The relationship between structure and function in locally observed complex networks

In order to apply our methodology we begin with a network having N nodes, which will be fixed through the entire process, and execute M times a dynamics on it. Each execution starts with a randomly sorted initial condition. It is important to note that depending on the dynamics being studied we can have a specific set of initial conditions that take the system to a particular state, so this state will rarely be accessed by sampling. This is not a problem in our method because we are analyzing the dynamics for the set of initial conditions imposed, that is, we are studying the signals that are in fact observed.

In possession of the dynamic signals for the M realizations, we need a mechanism to represent them in order to compare their behavior. In our case we use dynamical measures that try to extract the most relevant information about the signals. Let F be one of such measures, after the many realizations we get a set of observed values f_i,r, where i is the node index and r the realization. These values can vary through four distinct mechanisms: (a) initial condition, (b) network topology, (c) particular dynamics and (d) stochasticity. Our objective here is to study only the relation between the initial condition and the topology, so we use dynamics having identical equations for every node and consider stochastic variations to be small. With these restrictions the only variations we can observe on the values of F are: (a) fluctuations on the dynamical values of a given node, which given the fact that the topology is static, can only be caused by the variation of the initial condition. (b) Differences on the mean values of F for distinct nodes, that because of the properties assumed can only be caused by the topological differences of the nodes. The term topological difference needs to be used with care, because unless in very specific cases where the network is perfectly symmetric (e.g. a lattice with toroidal boundary), the topology of two given nodes is rarely identical, that is, we can always find a structural characterization that will have distinct values for them. Nevertheless, since the networks we use are not regular, every significant dynamical difference we observe must be caused by the topology. It is also important to note that the reverse is not true, if the dynamics of two nodes appear to be the same, their topologies are still distinct, what happened is that both nodes felt the topology in the same manner. An example of this last case is the diffusion dynamics on graphs [19], in which the equilibrium behavior depends only on the degree of the nodes, that is, although the nodes possess distinct general topology, the localized characteristic of the dynamics allows only the degree to differentiate the nodes. The two cases we may come across are shown in figure 1.

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In order to quantify the difference of the values obtained for each node, we use a statistical test that will be defined on the next section. Through this statistic we can identify if the difference of the means of the dynamical values obtained are in fact significant, that is, we are quantifying the difference between the dynamics of the nodes normalized by the intrinsic fluctuations caused by the initial condition. The distance between every pair of nodes is then represented by the matrix Ξ, where each line i and column j represents the distance obtained between the nodes i and j, that is

$$\begin{equation} \Xi_{ij}=d_h(i,j), \end{equation} \tag{ 1 }$$

where d_h(i,j) is the distance defined by equation (6). Finally, we can define our mean dynamical differentiation measure, α, as the mean values of each line of this matrix

$$\begin{equation} \alpha_{i}=\frac{1}{N-1}\sum^{N}_{j=1}\Xi_{ij}. \end{equation} \tag{ 2 }$$

The standard procedure now would be to calculate the statistical significance of the observed values of α, but since we are concerned with the comparison between the distances and not their absolute values, this does not need to be performed. To carry out the comparison, we construct a histogram of the obtained values of α. Having in mind that α is relative to some dynamical characteristic, we can have distinct histograms relative to the desired characterization.

What we search for are particular behaviors of the histograms, for example, it is expected that a single node with very distinct dynamics compared to the rest of the network will have a very large α value. It is important to observe that although we presented the methodology for a single measure, nothing prevents us from calculating the distances using simultaneously various dynamical measures. In figure 2 we show an example application of the presented model.

