrees [6, 15]. The second one indicates that the shortest path from one vertex to another is generally rather small, typically of size .
2 Graph clustering
In order to extract information from a unique graph, unsupervised methods usually look for cluster of vertices sharing similar connection profiles, a particular case of general vertices clustering [63]. They differ in the way they define the topology on top of which clusters are built.
2.1 Community structure
Most graph clustering algorithms aim at uncovering specific types of clusters, so called communities, where there are more edges between vertices of the same community than between vertices of different communities. Thus, communities appear in the form of densely connected clusters of vertices, with sparser connections between groups. They are characterised by the friend of my friend is my friend effect, i.e. a transitivity property, also called assortative mixing effect. Two families of methods for community discovering can be singled out among a vast set of methods [24], depending on wether they maximize a score derived from the modularity score of Girvan and Newman [27] or rely on the latent position cluster model (LPCM) of Handcock, Raftery and Tantrum [34].
2.1.1 Modularity score
A series of community detection algorithms have been proposed (see for instance [27, 52, 53] and the survey [24]). They involve iterative removal of edges from the network to detect communities where candidate edges for removal are chosen according to betweenness measures. All measures rely on the same idea that two communities, by definition, are joined by a few edges and therefore, all paths from vertices in one community to vertices in the other are likely to path along these few edges. Therefore, the number of paths that go along an edge is expected to be larger for inter community edges. For instance, the edge betweenness of an edge does account for the number of shortest paths between all pairs of vertices that run along that edge. Moreover, the random walk betweenness evaluates the expected number of times a random walk would path along the edge, for all pairs of vertices.
The iterative removal of edges produces a dendrogram, describing a hierarchical structure, from a situation where each vertex belongs to a different cluster to the inverse scenario where all vertices are clustered within the same community. The modularity score [27] is then considered to select a particular division of the network into clusters. Denoting the fraction of edges in the network connecting vertices of communities and , as well as , the fraction of edges that connect with vertices of community , the modularity score is given by:
Such a criterion is computed for the different levels in the hierarchy and is chosen such that is maximised.
Rather that building the complete dendrogram, other algorithms have focused on optimising the modularity score directly, as it is beneficial both in computational terms and in the perceived quality of the obtained partitions. A very popular algorithm, the socalled Louvain method [10]
, proceeds by a series of greedy exchanges and merging that turns a fully refined partition into a coarser one that provides a (local) maximum of the modularity. Better solutions can be obtained using more sophisticated heuristics
[55] but maximising the modularity is a NPhard problem [12].Note that modularity approaches have been shown to be asymptotically biased [9]. To tackle this issue, degree corrected methods were introduced in order to take the degrees of nodes into account.
2.1.2 Latent position cluster model
Alternative approaches, looking for clusters of vertices with assortative mixing, usually rely on the LPCM model [34] which is a generalisation of the latent position model (LPM) [36]. In the original LPM model, each vertex is first assumed to be associated with a position in a Euclidean latent space . Each edge between a pair of vertices is then drawn depending on and
. Both maximum likelihood and Markov chain Monte Carlo (MCMC) techniques were considered to estimate the model parameters and the latent positions. The corresponding mapping of the vertices into the Euclidean latent space produces a representation of the network such that nodes which are likely to be connected have similar positions. Note that if the latent space is low dimensional, typically of dimension
, then the representation can be visualised which is feature appreciated by practitioners.The LPM model was extended in order to look for both a representation of the network and a clustering of the vertices. Thus, the corresponding LPCM model assumes that the positions are drawn from a Gaussian mixture model in the latent space such that each Gaussian distribution corresponds to a cluster. A two stage maximum likelihood approach along with a Bayesian MCMC scheme were proposed for inference purposes. Moreover, conditional Bayes factors were considered to estimate the number of clusters from the data. Finally variational bayesian inference is also possible
[61].2.2 Heterogeneous structures
So far, we have discussed methods looking exclusively for communities in networks. Other approaches usually derive from the stochastic block model (SBM) of Nowicki and Snijders [56]. They can also look for communities, but not only.
