Individual nodeʼs contribution to the mesoscale of complex networks

Before presenting our formalism to characterize the roles of nodes in modular networks, we first review two previous efforts.

2.1.1. Mapping of functional roles

The framework proposed in [18, 19] consists of a mapping of the nodes based on two parameters; one parameter evaluates the internal importance of the node within its module and the other one evaluates its external connectivity. The internal parameter, called the within-module degree, is defined as the z-score of the internal degree of the node:

$$\begin{eqnarray}&&{{z}_{i}}=\frac{k_{i}^{\prime }-\langle k^{\prime} \rangle }{\sigma ^{\prime} }\end{eqnarray} \tag{ 2 }$$

where k_im is the number of links the node makes with members of its community, ${{\langle k\rangle }_{m}}$ denotes the average internal degree of the nodes in the community and ${{\sigma }_{m}}$ their standard deviation. The external parameter, the participation coefficient, quantifies how distributed are the links of a node among all the communities:

$$\begin{eqnarray}&&{{P}_{i}}=1-\mathop{\sum }\limits_{m=1}^{M}{{\left( \frac{{{k}_{im}}}{{{k}_{i}}} \right)}^{2}}\;.\end{eqnarray} \tag{ 3 }$$

P_i = 0 if all the links of a node fall in the same community and P_i = 1 if its links are uniformly distributed among all the modules. Following these two parameters, the nodes of a network can be mapped into a z-P plane in which nodes of different roles occupy distinguished regions, similar to those illustrated in figure 1. This framework suffers from a few shortcomings that limit its application and the interpretation of the results obtained: (i) nodes are classified as hubs or non-hubs based exclusively on the information of their internal degrees when in the literature the term hub is associated with the degree to which a node, comprises the whole network. (ii) All communities are assumed to have identical statistical properties. For example, if two modules had different internal degree distributions their z_i nodes are evaluated over different statistical baselines. On the other hand, the formulation of the participation index assumes that all communities are of the same size. (iii) The largest value of P_i depends on the number of communities. In a partition with M communities the largest participation a node can take is ${{P}_{i}}=1-\frac{1}{M}\lt 1$ when according to [18], it should be P_i = 1. As a consequence P_i values across networks are only comparable if both networks contain the same number of communities.

2.1.2. Algebraic approach to characterize the modular skeleton

One of the most important features found in real networks is the presence of highly connected nodes or hubs. Despite their importance a formal definition of hubness is missing; they are informally defined as 'those nodes with many more connections than others'. The hallmark of networks containing hubs, scale-free networks, possess a power-law degree distribution meaning that most nodes make only few links and a few nodes have many connections. The degrees of hubs is usually orders of magnitude larger than that of the averagely connected nodes. The within-module degree in equation (2) is intended to capture this notion of hubness by estimating the significance of a nodeʼs degree compared to that of its neighbours.

Now, we note a paradoxical behaviour in the hubs of scale-free networks: despite having much larger degrees than other nodes, hubs are only connected to a small fraction of the network. This happens because typically studied scale-free networks, both synthetic and real, are sparse. For example, in a scale-free network with $N=100,\;000$ nodes and exponent $\gamma =3$ the most connected hubs are linked with only $2\%-5\%$ of all nodes. In contrast, many real networks are dense and therefore the loose definition of hubness above does not comply. The same applies to the internal connectivity of the modules which are, by definition, densely connected parts of a network. In dense (sub-)networks the mean degree is of the same order of magnitude as its size, $\langle k\rangle \gt \sim N$, and therefore the difference between the least and the most connected nodes is considerably reduced. Also, contrary to the hubs of scale-free networks, the hubs of dense networks are connected to a large fraction of the nodes. In dense networks it is usual to find hubs connected from $30\%$ up to $70\%$ of the network.