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In this section we present and motivate the statistical test that will be used throughout the paper. Suppose that we have a set of points inserted in a m-dimensional space and these points form two distinct groups (see figure 3). An immediate way to quantify the separation between the groups is by using the Euclidean distance between the center of mass of each group, given by

$$\begin{equation} d(r,s)=\sqrt{\sum^{m}_{i}(\langle x_{ri}\rangle-\langle x_{si}\rangle)^{2}}, \end{equation} \tag{ 3 }$$

where r and s are the indices of the groups and 〈x_ri〉 represents the mean (or center of mass) of the points in group r on the ith dimension. The disadvantage of the Euclidean distance is that it does not take into account the dispersions of the groups from which the distance is being measured. Following figure 3, if we have two random variables X_r and X_s with the realized values {x_r} and {x_s}, marked respectively in blue and red in the figure, clearly the two cases shown in figures 3(a) and (b) have a more significant distance between their mean than the case in figure 3(c), although the Euclidean distance is the same. There are many ways to take into account the dispersions of the groups, here we use the Hotelling statistic [20], which considers the variance of each group in the direction defined by the line that passes between the two means. The distance, h, between two groups is defined by

$$\begin{equation} h^{2}=\frac{n_{r}n_{s}}{n_{r}+n_{s}}(\langle\vec{x}_{r}\rangle-\langle\vec{x}_{s}\rangle)'\Sigma_{m}^{-1}(\langle\vec{x}_{r}\rangle-\langle\vec{x}_{s}\rangle), \end{equation} \tag{ 4 }$$

where $\langle \vec {x}_{r}\rangle$ is the average position of group r. The variable Σ_m is the estimation of the equivalent covariance matrix of each group, given by

$$\begin{equation} \Sigma_{m}=\frac{n_{r}\Sigma_{r}+n_{s}\Sigma_{s}}{n_{r}+n_{s}-2}, \end{equation} \tag{ 5 }$$

where Σ_r and Σ_s are the covariance matrix of the groups. In figure 3 we show the values of h for each case.

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The distance we considered is a hybrid version of both Euclidean and Hotelling metrics. This is required in order to avoid singularities for a set of nodes with zero variance. Since we are not interested in the absolute value of the distance, but in the comparison of values between groups, we can define a new hybrid distance, given by

$$\begin{equation} d_{h}(r,s)=\frac{h}{h+1}d(r,s), \end{equation}\noindent \tag{ 6 }$$

where d(r,s) is the usual Euclidean distance between the center of mass of groups r and s.

Finally, in cases where we have to calculate the distances through more than one variable, it is necessary to normalize them so as to give a fair comparison. In order to do so we calculate the standard score of each measure x, given by

$$\begin{equation} \hat{x}=\frac{x-\langle x\rangle}{\mathrm{std}(x)}, \end{equation}\noindent \tag{ 7 }$$

where std(x) is the standard deviation of x. Since throughout this paper we always use the standardized version of the values, we simplify the notation by calling $\hat {x}$ just by x.

In our study we use three network models to compare the methodology applied to three distinct scenarios. The first is the traditional ER [16] model that connects every possible pair of nodes with a probability p, originating a Poisson degree distribution representing a completely random graph.

The second scenario is when every connection has a cost associated with it, usually represented by a geographic network where the nodes have a spatial position. A commonly used mechanism to model this behavior is due to Waxman [17], who randomly placed the nodes in a [1,1] grid and, for every pair (i,j) of nodes, defined a probability of this pair being connected, given by

$$\begin{equation} P(i,j)=\beta \mathrm{e}^{-d(i,j)/d_{0}}, \end{equation}\noindent \tag{ 8 }$$

where d(i,j) is the Euclidean distance between nodes i and j, β tunes the density of edges, and d₀ sets the typical size of the connections. This generates a topology where a long range shortcut rarely occurs. Because of this, one of the main characteristics of this model is that, for the most commonly used values of β and d₀, it does not exhibit the small world behavior.

Another class of networks, usually called power-law, is greatly represented by the BA [18] model, where two principles, namely growth and referential attachment, are used to model the usual power-law degree distribution (P_k ≈ k⁻³) that is observed in many real systems [21]. Due to the power-law distribution, the degree of the nodes can show large fluctuations.

3.2.1. Integrate-and-fire dynamics

Our first application of the method will be related to the transmission of neuronal signals, modeled by the integrate-and-fire dynamics [22]. This model treats the neuron as an integrator with a hard threshold limit, $\mathcal {T}$. The actions of a given neuron i along the time is stored in the binary time series s_i(t), which indicates that the neuron spikes at instant t whether s_i(t) = 1. Using this time series, we define the spike rate of a neuron as

$$\begin{equation} r_{i}=\frac{1}{T_{\rm sim}-T_{\rm est}}\sum_{t=T_{\rm est}}^{T_{\rm sim}}s_i(t), \end{equation} \tag{ 9 }$$

where T_sim is the total simulation (or experiment) time and T_est is a long enough time, found empirically through prior simulations, so that r_i do not significantly change after T_est (we say that the dynamics stabilized). This measure corresponds to the average number of spikes during the considered interval, and it is widely used in neuroscience, since many neurons codify the stimulus amplitude through the rate of spikes [23].