The SBM models assumes that nodes are spread in unknown clusters and that the probability of a connection between two nodes
anddepends on their corresponding clusters. In practice, a latent vector
is drawn from a multinomial distribution with parameters , where is the proportion of cluster . Therefore, is a binary vector of size with a single , such that indicates that belongs to cluster , otherwise. If is in cluster and in , then the SBM model assumes that there is a probability of a connection between the two nodes. All connection probabilities are characterised by a matrix . Note that a community structure can be defined by setting values for the diagonal terms of to higher values than extra diagonal terms [37]. In practice, because no assumptions are made regarding , the SBM model can take heterogeneous structures into account [18, 42, 44].While generating a network with such a sampling scheme is straightforward, estimating the model parameters and as well as the set of all latent vectors is challenging. One of the key issue is that the posterior distribution of given the adjacency matrix and the model parameters cannot be factorised due to conditional dependency. Therefore, standard optimisation algorithms, such as the expectation maximisation (EM) algorithm, cannot be derived. To tackle this issue variational and stochastic approximations have been proposed. Thus, [18] relied on a variational EM (VEM) algorithm whereas [44] used a variational Bayes EM (VBEM) approach. Alternatively, [56] estimated the posterior distribution of the model parameters and , given , by considering Gibbs sampling.
A even more fondamental question concerns the estimation of the number of clusters present in the data. Unfortunately, since the likelihood is not tractable either, standard model selection criteria, like the Akaike information criterion (AIC) or the Bayesian IC (BIC) cannot be computed. Again, variational along with asymptotic Laplace approximations were derived to obtain approximate model selection criteria [18, 44].
2.3 Extensions
Since the original development of the SBM model, many extensions have been proposed to deal for instance with valued edges [48] or to take into account covariate information [76, 49]. The random subgraph model (RSM) [39] for instance assumes that a partition of the nodes into subgraphs is observed and that the subgraphs are made of (unknown) latent clusters, as in the SBM model, with various mixing proportions. The edges are typed. In parallel, strategies looking for overlapping clusters, where each node can belong to multiple clusters, have been derived. In [1], a vertex belongs a cluster in its relation with a given vertex . Because is involved in multiple relations in the network, it can belong to more than one cluster. In [43]
, the multinomial distribution of the SBM model is replaced with a product of Bernoulli distribution, allowing each vertex to belong to no, one, or several clusters.
In the last few years, a lot of attention has been paid on extending the approaches mentioned previously in order to deal with dynamic networks where nodes and/or edges can evolve through time. The main idea consists in introducing temporal processes, such as hidden Markov model (HMM) or linear dynamic systems
[72, 74, 73]. While models usually focus on modelling the dynamic of networks through the evolution of their latent structures, Heaukulani and Gharamani [35] chose to define how observed social interactions can affect future unobserved latent structures. We would also like to highlight the work of Dubois, Butts, and P. Smyth [21]. Contrary to most dynamic clustering approaches, they considered a non homogeneous Poisson process allowing to deal with a continuous time periods where events, i.e. the creation or removal of an edge, can occur one at a time. Another approach for graph clustering in the continuous time context is provided by [29] which builds a coclustering structure on the vertices of the graph and on the time stamps of the edges.3 Multiple graphs
While a large part of the graph related literature in machine learning targets the case of a single graph, numerous applications lead naturally to data sets made of graphs, that is situations in which each data point is a graph (or consists in several components including at least one graph). This is the case for instance in chemistry where molecules can be represented by undirected labelled graphs (see e.g. [57]) and in biology where the structure of a protein can be represented by a graph that encodes neighborhoods between it fragments as in [11]. In fact, the use of graphs as structured representations of complex data follows a long tradition with early examples appearing in the late seventies [60] and with a tendency to become pervasive in the last decade.
It should be noted that even in the case of a single global graph described in the first part of this paper, it is quite natural to study multiple graphs derived from the global one, in particular via the egocentered approach which is very common in social sciences (see e.g. [26]). The main idea is to extract from a large social network a set of small networks centered on each of the vertices under study. For real world social networks, it is in general the only possible course of action, the whole network being impossible to observe (see e.g. [19] for an example).
When dealing with multiple graphs, one tackles the traditional tasks of machine learning, from unsupervised problems (clustering, frequent patterns analysis, etc.) to supervised ones (classification, regression, etc.). There are two main tendencies in the literature: the design of specialized methods obtained by adapting classical ones to graphs and the use of distances and kernels coupled with generic methods.