We find that for a measure of hubness to be universal it requires the assessment of the degrees under a common statistical baseline. The simplest choice to define the hubness index h_i of a node such that it is comparable across different (sub-)networks is to normalize the degree by the size of the (sub-)network, ${{h}_{i}}=\frac{{{k}_{i}}m}{{{N}_{m}}}$ [23]. However, this normalization does not account for the link densities. Therefore, we define the hubness as the comparison of the nodeʼs degree k_i with the degree distribution of an equivalent random graph of the same size N and density ρ. Since random graphs possess a narrow degree distribution around their mean degree ${{\langle k\rangle }_{R}}$, the more a nodeʼs degree deviates from the degree distribution of an equivalent random graph, the more reasonable it is to be considered as a hub. The mean degree of a random graph is ${{\langle k\rangle }_{R}}=(N-1)\rho$ and the standard deviation of its degree distribution is ${{\sigma }_{R}}=\sqrt{(N-1)\rho \;(1-\rho )}$ [1]. Constraining to networks without self-loops, we define the hubness index of a node in a network of size N and density ρ as:

$$\begin{eqnarray}&&{{h}_{i}}=\frac{{{k}_{i}}-{{\langle k\rangle }_{R}}}{{{\sigma }_{R}}}=\frac{{{k}_{i}}-(N-1)\rho }{\sqrt{(N-1)\;\rho \;(1-\rho )}}.\end{eqnarray} \tag{ 4 }$$

The hubness is negative for nodes with ${{k}_{i}}\lt {{\langle k\rangle }_{R}}$ allowing us to also identify outliers that are significantly less connected than expected from randomness. This index differs from the one defined in equation (2) in that all networks are compared to the same null-model—the random graph—instead of using the internal statistics of each network and of each community as a baseline for itself. This makes the index universal and the results for different networks and modules comparable.

Once the hubness index is defined we need to interpret the results and to identify its extremal values. In a random graph the degree of 99% of the nodes lies in the range bounded by ${{\langle k\rangle }_{R}}\pm \;2.5\;{{\sigma }_{R}}$. In any network, nodes within this range shall not be considered outliers. The larger the h_i index grows above $2.5\ \sigma$ the more likely is the node a hub. The largest and smallest values that h_i can take depend on the size and the density of the network. In sparse networks it is not possible to find outliers on the negative side because $0\lt \langle k\rangle \ll N$ and even disconnected nodes lie in the range of statistically expected degrees. However, there is plenty of room for some nodes to have a degree much larger than the mean and achieve high significance. Figure 2(a) illustrates this for a network with N = 30 and $\rho =0.25$. A node with degree k_i = 1 gives ${{h}_{i}}=-3.1$ while a hub with k_i = 29 achieves h_i = 8.3. As the density increases the situation is reversed because $\langle k\rangle \to N$. The room for hubs decreases limited by the size of the network while from the bottom, occasional nodes with few connections become very significant. The hubness of a node with degree k_i = 0 is:

$$\begin{eqnarray}&&{{h}^{-}}(N,\rho )=-\frac{{{\left\langle k \right\rangle }_{R}}}{{{\sigma }_{R}}}=-\sqrt{\frac{(N-1)\;\rho }{1-\rho }}\end{eqnarray} \tag{ 5 }$$

and for a node with degree ${{k}_{i}}=N-1$:

$$\begin{eqnarray}&&{{h}^{+}}(N,\rho )=\sqrt{\frac{(N-1)\;(1-\rho )}{\rho }}\;.\end{eqnarray} \tag{ 6 }$$

Equations (5) and (6) represent the lower and upper boundaries for the hubness index. In figure 2(b) the boundaries are shown for networks of different size and varying density. When networks are sparse the lower limit ${{h}^{-}}\gt -2.5\;\sigma$ regardless the size N and the upper bound tends to infinity. When networks are very dense, the opposite happens, the lower bound goes to $-\infty$ while there is no room for significant hubs because ${{h}^{+}}\lt 2.5\;\sigma$. As highlighted in the inset, the limitations are stronger for smaller networks. For example, in networks of N = 10 there is no regime of density in which nodes with significant low and high degree can be found simultaneously.

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From now on we will characterize nodes by two hubness values: the global hubness h^g is the hubness of the node in the network and the local hubness h^l is hubness within the community it belongs to. In this case N and ρ are replaced by the size and density of the community $N^{\prime}$ and $\rho ^{\prime}$.