3.2.2. SIS model

We compare the differentiation relative to the spike rate feature, which we call α_r for different network topologies, namely ER [16], BA [18] and Waxman's geographic model [17]. In figure 4(a) we show the result obtained for the geographic model with N = 1000 and 〈k〉 = 10 and an integrate-and-fire dynamics with $\mathcal {T}=8$ taking place on the network. To obtain statistical significance we use 100 different generated networks; each network is subjected to M = 1000 realizations of the dynamics with different initial conditions. We construct histograms of α_r obtained for each generated network and show in figure 4(a) the mean value and standard deviation the set presents. We see that the frequency of nodes with small α_r has a large variation, which is caused by the intrinsic fluctuations in the dynamics of the many nodes with a similar spike rate present on the network. It is feasible to think that this fluctuation would decay as α_r increases, but this is true only for intermediate α_r, while at high values of α_r we observe a sudden increase of fluctuation. Additionally, the mean value stays at an almost constant value for α_r in the range [0.9,1.5]. This is caused by the high potential of the geographic model to display structural fluctuations, originating in regions with a higher density when compared to the rest of the network. These regions significantly alter the spike rate of the nodes and highly differentiated groups appear. In figure 4(b) we apply the same procedure to calculate α_r in the geographic model, only changing the dynamic threshold to $\mathcal {T}=10$. It is clear that the groups are no longer distinguishable. This is so because the threshold is now so large that even the topological fluctuations cannot differentiate a significant number of nodes, when compared to the ones with small α_r.

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We apply the same procedure used for the geographic model to the ER networks with N = 1000 and 〈k〉 = 10. In figure 4(c) we show the information about the obtained histograms for $\mathcal {T}=8$, which makes it clear that this model exhibits much smaller fluctuations. This is caused by the much shorter geodesic distances that the model exhibits, when compared to the geographic counterpart, which creates a more compact network. We also show in figure 4(d) the case $\mathcal {T}=10$ for the ER model, where we see a decaying behavior expected for a random Poissonian system.

The third investigated model is the BA with N = 1000 and 〈k〉 = 6. Figures 4(e) and (f) show the log-scale histogram for, respectively, $\mathcal {T}=8$ and 10. In both cases we observe a significantly high peak for large α_r, which we found to be related to the network hubs (nodes with a very high degree). This result was expected, given our observations of large fluctuations on the geographic network, but this is not the main result for the BA model. The important result is that although the power-law degree distribution of the model has a continuous decaying behavior, the histogram of α_r shows small values for intermediate α_r and increases for large α_r. This behavior can be interpreted as follows: the dynamical differentiation of a hub is, as expected, very large, but a node with an almost equal degree can end up with a much smaller differentiation, having dynamics more similar to the low degree nodes. This result confirms the important role that hubs have in complex systems, not only in the sense of being central, but also in having a different purpose to the network dynamics.

Although many interesting properties arise when studying random network models, it is on real networks that the dynamical differentiation analysis can show its real potential. To show this we now apply the methodology to the C. elegans neuronal network. In this network, each node represents a neuron and two nodes are connected if there exists some kind of directed communication between them (e.g. synapses, gap junctions, etc). The data was compiled by Chen et al [24, 25] and obtained from [26]. The network has 279 nodes and 〈k〉 = 22.4.

Although we motivate the method with a real network, it is important to note that our dynamics do not take into account many signal particularities that arise for real neurons [27], therefore we are looking for a coarse grained description of the neurons inside the network. We will show that, even with this simplified description, it is still possible to observe some interesting phenomena.

