Imparcialidad y objetividad.

The Graphlet Correlation Matrix is a new network statistic that encodes the topology of a network using the Spearman’s Correlation Coefficients among various node properties contained in graphlet degrees, over all nodes. Given a network G(V, E), first we compute graphlet degree vectors of all nodes, v₂V, and construct a matrix where each row represents the graphlet degree vector of a node,GDV(v). We exploit the existence of dependencies between orbits by computing the Spearman’s correlation coefficient among all pairs of orbits (i.e., among all columns of the matrix of graphlet degree vectors) and present them in an⇥nsymmetric matrix that we name as theGraphlet Correlation Matrix of network, GCMG. Graphlet correlation matrices can be defined using di↵erent sets of orbits. We focus on two particular orbit sets in our experiments: (1) 11 non-redundant orbits of 2- to 4-node graphlets (illustrated in Figure 2.3), (2) the complete set of 73 orbits of 2- to 5-node

graphlets (illustrated in Figure 1.4). In this way, we can encode the topology of a network of any size into ann_⇥nsymmetric matrix with values in the interval [ 1,1], wherenis the number of orbits that are used for computing theGCM. Graphlet Correlation Matrix computation is illustrated in Figure 2.5 on a random geometric graph with 500 nodes and 1% edge density.

Networks that have di↵erent topologies are expected to have di↵erent graphlet correlation matrices. For example, Figure 2.6 illustrate the graph- let correlation matrices of four di↵erent networks: a scale-free network that is generated by the preferential attachment (i.e., Barab`asi-Albert) model, a network generated by the geometric random network model, the world trade network of 2010, and the human metabolic network. In agreement with known properties of scale-free Barab´asi-Albert (SF-BA) networks, or- bits 0, 2, 5, and 7, which are characteristic to existence of hubs, form a cluster of dependent orbits with their correlation coefficients being close to 1 (Figure 2.6–A). Orbits 10 and 11, which are characteristic to existence of clustering “near” hubs, also form a cluster of correlated orbits. Finally, orbits 1, 4, 6, and 9, which are characteristic to existence of a large num- ber of degree 1 nodes, are dependent as well. The picture is quite di↵erent for geometric random graphs (GEO) of the same size, which have Poisson degree distributions, and hence the structure is not dominated by a large fraction of degree 1 nodes and a small number of hubs (Figure 2.6–B).

Uncovering orbit dependencies in real-world networks is much more inter- esting, since they can reveal currently unknown organizational principles of these networks. Indeed, the world-trade network of 2010 [34] contains two large clusters of dependent orbits,_{0,2,5,7,8,10,11_} and_{6,9,4,1_}, while there is anti-correlation between orbits{4,6,9}and orbits{0,2,5,7,8,10,11} (Figure 2.6-C). Investigating the implications of this, we notice that orbits 4,6 and 9 correspond toperipheral, degree 1 nodes that are “hanging” from graphlets G3, G4 and G6 (Figure 2.3), while members of the large cluster

of correlated orbits,_{0,2,5,7,8,10,11_}, correspond to higher degree, either clustered (in a dense neighbourhood), or broker-type (mediators between nodes that are not directly interacting) orbits. Since these two clusters are anti-correlated, we can conclude that countries are either clustered/bro- kers, or on the periphery of the world trade [44], but not both. Hence, GCM unveils a hidden structure of this network that can be further interpreted qualitatively: through further analysis presented below, we interpret this

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Figure 2.5: Graphlet Correlation Matrix computation is illustrated on a ge- ometric network G with 500 nodes and 1% edge density (the network on the left). In the matrix of graphlet degree vectors (shown on the left), each row represents the graphlet degree vec- tor of a node, and each column contains the graphlet degrees of all nodes for orbit i, di

G. The graphlet degrees of orbits 0 and 1, d0_G and d1_G are highlighted in red. The graphlet correlation between orbits iand j, GCMG[i, j], is the Spearman’s correla- tion coefficient betweendi

Gand djG. Computing theGCMG[i, j] for all pairs of orbits, we obtain the symmetric graphlet correla- tion matrix ofG,GCMG. The rows and columns of theGCMG are ordered based on the correlation similarities of orbits for visualising the orbit clustering patterns better.

(A) (B)

(C) (D)

Figure 2.6: Graphlet Correlation Matrices (GCMs) of di↵erent types of net- works: Panel A – a scale-free Barab`asi-Albert (SF-BA) network with 500 nodes and 1% edge-density; Panel B – a geometric random network (GEO) with 500 nodes and 1% edge-density; Panel C – the world trade network of 2010; and Panel D – the human metabolic network. The rows and columns of the GCMs are ordered based on the correlation similarities of orbits for visualising the orbit clustering patterns better.

observation on 49 world trade networks corresponding to trade data from 1962 to 2010. In contrast, the topology of the human metabolic network [98] is very di↵erent from the topology of world trade networks: the correlations between all orbits are high, indicating that constituent bio-molecules can be at the same time both peripheral and clustered/broker (Figure 2.6-D).

It is possible that a graphlet does not appear in a network. When this is the case, graphlet degrees of all nodes are equal to 0 for the corresponding orbits. Since the graphlet degrees are constant for all nodes, Spearman’s Correlation coefficient cannot be computed for these orbits. To overcome this problem, we include a dummy graphlet degree vector, [1,1, ...,1], into the matrix of graphlet degree vectors. This small amount of noise resolves the Spearman’s correlation coefficient computation problem. As a result, the problematic orbits correlate perfectly (having Spearman’s correlation coefficients of 1) while these orbits do not correlate with the rest of the non-zero orbits (having Spearman’s correlation coefficients close to 0).

The graphlet degrees of di↵erent orbits do not scale within the same in- tervals, due to the di↵erences in the search spaces of orbits. For example, graphlet degree of orbit 15 searches up to 4th neighbourhood of a node, while graphlet degree for orbit 7 is only dependent on the 1st _neighbour- hood, which causes the graphlet degrees of orbit 15 to span at a wider range. The graphlet degree ranges might even di↵er for orbits that search the same distance neighbourhoods, since the chances of each graphlet’s appearance are not distributed evenly and depend on the density of the network. Due to the di↵erences in the graphlet degree scales, a ranking based correla- tion coefficient that measures monotonic correlations between orbits (i.e., Spearman’s Correlation Coefficient) is preferable over a correlation coef- ficient that measures the linear correlations among graphlet degrees (i.e., Pearson’s Correlation Coefficient) for measuring the correlation between the graphlet degrees of di↵erent orbits. This is the reason for us to de- fine the Graphlet Correlation Matrices based on Spearman’s Correlation Coefficients rather than any other correlation coefficients.