Given a partition $\mathcal{L}$ with M disjoint communities our aim is now to quantify how distributed is a node among them. In our framework we acknowledge that the contribution of a node to each community depends also on the size of the community. Imagine a network with only two communities, one of size ${{N}_{1}}=20$ and another of size ${{N}_{2}}=40$. A node devoting ten links to each of them is contributing more to the small community than to the large one and therefore, it is more likely to belong to the small community. We introduce the concept of participation vector ${{{\boldsymbol{P}} }_{i}}$ whose elements P_im represent the probability that node i belongs to community C_m, where $m=1,2,\ldots ,M$. This probability is given by ${{P}_{im}}=\frac{{{k}_{im}}}{{{N}_{m}}}$ where N_m is the size of the community. Since we are only interested in the relative differences participation vectors are normalized such that $\sum _{m=1}^{M}{{P}_{im}}=1$. Otherwise the norm of ${{{\boldsymbol{P}} }_{i}}$ would be proportional to the degree k_i but this information shall be disentangled and captured only by the hubness index. The vector of a node devoting all its links to the second community of a network with M = 4 communities is ${{{\boldsymbol{P}} }_{i}}$ = $(0,1,0,0)$ and for a node whose links are all equally likely distributed among the four communities, ${{{\boldsymbol{P}} }_{i}}$ = ${{(}^{1}}{{/}_{4}},{{\;}^{1}}{{/}_{4}},{{\;}^{1}}{{/}_{4}},{{\;}^{1}}{{/}_{4}})$.

Once the participation vectors have been computed for all nodes, we want to reduce that information into scalar values to quantify how distributed are the links of a node among all the communities. For consistency with previous definitions p_i = 0 if the node devotes all its links to a single community and p_i = 1 if its links are equally likely to be distributed among all the modules. Therefore we evaluate the standard deviation $\sigma ({{{\bf P}}_{{\bf i}}})$ of the elements of the participation vector ${{{\boldsymbol{P}} }_{i}}$ and define the participation index as:

$$\begin{eqnarray}&&{{p}_{i}}=1-\frac{\sigma ({{{\bf P}}_{{\bf i}}})}{{{\sigma }_{{\rm max} }}(M)}=1-\frac{M}{\sqrt{M-1}}\ \sigma ({{{\bf P}}_{{\bf i}}}).\end{eqnarray} \tag{ 7 }$$

The normalization factor accounts for the fact that the standard deviation of an M dimensional vector with all elements equal to zero but one is ${{\sigma }_{{\rm max} }}(M)=\sqrt{M-1}\;/\;M$.

The larger its p_i the more difficult it is to classify the node into one and only one community. But for a node to be difficult to classify it is not necessary that it connects to all the communities. If a node is equally likely connected with only two of the communities, it is also difficult to classify. Such a node has for example a participation vector ${{{\boldsymbol{P}} }_{i}}=(0,{{\;}^{1}}{{/}_{2}},\;0,{{\;}^{1}}{{/}_{2}})$. To account for this information we define the dispersion of a node d_i equivalent to the participation index but considering only the non-zero entries of the participation vector:

$$\begin{eqnarray}&&{{d}_{i}}=1-\frac{\sigma ({\bf P}_{{\bf i}}^{\prime })}{{{\sigma }_{{\rm max} }}(M^{\prime} )}=1-\frac{M^{\prime} }{\sqrt{M^{\prime} -1}}\sigma ({\bf P}_{{\bf i}}^{\prime }),\end{eqnarray} \tag{ 8 }$$

where ${\boldsymbol{P}} _{i}^{\prime }$ is the subvector containing only the non-zero entries of ${{{\boldsymbol{P}} }_{i}}$, and $M^{\prime}$ its dimension. A node only connected within one community has d_i = 0 and a node equally likely connected among $M^{\prime} \leqslant M$ has d_i = 1. We note that in general ${{d}_{i}}\geqslant {{p}_{i}}$ with the equality only happening when $M^{\prime} =M$. The dispersion index shall be regarded as a measure of how difficult it is to classify a node into only one community and the participation index as the global reach of a nodeʼs links among all the communities.