3.1 Specialized methods
As graphs are not vector data, classical machine learning techniques do not apply directly. Numerous methods have been adapted in rather specific ways to handle graphs and other non vector data, especially in the neural network community
[32, 17], for instance via recursive neural networks as in [33, 30]. In those approaches, each graph is processed vertex by vertex, by leveraging the structure to build a form of abstract time. The recursive model maintains an implicit knowledge of the vertices already processed by means of its space state neurons.
3.2 Distances and kernels
A somewhat more generic solution consists in building distances (or dissimilarities) between graphs and then in using distances based methods (such as methods based on the socalled relational approach [31]). One difficulty is that graph isomorphism should be accounted for when two graphs are compared: two graphs are isomorphic if they are equal up to a relabeling of their vertices. Any sound dissimilarity/distance between graphs should detect isomorphic graphs. This is however far more complex than expected [23] up to a point that the actual complexity class of the graph isomorphism problem remains unknown (it belongs to the NP class but not to the NPcomplete class, for instance). While exact algorithms appear fast on real world graphs, their worst case complexities are exponential, with a best bound in [47]. In addition, subgraph isomorphism, i.e. determining whether a given graph contains a subgraph that is isomorphic to another graph is NPcomplete.
Nevertheless, numerous exact or approximate algorithms have been defined to try and solve the (sub)graph isomorphism problem (see e.g. [14]). In particular, it has been shown that those problems (and related ones) are special cases of the computation of the graph edit distance [13], a generalization of the string edit distance [46]. The graph edit distance is defined by first introducing edition operations on graph, such as insertion, deletion and substitution of edges and vertices (labels and weights included). Each operation is assigned a numeric cost. The total cost of a series of operations is simply the sum of the individual costs. Then the graph edit distance between two graphs is the cost of the least costly sequence of operations that transforms one of the graph into the other one (see [25] for a survey).
A rather different line of research has provided a set of tools to compare graphs by means of kernels (as in reproducing kernels [4]). Those symmetric positive definite similarity functions allow one to generalize any classical vector space method to non vector data in a straightforward way [64]. Numerous of such kernels have been defined to compare two graphs [69]. Most of them are based on random walks that take place on the product of the two graphs under comparison.
Once a kernel or a distance has been chosen, one can apply any of the kernel methods [65] or of the relational methods [31]
, which gives access to support vector machine, kernel ridge regression and kernel kmeans to cite only a few.
4 Conclusion
This paper has only scraped the surface of the vast literature about graphs in machine learning. Complete area of graph applications in machine learning were ignored.
For instance, it is well know that extracting a neighborhood graph from a classical vector data set is an efficient way to get insights on the topology of the data set. This has led to numerous interesting applications ranging from visualization (as in isomap [68]
and its successors) to semisupervised learning
[8], going through spectral clustering
[70] and exploratory analysis of labelled data sets [5].Another interesting area concerns the socalled relational data framework when a classical data set is augmented with a graph structure: the vertices of the graph are elements of a standard vector space and are thus traditional data points, but they are interconnected via a graph structure (or several ones in complex settings). The challenge consists here in taking into account the graph structure while processing the classical data or viceversa in taking into account the data point descriptions when processing the graph. Among other issues, those two different sources of information can be contradictory for a given task. A typical application of such a framework consists in annotating nodes on social media [38].
While we have presented some temporal extensions of classical graph related problems, we have ignored most of them. For instance, the issue of information propagation on graphs has received a lot of attention [58]. Among other tasks, machine learning can be used e.g. to predict the probability of passing information from one actor to another as in [19].
More generally, the massive spread in the last decade of online social networking has the obvious consequence of generating very large relational data sets. While non vector data have been studied for quite a long time, those new data sets push the complexity one step further by mixing several types of non vector data. Objects under study are now described by complex mixed data (texts, images, etc.) and are related by several networks (friendship, online discussion, etc.). In addition, the temporal dynamic of those data cannot be easily ignored or summarized. It seems therefore that the next set of problems faced by the machine learning community will include graphs in numerous forms, including dynamic ones.
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