We begin by showing in figure 5 the histogram of α relative to the spike rate, α_r. An immediate result observed is that there is a group with high differentiation, indicated with a red circle, which is somewhat similar to that observed for the BA random model. With this in mind, we plot in figure 6 the degree histogram of the network, indicating in red the nodes inside the observed group. We see that the nodes with high differentiation possess a high degree in the network, but there are some high degree nodes that do not show a distinct dynamics. It is clear that the topology influence does not occur merely by the degree of the nodes. There is a particular relation between these high differentiated nodes that make their dynamics very peculiar when compared to the rest of the network. The nodes inside the red circle in figure 5 are known as the interneurons of the ventral cord of the C. elegans. They are well recognized for possessing a high number of synapses [28], given that they make the bridge between sensory and motor neurons without much restriction on the type of transmitted signals (some classes of interneurons are known for receiving only a specific type of signal).

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In table 1 we present the traditional names of these neurons and some information about their spatial and topological distance. Position refers to the spacial localization of each neuron relative to the axis that goes from the head (value 0) to the tail (value 1) of the nematode. We see that the majority of the described neurons are on the head (more specifically, in the nerve ring of the nematode [29, 30]), with the exception of PVCL and PVCR that are on the tail. In the table, D1 is the mean topological distance between these neurons, that is, given a node i we measure how many edges we need to travel in order to go to node j and we take the mean of this distance for all j inside the differentiated group. D1 = 1 means that the neuron is a neighbor, or receives a direct signal, of all the other neurons shown in the table. The feature D2 complements D1 as it shows the topological distance between the given neuron and all other neurons, excluding those present in the table. We see that in all cases the differentiated nodes are closer between themselves than with the rest of the network, an effect partially provoked by their high degree. That is, besides having a high degree, these nodes are well connected between themselves, originating a high capacity of communication inside the group and rendering their dynamics very distinct in comparison to the rest of the network. In order to illustrate these neurons, we show in figure 1 of the supplementary material (available from stacks.iop.org/NJP/15/013048/mmedia) the position of each neuron inside the nematode.

In order to show that our approach can be applied to a completely different system, we also study the first infection time, I_f, of nodes going through an epidemics dynamic. The network we used is the world-wide airport network, which describes the flying routes between a large number of airports throughout the world. Each node is an airport (3302 in total) and an edge indicates that there is a flight between two airports. The data was obtained from www.openflights.org/data.html.

The value of I_f for a node strongly depends on its relative position with respect to where the epidemic has started, so for each initial condition the node will have many different I_f values. Still, the local topology of the node can influence the mean time it takes for the infection to arrive. Suppose we take two nodes with similar local topologies. It is expected that their I_f will be slightly different, but how different are they really? If the topologies of these nodes completely define their dynamics, i.e. I_f does not vary for the different initial conditions, then the difference in their dynamics, caused by the topology, is significant. On the contrary, if their mean values are close but the topology does not have a strong influence, then with different initial conditions it is possible to reach many different dynamical states. This means that the nodes are not significantly differentiated by their local topology. We reinforce that this is different from simply comparing their mean values, as we are taking into account both the variation caused by the initial condition and the network topology.

We performed 1000 simulations of the SIS epidemics dynamics and calculated the respective α values; the result is shown in figure 7. The first noticeable feature is the small number of nodes having a large value of α, these nodes are probably in regions of the network where the epidemic is far from its average behavior. They could be isolated from the network and so would have a very small chance of receiving the disease, or they could be hubs of the network: it does not matter where the disease started, they always catch it immediately. We found that the node types are actually mixed, with a little advantage to the hubs. In figure 8(a) we plot α versus the degree of the nodes, where we see that a node having high degree is guaranteed to have large α, while low degree nodes can have a range of α values. This means that high degree nodes depend only on their immediate neighborhood, i.e. the local topology completely defines their dynamics. On the other hand, as low degree nodes are strongly influenced by their higher order neighborhood, the local topology is not sufficient to define the dynamics. In order to show this more precisely, we plot in figure 8(b) the relationship of α with the closeness centrality of the nodes [31]. This topological measure takes into account all network nodes on its calculation, quantifying how central the node is. We see that the closeness defines much better the dynamics of the less connected nodes, showing that they are differentiated by the topology, but the difference is caused by the influence of the entire network.

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