We note also that not all combinations of dispersion and participation are possible. The highest difference between a nodeʼs d and p is reached when $M^{\prime} =2$ and follows:

$$\begin{eqnarray}&&{{p}^{+}}(d,M)=1-\sqrt{\frac{M}{M-1}\left[ {{\left( \frac{2-d}{2} \right)}^{2}}+{{\left( 1-\frac{2-d}{2} \right)}^{2}}-\frac{1}{M} \right]}.\end{eqnarray} \tag{ 9 }$$

In the following sections we apply the four indices here defined—local hubness, global hubness, dispersion and participation—to investigate the topological contribution nodes take in both synthetic and in empirical networks.

The graph in figure 3(a) consists of N = 47 nodes that are grouped into three modules. Each nodeʼs affiliation is represented by a red circle (community I), blue diamond (community II), or purple triangle (community III). The partitioning was derived using the Louvain method for community detection [31, 32]. Community III is the largest one with 23 nodes and I and II are equally sized with 12 nodes each. Communities have density 0.52, 0.20 and 0.29 respectively as summarized in table 1. Thus each community is distinct from the other two. We hand-picked seven nodes to illustrate the different roles that can be identified in the 4-dimensional space $({{h}^{g}},{{h}^{l}},p,d)$. Instead of showing all six possible relations we restrict ourselves to the three plots we find most informative: (${{h}^{l}},{{h}^{g}}$), (${{p}_{i}},{{h}^{g}}$) and (${{p}_{i}},{{d}_{i}}$) corresponding to figure 3(b)–(d). The participation vectors representing the probability of belonging to the communities I, II and III for the selected nodes are:

• Nodes (11) and (22) are connected only within their modules and their participation and dispersion are therefore $p=d=0;$ we call them peripheral nodes. Both are connected to five nodes in their communities and have a small global hubness ${{h}^{g}}\lt 0$. They are representative of peripheral non-hubs, see figure 1, which can functionally be identified with specialized or segregated function in the network. The examination of their local hubness shows a fundamental difference: (11) also shows ${{h}^{l}}\lt 0$ and does not fulfil a prominent position in its module since community I is dense but (22) is the node with highest internal degree in II, which is a sparse module and is therefore a local hub.
• Node (43) is also only connected within its module and its participation and dispersion are $p=d=0$. Since it belongs to the largest community it is also one of the nodes with highest global hubness h^g. It is both a local and a global hub with no participation. It is characteristic of the rare peripheral hub category; see figure 1.
• Node (44) is a member of community II but devotes the same number of links, six, to community I. It is therefore hard to determine to which of the two communities it belongs. This results in a dispersion index $d\approx 1$. However, it makes no links with community III and has a reduced participation index of p = 0.5. This node cannot be considered as a kinless hub but it is a typical connector hub. The reduced participation however restricts the nodeʼs ability to integrate information from all parts of the network.
• Node (45) instead has the same number of neighbours but those are spread out amongst all three modules. It is a kinless hub with the ability to integrate on a global scale. Since it belongs to community III, which is densely connected, its local hubness is ${{h}^{l}}\lt 0$.
• Node (46) has only three neighbours but they are each in one of the three communities. Therefore (46) is neither locally nor globally a hub but its high participation makes it a kinless node that is located between all communities.
• Node (47) has two neighbours in its own community III and one in the community II which is half the size. Therefore it has almost equal likelihood to belong to any of the two communities and achieves a large dispersion value close to one. Its lack of connectivity to the community III reduces its participation index to $p\approx 0.5;$ therefore we can classify this node as a non-hub connector.

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In figure 3(e) the results for this graph are shown using the framework of the functional roles. Most of the nodes, including (11), (22) and (43) are classified as ultra-peripheral (R1) but none as hubs. Since only the internal degree is considered for the classification of nodes into hubs and non-hubs, node (22) is almost regarded as a hub but node (43) is assigned lower hubness despite being connected to more than twice the number of nodes as node (22). This misclassification is the result of assigning global hubness to local hubness and of a lack of normalization that accounts for the size and density of each community. On the other hand, the dependence of P_i on the number of communities, the accessibility to the region (R3) of connector hubs is reduced and to the region of kinless nodes (R4), forbidden.

