Author Archives: Anatol

Network Comparison

Why network comparison?

Many complex systems can be represented as networks, including friendships (e.g. Facebook), the World Wide Web trade relations and biological interactions. For a friendship network, for example, individuals are represented as nodes and an edge between two nodes represents a friendship. The study of networks has thus been a very active area of research in recent years, and, in particular, network comparison has become increasingly relevant. Network comparison, itself, has many wide-ranging applications, for example, comparing protein-protein interaction networks could lead to increased understanding of underlying biological processes. Network comparison can also be used to study the evolution of networks over time and for identifying sudden changes and shocks.

net

An example of a network.

How do we compare networks?

There are numerous methods that can be used to compare networks, including alignment methods, fitting existing models,
global properties such as density of the network, and comparisons based on local structure. As a very simple example, one could base comparisons on a single summary statistic such as the number of triangles in each network. If there was a significant difference between these counts (relative to the number of nodes in each network) then we would conclude that the networks are different; for example, one may be a social network in which triangles are common – “friends of friends are friends”. However, this a very crude approach and is often not helpful to the problem of determining whether the two networks are similar. Real-world networks can be very large, are often deeply inhomogeneous and have multitude of properties, which makes the problem of network comparison very challenging.

A network comparison methodology: Netdis

Here, we describe a recently introduced network comparison methodology. At the heart of this methodology is a topology-based similarity measure between networks, Netdis [1]. The Netdis statistic assigns a value between 0 and 1 (close to 1 for very good matches between networks and close to 0 for similar networks) and, consequently, allows many networks to be compared simultaneously via their Netdis values.

The method

Let us now describe how the Netdis statistic is obtained and used for comparison of the networks G and H with n and m nodes respectively.

For a network G, pick a node i and obtain its two-step ego-network. That is, the network induced by collection of all nodes in G that are connected to i via a path containing at most two edges. By induced we mean that a edge is present in the two-step ego-network of i if and only if it is also present in the original network G. We then count the number of times that various subgraphs occur in the ego-network, which we denote by $latex N_{w,i}(G)$ for subgraph w. For computational reasons, this is typically restricted to subgraphs on 5 or fewer nodes. This processes is repeated for all nodes in G, for fixed k=3,4,5.

  1. Under an appropriately chosen null model, an expected value for the quantities $latex N_{w,i}(G)$ is given, denoted by $latex E_w^i(G)$. We omit some of these details here, but the idea is to centre the quantities $latex N_{w,i}(G)$ to remove background noise from an individual networks.
  2. Under an appropriately chosen null model, an expected value for the quantities $latex N_{w,i}(G)$ is given, denoted by $latex E_w^i(G)$.  We omit some of these details here, but the idea is to centre the quantities $latex N_{w,i}(G)$ to remove background noise from an individual networks.
  3. Calculate:  eq1
  4. To compare networks G and H, define:eq2where A(k) is the set of all subgraphs on k nodes andeq3is a normalising constant that ensures that the statistic $latex netD_2^S(k)$  takes values between -1 and 1.  The corresponding Netdis statistic is: eq4 which now takes values in the interval between 0 and 1.
  5. The pairwise Netdis values from the equation above are then used to build a similarity matrix for all query networks. This can be done for any $latex k \geq3$, but for computational reasons, this typically needs to be limited to $latex k\leq5$. Note that for     k=3,4,5 we obtain three different distance matrices.
  6. The performance of Netdis can be assessed by comparing the nearest neighbour    assignments of networks according to Netdis with a ‘ground truth’ or ‘reference’ clustering. A  network is said to have a correct nearest neighbour whenever its nearest neighbour according to Netdis is in the same cluster as the network itself.  The overall performance of   Netdis on a given data set can then be quantified using the nearest neighbour score (NN),   which for a given set of networks is defined to be the fraction of networks that are assigned correct nearest neighbours by Netdis.
tree

The phylogenetic tree obtained by Netdis for protein interaction networks. The tree agrees with the currently accepted phylogeny between these species.

Why Netdis?

The Netdis methodology has been shown to be effective at correctly clustering networks from a variety of data sets, including both model networks and real world networks, such Facebook. In particular, the methodology allowed for the correct phylogenetic tree for five species (human, yeast, fly, hpylori and ecoli) to be obtained from a Netdis comparison of their protein-protein interaction networks. Desirable properties of the Netdis methodology are the following:

\item The statistic is based on counts of small subgraphs (for example triangles) in local neighbourhoods of nodes. By taking into account a variety of subgraphs, we capture the topology more effectively than by just considering a single summary statistic (such as number of triangles). Also, by considering local neighbourhoods, rather than global summaries, we can often deal more effectively with inhomogeneous graphs.

  • The Netdis statistic contains a centring by subtracting background expectations from a null model. This ensures that the statistic is not dominating by noise from individual networks.
  • The statistic also contains a rescaling to ensure that counts of certain commonly represented subgraphs do not dominate the statistic. This also allows for effective comparison even when the networks we are comparing have a different number of nodes.
  • The statistic is normalised to take values between 0 and 1 (close to 1 for very good matches between networks and close to 0 for similar networks). The statistic gives values between 0 and 1 and based on this number, we can simultaneously compare many networks; networks with Netdis value close to one can be clustered together. This offers the possibility of network phylogeny reconstruction.
A new variant of Netdis: subsampling
sampling

The performance of Netdis under subsampling for a data set consisting of protein interaction networks. The performance of Netdis starts to deteriorate significantly only after less than 10% of ego networks are sampled.

Despite the power of Netdis as an effective network comparison method, like many other network comparison methods, it can become computationally expensive for large networks. In such situations the following variant of Netdis may be preferable (see [2]). This variant works by only querying a small subsample of the nodes in each network. An analogous Netdis statistic is then computed based on subgraph counts in the two-step ego networks of the sampled nodes. From numerous simulation studies and experimentations, it has been shown that this statistic based on subsampling is almost as effective as Netdis provided that at least 5 percent of the nodes in each network are sampled, with the new statistic only really dropping off significantly when fewer than 1 percent of nodes are sampled. Remarkably, this procedure works well for inhomogeneous real-world networks, and not just for networks realised from classical homogeneous random graphs, in which case one would not be surprised that the procedure works.

Other network comparison methods

Finally, we note that Netdis is one of many network comparison methodologies present in the literature Other popular network comparison methodologies include GCD [3], GDDA [4], GHOST [5], MI-Graal [6] and NETAL [7].

[1]  Ali W., Rito, T., Reinert, G., Sun, F. and Deane, C. M. Alignment-free protein
interaction network comparison. Bioinformatics 30 (2014), pp. i430–i437.

[2] Ali, W., Wegner, A. E., Gaunt, R. E., Deane, C. M. and Reinert, G. Comparison of
large networks with sub-sampling strategies. Submitted, 2015.

[3] Yaveroglu, O. N., Malod-Dognin, N., Davis, D., Levnajic, Z., Janjic, V., Karapandza,
R., Stojmirovic, A. and Prˇzulj, N. Revealing the hidden language of complex networks. Scientific Reports 4 Article number: 4547, (2014)

[4] Przulj, N. Biological network comparison using graphlet degree distribution. Bioinformatics 23 (2007), pp. e177–e183.

[5] Patro, R. and Kingsford, C. Global network alignment using multiscale spectral
signatures. Bioinformatics 28 (2012), pp. 3105–3114.

[6] Kuchaiev, O. and Przulj, N. Integrative network alignment reveals large regions of
global network similarity in yeast and human. Bioinformatics 27 (2011), pp. 1390–
1396.

[7] Neyshabur, B., Khadem, A., Hashemifar, S. and Arab, S. S. NETAL: a new graph-
based method for global alignment of protein–protein interaction networks. Bioinformatics 27 (2013), pp. 1654–1662.

Identifying basic building blocks/motifs of networks

elec3p

The optimal subgraph decomposition of an electronic circuit.