The participation vectorʼs reduction onto the leading two dimensions with the SVD is able to detect two main insights: a larger radius R stands for a mixture between higher degree/hubness and participation. The intramodular projection vector for each module (dashed lines) $\tilde{e}$ is close to the modular projection (solid lines) $\tilde{m}$ if the module is mostly internally connected. The projection on this vector gives for each node a measure of the number of neighbours in this particular module. The angular distance then is a measure of how strongly a node participates in each module. Node (45) for example has a large angular distance from all modular vectors since it is a kinless node. However the SVD is strongly dominated by the largest module III and modules I and II are almost not distinguishable from each other. Therefore the method, in its 2-dimensional representation, is not suited for our purposes.

We have seen that our approach is capable of detecting a wide variety of node roles in the mesoscale. However, the usage of four different metrics makes the illustration and interpretation a non-trivial task. Therefore we proceed with the illustration of the method applied to four artificial networks, followed by examples of biological networks. In the synthetic network examples we show results for a single realization since the number of communities usually differs from realization to realization. Still, the results remain general.

We now analyze two fundamental network models: the Erőds-Renyi (ER) graph and the scale-free (SF) graph. These models lack intrinsic community structure and serve to illustrate the case in which the analysis of the mesoscale fails to return reasonable results due to the absence of structure in the mesoscale, or due to poor partitioning. Ensembles of ER and SF graphs of size N = 1000 nodes and density $\rho =0.1$ were generated. The SF networks were created using the method introduced in [37] and they were tuned to follow a scale-free degree distribution $P(k)={{k}^{-\gamma }}$ with $\gamma =3$.

The upper panel of figure 4 shows the distribution, as density plots, of the four metrics across realizations for the ER graph partitioned into nine modules. Firstly we note that the distribution of both local and global hubness is rather small as expected. There is a correlated trend between both since nodes with a large number of neighbours inside their module are, in general, those with high number of neighbours in the whole network. The high values of participation p and dispersion d mean that all nodes have a significant number of neighbours in other modules. Mostly in all nine modules as indicated by the fact that most nodes lie on the diagonal p = d. Nodes directly above the diagonal are linked to eight modules. There are no peripheral nodes having neighbours only in their own community ($p=d=0$). These observations indicate that the community structure found is very poor, as it is expected for ER graphs. The (p, h^g) plot is often useful to detect the global roles and functionality of the nodes since it closely resembles the classification framework illustrated in figure 1. Here we see that all nodes have a kinless contribution to all different modules due to the lack of structure at the mesoscale.

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The results for the SF graph are very similar (lower panel of figure 4), especially in terms of participation and dispersion. This also demonstrates the lack of structure at the mesoscale in random SF graphs. The difference arises in the hubness indices which reach much higher values due to the heavy tailed degree distribution. In the (p, h^g) plot it is seen that all nodes are again kinless, now with a clear hub structure expanding vertically.

Summarizing we find that networks with no intrinsic community structure do not show a diversity of roles when analyzing their mesoscale. All nodes tend to be connected with all modules and have a high participation. The hubs are detectable in the SF case but they are not distinguishable in participation and dispersion from the other nodes.

Now we discuss graphs with an intrinsic modular structure. We analyze two modular graphs models, both consisting of N = 1000 nodes that are grouped into four equally sized modules. Inside the modules pairs of nodes are connected with probability 0.3 and externally with probability 0.05. In the first model version all connections are created at random and we refer to it as the random modular (RM) graph. We also introduce a modified version in which the external links between modules are created following a preferential attachment rule. This leads to a centralized modular (CM) graph. Following the SF network generation method in [37], the nodes of each modules of size M_m are indexed as i ($i=1,2,...,{{M}_{m}}$) and are assigned a weight ${{p}_{i}}={{i}^{-\alpha }}$, where α is a control parameter in $[0,1)$. When connecting two different modules, nodes from each module are selected with a probability proportional to their weight. The degree distribution of network generated with this method follows a power law with $\gamma =(1+\alpha )/\alpha$. In this case, the hubs from the different modules become interconnected. This strong connection between the hub nodes is called a rich-club since it was first discovered in social networks accounting for the fact that rich people tend to gather in social clubs [38]. Its first sound quantification was introduced in the study of the topological structure of the internet [39] and later found also in neuronal networks [25, 40, 41]. This combination of characteristics, modular structure with centralized interconnectivity occur in neuronal networks and so we want to understand their influence on the mesoscale.