There are many verbal descriptions for network motifs: characteristic connectivity patterns, over represented subgraphs, recurrent circuits, basic building-blocks of networks just to name a few. However, as with most concepts in network science network motifs are maybe best explained in terms of empirical observations. For instance the most basic example of a network motif is the motif consisting of tree mutually connected nodes that is: a triangle. Many real world networks ranging from the internet to social networks to biological networks contain many more triangles than one would expect if they were wired randomly. In certain cases there exist good explanations for the large number of triangles found in the network. For instance, the presence of many triangles in friendship networks simply tell us that we are more likely to be friends with the friends of our friends. In biological networks triangles and other motifs are believed to contribute to the overall function of the network by performing modular tasks such as information processing and therefore are believed to be favoured by natural selection.

The predominant definition of network motifs is due to Milo et al. [1]  and defines network motifs on the basis of how surprising their frequency in the network is when compared to randomized version of the network. The randomized version of the network is usually taken to be the configuration model i.e. the ensemble of all networks that have the same degree distribution as the original network. Following this definition motifs are identified by comparing their counts in the original network with a large sample of this null model. The approach of Milo et al. formalizes the concept of network motifs as over represented connectivity patterns. However, the results of the method are highly dependent on the choice of null model.

In my talk I presented an alternative approach to motif analysis [2] that seeks to formalize the network motifs from the perspective of simple building blocks. The approach is based on finding an optimal decomposition of the network into subgraphs. Here, subgraph decompositions are defined as subgraph covers which are sets of subgraphs such that every edge of the network is contained in at least one of the subgraphs in the cover. It follows from this definition that a subgraph cover is a representation of the network in the sense that given a subgraph cover the network can be recovered fully by simply taking the union of the edge sets of the subgraphs in the cover. In fact many network representations including edge lists, adjacency lists, bipartite representations and power-graphs fall into the category of subgraph covers. For instance, the edge list representation is equivalent to the cover consisting of all single edge subgraphs of the network and bipartite representations are simply covers which consist of cliques of various sizes.

Given that there are many competing ways of representing a network as a subgraph cover the question of how one picks one of the covers over the other arises. In order to address this problem we consider the total information of subgraph covers as a measure of optimality. The total information is a information measure introduced by Gell-Mann and Hartle [3] which given a model for a certain entity e is defined to be sum of the entropy and effective complexity of M. While the entropy measures the information required to describe e given M and the effective complexity measures the amount of information required to specify M which is given by its algorithmic information content. The total information also provides a framework for model selection:  given two or more models for the same entity one picks the one that has lowest total information and if two models have the same total information one picks the one that has lower effective complexity i.e. the simpler one. This essentially tells us how to trade off goodness of fit and model complexity.

In the context of subgraph covers the entropy of a cover corresponds to the information required to give the positions of the subgraphs in the cover given the different motifs that occur in C and their respective frequencies in C. On the other hand the effective complexity of C corresponds to the information required to describe the set of motifs occurring in the cover together with their respective frequencies. While the entropy of subgraph covers can be calculated analytically their effective complexity is not computable due to the halting problem. However, in practice one can use approximations in the form of upper bounds.

Following the total information approach we now define an optimal subgraph cover of network G to be a subgraph cover that minimizes the total information and the network motifs of G to be the motifs/connectivity patterns that occur in such an optimal cover.
The problem of finding an optimal cover turns out to be computationally rather challenging. Besides the usual difficulties associated to counting subgraphs  (subgraph isomorphism problem-NP complete) and classifying subgraphs (graph isomorphism problem-complexity unknown) the problem is a non-linear set covering problem and therefore NP-hard. Consequently, we construct a greedy heuristic for the problem.

When applied to real world networks the method finds very similar motifs in networks representing similar systems. Moreover, the counts of the motifs in networks of the same type scale approximately with network size. Consequently, the method can also be used to classify networks according to their subgraph structure.

 

References:

[1] R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii, and U. Alon, Network Motifs: Simple Building Blocks of Complex Networks, Science 298, 824 (2002)

[2] Wegner, A. E. Subgraph covers: An information-theoretic approach to motif analysis in
networks. Phys. Rev. X, 4:041026, Nov 2014

[3] M. Gell-Mann and S. Lloyd, Information Measures, Effective Complexity, and Total Information, Complexity 2, 44 (1996).