We note that the local and the global hubness are strongly correlated; see figure 5 plot $({{h}^{l}},{{h}^{g}})$, mainly because they reflect the internal degrees of the nodes. The small amount of external connections per node has a weak influence on the relation ship between local and global hubness, especially in the RM model. Overall, due to the preferential attachment, global hubness takes larger values in the CM than in the RM model.

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Focusing on the measures of participation and dispersion, plots (p, d), we find that their values are much lower than those observed before in ER and SF networks. This happens because the modules are well defined with most nodes densely connected inside their module and weakly connected externally. The communities are well defined in this case. However, only a small amount of nodes have neighbours in only one module and thus $p=d=0$. This shows that the strong community structure does not go hand in hand with a strong core of nodes inside each module. In contrast the small amount of external edges is evenly distributed amongst all nodes and therefore almost all of them show external affiliations. In the RM graph there is no dependence between h^g and p. This occurs because external connections are created at random and therefore all nodes have the same probability of connecting to other modules. The ($p,{{h}^{g}}$) plot for the CM model is very similar but the hubs clearly stand out—they have larger hubness and participation. In the ($p,d$)-plane, the hubs lie on the p = d line because they connect with the four modules. Due to this centralization non-hub nodes receive fewer external links and see their participation lessened.

In real networks the community organization happens as the consequence of functionally related nodes being gathered together. In this manner groups of specialized function are segregated, restricting the influence of nodes with different function on them. In the RM model the communication between the modules is limited due to the overall small dispersion. In the CM model the hubs connect with all the modules allowing them to integrate information from the distinct functional groups. In the following we analyze neural networks for which the ability to maintain a balance between the amount of segregation between functional groups and an efficient communication between them is fundamental for their optimal working.

The framework here proposed builds upon previous efforts to characterize the roles that nodes take in modular networks: the functional roles framework introduced in Guimerá and Amaral [18, 19] and the SVD approach by Arenas et al [20]. A common limitation of the two frameworks is that both assume all communities to be of the same size. While this is a reasonable approximation for large networks containing several large and near-to-homogeneous communities, many real networks are small or contain inhomogeneous communities.

Here, we have taken a probabilistic approach and evaluated the likelihood of a node to belong to a community that depends not only on the degree of the node but also on the size of the community. Stacking our participation vectors would lead to a matrix similar to the contribution matrix which would additionally account for the inhomogeneity of communities. We have also proposed a measure of hubness that is consistent with the common and original understanding of hubs, say, that the degree of hubs in scale-free networks largely deviates from the expected, narrow degree distribution of random graphs. We have defined the hubness of a node as the difference between its degree and the typical degree distribution in equivalent random graphs. This definition allows us to compare the results out of different networks under a common statistical baseline. Additionally, we have shown that the hubness is bounded by finite-size effects depending on the size and on the density of the (sub-)networks.

Future challenges of the formalism are its application to weighted or signed networks. From a technical point of view, we note the potential of participation vectors to improve community detection methods because they are ideal tools to identify misclassified nodes. This information can be used to switch nodes accordingly until the partition minimizing the number of misclassified nodes is reached. From a practical point of view, we foresee the interest of applying our formalism to multiplex and to multilayer networks. Our measures would allow one to one characterize the importance of nodes in relation to the different slices of the network in a statistically qualified way. We expect these tools to aid in uncovering general principles of organization in biological networks as the modular and hierarchical relations we found here for neural and transcriptional networks. In order to further analyze hierarchical organization recent multi-resolution community detection methods can be applied that return optimal partitions at different levels of mesoscale [55, 56]. For three of the discussed networks, C. elegans [57], human cortex connectivity [58] and Mycobacterium tuberculosis [59] such multiresolution approaches have been undertaken and the evaluation and discussion of the roles of nodes across scales will be a challenging and interesting task. Also, as observed for the transcriptional network of the Mycobacterium tuberculosis analyzed here, the function of many genes is still an enigma even for rather simple and small organisms, especially the function of those genes involved in multi-contextual processes. The examination of the individual geneʼs topological position within its network can largely help understand its purpose and to create predictions about its function that are experimentally testable a posteriori.